Chapter 5

 

Analytical techniques and solutions for linear elastic solids

 

 

 

5.3 Complex Variable Solution to Plane Strain Static Linear Elastic Problems

 

Airy functions have been used to find many useful solutions to plane elastostatic boundary value problems.  The method does have some limitations, however.  The biharmonic equation is not the easiest field equation to solve, for one thing. Another limitation is that displacement components are difficult to determine from Airy functions, so that the method is not well suited to displacement boundary value problems.

 

In this section we outline a more versatile representation for 2D static linear elasticity problems, based on complex potentials.  The main goal is to provide you with enough background to be able to interpret solutions that use the complex variable formulation.  The techniques to derive the complex potentials are beyond the scope of this book, but can be found in most linear elasticity texts.

 

A typical plane elasticity problem is illustrated in the picture.  Just as in the preceding section, 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 hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaahwgadaWgaaWcbaGaaGOmaaqabaGccaGG9baaaa@36E7@  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 hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaaG4maaqabaaaaa@324F@  direction, is prevented from stretching parallel to the e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwgadaWgaaWcbaGaaG4maaqabaaaaa@324F@  axis, and every cross section is loaded identically and only in the { e 1 , e 2 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaahwgadaWgaaWcbaGaaGOmaaqabaGccaGG9baaaa@36E7@  plane.  In this case, the plane strain solution is applicable.

 

Some additional basic assumptions and restrictions are:

 The complex variable method outlined below 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 hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe27aUbaa@3230@  and mass density ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYnaaBaaaleaacaaIWaaabeaaaa a@331E@

 We will assume no body forces, and constant temperature

 

 

5.3.1 Complex variable solutions to elasticity problems

 

The picture shows a 2D solid.  In the complex variable formalism,

 The position of a point in the solid is specified by a complex number z= x 1 +i x 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG6bGaeyypa0JaamiEamaaBaaale aacaaIXaaabeaakiabgUcaRiaadMgacaWG4bWaaSbaaSqaaiaaikda aeqaaaaa@3A6E@

 The position of a point can also be expressed as z=r e iθ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG6bGaeyypa0JaamOCaiaadwgada ahaaWcbeqaaiaadMgacqaH4oqCaaaaaa@397D@  where r= x 1 2 + x 2 2 θ= tan 1 x 2 / x 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbGaeyypa0ZaaOaaaeaacaWG4b Waa0baaSqaaiaaigdaaeaacaaIYaaaaOGaey4kaSIaamiEamaaDaaa leaacaaIYaaabaGaaGOmaaaaaeqaaOGaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaeqiUdeNaeyypa0JaciiDaiaacggacaGGUbWa aWbaaSqabeaacqGHsislcaaIXaaaaOGaamiEamaaBaaaleaacaaIYa aabeaakiaac+cacaWG4bWaaSbaaSqaaiaaigdaaeqaaaaa@5040@

You can show that these are equivalent using Euler’s formula e iθ =cosθ+isinθ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGLbWaaWbaaSqabeaacaWGPbGaeq iUdehaaOGaeyypa0Jaci4yaiaac+gacaGGZbGaeqiUdeNaey4kaSIa amyAaiGacohacaGGPbGaaiOBaiabeI7aXbaa@4278@ , which gives

z=r e iθ = x 1 2 + x 2 2 ( cosθ+isinθ ) = x 1 2 + x 2 2 ( x 1 x 1 2 + x 2 2 + i x 2 x 1 2 + x 2 2 )= x 1 +i x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaadQhacqGH9aqpcaWGYbGaam yzamaaCaaaleqabaGaamyAaiabeI7aXbaakiabg2da9maakaaabaGa amiEamaaDaaaleaacaaIXaaabaGaaGOmaaaakiabgUcaRiaadIhada qhaaWcbaGaaGOmaaqaaiaaikdaaaaabeaakmaabmaabaGaci4yaiaa c+gacaGGZbGaeqiUdeNaey4kaSIaamyAaiGacohacaGGPbGaaiOBai abeI7aXbGaayjkaiaawMcaaaqaaiaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqGH9aqp daGcaaqaaiaadIhadaqhaaWcbaGaaGymaaqaaiaaikdaaaGccqGHRa WkcaWG4bWaa0baaSqaaiaaikdaaeaacaaIYaaaaaqabaGcdaqadaqa amaalaaabaGaamiEamaaBaaaleaacaaIXaaabeaaaOqaamaakaaaba GaamiEamaaDaaaleaacaaIXaaabaGaaGOmaaaakiabgUcaRiaadIha daqhaaWcbaGaaGOmaaqaaiaaikdaaaaabeaaaaGccqGHRaWkdaWcaa qaaiaadMgacaWG4bWaaSbaaSqaaiaaikdaaeqaaaGcbaWaaOaaaeaa caWG4bWaa0baaSqaaiaaigdaaeaacaaIYaaaaOGaey4kaSIaamiEam aaDaaaleaacaaIYaaabaGaaGOmaaaaaeqaaaaaaOGaayjkaiaawMca aiaaykW7caaMc8UaaGPaVlabg2da9iaadIhadaWgaaWcbaGaaGymaa qabaGccqGHRaWkcaWGPbGaamiEamaaBaaaleaacaaIYaaabeaaaaaa @911E@

 The displacement of a point is specified using a second complex number D= u 1 +i u 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGebGaeyypa0JaamyDamaaBaaale aacaaIXaaabeaakiabgUcaRiaadMgacaWG1bWaaSbaaSqaaiaaikda aeqaaaaa@3A32@

 The displacement and stress fields in rectangular coordinates are generated from two complex potentials Ω(z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHPoWvcaGGOaGaamOEaiaacMcaaa a@36AC@  and ω(z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHjpWDcaGGOaGaamOEaiaacMcaaa a@36EB@ , which are differentiable (also called `analytic’ or `holomorphic’) functions of z  (e.g. a polynomial), using the following formulas

E (1+ν) D=(34ν)Ω(z)z Ω (z) ¯ ω(z) ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiqacaWaaaGcbaWaaSaaaeaacaWGfbaabaGaaiikaiaaig dacqGHRaWkcqaH9oGBcaGGPaaaaiaadseacqGH9aqpcaGGOaGaaG4m aiabgkHiTiaaisdacqaH9oGBcaGGPaGaeuyQdCLaaiikaiaadQhaca GGPaGaeyOeI0IaamOEaiaaykW7caaMc8+aa0aaaeaacuqHPoWvgaqb aiaacIcacaWG6bGaaiykaaaacaaMcSUaeyOeI0IaaGPaVpaanaaaba GaeqyYdCNaaiikaiaadQhacaGGPaaaaaaa@52BC@

σ 11 + σ 22 =2( Ω (z)+ Ω (z) ¯ ) σ 11 σ 22 +2i σ 12 =2( z Ω (z) ¯ + ω (z) ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiqacaWaaaGceaqabeaacqaHdpWCdaWgaaWcbaGaaGymai aaigdaaeqaaOGaey4kaSIaeq4Wdm3aaSbaaSqaaiaaikdacaaIYaaa beaakiabg2da9iaaikdadaqadaqaaiqbfM6axzaafaGaaiikaiaadQ hacaGGPaGaey4kaSYaa0aaaeaacuqHPoWvgaqbaiaacIcacaWG6bGa aiykaaaaaiaawIcacaGLPaaaaeaacqaHdpWCdaWgaaWcbaGaaGymai aaigdaaeqaaOGaeyOeI0Iaeq4Wdm3aaSbaaSqaaiaaikdacaaIYaaa beaakiabgUcaRiaaikdacaWGPbGaeq4Wdm3aaSbaaSqaaiaaigdaca aIYaaabeaakiabg2da9iabgkHiTiaaikdadaqadaqaaiaadQhacaaM c8+aa0aaaeaacuqHPoWvgaGbaiaacIcacaWG6bGaaiykaaaacqGHRa WkdaqdaaqaaiqbeM8a3zaafaGaaiikaiaadQhacaGGPaaaaaGaayjk aiaawMcaaaaaaa@626A@

Here, Ω (z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuqHPoWvgaqbaiaacIcacaWG6bGaai ykaaaa@36B8@  denotes the derivative of Ω(z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHPoWvcaGGOaGaamOEaiaacMcaaa a@36AC@  with respect to z, and Ω(z) ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaqdaaqaaiabfM6axjaacIcacaWG6b Gaaiykaaaaaaa@36BD@  denotes the complex conjugate of Ω(z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHPoWvcaGGOaGaamOEaiaacMcaaa a@36AC@ .  Recall that to calculate the complex conjugate of a complex number, you simply change the sign of its imaginary part, i.e. a+ib ¯ =aib MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaqdaaqaaiaadggacqGHRaWkcaWGPb GaamOyaaaacqGH9aqpcaWGHbGaeyOeI0IaamyAaiaadkgaaaa@3B22@ .

 

 The displacement and stress in polar coordinates can be derived as

E (1+ν) ( u r +i u θ )=[ (34ν)Ω(z)z Ω (z) ¯ ω(z) ¯ ] e iθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiqacaWaaaGcbaWaaSaaaeaacaWGfbaabaGaaiikaiaaig dacqGHRaWkcqaH9oGBcaGGPaaaamaabmaabaGaamyDamaaBaaaleaa caWGYbaabeaakiabgUcaRiaadMgacaWG1bWaaSbaaSqaaiabeI7aXb qabaaakiaawIcacaGLPaaacqGH9aqpdaWadaqaaiaacIcacaaIZaGa eyOeI0IaaGinaiabe27aUjaacMcacqqHPoWvcaGGOaGaamOEaiaacM cacqGHsislcaWG6bGaaGPaVlaaykW7daqdaaqaaiqbfM6axzaafaGa aiikaiaadQhacaGGPaaaaiaaykW6cqGHsislcaaMc8+aa0aaaeaacq aHjpWDcaGGOaGaamOEaiaacMcaaaaacaGLBbGaayzxaaGaamyzamaa CaaaleqabaGaeyOeI0IaamyAaiabeI7aXbaaaaa@60F5@

σ rr + σ θθ =2( Ω (z)+ Ω (z) ¯ ) σ rr σ θθ +2i σ rθ =2( z Ω (z) ¯ + ω (z) ¯ ) e 2iθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiqacaWaaaGceaqabeaacqaHdpWCdaWgaaWcbaGaamOCai aadkhaaeqaaOGaey4kaSIaeq4Wdm3aaSbaaSqaaiabeI7aXjabeI7a XbqabaGccqGH9aqpcaaIYaWaaeWaaeaacuqHPoWvgaqbaiaacIcaca WG6bGaaiykaiabgUcaRmaanaaabaGafuyQdCLbauaacaGGOaGaamOE aiaacMcaaaaacaGLOaGaayzkaaaabaGaeq4Wdm3aaSbaaSqaaiaadk hacaWGYbaabeaakiabgkHiTiabeo8aZnaaBaaaleaacqaH4oqCcqaH 4oqCaeqaaOGaey4kaSIaaGOmaiaadMgacqaHdpWCdaWgaaWcbaGaam OCaiabeI7aXbqabaGccqGH9aqpcqGHsislcaaIYaWaaeWaaeaacaWG 6bGaaGPaVpaanaaabaGafuyQdCLbauGbauaacaGGOaGaamOEaiaacM caaaGaey4kaSYaa0aaaeaacuaHjpWDgaqbaiaacIcacaWG6bGaaiyk aaaaaiaawIcacaGLPaaacaWGLbWaaWbaaSqabeaacqGHsislcaaIYa GaamyAaiabeI7aXbaaaaaa@6DE8@

 The formulas given here for displacements and stresses are the most general representation, but other special formulas are sometimes used for particular problems.  For example, if the solid is a half-space in the region   x 2 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaikdaaeqaaO GaeyyzImRaaGimaaaa@3735@  with a boundary at x 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaikdaaeqaaO Gaeyypa0JaaGimaaaa@3675@  the solution can be generated from a single complex potential Ω(z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHPoWvcaGGOaGaamOEaiaacMcaaa a@36BC@ , using the formulas

2μD=(34ν)Ω(z)+Ω( z ¯ )+( z ¯ z) Ω'(z) ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaaIYaGaeqiVd0Maamiraiabg2da9i aacIcacaaIZaGaeyOeI0IaaGinaiabe27aUjaacMcacqqHPoWvcaGG OaGaamOEaiaacMcacqGHRaWkcqqHPoWvcaGGOaGabmOEayaaraGaai ykaiabgUcaRiaacIcaceWG6bGbaebacqGHsislcaWG6bGaaiykamaa naaabaGaeuyQdCLaai4jaiaacIcacaWG6bGaaiykaaaaaaa@4F25@

σ 11 + σ 22 =2( Ω (z)+ Ω (z) ¯ ) σ 22 i σ 12 = Ω (z) Ω ( z ¯ )+(z z ¯ ) Ω (z) ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabeo8aZnaaBaaaleaacaaIXa GaaGymaaqabaGccqGHRaWkcqaHdpWCdaWgaaWcbaGaaGOmaiaaikda aeqaaOGaeyypa0JaaGOmamaabmaabaGafuyQdCLbauaacaGGOaGaam OEaiaacMcacqGHRaWkdaqdaaqaaiqbfM6axzaafaGaaiikaiaadQha caGGPaaaaaGaayjkaiaawMcaaaqaaiabeo8aZnaaBaaaleaacaaIYa GaaGOmaaqabaGccqGHsislcaWGPbGaeq4Wdm3aaSbaaSqaaiaaigda caaIYaaabeaakiabg2da9iqbfM6axzaafaGaaiikaiaadQhacaGGPa GaeyOeI0IafuyQdCLbauaacaGGOaGabmOEayaaraGaaiykaiabgUca RiaacIcacaWG6bGaeyOeI0IabmOEayaaraGaaiykamaanaaabaGafu yQdCLbayaacaGGOaGaamOEaiaacMcaaaaaaaa@629D@

For example, you can use these formulas to calculate stresses from the potentials given in Sections 5.3.7-5.3.9.  The conventional representation gives the same results, of course.

 

 

 

5.3.2 Demonstration that the complex variable solution satisfies the governing equations

 

We need to show two things:

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

1 12ν 2 u k x k x i + 2 u i x k x k =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVKI8FfYJH8sipiYdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaaG ymaaqaaiaaigdacqGHsislcaaIYaGaeqyVd4gaaiaaykW6daWcaaqa aiabgkGi2oaaCaaaleqabaGaaGOmaaaakiaadwhadaWgaaWcbaGaam 4AaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadUgaaeqaaOGa eyOaIyRaamiEamaaBaaaleaacaWGPbaabeaaaaGccqGHRaWkcaaMcS +aaSaaaeaacqGHciITdaahaaWcbeqaaiaaikdaaaGccaWG1bWaaSba aSqaaiaadMgaaeqaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaWGRb aabeaakiabgkGi2kaadIhadaWgaaWcbaGaam4AaaqabaaaaOGaeyyp a0JaaGimaaaa@56E0@

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

 

To do this, we need to review some basic results from the theory of complex variables.  Recall that we have set z= x 1 +i x 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG6bGaeyypa0JaamiEamaaBaaale aacaaIXaaabeaakiabgUcaRiaadMgacaWG4bWaaSbaaSqaaiaaikda aeqaaaaa@3A6E@ , so that a differentiable function f(z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGMbGaaiikaiaadQhacaGGPaaaaa@3609@  can be decomposed into real and imaginary parts, each of which are functions of x 1 , x 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaigdaaeqaaO GaaiilaiaadIhadaWgaaWcbaGaaGOmaaqabaaaaa@3749@ , as

f(z)=v( x 1 , x 2 )+iw( x 1 , x 2 )=v( z+ z ¯ 2 ,i z ¯ z 2 )+iw( z+ z ¯ 2 ,i z ¯ z 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGMbGaaiikaiaadQhacaGGPaGaey ypa0JaamODaiaacIcacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaaiil aiaadIhadaWgaaWcbaGaaGOmaaqabaGccaGGPaGaey4kaSIaamyAai aadEhacaGGOaGaamiEamaaBaaaleaacaaIXaaabeaakiaacYcacaWG 4bWaaSbaaSqaaiaaikdaaeqaaOGaaiykaiaaykW7caaMc8UaaGPaVl abg2da9iaaykW7caaMc8UaaGPaVlaadAhadaqadaqaamaalaaabaGa amOEaiabgUcaRiqadQhagaqeaaqaaiaaikdaaaGaaiilaiaadMgada WcaaqaaiqadQhagaqeaiabgkHiTiaadQhaaeaacaaIYaaaaaGaayjk aiaawMcaaiabgUcaRiaadMgacaWG3bWaaeWaaeaadaWcaaqaaiaadQ hacqGHRaWkceWG6bGbaebaaeaacaaIYaaaaiaacYcacaWGPbWaaSaa aeaaceWG6bGbaebacqGHsislcaWG6baabaGaaGOmaaaaaiaawIcaca GLPaaaaaa@6A24@

This shows that

z 1 2 ( x 1 i x 2 ) z ¯ 1 2 ( x 1 +i x 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2cqaaiabgkGi2k aadQhaaaGaeyyyIO7aaSaaaeaacaaIXaaabaGaaGOmaaaadaqadaqa amaalaaabaGaeyOaIylabaGaeyOaIyRaamiEamaaBaaaleaacaaIXa aabeaaaaGccqGHsislcaWGPbWaaSaaaeaacqGHciITaeaacqGHciIT caWG4bWaaSbaaSqaaiaaikdaaeqaaaaaaOGaayjkaiaawMcaaiaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7daWcaa qaaiabgkGi2cqaaiabgkGi2kqadQhagaqeaaaacqGHHjIUdaWcaaqa aiaaigdaaeaacaaIYaaaamaabmaabaWaaSaaaeaacqGHciITaeaacq GHciITcaWG4bWaaSbaaSqaaiaaigdaaeqaaaaakiabgUcaRiaadMga daWcaaqaaiabgkGi2cqaaiabgkGi2kaadIhadaWgaaWcbaGaaGOmaa qabaaaaaGccaGLOaGaayzkaaaaaa@73CA@

Next, recall that if f(z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGMbGaaiikaiaadQhacaGGPaaaaa@3609@  is differentiable with respect to z, its real and imaginary parts must satisfy the Cauchy-Riemann equations

v x 1 = w x 2 w x 1 = v x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2kaadAhaaeaacq GHciITcaWG4bWaaSbaaSqaaiaaigdaaeqaaaaakiabg2da9maalaaa baGaeyOaIyRaam4DaaqaaiabgkGi2kaadIhadaWgaaWcbaGaaGOmaa qabaaaaOGaaGPaRlaaykW6caaMcSUaaGPaRlaaykW6caaMcSUaaGPa RlaaykW6caaMcSUaaGPaRlaaykW6caaMcSUaaGPaRlaaykW6daWcaa qaaiabgkGi2kaadEhaaeaacqGHciITcaWG4bWaaSbaaSqaaiaaigda aeqaaaaakiabg2da9iabgkHiTmaalaaabaGaeyOaIyRaamODaaqaai abgkGi2kaadIhadaWgaaWcbaGaaGOmaaqabaaaaaaa@622E@

We can then show that the derivative of f(z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGMbGaaiikaiaadQhacaGGPaaaaa@3609@  with respect to z ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWG6bGbaebaaaa@33ED@  is zero, and similarly, the derivative of f(z) ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaqdaaqaaiaadAgacaGGOaGaamOEai aacMcaaaaaaa@361A@  with respect to z is zero.  To see these, use the definitions and the Cauchy-Riemann equations

2 f(z) z ¯ =( x 1 +i x 2 )(v+iw)= v x 1 w x 2 +i( v x 2 + w x 1 )=0 2 f(z) ¯ z =( x 1 i x 2 )(viw)= v x 1 w x 2 i( v x 2 + w x 1 )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaaikdadaWcaaqaaiabgkGi2k aadAgacaGGOaGaamOEaiaacMcaaeaacqGHciITceWG6bGbaebaaaGa eyypa0ZaaeWaaeaadaWcaaqaaiabgkGi2cqaaiabgkGi2kaadIhada WgaaWcbaGaaGymaaqabaaaaOGaey4kaSIaamyAamaalaaabaGaeyOa IylabaGaeyOaIyRaamiEamaaBaaaleaacaaIYaaabeaaaaaakiaawI cacaGLPaaacaGGOaGaamODaiabgUcaRiaadMgacaWG3bGaaiykaiab g2da9maalaaabaGaeyOaIyRaamODaaqaaiabgkGi2kaadIhadaWgaa WcbaGaaGymaaqabaaaaOGaeyOeI0YaaSaaaeaacqGHciITcaWG3baa baGaeyOaIyRaamiEamaaBaaaleaacaaIYaaabeaaaaGccqGHRaWkca WGPbGaaiikamaalaaabaGaeyOaIyRaamODaaqaaiabgkGi2kaadIha daWgaaWcbaGaaGOmaaqabaaaaOGaey4kaSYaaSaaaeaacqGHciITca WG3baabaGaeyOaIyRaamiEamaaBaaaleaacaaIXaaabeaaaaGccaGG PaGaeyypa0JaaGimaaqaaiaaikdadaWcaaqaaiabgkGi2oaanaaaba GaamOzaiaacIcacaWG6bGaaiykaaaaaeaacqGHciITcaWG6baaaiab g2da9maabmaabaWaaSaaaeaacqGHciITaeaacqGHciITcaWG4bWaaS baaSqaaiaaigdaaeqaaaaakiabgkHiTiaadMgadaWcaaqaaiabgkGi 2cqaaiabgkGi2kaadIhadaWgaaWcbaGaaGOmaaqabaaaaaGccaGLOa GaayzkaaGaaiikaiaadAhacqGHsislcaWGPbGaam4DaiaacMcacqGH 9aqpdaWcaaqaaiabgkGi2kaadAhaaeaacqGHciITcaWG4bWaaSbaaS qaaiaaigdaaeqaaaaakiabgkHiTmaalaaabaGaeyOaIyRaam4Daaqa aiabgkGi2kaadIhadaWgaaWcbaGaaGOmaaqabaaaaOGaeyOeI0Iaam yAaiaacIcadaWcaaqaaiabgkGi2kaadAhaaeaacqGHciITcaWG4bWa aSbaaSqaaiaaikdaaeqaaaaakiabgUcaRmaalaaabaGaeyOaIyRaam 4DaaqaaiabgkGi2kaadIhadaWgaaWcbaGaaGymaaqabaaaaOGaaiyk aiabg2da9iaaicdaaaaa@A69E@

 

We can now proceed with the proof.   The equilibrium equations for plane deformation reduce to

( 2 x 1 2 + 2 x 2 2 ) u 1 + 1 12ν x 1 ( u 1 x 1 + u 2 x 2 )=0 ( 2 x 1 2 + 2 x 2 2 ) u 2 + 1 12ν x 2 ( u 1 x 1 + u 2 x 2 )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaamaabmaabaWaaSaaaeaacqGHci ITdaahaaWcbeqaaiaaikdaaaaakeaacqGHciITcaWG4bWaa0baaSqa aiaaigdaaeaacaaIYaaaaaaakiabgUcaRmaalaaabaGaeyOaIy7aaW baaSqabeaacaaIYaaaaaGcbaGaeyOaIyRaamiEamaaDaaaleaacaaI YaaabaGaaGOmaaaaaaaakiaawIcacaGLPaaacaWG1bWaaSbaaSqaai aaigdaaeqaaOGaey4kaSYaaSaaaeaacaaIXaaabaGaaGymaiabgkHi TiaaikdacqaH9oGBaaWaaSaaaeaacqGHciITaeaacqGHciITcaWG4b Waa0baaSqaaiaaigdaaeaaaaaaaOWaaeWaaeaadaWcaaqaaiabgkGi 2kaadwhadaWgaaWcbaGaaGymaaqabaaakeaacqGHciITcaWG4bWaa0 baaSqaaiaaigdaaeaaaaaaaOGaey4kaSYaaSaaaeaacqGHciITcaWG 1bWaaSbaaSqaaiaaikdaaeqaaaGcbaGaeyOaIyRaamiEamaaDaaale aacaaIYaaabaaaaaaaaOGaayjkaiaawMcaaiabg2da9iaaicdaaeaa daqadaqaamaalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaaGcba GaeyOaIyRaamiEamaaDaaaleaacaaIXaaabaGaaGOmaaaaaaGccqGH RaWkdaWcaaqaaiabgkGi2oaaCaaaleqabaGaaGOmaaaaaOqaaiabgk Gi2kaadIhadaqhaaWcbaGaaGOmaaqaaiaaikdaaaaaaaGccaGLOaGa ayzkaaGaamyDamaaBaaaleaacaaIYaaabeaakiabgUcaRmaalaaaba GaaGymaaqaaiaaigdacqGHsislcaaIYaGaeqyVd4gaamaalaaabaGa eyOaIylabaGaeyOaIyRaamiEamaaDaaaleaacaaIYaaabaaaaaaakm aabmaabaWaaSaaaeaacqGHciITcaWG1bWaaSbaaSqaaiaaigdaaeqa aaGcbaGaeyOaIyRaamiEamaaDaaaleaacaaIXaaabaaaaaaakiabgU caRmaalaaabaGaeyOaIyRaamyDamaaBaaaleaacaaIYaaabeaaaOqa aiabgkGi2kaadIhadaqhaaWcbaGaaGOmaaqaaaaaaaaakiaawIcaca GLPaaacqGH9aqpcaaIWaaaaaa@8DBE@

These equations can be written in a combined, complex, form as

( 2 x 1 2 + 2 x 2 2 )( u 1 +i u 2 )+ 1 12ν ( x 1 +i x 2 )( u 1 x 1 + u 2 x 2 )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaqadaqaamaalaaabaGaeyOaIy7aaW baaSqabeaacaaIYaaaaaGcbaGaeyOaIyRaamiEamaaDaaaleaacaaI XaaabaGaaGOmaaaaaaGccqGHRaWkdaWcaaqaaiabgkGi2oaaCaaale qabaGaaGOmaaaaaOqaaiabgkGi2kaadIhadaqhaaWcbaGaaGOmaaqa aiaaikdaaaaaaaGccaGLOaGaayzkaaGaaiikaiaadwhadaWgaaWcba GaaGymaaqabaGccqGHRaWkcaWGPbGaamyDamaaBaaaleaacaaIYaaa beaakiaacMcacqGHRaWkdaWcaaqaaiaaigdaaeaacaaIXaGaeyOeI0 IaaGOmaiabe27aUbaadaqadaqaamaalaaabaGaeyOaIylabaGaeyOa IyRaamiEamaaDaaaleaacaaIXaaabaaaaaaakiabgUcaRiaadMgada WcaaqaaiabgkGi2cqaaiabgkGi2kaadIhadaqhaaWcbaGaaGOmaaqa aaaaaaaakiaawIcacaGLPaaadaqadaqaamaalaaabaGaeyOaIyRaam yDamaaBaaaleaacaaIXaaabeaaaOqaaiabgkGi2kaadIhadaqhaaWc baGaaGymaaqaaaaaaaGccqGHRaWkdaWcaaqaaiabgkGi2kaadwhada WgaaWcbaGaaGOmaaqabaaakeaacqGHciITcaWG4bWaa0baaSqaaiaa ikdaaeaaaaaaaaGccaGLOaGaayzkaaGaeyypa0JaaGimaaaa@6D77@

It is easy to show (simply substitute D= u 1 +i u 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGebGaeyypa0JaamyDamaaBaaale aacaaIXaaabeaakiabgUcaRiaadMgacaWG1bWaaSbaaSqaaiaaikda aeqaaaaa@3A32@  and use the definitions of differentiation with respect to z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG6baaaa@33D5@  and z ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWG6bGbaebaaaa@33ED@  ) that this can be re-written as

4 2 z z ¯ D+ 2 12ν z ¯ ( D z + D ¯ z ¯ )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaaI0aWaaSaaaeaacqGHciITdaahaa WcbeqaaiaaikdaaaaakeaacqGHciITcaWG6bGaeyOaIyRabmOEayaa raaaaiaadseacqGHRaWkdaWcaaqaaiaaikdaaeaacaaIXaGaeyOeI0 IaaGOmaiabe27aUbaadaWcaaqaaiabgkGi2cqaaiabgkGi2kqadQha gaqeaaaadaqadaqaamaalaaabaGaeyOaIyRaamiraaqaaiabgkGi2k aadQhaaaGaey4kaSYaaSaaaeaacqGHciITdaqdaaqaaiaadseaaaaa baGaeyOaIyRabmOEayaaraaaaaGaayjkaiaawMcaaiabg2da9iaaic daaaa@52F0@

Finally, substituting

D= ( 1+ν ) E { (34ν)Ω(z)z Ω (z) ¯ ω(z) ¯ } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiqacaWaaaGcbaGaamiraiabg2da9maalaaabaWaaeWaae aacaaIXaGaey4kaSIaeqyVd4gacaGLOaGaayzkaaaabaGaamyraaaa daGadaqaaiaacIcacaaIZaGaeyOeI0IaaGinaiabe27aUjaacMcacq qHPoWvcaGGOaGaamOEaiaacMcacqGHsislcaWG6bGaaGPaVlaaykW7 daqdaaqaaiqbfM6axzaafaGaaiikaiaadQhacaGGPaaaaiaaykW6cq GHsislcaaMc8+aa0aaaeaacqaHjpWDcaGGOaGaamOEaiaacMcaaaaa caGL7bGaayzFaaaaaa@551D@

and noting that Ω/ z ¯ =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITcqqHPoWvcaGGVaGaeyOaIy RabmOEayaaraGaeyypa0JaaGimaaaa@3AAA@  and Ω ¯ /z= ω ¯ /z=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITdaqdaaqaaiabfM6axbaaca GGVaGaeyOaIyRaamOEaiabg2da9iabgkGi2oaanaaabaGaeqyYdCha aiaac+cacqGHciITcaWG6bGaeyypa0JaaGimaaaa@4205@  shows that this equation is indeed satisfied.

 

To show that the stress-strain relations are satisfied, note that the stress-strain relations for plane strain deformation (Section 3.1.4) can be written as

σ 11 + σ 22 = E (1+ν)(12ν) ( u 1 x 1 + u 2 x 2 )= E (1+ν)(12ν) ( D z + D ¯ z ¯ ) σ 11 σ 22 +2i σ 12 = E (1+ν) ( x 1 +i x 2 )( u 1 +i u 2 )= 2E (1+ν) D z ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabeo8aZnaaBaaaleaacaaIXa GaaGymaaqabaGccqGHRaWkcqaHdpWCdaWgaaWcbaGaaGOmaiaaikda aeqaaOGaeyypa0ZaaSaaaeaacaWGfbaabaGaaiikaiaaigdacqGHRa WkcqaH9oGBcaGGPaGaaiikaiaaigdacqGHsislcaaIYaGaeqyVd4Ma aiykaaaadaqadaqaamaalaaabaGaeyOaIyRaamyDamaaBaaaleaaca aIXaaabeaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaaGymaaqabaaa aOGaey4kaSYaaSaaaeaacqGHciITcaWG1bWaaSbaaSqaaiaaikdaae qaaaGcbaGaeyOaIyRaamiEamaaBaaaleaacaaIYaaabeaaaaaakiaa wIcacaGLPaaacqGH9aqpdaWcaaqaaiaadweaaeaacaGGOaGaaGymai abgUcaRiabe27aUjaacMcacaGGOaGaaGymaiabgkHiTiaaikdacqaH 9oGBcaGGPaaaamaabmaabaWaaSaaaeaacqGHciITcaWGebaabaGaey OaIyRaamOEaaaacqGHRaWkdaWcaaqaaiabgkGi2oaanaaabaGaamir aaaaaeaacqGHciITceWG6bGbaebaaaaacaGLOaGaayzkaaaabaGaeq 4Wdm3aaSbaaSqaaiaaigdacaaIXaaabeaakiabgkHiTiabeo8aZnaa BaaaleaacaaIYaGaaGOmaaqabaGccqGHRaWkcaaIYaGaamyAaiabeo 8aZnaaBaaaleaacaaIXaGaaGOmaaqabaGccqGH9aqpdaWcaaqaaiaa dweaaeaacaGGOaGaaGymaiabgUcaRiabe27aUjaacMcaaaWaaeWaae aadaWcaaqaaiabgkGi2cqaaiabgkGi2kaadIhadaWgaaWcbaGaaGym aaqabaaaaOGaey4kaSIaamyAamaalaaabaGaeyOaIylabaGaeyOaIy RaamiEamaaBaaaleaacaaIYaaabeaaaaaakiaawIcacaGLPaaacaGG OaGaamyDamaaBaaaleaacaaIXaaabeaakiabgUcaRiaadMgacaWG1b WaaSbaaSqaaiaaikdaaeqaaOGaaiykaiabg2da9maalaaabaGaaGOm aiaadweaaeaacaGGOaGaaGymaiabgUcaRiabe27aUjaacMcaaaWaaS aaaeaacqGHciITcaWGebaabaGaeyOaIyRabmOEayaaraaaaaaaaa@A293@

Substituting for D in terms of the complex potentials and evaluating the derivatives gives the required results.

 

 

 

5.3.3 Complex variable solution for a line force in an infinite solid (plane strain deformation)

 

The displacements and stresses induced by a line load with force per unit out of plane distance F= F 1 e 1 + F 2 e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHgbGaeyypa0JaamOramaaBaaale aacaaIXaaabeaakiaahwgadaWgaaWcbaGaaGymaaqabaGccqGHRaWk caWGgbWaaSbaaSqaaiaaikdaaeqaaOGaaCyzamaaBaaaleaacaaIYa aabeaaaaa@3CAB@  acting at the origin of a large (infinite) solid are calculated from the complex potentials

Ω(z)= F 1 +i F 2 8π(1ν) log(z)ω(z)= (34ν)( F 1 i F 2 ) 8π(1ν) log(z) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiqacaWaaaGcbaGaeuyQdCLaaiikaiaadQhacaGGPaGaey ypa0JaeyOeI0YaaSaaaeaacaWGgbWaaSbaaSqaaiaaigdaaeqaaOGa ey4kaSIaamyAaiaadAeadaWgaaWcbaGaaGOmaaqabaaakeaacaaI4a GaeqiWdaNaaiikaiaaigdacqGHsislcqaH9oGBcaGGPaaaaiGacYga caGGVbGaai4zaiaacIcacaWG6bGaaiykaiaaykW6caaMcSUaaGPaRl aaykW6caaMcSUaaGPaRlaaykW6cqaHjpWDcaGGOaGaamOEaiaacMca cqGH9aqpdaWcaaqaaiaacIcacaaIZaGaeyOeI0IaaGinaiabe27aUj aacMcacaGGOaGaamOramaaBaaaleaacaaIXaaabeaakiabgkHiTiaa dMgacaWGgbWaaSbaaSqaaiaaikdaaeqaaOGaaiykaaqaaiaaiIdacq aHapaCcaGGOaGaaGymaiabgkHiTiabe27aUjaacMcaaaGaciiBaiaa c+gacaGGNbGaaiikaiaadQhacaGGPaaaaa@70D0@

 

The displacements can be calculated from these potentials as

u 1 = (1+ν) F 1 8πE(1ν) { 2(34ν)log(r)+cos2θ } (1+ν) F 2 8πE(1ν) sin2θ u 2 = (1+ν) F 2 8πE(1ν) { 2(34ν)log(r)sin2θ }+ (1+ν) F 1 8πE(1ν) cos2θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaadwhadaWgaaWcbaGaaGymaa qabaGccqGH9aqpcqGHsisldaWcaaqaaiaacIcacaaIXaGaey4kaSIa eqyVd4MaaiykaiaadAeadaWgaaWcbaGaaGymaaqabaaakeaacaaI4a GaeqiWdaNaamyraiaacIcacaaIXaGaeyOeI0IaeqyVd4Maaiykaaaa daGadaqaaiaaikdacaGGOaGaaG4maiabgkHiTiaaisdacqaH9oGBca GGPaGaciiBaiaac+gacaGGNbGaaiikaiaadkhacaGGPaGaey4kaSIa ci4yaiaac+gacaGGZbGaaGOmaiabeI7aXbGaay5Eaiaaw2haaiabgk HiTmaalaaabaGaaiikaiaaigdacqGHRaWkcqaH9oGBcaGGPaGaamOr amaaBaaaleaacaaIYaaabeaaaOqaaiaaiIdacqaHapaCcaWGfbGaai ikaiaaigdacqGHsislcqaH9oGBcaGGPaaaaiGacohacaGGPbGaaiOB aiaaikdacqaH4oqCaeaacaWG1bWaaSbaaSqaaiaaikdaaeqaaOGaey ypa0JaeyOeI0YaaSaaaeaacaGGOaGaaGymaiabgUcaRiabe27aUjaa cMcacaWGgbWaaSbaaSqaaiaaikdaaeqaaaGcbaGaaGioaiabec8aWj aadweacaGGOaGaaGymaiabgkHiTiabe27aUjaacMcaaaWaaiWaaeaa caaIYaGaaiikaiaaiodacqGHsislcaaI0aGaeqyVd4MaaiykaiGacY gacaGGVbGaai4zaiaacIcacaWGYbGaaiykaiabgkHiTiGacohacaGG PbGaaiOBaiaaikdacqaH4oqCaiaawUhacaGL9baacqGHRaWkdaWcaa qaaiaacIcacaaIXaGaey4kaSIaeqyVd4MaaiykaiaadAeadaWgaaWc baGaaGymaaqabaaakeaacaaI4aGaeqiWdaNaamyraiaacIcacaaIXa GaeyOeI0IaeqyVd4MaaiykaaaaciGGJbGaai4BaiaacohacaaIYaGa eqiUdehaaaa@A844@

σ rr = 32ν 4π(1ν)r ( F 1 cosθ+ F 2 sinθ ) σ θθ = (12ν) 4π(1ν)r ( F 1 cosθ+ F 2 sinθ ) σ rθ = (12ν) 4π(1ν)r ( F 1 sinθ F 2 cosθ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabeo8aZnaaBaaaleaacaWGYb GaamOCaaqabaGccqGH9aqpcqGHsisldaWcaaqaaiaaiodacqGHsisl caaIYaGaeqyVd4gabaGaaGinaiabec8aWjaacIcacaaIXaGaeyOeI0 IaeqyVd4MaaiykaiaadkhaaaWaaeWaaeaacaWGgbWaaSbaaSqaaiaa igdaaeqaaOGaci4yaiaac+gacaGGZbGaeqiUdeNaey4kaSIaamOram aaBaaaleaacaaIYaaabeaakiGacohacaGGPbGaaiOBaiabeI7aXbGa ayjkaiaawMcaaaqaaiabeo8aZnaaBaaaleaacqaH4oqCcqaH4oqCae qaaOGaeyypa0ZaaSaaaeaacaGGOaGaaGymaiabgkHiTiaaikdacqaH 9oGBcaGGPaaabaGaaGinaiabec8aWjaacIcacaaIXaGaeyOeI0Iaeq yVd4MaaiykaiaadkhaaaWaaeWaaeaacaWGgbWaaSbaaSqaaiaaigda aeqaaOGaci4yaiaac+gacaGGZbGaeqiUdeNaey4kaSIaamOramaaBa aaleaacaaIYaaabeaakiGacohacaGGPbGaaiOBaiabeI7aXbGaayjk aiaawMcaaaqaaiabeo8aZnaaBaaaleaacaWGYbGaeqiUdehabeaaki abg2da9maalaaabaGaaiikaiaaigdacqGHsislcaaIYaGaeqyVd4Ma aiykaaqaaiaaisdacqaHapaCcaGGOaGaaGymaiabgkHiTiabe27aUj aacMcacaWGYbaaamaabmaabaGaamOramaaBaaaleaacaaIXaaabeaa kiGacohacaGGPbGaaiOBaiabeI7aXjabgkHiTiaadAeadaWgaaWcba GaaGOmaaqabaGcciGGJbGaai4BaiaacohacqaH4oqCaiaawIcacaGL Paaaaaaa@9974@

σ 11 = F 1 cosθ 4π(1ν)r ( 12ν+2 cos 2 θ )+ F 2 sinθ 4π(1ν)r ( 12ν2 cos 2 θ ) σ 22 = F 1 cosθ 4π(1ν)r ( 1+2ν2 cos 2 θ ) F 2 sinθ 4π(1ν)r ( 32ν2 cos 2 θ ) σ 12 = F 1 sinθ 4π(1ν)r ( 12ν+2 cos 2 θ ) F 2 cosθ 4π(1ν)r ( 32ν2 cos 2 θ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabeo8aZnaaBaaaleaacaaIXa GaaGymaaqabaGccqGH9aqpcqGHsisldaWcaaqaaiaadAeadaWgaaWc baGaaGymaaqabaGcciGGJbGaai4BaiaacohacqaH4oqCaeaacaaI0a GaeqiWdaNaaiikaiaaigdacqGHsislcqaH9oGBcaGGPaGaamOCaaaa daqadaqaaiaaigdacqGHsislcaaIYaGaeqyVd4Maey4kaSIaaGOmai GacogacaGGVbGaai4CamaaCaaaleqabaGaaGOmaaaakiabeI7aXbGa ayjkaiaawMcaaiabgUcaRmaalaaabaGaamOramaaBaaaleaacaaIYa aabeaakiGacohacaGGPbGaaiOBaiabeI7aXbqaaiaaisdacqaHapaC caGGOaGaaGymaiabgkHiTiabe27aUjaacMcacaWGYbaaamaabmaaba GaaGymaiabgkHiTiaaikdacqaH9oGBcqGHsislcaaIYaGaci4yaiaa c+gacaGGZbWaaWbaaSqabeaacaaIYaaaaOGaeqiUdehacaGLOaGaay zkaaaabaGaeq4Wdm3aaSbaaSqaaiaaikdacaaIYaaabeaakiabg2da 9iabgkHiTmaalaaabaGaamOramaaBaaaleaacaaIXaaabeaakiGaco gacaGGVbGaai4CaiabeI7aXbqaaiaaisdacqaHapaCcaGGOaGaaGym aiabgkHiTiabe27aUjaacMcacaWGYbaaamaabmaabaGaaGymaiabgU caRiaaikdacqaH9oGBcqGHsislcaaIYaGaci4yaiaac+gacaGGZbWa aWbaaSqabeaacaaIYaaaaOGaeqiUdehacaGLOaGaayzkaaGaeyOeI0 YaaSaaaeaacaWGgbWaaSbaaSqaaiaaikdaaeqaaOGaci4CaiaacMga caGGUbGaeqiUdehabaGaaGinaiabec8aWjaacIcacaaIXaGaeyOeI0 IaeqyVd4MaaiykaiaadkhaaaWaaeWaaeaacaaIZaGaeyOeI0IaaGOm aiabe27aUjabgkHiTiaaikdaciGGJbGaai4BaiaacohadaahaaWcbe qaaiaaikdaaaGccqaH4oqCaiaawIcacaGLPaaaaeaacqaHdpWCdaWg aaWcbaGaaGymaiaaikdaaeqaaOGaeyypa0JaeyOeI0YaaSaaaeaaca WGgbWaaSbaaSqaaiaaigdaaeqaaOGaci4CaiaacMgacaGGUbGaeqiU dehabaGaaGinaiabec8aWjaacIcacaaIXaGaeyOeI0IaeqyVd4Maai ykaiaadkhaaaWaaeWaaeaacaaIXaGaeyOeI0IaaGOmaiabe27aUjab gUcaRiaaikdaciGGJbGaai4BaiaacohadaahaaWcbeqaaiaaikdaaa GccqaH4oqCaiaawIcacaGLPaaacqGHsisldaWcaaqaaiaadAeadaWg aaWcbaGaaGOmaaqabaGcciGGJbGaai4BaiaacohacqaH4oqCaeaaca aI0aGaeqiWdaNaaiikaiaaigdacqGHsislcqaH9oGBcaGGPaGaamOC aaaadaqadaqaaiaaiodacqGHsislcaaIYaGaeqyVd4MaeyOeI0IaaG OmaiGacogacaGGVbGaai4CamaaCaaaleqabaGaaGOmaaaakiabeI7a XbGaayjkaiaawMcaaaaaaa@E964@

 

We will work through the algebra required to calculate these formulae for displacement and stress as a representative example.  In practice a symbolic manipulation program makes the calculations painless.  To begin, note that

log(z)=log(r e iθ )=log(r)+iθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiqacaWaaaGcbaGaciiBaiaac+gacaGGNbGaaiikaiaadQ hacaGGPaGaeyypa0JaciiBaiaac+gacaGGNbGaaiikaiaadkhacaWG LbWaaWbaaSqabeaacaWGPbGaeqiUdehaaOGaaiykaiabg2da9iGacY gacaGGVbGaai4zaiaacIcacaWGYbGaaiykaiabgUcaRiaadMgacqaH 4oqCcaaMcSoaaa@4B14@

and

d dz (log(z))= 1 z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadsgaaeaacaWGKbGaam OEaaaacaGGOaGaciiBaiaac+gacaGGNbGaaiikaiaadQhacaGGPaGa aiykaiabg2da9maalaaabaGaaGymaaqaaiaadQhaaaaaaa@3EF8@

The displacements are thus

E (1+ν) D=(34ν)Ω(z) ω(z) ¯ z Ω (z) ¯ = (34ν) 8π(1ν) ( ( F 1 +i F 2 )log(z)+ ( F 1 i F 2 ) ¯ log(z) ¯ ) F 1 +i F 2 ¯ 8π(1ν) z z ¯ = 2(34ν) 8π(1ν) ( F 1 +i F 2 )log(r) ( F 1 i F 2 ) 8π(1ν) r e iθ r e iθ = 2(34ν) 8π(1ν) ( F 1 +i F 2 )log(r) ( F 1 i F 2 ) 8π(1ν) e 2iθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiqacaWaaaGceaqabeaadaWcaaqaaiaadweaaeaacaGGOa GaaGymaiabgUcaRiabe27aUjaacMcaaaGaamiraiabg2da9iaacIca caaIZaGaeyOeI0IaaGinaiabe27aUjaacMcacqqHPoWvcaGGOaGaam OEaiaacMcacqGHsislcaaMc8+aa0aaaeaacqaHjpWDcaGGOaGaamOE aiaacMcaaaGaeyOeI0IaamOEaiaaykW7caaMc8+aa0aaaeaacuqHPo WvgaqbaiaacIcacaWG6bGaaiykaaaaaeaacaaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7cqGH9aqpcaaMcSUaeyOeI0YaaSaaaeaacaGGOaGaaG4maiab gkHiTiaaisdacqaH9oGBcaGGPaaabaGaaGioaiabec8aWjaacIcaca aIXaGaeyOeI0IaeqyVd4MaaiykaaaadaqadaqaamaabmaabaGaamOr amaaBaaaleaacaaIXaaabeaakiabgUcaRiaadMgacaWGgbWaaSbaaS qaaiaaikdaaeqaaaGccaGLOaGaayzkaaGaciiBaiaac+gacaGGNbGa aiikaiaadQhacaGGPaGaey4kaSYaa0aaaeaadaqadaqaaiaadAeada WgaaWcbaGaaGymaaqabaGccqGHsislcaWGPbGaamOramaaBaaaleaa caaIYaaabeaaaOGaayjkaiaawMcaaaaadaqdaaqaaiGacYgacaGGVb Gaai4zaiaacIcacaWG6bGaaiykaaaaaiaawIcacaGLPaaacqGHsisl daWcaaqaamaanaaabaGaamOramaaBaaaleaacaaIXaaabeaakiabgU caRiaadMgacaWGgbWaaSbaaSqaaiaaikdaaeqaaaaaaOqaaiaaiIda cqaHapaCcaGGOaGaaGymaiabgkHiTiabe27aUjaacMcaaaWaaSaaae aacaWG6baabaGabmOEayaaraaaaaqaaiaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7cqGH9aqpcqGHsisldaWcaaqaaiaaikdacaGGOaGaaG4m aiabgkHiTiaaisdacqaH9oGBcaGGPaaabaGaaGioaiabec8aWjaacI cacaaIXaGaeyOeI0IaeqyVd4MaaiykaaaadaqadaqaaiaadAeadaWg aaWcbaGaaGymaaqabaGccqGHRaWkcaWGPbGaamOramaaBaaaleaaca aIYaaabeaaaOGaayjkaiaawMcaaiGacYgacaGGVbGaai4zaiaacIca caWGYbGaaiykaiabgkHiTmaalaaabaWaaeWaaeaacaWGgbWaaSbaaS qaaiaaigdaaeqaaOGaeyOeI0IaamyAaiaadAeadaWgaaWcbaGaaGOm aaqabaaakiaawIcacaGLPaaaaeaacaaI4aGaeqiWdaNaaiikaiaaig dacqGHsislcqaH9oGBcaGGPaaaamaalaaabaGaamOCaiaadwgadaah aaWcbeqaaiaadMgacqaH4oqCaaaakeaacaWGYbGaamyzamaaCaaale qabaGaeyOeI0IaamyAaiabeI7aXbaaaaaakeaacaaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlabg2da9iabgkHiTmaalaaabaGaaGOmaiaacIcacaaIZaGaeyOe I0IaaGinaiabe27aUjaacMcaaeaacaaI4aGaeqiWdaNaaiikaiaaig dacqGHsislcqaH9oGBcaGGPaaaamaabmaabaGaamOramaaBaaaleaa caaIXaaabeaakiabgUcaRiaadMgacaWGgbWaaSbaaSqaaiaaikdaae qaaaGccaGLOaGaayzkaaGaciiBaiaac+gacaGGNbGaaiikaiaadkha caGGPaGaeyOeI0YaaSaaaeaadaqadaqaaiaadAeadaWgaaWcbaGaaG ymaaqabaGccqGHsislcaWGPbGaamOramaaBaaaleaacaaIYaaabeaa aOGaayjkaiaawMcaaaqaaiaaiIdacqaHapaCcaGGOaGaaGymaiabgk HiTiabe27aUjaacMcaaaGaamyzamaaCaaaleqabaGaaGOmaiaadMga cqaH4oqCaaaaaaa@271C@

Finally, using Euler’s formula and taking real and imaginary parts gives the answer listed earlier.  Similarly, the formulas for stress give

σ rr + σ θθ =2( Ω (z)+ Ω (z) ¯ )= 1 4π(1ν) ( F 1 +i F 2 z + F 1 i F 2 z ¯ )= ( F 1 +i F 2 ) e iθ +( F 1 i F 2 ) e iθ 4π(1ν)r σ rr σ θθ +2i σ rθ =2( z Ω (z) ¯ + ω (z) ¯ ) e 2iθ = e 2iθ 4π(1ν) { ( F 1 i F 2 ) r e iθ r 2 e 2iθ +(34ν)( F 1 +i F 2 ) 1 r e iθ } = 1 4π(1ν)r { ( F 1 i F 2 ) e iθ +(34ν)( F 1 +i F 2 ) e iθ } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiqacaWaaaGceaqabeaacqaHdpWCdaWgaaWcbaGaamOCai aadkhaaeqaaOGaey4kaSIaeq4Wdm3aaSbaaSqaaiabeI7aXjabeI7a XbqabaGccqGH9aqpcaaIYaWaaeWaaeaacuqHPoWvgaqbaiaacIcaca WG6bGaaiykaiabgUcaRmaanaaabaGafuyQdCLbauaacaGGOaGaamOE aiaacMcaaaaacaGLOaGaayzkaaGaeyypa0JaeyOeI0YaaSaaaeaaca aIXaaabaGaaGinaiabec8aWjaacIcacaaIXaGaeyOeI0IaeqyVd4Ma aiykaaaadaqadaqaamaalaaabaGaamOramaaBaaaleaacaaIXaaabe aakiabgUcaRiaadMgacaWGgbWaaSbaaSqaaiaaikdaaeqaaaGcbaGa amOEaaaacqGHRaWkdaWcaaqaaiaadAeadaWgaaWcbaGaaGymaaqaba GccqGHsislcaWGPbGaamOramaaBaaaleaacaaIYaaabeaaaOqaaiqa dQhagaqeaaaaaiaawIcacaGLPaaacqGH9aqpcqGHsisldaWcaaqaai aacIcacaWGgbWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaamyAaiaa dAeadaWgaaWcbaGaaGOmaaqabaGccaGGPaGaamyzamaaCaaaleqaba GaeyOeI0IaamyAaiabeI7aXbaakiabgUcaRiaacIcacaWGgbWaaSba aSqaaiaaigdaaeqaaOGaeyOeI0IaamyAaiaadAeadaWgaaWcbaGaaG OmaaqabaGccaGGPaGaamyzamaaCaaaleqabaGaamyAaiabeI7aXbaa aOqaaiaaisdacqaHapaCcaGGOaGaaGymaiabgkHiTiabe27aUjaacM cacaWGYbaaaaqaaiabeo8aZnaaBaaaleaacaWGYbGaamOCaaqabaGc cqGHsislcqaHdpWCdaWgaaWcbaGaeqiUdeNaeqiUdehabeaakiabgU caRiaaikdacaWGPbGaeq4Wdm3aaSbaaSqaaiaadkhacqaH4oqCaeqa aOGaeyypa0JaeyOeI0IaaGOmamaabmaabaGaamOEaiaaykW7daqdaa qaaiqbfM6axzaagaGaaiikaiaadQhacaGGPaaaaiabgUcaRmaanaaa baGafqyYdCNbauaacaGGOaGaamOEaiaacMcaaaaacaGLOaGaayzkaa GaamyzamaaCaaaleqabaGaeyOeI0IaaGOmaiaadMgacqaH4oqCaaaa keaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqGH9a qpcqGHsisldaWcaaqaaiaadwgadaahaaWcbeqaaiabgkHiTiaaikda caWGPbGaeqiUdehaaaGcbaGaaGinaiabec8aWjaacIcacaaIXaGaey OeI0IaeqyVd4MaaiykaaaadaGadaqaaiaacIcacaWGgbWaaSbaaSqa aiaaigdaaeqaaOGaeyOeI0IaamyAaiaadAeadaWgaaWcbaGaaGOmaa qabaGccaGGPaWaaSaaaeaacaWGYbGaamyzamaaCaaaleqabaGaamyA aiabeI7aXbaaaOqaaiaadkhadaahaaWcbeqaaiaaikdaaaGccaWGLb WaaWbaaSqabeaacqGHsislcaaIYaGaamyAaiabeI7aXbaaaaGccqGH RaWkcaGGOaGaaG4maiabgkHiTiaaisdacqaH9oGBcaGGPaGaaiikai aadAeadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcaWGPbGaamOramaa BaaaleaacaaIYaaabeaakiaacMcadaWcaaqaaiaaigdaaeaacaWGYb GaamyzamaaCaaaleqabaGaeyOeI0IaamyAaiabeI7aXbaaaaaakiaa wUhacaGL9baaaeaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8Uaeyypa0JaeyOeI0YaaSaaaeaacaaIXaaabaGaaGin aiabec8aWjaacIcacaaIXaGaeyOeI0IaeqyVd4Maaiykaiaadkhaaa WaaiWaaeaacaGGOaGaamOramaaBaaaleaacaaIXaaabeaakiabgkHi TiaadMgacaWGgbWaaSbaaSqaaiaaikdaaeqaaOGaaiykaiaadwgada ahaaWcbeqaaiaadMgacqaH4oqCaaGccqGHRaWkcaGGOaGaaG4maiab gkHiTiaaisdacqaH9oGBcaGGPaGaaiikaiaadAeadaWgaaWcbaGaaG ymaaqabaGccqGHRaWkcaWGPbGaamOramaaBaaaleaacaaIYaaabeaa kiaacMcacaWGLbWaaWbaaSqabeaacqGHsislcaWGPbGaeqiUdehaaa GccaGL7bGaayzFaaaaaaa@8228@

Adding the two formulas for stress shows that

2 σ rr +2i σ rθ = 1 4π(1ν)r { 2( F 1 i F 2 ) e iθ +4(1ν)( F 1 +i F 2 ) e iθ } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaaIYaGaeq4Wdm3aaSbaaSqaaiaadk hacaWGYbaabeaakiabgUcaRiaaikdacaWGPbGaeq4Wdm3aaSbaaSqa aiaadkhacqaH4oqCaeqaaOGaeyypa0JaeyOeI0YaaSaaaeaacaaIXa aabaGaaGinaiabec8aWjaacIcacaaIXaGaeyOeI0IaeqyVd4Maaiyk aiaadkhaaaWaaiWaaeaacaaIYaGaaiikaiaadAeadaWgaaWcbaGaaG ymaaqabaGccqGHsislcaWGPbGaamOramaaBaaaleaacaaIYaaabeaa kiaacMcacaWGLbWaaWbaaSqabeaacaWGPbGaeqiUdehaaOGaey4kaS IaaGinaiaacIcacaaIXaGaeyOeI0IaeqyVd4MaaiykaiaacIcacaWG gbWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaamyAaiaadAeadaWgaa WcbaGaaGOmaaqabaGccaGGPaGaamyzamaaCaaaleqabaGaeyOeI0Ia amyAaiabeI7aXbaaaOGaay5Eaiaaw2haaaaa@6890@

Using Euler’s formula and taking real and imaginary parts of this expression gives the formulas for σ rr MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOCaiaadk haaeqaaaaa@36B3@  and σ rθ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOCaiabeI 7aXbqabaaaaa@3772@

 

Finally, we need to verify that the stresses are consistent with a point force acting at the origin.  To do this, we can evaluate the resultant force exerted by tractions acting on a circle enclosing the point force.  Since the solid is in static equilibrium, the total force acting on this circular region must sum to zero.  Recall that the resultant force exerted by stresses on an internal surface can be calculated as

R= A nσdA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHsbGaeyypa0Zaa8quaeaacaWHUb GaeyyXICTaaC4WdiaadsgacaWGbbaaleaacaWGbbaabeqdcqGHRiI8 aaaa@3DF6@

A unit normal to the circle is n=cosθ e 1 +sinθ e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHUbGaeyypa0Jaci4yaiaac+gaca GGZbGaeqiUdeNaaCyzamaaBaaaleaacaaIXaaabeaakiabgUcaRiGa cohacaGGPbGaaiOBaiabeI7aXjaahwgadaWgaaWcbaGaaGOmaaqaba aaaa@4271@ ; multiplying by the stress tensor (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) gives

R 1 = 0 2π ( σ 11 cosθ+ σ 12 sinθ ) rdθ R 2 = 0 2π ( σ 22 sinθ+ σ 12 cosθ ) rdθ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGsbWaaSbaaSqaaiaaigdaaeqaaO Gaeyypa0Zaa8qCaeaadaqadaqaaiabeo8aZnaaBaaaleaacaaIXaGa aGymaaqabaGcciGGJbGaai4BaiaacohacqaH4oqCcqGHRaWkcqaHdp WCdaWgaaWcbaGaaGymaiaaikdaaeqaaOGaci4CaiaacMgacaGGUbGa eqiUdehacaGLOaGaayzkaaaaleaacaaIWaaabaGaaGOmaiabec8aWb qdcqGHRiI8aOGaamOCaiaadsgacqaH4oqCcaaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaamOuamaaBaaaleaacaaIYaaabe aakiabg2da9maapehabaWaaeWaaeaacqaHdpWCdaWgaaWcbaGaaGOm aiaaikdaaeqaaOGaci4CaiaacMgacaGGUbGaeqiUdeNaey4kaSIaeq 4Wdm3aaSbaaSqaaiaaigdacaaIYaaabeaakiGacogacaGGVbGaai4C aiabeI7aXbGaayjkaiaawMcaaaWcbaGaaGimaaqaaiaaikdacqaHap aCa0Gaey4kIipakiaadkhacaWGKbGaeqiUdehaaa@7A6C@

Evaluating the integrals shows that R 1 = F 1 R 2 = F 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGsbWaaSbaaSqaaiaaigdaaeqaaO Gaeyypa0JaeyOeI0IaamOramaaBaaaleaacaaIXaaabeaakiaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaamOuamaaBaaaleaacaaIYaaabe aakiabg2da9iabgkHiTiaadAeadaWgaaWcbaGaaGOmaaqabaaaaa@4563@ , so R+ F 1 e 1 + F 2 e 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHsbGaey4kaSIaamOramaaBaaale aacaaIXaaabeaakiaahwgadaWgaaWcbaGaaGymaaqabaGccqGHRaWk caWGgbWaaSbaaSqaaiaaikdaaeqaaOGaaCyzamaaBaaaleaacaaIYa aabeaakiabg2da9iaahcdaaaa@3E5C@  as required.

 

 

 

 

5.3.4 Complex variable solution for an edge dislocation in an infinite solid

 

A dislocation is an atomic-scale defect in a crystal.  The defect can be detected directly in high-resolution transmission electron microscope pictures, which can show the positions of individual atoms in a crystal.  The picture shows a typical example (a Lomer dislocation at the interface between InGaAs and GaAs, from Tillmann et al Microsc. Microanal. 10, 185–198, 2004).  The dislocation is not easy to see, but can be identified by describing a `burger’s circuit’ around the dislocation, as shown by the yellow line.  Each straight portion of the circuit connects seven atoms.   In a perfect crystal, the circuit would start and end at the same atom.  (Try this for yourself for any path that does not encircle the dislocation).  Since the yellow curve encircles the dislocation, it does not start and end on the same atom.  The `Burger’s vector’ for the dislocation is the difference in position vector of the start and end atom, as shown in the picture.

 

A continuum model of a dislocation can be created using the procedure illustrated in the picture.  Take an elastic solid, and cut part-way through it.  The edge of the cut defines a dislocation line ξ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH+oaaaa@3420@ .  Next, displace the two material surfaces created by the cut by the burger’s vector b, and fill in the (infinitesimal) gap. Note that (by convention) the burger’s vector specifies the displacement of a point at the end of the Burger’s circuit as seen by an observer who sits on the start of the circuit, as shown in the picture.

 

HEALTH WARNING: Some texts define the Burger’s vector to be the negative of the vector defined here MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  that is to say, the vector pointing from the end of the circuit back to the start.

 

A general Burger’s vector has three components MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  the component b s =bξ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGIbWaaSbaaSqaaiaadohaaeqaaO Gaeyypa0JaaCOyaiabgwSixlaah67aaaa@3A60@  parallel to the dislocation line is known as the screw component of b, while the two remaining components b e =b b s ξ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHIbWaaSbaaSqaaiaadwgaaeqaaO Gaeyypa0JaaCOyaiabgkHiTiaadkgadaWgaaWcbaGaam4CaaqabaGc caWH+oaaaa@3B0E@  are known as the edge components of b. The stress field induced by the dislocation depends only on ξ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH+oaaaa@3420@  and b, and is independent of the cut that created it.

 

The displacement and stress field induced by a pure edge dislocation, with line direction parallel to the e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaiodaaeqaaa aa@34AD@  axis and burgers vector b= b 1 e 1 + b 2 e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHIbGaeyypa0JaamOyamaaBaaale aacaaIXaaabeaakiaahwgadaWgaaWcbaGaaGymaaqabaGccqGHRaWk caWGIbWaaSbaaSqaaiaaikdaaeqaaOGaaCyzamaaBaaaleaacaaIYa aabeaaaaa@3CFF@  at the origin of an infinite solid can be derived from the complex potentials

Ω(z)=i E( b 1 +i b 2 ) 8π(1 ν 2 ) log(z)ω(z)=i E( b 1 i b 2 ) 8π(1 ν 2 ) log(z) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiqacaWaaaGcbaGaeuyQdCLaaiikaiaadQhacaGGPaGaey ypa0JaeyOeI0IaamyAamaalaaabaGaamyramaabmaabaGaamOyamaa BaaaleaacaaIXaaabeaakiabgUcaRiaadMgacaWGIbWaaSbaaSqaai aaikdaaeqaaaGccaGLOaGaayzkaaaabaGaaGioaiabec8aWjaacIca caaIXaGaeyOeI0IaeqyVd42aaWbaaSqabeaacaaIYaaaaOGaaiykaa aaciGGSbGaai4BaiaacEgacaGGOaGaamOEaiaacMcacaaMcSUaaGPa RlaaykW6caaMcSUaaGPaRlaaykW6caaMcSUaaGPaRlaaykW6caaMcS UaaGPaRlaaykW6caaMcSUaaGPaRlaaykW6caaMcSUaaGPaRlaaykW6 caaMcSUaaGPaRlabeM8a3jaacIcacaWG6bGaaiykaiabg2da9iaadM gadaWcaaqaaiaadweacaGGOaGaamOyamaaBaaaleaacaaIXaaabeaa kiabgkHiTiaadMgacaWGIbWaaSbaaSqaaiaaikdaaeqaaOGaaiykaa qaaiaaiIdacqaHapaCcaGGOaGaaGymaiabgkHiTiabe27aUnaaCaaa leqabaGaaGOmaaaakiaacMcaaaGaciiBaiaac+gacaGGNbGaaiikai aadQhacaGGPaaaaa@8681@

 

The displacement and stresses (in polar coordinates) can be derived from these potentials as

u 1 = θ b 1 2π + 12ν 4π(1ν) b 2 log(r) 1 8π(1ν) ( b 2 cos2θ b 1 sin2θ ) u 2 = θ b 2 2π 12ν 4π(1ν) b 1 log(r) 1 8π(1ν) ( b 1 cos2θ+ b 2 sin2θ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaadwhadaWgaaWcbaGaaGymaa qabaGccqGH9aqpdaWcaaqaaiabeI7aXjaadkgadaWgaaWcbaGaaGym aaqabaaakeaacaaIYaGaeqiWdahaaiabgUcaRmaalaaabaGaaGymai abgkHiTiaaikdacqaH9oGBaeaacaaI0aGaeqiWdaNaaiikaiaaigda cqGHsislcqaH9oGBcaGGPaaaaiaadkgadaWgaaWcbaGaaGOmaaqaba GcciGGSbGaai4BaiaacEgacaGGOaGaamOCaiaacMcacqGHsisldaWc aaqaaiaaigdaaeaacaaI4aGaeqiWdaNaaiikaiaaigdacqGHsislcq aH9oGBcaGGPaaaamaabmaabaGaamOyamaaBaaaleaacaaIYaaabeaa kiGacogacaGGVbGaai4CaiaaikdacqaH4oqCcqGHsislcaWGIbWaaS baaSqaaiaaigdaaeqaaOGaci4CaiaacMgacaGGUbGaaGOmaiabeI7a XbGaayjkaiaawMcaaaqaaiaadwhadaWgaaWcbaGaaGOmaaqabaGccq GH9aqpdaWcaaqaaiabeI7aXjaadkgadaWgaaWcbaGaaGOmaaqabaaa keaacaaIYaGaeqiWdahaaiabgkHiTmaalaaabaGaaGymaiabgkHiTi aaikdacqaH9oGBaeaacaaI0aGaeqiWdaNaaiikaiaaigdacqGHsisl cqaH9oGBcaGGPaaaaiaadkgadaWgaaWcbaGaaGymaaqabaGcciGGSb Gaai4BaiaacEgacaGGOaGaamOCaiaacMcacqGHsisldaWcaaqaaiaa igdaaeaacaaI4aGaeqiWdaNaaiikaiaaigdacqGHsislcqaH9oGBca GGPaaaamaabmaabaGaamOyamaaBaaaleaacaaIXaaabeaakiGacoga caGGVbGaai4CaiaaikdacqaH4oqCcqGHRaWkcaWGIbWaaSbaaSqaai aaikdaaeqaaOGaci4CaiaacMgacaGGUbGaaGOmaiabeI7aXbGaayjk aiaawMcaaaaaaa@9EA0@

σ rr = σ θθ = E( b 1 sinθ b 2 cosθ) 4π(1 ν 2 )r σ rθ = E( b 1 cosθ+ b 2 sinθ) 4π(1 ν 2 )r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamOCaiaadk haaeqaaOGaeyypa0Jaeq4Wdm3aaSbaaSqaaiabeI7aXjabeI7aXbqa baGccqGH9aqpcqGHsisldaWcaaqaaiaadweacaGGOaGaamOyamaaBa aaleaacaaIXaaabeaakiGacohacaGGPbGaaiOBaiabeI7aXjabgkHi TiaadkgadaWgaaWcbaGaaGOmaaqabaGcciGGJbGaai4Baiaacohacq aH4oqCcaGGPaaabaGaaGinaiabec8aWjaacIcacaaIXaGaeyOeI0Ia eqyVd42aaWbaaSqabeaacaaIYaaaaOGaaiykaiaadkhaaaGaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7cqaHdpWCdaWgaaWcbaGaamOCaiabeI7aXbqabaGccq GH9aqpdaWcaaqaaiaadweacaGGOaGaamOyamaaBaaaleaacaaIXaaa beaakiGacogacaGGVbGaai4CaiabeI7aXjabgUcaRiaadkgadaWgaa WcbaGaaGOmaaqabaGcciGGZbGaaiyAaiaac6gacqaH4oqCcaGGPaaa baGaaGinaiabec8aWjaacIcacaaIXaGaeyOeI0IaeqyVd42aaWbaaS qabeaacaaIYaaaaOGaaiykaiaadkhaaaaaaa@87B0@

σ 11 = E b 1 (3sinθ+sin3θ) 8π(1 ν 2 )r + E b 2 (cosθ+cos3θ) 8π(1 ν 2 )r σ 22 = E b 1 (sinθsin3θ) 8π(1 ν 2 )r + E b 2 (3cosθcos3θ) 8π(1 ν 2 )r σ 12 =+ E b 1 (cosθ+cos3θ) 8π(1 ν 2 )r E b 2 (sinθsin3θ) 8π(1 ν 2 )r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabeo8aZnaaBaaaleaacaaIXa GaaGymaaqabaGccqGH9aqpcqGHsisldaWcaaqaaiaadweacaWGIbWa aSbaaSqaaiaaigdaaeqaaOGaaiikaiaaiodaciGGZbGaaiyAaiaac6 gacqaH4oqCcqGHRaWkciGGZbGaaiyAaiaac6gacaaIZaGaeqiUdeNa aiykaaqaaiaaiIdacqaHapaCcaGGOaGaaGymaiabgkHiTiabe27aUn aaCaaaleqabaGaaGOmaaaakiaacMcacaWGYbaaaiabgUcaRmaalaaa baGaamyraiaadkgadaWgaaWcbaGaaGOmaaqabaGccaGGOaGaci4yai aac+gacaGGZbGaeqiUdeNaey4kaSIaci4yaiaac+gacaGGZbGaaG4m aiabeI7aXjaacMcaaeaacaaI4aGaeqiWdaNaaiikaiaaigdacqGHsi slcqaH9oGBdaahaaWcbeqaaiaaikdaaaGccaGGPaGaamOCaaaaaeaa cqaHdpWCdaWgaaWcbaGaaGOmaiaaikdaaeqaaOGaeyypa0JaeyOeI0 YaaSaaaeaacaWGfbGaamOyamaaBaaaleaacaaIXaaabeaakiaacIca ciGGZbGaaiyAaiaac6gacqaH4oqCcqGHsislciGGZbGaaiyAaiaac6 gacaaIZaGaeqiUdeNaaiykaaqaaiaaiIdacqaHapaCcaGGOaGaaGym aiabgkHiTiabe27aUnaaCaaaleqabaGaaGOmaaaakiaacMcacaWGYb aaaiabgUcaRmaalaaabaGaamyraiaadkgadaWgaaWcbaGaaGOmaaqa baGccaGGOaGaaG4maiGacogacaGGVbGaai4CaiabeI7aXjabgkHiTi GacogacaGGVbGaai4CaiaaiodacqaH4oqCcaGGPaaabaGaaGioaiab ec8aWjaacIcacaaIXaGaeyOeI0IaeqyVd42aaWbaaSqabeaacaaIYa aaaOGaaiykaiaadkhaaaaabaGaeq4Wdm3aaSbaaSqaaiaaigdacaaI Yaaabeaakiabg2da9iabgUcaRmaalaaabaGaamyraiaadkgadaWgaa WcbaGaaGymaaqabaGccaGGOaGaci4yaiaac+gacaGGZbGaeqiUdeNa ey4kaSIaci4yaiaac+gacaGGZbGaaG4maiabeI7aXjaacMcaaeaaca aI4aGaeqiWdaNaaiikaiaaigdacqGHsislcqaH9oGBdaahaaWcbeqa aiaaikdaaaGccaGGPaGaamOCaaaacqGHsisldaWcaaqaaiaadweaca WGIbWaaSbaaSqaaiaaikdaaeqaaOGaaiikaiGacohacaGGPbGaaiOB aiabeI7aXjabgkHiTiGacohacaGGPbGaaiOBaiaaiodacqaH4oqCca GGPaaabaGaaGioaiabec8aWjaacIcacaaIXaGaeyOeI0IaeqyVd42a aWbaaSqabeaacaaIYaaaaOGaaiykaiaadkhaaaaaaaa@D68D@

The displacement components are plotted in the picture below, for a dislocation with b 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGIbWaaSbaaSqaaiaaikdaaeqaaO Gaeyypa0JaaGimaaaa@365F@ .  The contours show a sudden jump in u 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaaigdaaeqaaa aa@34B7@  at x 2 =0, x 1 >0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaikdaaeqaaO Gaeyypa0JaaGimaiaacYcacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGa eyOpa4JaaGimaaaa@3AD5@  (This is caused by the term involving θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH4oqCaaa@348C@  in the formula for u 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaaigdaaeqaaa aa@34A7@  - we assumed that 0<θ<2π MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaaIWaGaeyipaWJaeqiUdeNaeyipaW JaaGOmaiabec8aWbaa@39B7@  when plotting the displacement contours). Physically, the plane x 2 =0, x 1 >0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaikdaaeqaaO Gaeyypa0JaaGimaiaacYcacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGa eyOpa4JaaGimaaaa@3AD5@  corresponds to the `cut’ that created the dislocation, and the jump in displacement across the cut is equal to the Burger’s vector.

   

Contours of stress are plotted in the figure below.  The radial and hoop stresses are compressive above the dislocation, and tensile below it, as one would expect.  Shear stress is positive to the right of the dislocation and negative to the left, again, in concord with our physical intuition.  The stresses are infinite at the dislocation itself, but of course in this region linear elasticity does not accurately model material behavior, because the atomic bonds are very severely distorted.

        

Radial or hoop stress                                                            Shear stress

 

 

 

5.3.5 Cylindrical hole in an infinite solid under remote loading

 

The figure shows a circular cylindrical cavity with radius a in an infinite, isotropic linear elastic solid. Far from the cavity, the solid is subjected to a tensile stress σ 11 = σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIXaGaaGymaa qabaGccqGH9aqpcqaHdpWCdaWgaaWcbaGaaGimaaqabaaaaa@3786@ , with all other stress components zero.

 

The solution is generated by complex potentials

Ω(z)= σ 0 4 ( z+ 2 a 2 z )ω(z)= σ 0 2 ( z+ a 2 z a 4 z 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfM6axjaacIcacaWG6bGaaiykaiabg2 da9maalaaabaGaeq4Wdm3aaSbaaSqaaiaaicdaaeqaaaGcbaGaaGin aaaadaqadaqaaiaadQhacqGHRaWkdaWcaaqaaiaaikdacaWGHbWaaW baaSqabeaacaaIYaaaaaGcbaGaamOEaaaaaiaawIcacaGLPaaacaaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaeqyYdCNaaiikaiaadQhacaGGPaGaeyypa0ZaaSaaaeaa cqGHsislcqaHdpWCdaWgaaWcbaGaaGimaaqabaaakeaacaaIYaaaam aabmaabaGaamOEaiabgUcaRmaalaaabaGaamyyamaaCaaaleqabaGa aGOmaaaaaOqaaiaadQhaaaGaeyOeI0YaaSaaaeaacaWGHbWaaWbaaS qabeaacaaI0aaaaaGcbaGaamOEamaaCaaaleqabaGaaG4maaaaaaaa kiaawIcacaGLPaaaaaa@6405@

 

The displacement and stress state is easily calculated as

u 1 = σ 0 (1+ν)a 2E { 2( 1ν )( r a + 2a r )cosθ+( a r a 3 r 3 )cos3θ } u 2 = σ 0 (1+ν)a 2E { 2( 12ν ) a r sinθ2ν r a sinθ+( a r a 3 r 3 )sin3θ } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamyDamaaBaaaleaacaaIXaaabe aakiabg2da9maalaaabaGaeq4Wdm3aaSbaaSqaaiaaicdaaeqaaOGa aiikaiaaigdacqGHRaWkcqaH9oGBcaGGPaGaamyyaaqaaiaaikdaca WGfbaaamaacmaabaGaaGOmamaabmaabaGaaGymaiabgkHiTiabe27a UbGaayjkaiaawMcaamaabmaabaWaaSaaaeaacaWGYbaabaGaamyyaa aacqGHRaWkdaWcaaqaaiaaikdacaWGHbaabaGaamOCaaaaaiaawIca caGLPaaaciGGJbGaai4BaiaacohacqaH4oqCcqGHRaWkdaqadaqaam aalaaabaGaamyyaaqaaiaadkhaaaGaeyOeI0YaaSaaaeaacaWGHbWa aWbaaSqabeaacaaIZaaaaaGcbaGaamOCamaaCaaaleqabaGaaG4maa aaaaaakiaawIcacaGLPaaaciGGJbGaai4BaiaacohacaaIZaGaeqiU dehacaGL7bGaayzFaaaabaGaamyDamaaBaaaleaacaaIYaaabeaaki abg2da9maalaaabaGaeq4Wdm3aaSbaaSqaaiaaicdaaeqaaOGaaiik aiaaigdacqGHRaWkcqaH9oGBcaGGPaGaamyyaaqaaiaaikdacaWGfb aaamaacmaabaGaeyOeI0IaaGOmamaabmaabaGaaGymaiabgkHiTiaa ikdacqaH9oGBaiaawIcacaGLPaaadaWcaaqaaiaadggaaeaacaWGYb aaaiGacohacaGGPbGaaiOBaiabeI7aXjabgkHiTiaaikdacqaH9oGB daWcaaqaaiaadkhaaeaacaWGHbaaaiGacohacaGGPbGaaiOBaiabeI 7aXjabgUcaRmaabmaabaWaaSaaaeaacaWGHbaabaGaamOCaaaacqGH sisldaWcaaqaaiaadggadaahaaWcbeqaaiaaiodaaaaakeaacaWGYb WaaWbaaSqabeaacaaIZaaaaaaaaOGaayjkaiaawMcaaiGacohacaGG PbGaaiOBaiaaiodacqaH4oqCaiaawUhacaGL9baaaaaa@9407@

σ 11 = σ 0 ( 1+( 3 a 4 2 r 4 a 2 r 2 )cos4θ 3 a 2 2 r 2 cos2θ ) σ 22 = σ 0 ( ( a 2 r 2 3 a 4 2 r 4 )cos4θ a 2 2 r 2 cos2θ ) σ 12 = σ 0 ( ( 3 a 4 2 r 4 a 2 r 2 )sin4θ a 2 2 r 2 sin2θ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaeq4Wdm3aaSbaaSqaaiaaigdaca aIXaaabeaakiabg2da9iabeo8aZnaaBaaaleaacaaIWaaabeaakmaa bmaabaGaaGymaiabgUcaRmaabmaabaWaaSaaaeaacaaIZaGaamyyam aaCaaaleqabaGaaGinaaaaaOqaaiaaikdacaWGYbWaaWbaaSqabeaa caaI0aaaaaaakiabgkHiTmaalaaabaGaamyyamaaCaaaleqabaGaaG OmaaaaaOqaaiaadkhadaahaaWcbeqaaiaaikdaaaaaaaGccaGLOaGa ayzkaaGaci4yaiaac+gacaGGZbGaaGinaiabeI7aXjabgkHiTmaala aabaGaaG4maiaadggadaahaaWcbeqaaiaaikdaaaaakeaacaaIYaGa amOCamaaCaaaleqabaGaaGOmaaaaaaGcciGGJbGaai4Baiaacohaca aIYaGaeqiUdehacaGLOaGaayzkaaGaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 Uaeq4Wdm3aaSbaaSqaaiaaikdacaaIYaaabeaakiabg2da9iabeo8a ZnaaBaaaleaacaaIWaaabeaakmaabmaabaWaaeWaaeaadaWcaaqaai aadggadaahaaWcbeqaaiaaikdaaaaakeaacaWGYbWaaWbaaSqabeaa caaIYaaaaaaakiabgkHiTmaalaaabaGaaG4maiaadggadaahaaWcbe qaaiaaisdaaaaakeaacaaIYaGaamOCamaaCaaaleqabaGaaGinaaaa aaaakiaawIcacaGLPaaaciGGJbGaai4BaiaacohacaaI0aGaeqiUde NaeyOeI0YaaSaaaeaacaWGHbWaaWbaaSqabeaacaaIYaaaaaGcbaGa aGOmaiaadkhadaahaaWcbeqaaiaaikdaaaaaaOGaci4yaiaac+gaca GGZbGaaGOmaiabeI7aXbGaayjkaiaawMcaaaqaaiabeo8aZnaaBaaa leaacaaIXaGaaGOmaaqabaGccqGH9aqpcqaHdpWCdaWgaaWcbaGaaG imaaqabaGcdaqadaqaamaabmaabaWaaSaaaeaacaaIZaGaamyyamaa CaaaleqabaGaaGinaaaaaOqaaiaaikdacaWGYbWaaWbaaSqabeaaca aI0aaaaaaakiabgkHiTmaalaaabaGaamyyamaaCaaaleqabaGaaGOm aaaaaOqaaiaadkhadaahaaWcbeqaaiaaikdaaaaaaaGccaGLOaGaay zkaaGaci4CaiaacMgacaGGUbGaaGinaiabeI7aXjabgkHiTmaalaaa baGaamyyamaaCaaaleqabaGaaGOmaaaaaOqaaiaaikdacaWGYbWaaW baaSqabeaacaaIYaaaaaaakiGacohacaGGPbGaaiOBaiaaikdacqaH 4oqCaiaawIcacaGLPaaaaaaa@B250@

 

 

 

 

5.3.6 Crack in an infinite elastic solid under remote loading

 

The figure shows a 2D crack with length 2a in an infinite solid, which is subjected to a uniform state of stress σ 22 , σ 12 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaqhaaWcbaGaaGOmaiaaik daaeaacqGHEisPaaGccaGGSaGaeq4Wdm3aa0baaSqaaiaaigdacaaI YaaabaGaeyOhIukaaaaa@3D41@  at infinity. The solution can be generated by complex potentials

Ω(z)= 1 4 ( σ 22 )z+ 1 2 ( σ 22 i σ 12 )( z 2 a 2 z ) ω(z)= Ω( z ¯ ) ¯ zΩ'(z)+ σ 22 z/2+i σ 12 z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqqHPo WvcaGGOaGaamOEaiaacMcacqGH9aqpdaWcaaqaaiaaigdaaeaacaaI 0aaaamaabmaabaGaeq4Wdm3aa0baaSqaaiaaikdacaaIYaaabaGaey OhIukaaaGccaGLOaGaayzkaaGaamOEaiabgUcaRmaalaaabaGaaGym aaqaaiaaikdaaaWaaeWaaeaacqaHdpWCdaqhaaWcbaGaaGOmaiaaik daaeaacqGHEisPaaGccqGHsislcaWGPbGaeq4Wdm3aa0baaSqaaiaa igdacaaIYaaabaGaeyOhIukaaaGccaGLOaGaayzkaaWaaeWaaeaada GcaaqaaiaadQhadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWGHbWa aWbaaSqabeaacaaIYaaaaaqabaGccqGHsislcaWG6baacaGLOaGaay zkaaaabaGaeqyYdCNaaiikaiaadQhacaGGPaGaeyypa0Zaa0aaaeaa cqqHPoWvcaGGOaGabmOEayaaraGaaiykaaaacqGHsislcaWG6bGaeu yQdCLaai4jaiaacIcacaWG6bGaaiykaiabgUcaRiabeo8aZnaaDaaa leaacaaIYaGaaGOmaaqaaiabg6HiLcaakiaadQhacaGGVaGaaGOmai abgUcaRiaadMgacqaHdpWCdaqhaaWcbaGaaGymaiaaikdaaeaacqGH EisPaaGccaWG6baaaaa@7C4E@

Some care is required to evaluate the square root in the complex potentials properly (square roots are multiple valued, and you need to know which value, or `branch’ to use.  Multiple valued functions are made single valued by introducing a `branch cut’ where the function is discontinuous.  In crack problems the branch cut is always along the line of the crack).  For this purpose, it is helpful to note that the appropriate branch can be obtained by setting

z 2 a 2 = (za)(z+a) = r 1 e i θ 1 /2 r 2 e i θ 2 /2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaakaaabaGaam OEamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaadggadaahaaWcbeqa aiaaikdaaaaabeaakiabg2da9maakaaabaGaaiikaiaadQhacqGHsi slcaWGHbGaaiykaiaacIcacaWG6bGaey4kaSIaamyyaiaacMcaaSqa baGccqGH9aqpdaGcaaqaaiaadkhadaWgaaWcbaGaaGymaaqabaaabe aakiaadwgadaahaaWcbeqaaiaadMgacqaH4oqCdaWgaaadbaGaaGym aaqabaWccaGGVaGaaGOmaaaakmaakaaabaGaamOCamaaBaaaleaaca aIYaaabeaaaeqaaOGaamyzamaaCaaaleqabaGaamyAaiabeI7aXnaa BaaameaacaaIYaaabeaaliaac+cacaaIYaaaaaaa@55B3@

where the angles and distances r 1 , θ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbWaaSbaaSqaaiaaigdaaeqaaO GaaiilaiabeI7aXnaaBaaaleaacaaIXaaabeaaaaa@37FB@  and r 2 , θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbWaaSbaaSqaaiaaikdaaeqaaO GaaiilaiabeI7aXnaaBaaaleaacaaIYaaabeaaaaa@380D@  are shown in the figure, and the angles θ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH4oqCdaWgaaWcbaGaaGymaaqaba aaaa@3573@  and θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH4oqCdaWgaaWcbaGaaGOmaaqaba aaaa@3574@  must lie in the ranges π θ 1 π0 θ 2 2π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaeq iWdaNaeyizImQaeqiUde3aaSbaaSqaaiaaigdaaeqaaOGaeyizImQa eqiWdaNaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaIWaGaeyizImQaeqiUde3aaSba aSqaaiaaikdaaeqaaOGaeyizImQaaGOmaiabec8aWbaa@5BB3@ , respectively to select the correct branch.

 

The solution is most conveniently expressed in terms of the polar coordinates (r,θ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamOCaiaacYcacqaH4oqCca GGPaaaaa@378C@  centered at the origin, together with the auxiliary angles r 1 , θ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbWaaSbaaSqaaiaaigdaaeqaaO GaaiilaiabeI7aXnaaBaaaleaacaaIXaaabeaaaaa@380B@  and r 2 , θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbWaaSbaaSqaaiaaikdaaeqaaO GaaiilaiabeI7aXnaaBaaaleaacaaIYaaabeaaaaa@380D@ . If I got the algebra correct, (which is unlikely MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  the algebra involved in getting these results from the complex potentials is unbelievably tedious and unfortunately beyond the capabilities of MAPLE) the displacement and stress fields are

u 1 = (1+ν) σ 22 r 1 r 2 4E { 4(12ν)cos( θ 1 + θ 2 )/2 4r(1ν) r 1 r 2 cosθ 2 r 2 r 1 r 2 ( cos( θ 1 + θ 2 )/2cos(2θ θ 1 /2 θ 2 /2) ) } + (1+ν) σ 12 r 1 r 2 E { 2(1ν)sin( θ 1 + θ 2 )/22(1ν) r r 1 r 2 sinθ + r 2 r 1 r 2 sinθcos(θ θ 1 /2 θ 2 /2) } u 2 = (1+ν) σ 22 r 1 r 2 4E { 8(1ν)sin( θ 1 + θ 2 )/2+ 4νr r 1 r 2 sinθ 2 r 2 r 1 r 2 ( sin( θ 1 + θ 2 )/2+sin(2θ θ 1 /2 θ 2 /2) ) } + (1+ν) σ 12 r 1 r 2 E { (12ν)cos( θ 1 + θ 2 )/2+2(1ν) r r 1 r 2 cosθ r 2 r 1 r 2 sinθsin(θ θ 1 /2 θ 2 /2) } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaadwhadaWgaaWcbaGaaGymaa qabaGccqGH9aqpdaWcaaqaaiaacIcacaaIXaGaey4kaSIaeqyVd4Ma aiykaiabeo8aZnaaDaaaleaacaaIYaGaaGOmaaqaaiabg6HiLcaakm aakaaabaGaamOCamaaBaaaleaacaaIXaaabeaakiaadkhadaWgaaWc 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σ 11 = σ 22 r r 1 r 2 { cos( θ θ 1 /2 θ 2 /2 )1 a 2 r 1 r 2 sinθsin3( θ 1 + θ 2 )/2 } + σ 12 r r 1 r 2 { 2sin( θ θ 1 /2 θ 2 /2 ) a 2 r 1 r 2 sinθcos3( θ 1 + θ 2 )/2 } σ 22 = σ 22 r r 1 r 2 { cos( θ θ 1 /2 θ 2 /2 )+ a 2 r 1 r 2 sinθsin3( θ 1 + θ 2 )/2 } + σ 12 r r 1 r 2 a 2 r 1 r 2 sinθcos3( θ 1 + θ 2 )/2 σ 12 = σ 22 r r 1 r 2 a 2 r 1 r 2 sinθcos3( θ 1 + θ 2 )/2 + σ 12 r r 1 r 2 { cos( θ θ 1 /2 θ 2 /2 )+ a 2 r 1 r 2 sinθsin3( θ 1 + θ 2 )/2 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabeo8aZnaaBaaaleaacaaIXa GaaGymaaqabaGccqGH9aqpdaWcaaqaaiabeo8aZnaaDaaaleaacaaI YaGaaGOmaaqaaiabg6HiLcaakiaadkhaaeaadaGcaaqaaiaadkhada WgaaWcbaGaaGymaaqabaGccaWGYbWaaSbaaSqaaiaaikdaaeqaaaqa baaaaOWaaiWaaeaaciGGJbGaai4BaiaacohadaqadaqaaiabeI7aXj abgkHiTiabeI7aXnaaBaaaleaacaaIXaaabeaakiaac+cacaaIYaGa eyOeI0IaeqiUde3aaSbaaSqaaiaaikdaaeqaaOGaai4laiaaikdaai aawIcacaGLPaaacqGHsislcaaIXaGaeyOeI0YaaSaaaeaacaWGHbWa aWbaaSqabeaacaaIYaaaaaGcbaGaamOCamaaBaaaleaacaaIXaaabe aakiaadkhadaWgaaWcbaGaaGOmaaqabaaaaOGaci4CaiaacMgacaGG UbGaeqiUdeNaci4CaiaacMgacaGGUbGaaG4mamaabmaabaGaeqiUde 3aaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaeqiUde3aaSbaaSqaaiaa ikdaaeqaaaGccaGLOaGaayzkaaGaai4laiaaikdaaiaawUhacaGL9b aaaeaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlabgUcaRmaalaaabaGaeq4Wdm3aa0baaSqaaiaaigdacaaIYa aabaGaeyOhIukaaOGaamOCaaqaamaakaaabaGaamOCamaaBaaaleaa caaIXaaabeaakiaadkhadaWgaaWcbaGaaGOmaaqabaaabeaaaaGcda GadaqaaiaaikdaciGGZbGaaiyAaiaac6gadaqadaqaaiabeI7aXjab gkHiTiabeI7aXnaaBaaaleaacaaIXaaabeaakiaac+cacaaIYaGaey OeI0IaeqiUde3aaSbaaSqaaiaaikdaaeqaaOGaai4laiaaikdaaiaa wIcacaGLPaaacqGHsisldaWcaaqaaiaadggadaahaaWcbeqaaiaaik daaaaakeaacaWGYbWaaSbaaSqaaiaaigdaaeqaaOGaamOCamaaBaaa leaacaaIYaaabeaaaaGcciGGZbGaaiyAaiaac6gacqaH4oqCciGGJb Gaai4BaiaacohacaaIZaWaaeWaaeaacqaH4oqCdaWgaaWcbaGaaGym aaqabaGccqGHRaWkcqaH4oqCdaWgaaWcbaGaaGOmaaqabaaakiaawI cacaGLPaaacaGGVaGaaGOmaaGaay5Eaiaaw2haaaqaaiabeo8aZnaa BaaaleaacaaIYaGaaGOmaaqabaGccqGH9aqpdaWcaaqaaiabeo8aZn aaDaaaleaacaaIYaGaaGOmaaqaaiabg6HiLcaakiaadkhaaeaadaGc aaqaaiaadkhadaWgaaWcbaGaaGymaaqabaGccaWGYbWaaSbaaSqaai aaikdaaeqaaaqabaaaaOWaaiWaaeaaciGGJbGaai4Baiaacohadaqa daqaaiabeI7aXjabgkHiTiabeI7aXnaaBaaaleaacaaIXaaabeaaki aac+cacaaIYaGaeyOeI0IaeqiUde3aaSbaaSqaaiaaikdaaeqaaOGa ai4laiaaikdaaiaawIcacaGLPaaacqGHRaWkdaWcaaqaaiaadggada ahaaWcbeqaaiaaikdaaaaakeaacaWGYbWaaSbaaSqaaiaaigdaaeqa aOGaamOCamaaBaaaleaacaaIYaaabeaaaaGcciGGZbGaaiyAaiaac6 gacqaH4oqCciGGZbGaaiyAaiaac6gacaaIZaWaaeWaaeaacqaH4oqC daWgaaWcbaGaaGymaaqabaGccqGHRaWkcqaH4oqCdaWgaaWcbaGaaG OmaaqabaaakiaawIcacaGLPaaacaGGVaGaaGOmaaGaay5Eaiaaw2ha aaqaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8Uaey4kaSYaaSaaaeaacqaHdpWCdaqh aaWcbaGaaGymaiaaikdaaeaacqGHEisPaaGccaWGYbaabaWaaOaaae aacaWGYbWaaSbaaSqaaiaaigdaaeqaaOGaamOCamaaBaaaleaacaaI YaaabeaaaeqaaaaakmaalaaabaGaamyyamaaCaaaleqabaGaaGOmaa aaaOqaaiaadkhadaWgaaWcbaGaaGymaaqabaGccaWGYbWaaSbaaSqa aiaaikdaaeqaaaaakiGacohacaGGPbGaaiOBaiabeI7aXjGacogaca GGVbGaai4CaiaaiodadaqadaqaaiabeI7aXnaaBaaaleaacaaIXaaa beaakiabgUcaRiabeI7aXnaaBaaaleaacaaIYaaabeaaaOGaayjkai aawMcaaiaac+cacaaIYaaabaGaeq4Wdm3aaSbaaSqaaiaaigdacaaI Yaaabeaakiabg2da9maalaaabaGaeq4Wdm3aa0baaSqaaiaaikdaca aIYaaabaGaeyOhIukaaOGaamOCaaqaamaakaaabaGaamOCamaaBaaa leaacaaIXaaabeaakiaadkhadaWgaaWcbaGaaGOmaaqabaaabeaaaa GcdaWcaaqaaiaadggadaahaaWcbeqaaiaaikdaaaaakeaacaWGYbWa aSbaaSqaaiaaigdaaeqaaOGaamOCamaaBaaaleaacaaIYaaabeaaaa GcciGGZbGaaiyAaiaac6gacqaH4oqCciGGJbGaai4BaiaacohacaaI ZaWaaeWaaeaacqaH4oqCdaWgaaWcbaGaaGymaaqabaGccqGHRaWkcq aH4oqCdaWgaaWcbaGaaGOmaaqabaaakiaawIcacaGLPaaacaGGVaGa aGOmaaqaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7cqGHRaWkdaWcaaqaaiabeo8aZnaaDaaa leaacaaIXaGaaGOmaaqaaiabg6HiLcaakiaadkhaaeaadaGcaaqaai aadkhadaWgaaWcbaGaaGymaaqabaGccaWGYbWaaSbaaSqaaiaaikda aeqaaaqabaaaaOWaaiWaaeaaciGGJbGaai4Baiaacohadaqadaqaai abeI7aXjabgkHiTiabeI7aXnaaBaaaleaacaaIXaaabeaakiaac+ca caaIYaGaeyOeI0IaeqiUde3aaSbaaSqaaiaaikdaaeqaaOGaai4lai aaikdaaiaawIcacaGLPaaacqGHRaWkdaWcaaqaaiaadggadaahaaWc beqaaiaaikdaaaaakeaacaWGYbWaaSbaaSqaaiaaigdaaeqaaOGaam OCamaaBaaaleaacaaIYaaabeaaaaGcciGGZbGaaiyAaiaac6gacqaH 4oqCciGGZbGaaiyAaiaac6gacaaIZaWaaeWaaeaacqaH4oqCdaWgaa WcbaGaaGymaaqabaGccqGHRaWkcqaH4oqCdaWgaaWcbaGaaGOmaaqa baaakiaawIcacaGLPaaacaGGVaGaaGOmaaGaay5Eaiaaw2haaaaaaa@839B@

 

 

 

5.3.7 Fields near the tip of a crack on bimaterial interface

 

The figure shows a semi-infinite crack, which lies in the x 1 , x 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaigdaaeqaaO GaaiilaiaadIhadaWgaaWcbaGaaG4maaqabaaaaa@374A@  plane, with crack tip aligned with the x 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaaiodaaeqaaa aa@34AC@  axis.  The material above the crack has shear modulus and Poisson’s ratio μ 1 , ν 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaigdaaeqaaOGaaiilaiabe27aUnaaBaaaleaacaaIXaaa beaaaaa@3CF3@ ; the material below the crack has shear modulus and Poisson’s ratio μ 2 , ν 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaikdaaeqaaOGaaiilaiabe27aUnaaBaaaleaacaaIYaaa beaakiaaykW7aaa@3E8A@ .  In this section we give the complex variable solution that governs the variation of stress and displacement near the crack tip.  The solution is significant because all interface cracks (regardless of their geometry and the way the solid is loaded) have the same stress and displacement distribution near the crack tip.

 

Additional elastic constants for bimaterial problems

 

To simplify the solution, we define additional elastic constants as follows

1.      Plane strain moduli E 1 =2 μ 1 /(1 ν 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyrayaafa WaaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaaGOmaiabeY7aTnaaBaaa leaacaaIXaaabeaakiaac+cacaGGOaGaaGymaiabgkHiTiabe27aUn aaBaaaleaacaaIXaaabeaakiaacMcaaaa@438A@ , E 2 =2 μ 2 /(1 ν 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyrayaafa WaaSbaaSqaaiaaikdaaeqaaOGaeyypa0JaaGOmaiabeY7aTnaaBaaa leaacaaIYaaabeaakiaac+cacaGGOaGaaGymaiabgkHiTiabe27aUn aaBaaaleaacaaIYaaabeaakiaacMcaaaa@438D@

2.      Bimaterial modulus 1 E * ={ 1 E 1 + 1 E 2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaamyramaaCaaaleqabaGaaiOkaaaaaaGccqGH9aqpdaGa daqaamaalaaabaGaaGymaaqaaiqadweagaqbamaaBaaaleaacaaIXa aabeaaaaGccqGHRaWkdaWcaaqaaiaaigdaaeaaceWGfbGbauaadaWg aaWcbaGaaGOmaaqabaaaaaGccaGL7bGaayzFaaaaaa@42B5@

3.      Dundur’s elastic constants

α= E 1 E 2 E 1 + E 2 β= ( 12 ν 2 )/ μ 2 ( 12 ν 1 )/ μ 1 2( 1 ν 2 )/ μ 2 +2( 1 ν 1 )/ μ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey ypa0ZaaSaaaeaaceWGfbGbauaadaWgaaWcbaGaaGymaaqabaGccqGH sislceWGfbGbauaadaWgaaWcbaGaaGOmaaqabaaakeaaceWGfbGbau aadaWgaaWcbaGaaGymaaqabaGccqGHRaWkceWGfbGbauaadaWgaaWc baGaaGOmaaqabaaaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7cqaHYoGycqGH9aqpdaWcaaqaamaabm aabaGaaGymaiabgkHiTiaaikdacqaH9oGBdaWgaaWcbaGaaGOmaaqa baaakiaawIcacaGLPaaacaGGVaGaeqiVd02aaSbaaSqaaiaaikdaae qaaOGaeyOeI0YaaeWaaeaacaaIXaGaeyOeI0IaaGOmaiabe27aUnaa BaaaleaacaaIXaaabeaaaOGaayjkaiaawMcaaiaac+cacqaH8oqBda WgaaWcbaGaaGymaaqabaaakeaacaaIYaWaaeWaaeaacaaIXaGaeyOe I0IaeqyVd42aaSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaGaai 4laiabeY7aTnaaBaaaleaacaaIYaaabeaakiabgUcaRiaaikdadaqa daqaaiaaigdacqGHsislcqaH9oGBdaWgaaWcbaGaaGymaaqabaaaki aawIcacaGLPaaacaGGVaGaeqiVd02aaSbaaSqaaiaaigdaaeqaaaaa aaa@891A@

Evidently α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdegaaa@3785@  is a measure of the relative stiffness of the two materials.  It must lie in the range 1<α<1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG ymaiabgYda8iabeg7aHjabgYda8iaaigdaaaa@3BF0@  for all possible material combinations, with α=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey ypa0JaaGymaaaa@3946@  signifying that material 1 is rigid, while α=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey ypa0JaeyOeI0IaaGymaaaa@3A33@  signifies that material 2 is rigid.  The second parameter does not have such a nice physical interpretation MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  it is a rough measure of the relative compressibilities of the two materials.  For Poisson’s ratios in the range 0<ν<1/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgY da8iabe27aUjabgYda8iaaigdacaGGVaGaaGOmaaaa@3C8A@ , one can show that that 1<α4β<1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG ymaiabgYda8iabeg7aHjabgkHiTiaaisdacqaHYoGycqGH8aapcaaI Xaaaaa@3F3C@ .

 

4.      Crack tip singularity parameter

ε= 1 2π log( 1β 1+β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTduMaey ypa0ZaaSaaaeaacaaIXaaabaGaaGOmaiabec8aWbaaciGGSbGaai4B aiaacEgadaqadaqaamaalaaabaGaaGymaiabgkHiTiabek7aIbqaai aaigdacqGHRaWkcqaHYoGyaaaacaGLOaGaayzkaaaaaa@47DE@

For most material combinations the value of ε MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdugaaa@379D@  is very small MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  typically of order 0.01 or so.

 

The full displacement and stress fields in the two materials are calculated from two sets of complex potentials

Ω 1 (z)= 1+β (12iε) 2π ( K 1 i K 2 ) z (12iε)/2 Im(z)>0 Ω 2 (z)= 1β (12iε) 2π ( K 1 i K 2 ) z (12iε)/2 Im(z)<0 ω 1 (z)= Ω 2 ( z ¯ ) ¯ z Ω 1 (z)Im(z)>0 ω 2 (z)= Ω 1 ( z ¯ ) ¯ z Ω 2 (z)Im(z)<0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabfM6axnaaBaaaleaacaaIXa aabeaakiaacIcacaWG6bGaaiykaiabg2da9maalaaabaGaaGymaiab gUcaRiabek7aIbqaaiaacIcacaaIXaGaeyOeI0IaaGOmaiaadMgacq aH1oqzcaGGPaWaaOaaaeaacaaIYaGaeqiWdahaleqaaaaakiaacIca caWGlbWaaSbaaSqaaiaaigdaaeqaaOGaeyOeI0IaamyAaiaadUeada WgaaWcbaGaaGOmaaqabaGccaGGPaGaamOEamaaCaaaleqabaGaaiik aiaaigdacqGHsislcaaIYaGaamyAaiabew7aLjaacMcacaGGVaGaaG OmaaaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8Uaciysaiaac2gacaGGOaGaamOEai aacMcacqGH+aGpcaaIWaaabaGaeuyQdC1aaSbaaSqaaiaaikdaaeqa aOGaaiikaiaadQhacaGGPaGaeyypa0ZaaSaaaeaacaaIXaGaeyOeI0 IaeqOSdigabaGaaiikaiaaigdacqGHsislcaaIYaGaamyAaiabew7a LjaacMcadaGcaaqaaiaaikdacqaHapaCaSqabaaaaOGaaiikaiaadU eadaWgaaWcbaGaaGymaaqabaGccqGHsislcaWGPbGaam4samaaBaaa leaacaaIYaaabeaakiaacMcacaWG6bWaaWbaaSqabeaacaGGOaGaaG ymaiabgkHiTiaaikdacaWGPbGaeqyTduMaaiykaiaac+cacaaIYaaa aOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7ciGGjbGaaiyBaiaacIcacaWG6bGaaiyk aiabgYda8iaaicdaaeaacqaHjpWDdaWgaaWcbaGaaGymaaqabaGcca GGOaGaamOEaiaacMcacqGH9aqpdaqdaaqaaiabfM6axnaaBaaaleaa caaIYaaabeaakiaacIcaceWG6bGbaebacaGGPaaaaiabgkHiTiaadQ hacuqHPoWvgaqbamaaBaaaleaacaaIXaaabeaakiaacIcacaWG6bGa aiykaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlGacMeacaGGTbGaaiikaiaadQhacaGGPaGaeyOpa4JaaG imaaqaaiabeM8a3naaBaaaleaacaaIYaaabeaakiaacIcacaWG6bGa aiykaiabg2da9maanaaabaGaeuyQdC1aaSbaaSqaaiaaigdaaeqaaO GaaiikaiqadQhagaqeaiaacMcaaaGaeyOeI0IaamOEaiqbfM6axzaa faWaaSbaaSqaaiaaikdaaeqaaOGaaiikaiaadQhacaGGPaGaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ciGGjbGaai yBaiaacIcacaWG6bGaaiykaiabgYda8iaaicdaaaaa@37E3@

where K 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaam4samaaBaaaleaacaaIXaaabeaaaa a@3440@  and K 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaam4samaaBaaaleaacaaIYaaabeaaaa a@3441@  are parameters that resemble the mode I and mode II stress intensity factors that characterize the crack-tip stresses in a homogeneous solid.  In practice these parameters are not usually used in fracture criteria for interface cracks MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  instead, the crack tip loading is characterized the magnitude of the stress intensity factor | K | MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaWaaqWaaeaacaWGlbaacaGLhWUaayjcSd aaaa@367B@ , a characteristic length L, and a phase angle ψ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaeqiYdKhaaa@3457@ , defined as

| K |= K 1 2 + K 2 2 ψ= tan 1 Im[ ( K 1 +i K 2 ) L iε ] Re[ ( K 1 +i K 2 ) L iε ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaWaaqWaaeaacaWGlbaacaGLhWUaayjcSd Gaeyypa0ZaaOaaaeaacaWGlbWaa0baaSqaaiaaigdaaeaacaaIYaaa aOGaey4kaSIaam4samaaDaaaleaacaaIYaaabaGaaGOmaaaaaeqaaO GaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabeI8a5jabg2da9iGacs hacaGGHbGaaiOBamaaCaaaleqabaGaeyOeI0IaaGymaaaakmaalaaa baGaciysaiaac2gadaWadaqaaiaacIcacaWGlbWaaSbaaSqaaiaaig daaeqaaOGaey4kaSIaamyAaiaadUeadaWgaaWcbaGaaGOmaaqabaGc caGGPaGaamitamaaCaaaleqabaGaamyAaiabew7aLbaaaOGaay5wai aaw2faaaqaaiGackfacaGGLbWaamWaaeaacaGGOaGaam4samaaBaaa leaacaaIXaaabeaakiabgUcaRiaadMgacaWGlbWaaSbaaSqaaiaaik daaeqaaOGaaiykaiaadYeadaahaaWcbeqaaiaadMgacqaH1oqzaaaa kiaawUfacaGLDbaaaaaaaa@74FA@

This means that ( K 1 +i K 2 ) L iε =| K | e iψ ( K 1 i K 2 )=| K | e iψ L iε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaWaaeWaaeaacaWGlbWaaSbaaSqaaiaaig daaeqaaOGaey4kaSIaamyAaiaadUeadaWgaaWcbaGaaGOmaaqabaaa kiaawIcacaGLPaaacaWGmbWaaWbaaSqabeaacaWGPbGaeqyTdugaaO Gaeyypa0ZaaqWaaeaacaWGlbaacaGLhWUaayjcSdGaamyzamaaCaaa leqabaGaamyAaiabeI8a5baakiabgkDiEpaabmaabaGaam4samaaBa aaleaacaaIXaaabeaakiabgkHiTiaadMgacaWGlbWaaSbaaSqaaiaa ikdaaeqaaaGccaGLOaGaayzkaaGaeyypa0ZaaqWaaeaacaWGlbaaca GLhWUaayjcSdGaamyzamaaCaaaleqabaGaeyOeI0IaamyAaiabeI8a 5baakiaadYeadaahaaWcbeqaaiaadMgacqaH1oqzaaaaaa@5C60@ .

 

Complete expressions for the displacement components and stress components at a point r,θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8skY=vi pgea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaamOCaiaacYcacqaH4oqCaaa@35E6@  in the solid can be calculated from these potentials.  To simplify the results, it is helpful to note that

cosh(πε)= 1 2 ( e πε + e πε )= 1 2 ( 1β 1+β + 1+β 1β )= 1 (1β ) (1+β) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGcbaGaci4yaiaac+gacaGGZbGaaiiAaiaacI cacqaHapaCcqaH1oqzcaGGPaGaeyypa0ZaaSaaaeaacaaIXaaabaGa aGOmaaaadaqadaqaaiaadwgadaahaaWcbeqaaiabec8aWjabew7aLb aakiabgUcaRiaadwgadaahaaWcbeqaaiabgkHiTiabec8aWjabew7a LbaaaOGaayjkaiaawMcaaiabg2da9maalaaabaGaaGymaaqaaiaaik daaaWaaeWaaeaadaGcaaqaamaalaaabaGaaGymaiabgkHiTiabek7a IbqaaiaaigdacqGHRaWkcqaHYoGyaaaaleqaaOGaey4kaSYaaOaaae aadaWcaaqaaiaaigdacqGHRaWkcqaHYoGyaeaacaaIXaGaeyOeI0Ia eqOSdigaaaWcbeaaaOGaayjkaiaawMcaaiabg2da9maalaaabaGaaG ymaaqaamaakaaabaGaaiikaiaaigdacqGHsislcqaHYoGyaSqabaGc caGGPaWaaOaaaeaacaGGOaGaaGymaiabgUcaRiabek7aIjaacMcaaS qabaaaaaaa@679E@

Then, in material 1

2 μ 1 ( u 1 +i u 2 )= | K | cosh(πε) r 2π { 1 12iε ( r L ) iε [ (34 ν 1 ) e i(θ/2ψ) e ε(θπ) e i(θ/2+ψ) e ε(θπ) ] ( r L ) iε isinθ e i(θ/2+ψ) e ε(θπ) } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGceaqabeaacaaIYaGaeqiVd02aaSbaaSqaai aaigdaaeqaaOGaaiikaiaadwhadaWgaaWcbaGaaGymaaqabaGccqGH RaWkcaWGPbGaamyDamaaBaaaleaacaaIYaaabeaakiaacMcacqGH9a qpdaWcaaqaamaaemaabaGaam4saaGaay5bSlaawIa7aaqaaiGacoga caGGVbGaai4CaiaacIgacaGGOaGaeqiWdaNaeqyTduMaaiykaaaada GcaaqaamaalaaabaGaamOCaaqaaiaaikdacqaHapaCaaaaleqaaOWa aiqaaeaadaWcaaqaaiaaigdaaeaacaaIXaGaeyOeI0IaaGOmaiaadM gacqaH1oqzaaWaaeWaaeaadaWcaaqaaiaadkhaaeaacaWGmbaaaaGa ayjkaiaawMcaamaaCaaaleqabaGaeyOeI0IaamyAaiabew7aLbaakm aadmaabaGaaiikaiaaiodacqGHsislcaaI0aGaeqyVd42aaSbaaSqa aiaaigdaaeqaaOGaaiykaiaadwgadaahaaWcbeqaaiaadMgacaGGOa GaeqiUdeNaai4laiaaikdacqGHsislcqaHipqEcaGGPaaaaOGaamyz amaaCaaaleqabaGaeqyTduMaaiikaiabeI7aXjabgkHiTiabec8aWj aacMcaaaGccqGHsislcaWGLbWaaWbaaSqabeaacqGHsislcaWGPbGa aiikaiabeI7aXjaac+cacaaIYaGaey4kaSIaeqiYdKNaaiykaaaaki aadwgadaahaaWcbeqaaiabgkHiTiabew7aLjaacIcacqaH4oqCcqGH sislcqaHapaCcaGGPaaaaaGccaGLBbGaayzxaaaacaGL7baaaeaaca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7daGacaqaaiabgkHiTmaa bmaabaWaaSaaaeaacaWGYbaabaGaamitaaaaaiaawIcacaGLPaaada ahaaWcbeqaaiaadMgacqaH1oqzaaGccaWGPbGaci4CaiaacMgacaGG UbGaeqiUdeNaamyzamaaCaaaleqabaGaamyAaiaacIcacqaH4oqCca GGVaGaaGOmaiabgUcaRiabeI8a5jaacMcaaaGccaWGLbWaaWbaaSqa beaacqaH1oqzcaGGOaGaeqiUdeNaeyOeI0IaeqiWdaNaaiykaaaaaO GaayzFaaaaaaa@7A6C@

σ 11 + σ 22 = | K | 2πr ( 1+β ) e εθ { ( r L ) iε e i(θ/2+ψ) + ( r L ) iε e i(θ/2+ψ) } σ 11 σ 22 +2i σ 12 = | K | e iθ 2πr { ( r L ) iε e iψ e εθ e iθ/2 ( 1+β )( cosθ+2εsinθ ) ( r L ) iε e iψ e εθ e iθ/2 ( 1β ) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiqacaWaaaGceaqabeaacqaHdpWCdaWgaaWcbaGaaGymai aaigdaaeqaaOGaey4kaSIaeq4Wdm3aaSbaaSqaaiaaikdacaaIYaaa beaakiabg2da9maalaaabaWaaqWaaeaacaWGlbaacaGLhWUaayjcSd aabaWaaOaaaeaacaaIYaGaeqiWdaNaamOCaaWcbeaaaaGcdaqadaqa aiaaigdacqGHRaWkcqaHYoGyaiaawIcacaGLPaaacaWGLbWaaWbaaS qabeaacqaH1oqzcqaH4oqCaaGcdaGadaqaamaabmaabaWaaSaaaeaa caWGYbaabaGaamitaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiabgk HiTiaadMgacqaH1oqzaaGccaWGLbWaaWbaaSqabeaacqGHsislcaWG PbGaaiikaiabeI7aXjaac+cacaaIYaGaey4kaSIaeqiYdKNaaiykaa aakiabgUcaRmaabmaabaWaaSaaaeaacaWGYbaabaGaamitaaaaaiaa wIcacaGLPaaadaahaaWcbeqaaiaadMgacqaH1oqzaaGccaWGLbWaaW baaSqabeaacaWGPbGaaiikaiabeI7aXjaac+cacaaIYaGaey4kaSIa eqiYdKNaaiykaaaaaOGaay5Eaiaaw2haaaqaaiabeo8aZnaaBaaale aacaaIXaGaaGymaaqabaGccqGHsislcqaHdpWCdaWgaaWcbaGaaGOm aiaaikdaaeqaaOGaey4kaSIaaGOmaiaadMgacqaHdpWCdaWgaaWcba GaaGymaiaaikdaaeqaaOGaeyypa0ZaaSaaaeaadaabdaqaaiaadUea aiaawEa7caGLiWoacaWGLbWaaWbaaSqabeaacaWGPbGaeqiUdehaaa GcbaWaaOaaaeaacaaIYaGaeqiWdaNaamOCaaWcbeaaaaGcdaGabaqa amaabmaabaWaaSaaaeaacaWGYbaabaGaamitaaaaaiaawIcacaGLPa aadaahaaWcbeqaaiaadMgacqaH1oqzaaGccaWGLbWaaWbaaSqabeaa caWGPbGaeqiYdKhaaOGaamyzamaaCaaaleqabaGaeqyTduMaeqiUde haaOGaamyzamaaCaaaleqabaGaamyAaiabeI7aXjaac+cacaaIYaaa aOWaaeWaaeaacaaIXaGaey4kaSIaeqOSdigacaGLOaGaayzkaaWaae WaaeaaciGGJbGaai4BaiaacohacqaH4oqCcqGHRaWkcaaIYaGaeqyT duMaci4CaiaacMgacaGGUbGaeqiUdehacaGLOaGaayzkaaaacaGL7b aaaeaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8+a aiGaaeaacqGHsisldaqadaqaamaalaaabaGaamOCaaqaaiaadYeaaa aacaGLOaGaayzkaaWaaWbaaSqabeaacqGHsislcaWGPbGaeqyTduga aOGaamyzamaaCaaaleqabaGaeyOeI0IaamyAaiabeI8a5baakiaadw gadaahaaWcbeqaaiabgkHiTiabew7aLjabeI7aXbaakiaadwgadaah aaWcbeqaaiabgkHiTiaadMgacqaH4oqCcaGGVaGaaGOmaaaakmaabm aabaGaaGymaiabgkHiTiabek7aIbGaayjkaiaawMcaaaGaayzFaaaa aaa@55FC@

while in material 2

2 μ 2 ( u 1 +i u 2 )= | K | cosh(πε) r 2π { 1 12iε ( r L ) iε [ (34 ν 2 ) e i(θ/2ψ) e ε(θ+π) e i(θ/2+ψ) e ε(θ+π) ] ( r L ) iε isinθ e i(θ/2+ψ) e ε(θ+π) } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8 fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiWabaWaaaGceaqabeaacaaIYaGaeqiVd02aaSbaaSqaai aaikdaaeqaaOGaaiikaiaadwhadaWgaaWcbaGaaGymaaqabaGccqGH RaWkcaWGPbGaamyDamaaBaaaleaacaaIYaaabeaakiaacMcacqGH9a qpdaWcaaqaamaaemaabaGaam4saaGaay5bSlaawIa7aaqaaiGacoga caGGVbGaai4CaiaacIgacaGGOaGaeqiWdaNaeqyTduMaaiykaaaada GcaaqaamaalaaabaGaamOCaaqaaiaaikdacqaHapaCaaaaleqaaOWa aiqaaeaadaWcaaqaaiaaigdaaeaacaaIXaGaeyOeI0IaaGOmaiaadM gacqaH1oqzaaWaaeWaaeaadaWcaaqaaiaadkhaaeaacaWGmbaaaaGa ayjkaiaawMcaamaaCaaaleqabaGaeyOeI0IaamyAaiabew7aLbaakm aadmaabaGaaiikaiaaiodacqGHsislcaaI0aGaeqyVd42aaSbaaSqa aiaaikdaaeqaaOGaaiykaiaadwgadaahaaWcbeqaaiaadMgacaGGOa GaeqiUdeNaai4laiaaikdacqGHsislcqaHipqEcaGGPaaaaOGaamyz amaaCaaaleqabaGaeqyTduMaaiikaiabeI7aXjabgUcaRiabec8aWj aacMcaaaGccqGHsislcaWGLbWaaWbaaSqabeaacqGHsislcaWGPbGa aiikaiabeI7aXjaac+cacaaIYaGaey4kaSIaeqiYdKNaaiykaaaaki aadwgadaahaaWcbeqaaiabgkHiTiabew7aLjaacIcacqaH4oqCcqGH RaWkcqaHapaCcaGGPaaaaaGccaGLBbGaayzxaaaacaGL7baaaeaaca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7daGacaqaaiabgkHiTmaa bmaabaWaaSaaaeaacaWGYbaabaGaamitaaaaaiaawIcacaGLPaaada ahaaWcbeqaaiaadMgacqaH1oqzaaGccaWGPbGaci4CaiaacMgacaGG UbGaeqiUdeNaamyzamaaCaaaleqabaGaamyAaiaacIcacqaH4oqCca GGVaGaaGOmaiabgUcaRiabeI8a5jaacMcaaaGccaWGLbWaaWbaaSqa beaacqaH1oqzcaGGOaGaeqiUdeNaey4kaSIaeqiWdaNaaiykaaaaaO GaayzFaaaaaaa@7A4D@

σ 11 + σ 22 = | K | 2πr ( 1β ) e εθ { ( r L ) iε e i(θ/2+ψ) + ( r L ) iε e i(θ/2+ψ) } σ 11 σ 22 +2i σ 12 = | K | e iθ 2πr { ( r L ) iε e iψ e εθ e iθ/2 ( 1β )( cosθ+2εsinθ ) ( r L ) iε e iψ e εθ e iθ/2 ( 1+β ) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiqacaWaaaGceaqabeaacqaHdpWCdaWgaaWcbaGaaGymai aaigdaaeqaaOGaey4kaSIaeq4Wdm3aaSbaaSqaaiaaikdacaaIYaaa beaakiabg2da9maalaaabaWaaqWaaeaacaWGlbaacaGLhWUaayjcSd aabaWaaOaaaeaacaaIYaGaeqiWdaNaamOCaaWcbeaaaaGcdaqadaqa aiaaigdacqGHsislcqaHYoGyaiaawIcacaGLPaaacaWGLbWaaWbaaS qabeaacqaH1oqzcqaH4oqCaaGcdaGadaqaamaabmaabaWaaSaaaeaa caWGYbaabaGaamitaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiabgk HiTiaadMgacqaH1oqzaaGccaWGLbWaaWbaaSqabeaacqGHsislcaWG PbGaaiikaiabeI7aXjaac+cacaaIYaGaey4kaSIaeqiYdKNaaiykaa aakiabgUcaRmaabmaabaWaaSaaaeaacaWGYbaabaGaamitaaaaaiaa wIcacaGLPaaadaahaaWcbeqaaiaadMgacqaH1oqzaaGccaWGLbWaaW baaSqabeaacaWGPbGaaiikaiabeI7aXjaac+cacaaIYaGaey4kaSIa eqiYdKNaaiykaaaaaOGaay5Eaiaaw2haaaqaaiabeo8aZnaaBaaale aacaaIXaGaaGymaaqabaGccqGHsislcqaHdpWCdaWgaaWcbaGaaGOm aiaaikdaaeqaaOGaey4kaSIaaGOmaiaadMgacqaHdpWCdaWgaaWcba GaaGymaiaaikdaaeqaaOGaeyypa0ZaaSaaaeaadaabdaqaaiaadUea aiaawEa7caGLiWoacaWGLbWaaWbaaSqabeaacaWGPbGaeqiUdehaaa GcbaWaaOaaaeaacaaIYaGaeqiWdaNaamOCaaWcbeaaaaGcdaGabaqa amaabmaabaWaaSaaaeaacaWGYbaabaGaamitaaaaaiaawIcacaGLPa aadaahaaWcbeqaaiaadMgacqaH1oqzaaGccaWGLbWaaWbaaSqabeaa caWGPbGaeqiYdKhaaOGaamyzamaaCaaaleqabaGaeqyTduMaeqiUde haaOGaamyzamaaCaaaleqabaGaamyAaiabeI7aXjaac+cacaaIYaaa aOWaaeWaaeaacaaIXaGaeyOeI0IaeqOSdigacaGLOaGaayzkaaWaae WaaeaaciGGJbGaai4BaiaacohacqaH4oqCcqGHRaWkcaaIYaGaeqyT duMaci4CaiaacMgacaGGUbGaeqiUdehacaGLOaGaayzkaaaacaGL7b aaaeaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8+a aiGaaeaacqGHsisldaqadaqaamaalaaabaGaamOCaaqaaiaadYeaaa aacaGLOaGaayzkaaWaaWbaaSqabeaacqGHsislcaWGPbGaeqyTduga aOGaamyzamaaCaaaleqabaGaeyOeI0IaamyAaiabeI8a5baakiaadw gadaahaaWcbeqaaiabgkHiTiabew7aLjabeI7aXbaakiaadwgadaah aaWcbeqaaiabgkHiTiaadMgacqaH4oqCcaGGVaGaaGOmaaaakmaabm aabaGaaGymaiabgUcaRiabek7aIbGaayjkaiaawMcaaaGaayzFaaaa aaa@5607@

The individual stress components can be determined by adding/subtracting the last two equations and taking real and imaginary parts.  Note that (r/L) iε =exp(iεlog(r/L))=cos(εlogr/L)+isin(εlogr/L) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaamOCaiaac+cacaWGmbGaai ykamaaCaaaleqabaGaamyAaiabew7aLbaakiabg2da9iGacwgacaGG 4bGaaiiCaiaacIcacaWGPbGaeqyTduMaciiBaiaac+gacaGGNbGaai ikaiaadkhacaGGVaGaamitaiaacMcacaGGPaGaeyypa0Jaci4yaiaa c+gacaGGZbGaaiikaiabew7aLjGacYgacaGGVbGaai4zaiaadkhaca GGVaGaamitaiaacMcacqGHRaWkcaWGPbGaci4CaiaacMgacaGGUbGa aiikaiabew7aLjGacYgacaGGVbGaai4zaiaadkhacaGGVaGaamitai aacMcaaaa@60EF@ . Features of this solution are discussed in more detail in Section 9.6.1.

 

 

 

5.3.8 Frictionless rigid flat indenter in contact with a half-space

 

A rigid, flat punch with width 2a and infinite length perpendicular to the plane of the figure is pushed into an elastic half-space with a force F 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGgbWaaSbaaSqaaiaaikdaaeqaaa aa@3489@  per unit out of plane distance. The half-space is a linear elastic solid with shear modulus μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH8oqBaaa@347C@  and Poisson’s ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH9oGBaaa@347E@ . The interface between the two solids is frictionless. 

 

The solution is generated from the following complex potentials

Ω(z)= i F 2 2π log( z+ z 2 a 2 )+ μi d 2 2(1ν) ω(z)= Ω( z ¯ ) ¯ z Ω (z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHPoWvcaGGOaGaamOEaiaacMcacq GH9aqpdaWcaaqaaiabgkHiTiaadMgacaWGgbWaaSbaaSqaaiaaikda aeqaaaGcbaGaaGOmaiabec8aWbaaciGGSbGaai4BaiaacEgadaqada qaaiaadQhacqGHRaWkdaGcaaqaaiaadQhadaahaaWcbeqaaiaaikda aaGccqGHsislcaWGHbWaaWbaaSqabeaacaaIYaaaaaqabaaakiaawI cacaGLPaaacaaMc8Uaey4kaSYaaSaaaeaacqaH8oqBcaWGPbGaamiz amaaBaaaleaacaaIYaaabeaaaOqaaiaaikdacaGGOaGaaGymaiabgk HiTiabe27aUjaacMcaaaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7cqaHjpWDcaGGOaGaamOEaiaacMcacqGH9aqpcqGHsisldaqd aaqaaiabfM6axjaacIcaceWG6bGbaebacaGGPaaaaiabgkHiTiaadQ hacuqHPoWvgaqbaiaacIcacaWG6bGaaiykaaaa@7AE5@

where d 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbWaaSbaaSqaaiaaikdaaeqaaa aa@3497@  is an arbitrary constant, representing an unknown rigid displacement. Note that the solution is valid only for Im(z)>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaciGGjbGaaiyBaiaacIcacaWG6bGaai ykaiabg6da+iaaicdaaaa@38A0@ .

 

Stresses and displacements can be determined by substituting for Ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHPoWvaaa@3454@  and ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHjpWDaaa@3493@  into the general formulas, or alternatively, by substituting Ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHPoWvaaa@3454@  into the simplified representation for half-space problems given in 5.3.1. Some care is required to evaluate the square root in the complex potentials, particularly when calculating Ω( z ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHPoWvcaGGOaGabmOEayaaraGaai ykaaaa@36C4@  and Ω ( z ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuqHPoWvgaqbaiaacIcaceWG6bGbae bacaGGPaaaaa@36D0@ . The solution assumes that

z 2 a 2 = (za)(z+a) = r 1 e i θ 1 /2 r 2 e i θ 2 /2 z 2 a 2 ¯ = (za)(z+a) ¯ = r 1 e i θ 1 /2 r 2 e i θ 2 /2 z ¯ 2 a 2 = ( z ¯ a)( z ¯ +a) = r 1 e i θ 1 /2 r 2 e i θ 2 /2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaWaaOaaae aacaWG6bWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaamyyamaaCaaa leqabaGaaGOmaaaaaeqaaOGaeyypa0ZaaOaaaeaacaGGOaGaamOEai abgkHiTiaadggacaGGPaGaaiikaiaadQhacqGHRaWkcaWGHbGaaiyk aaWcbeaakiabg2da9maakaaabaGaamOCamaaBaaaleaacaaIXaaabe aaaeqaaOGaamyzamaaCaaaleqabaGaamyAaiabeI7aXnaaBaaameaa caaIXaaabeaaliaac+cacaaIYaaaaOWaaOaaaeaacaWGYbWaaSbaaS qaaiaaikdaaeqaaaqabaGccaWGLbWaaWbaaSqabeaacaWGPbGaeqiU de3aaSbaaWqaaiaaikdaaeqaaSGaai4laiaaikdaaaaakeaadaqdaa qaamaakaaabaGaamOEamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaa dggadaahaaWcbeqaaiaaikdaaaaabeaaaaGccqGH9aqpdaqdaaqaam aakaaabaGaaiikaiaadQhacqGHsislcaWGHbGaaiykaiaacIcacaWG 6bGaey4kaSIaamyyaiaacMcaaSqabaaaaOGaeyypa0ZaaOaaaeaaca WGYbWaaSbaaSqaaiaaigdaaeqaaaqabaGccaWGLbWaaWbaaSqabeaa cqGHsislcaWGPbGaeqiUde3aaSbaaWqaaiaaigdaaeqaaSGaai4lai aaikdaaaGcdaGcaaqaaiaadkhadaWgaaWcbaGaaGOmaaqabaaabeaa kiaadwgadaahaaWcbeqaaiabgkHiTiaadMgacqaH4oqCdaWgaaadba GaaGOmaaqabaWccaGGVaGaaGOmaaaaaOqaamaakaaabaGabmOEayaa raWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaamyyamaaCaaaleqaba GaaGOmaaaaaeqaaOGaeyypa0ZaaOaaaeaacaGGOaGabmOEayaaraGa eyOeI0IaamyyaiaacMcacaGGOaGabmOEayaaraGaey4kaSIaamyyai aacMcaaSqabaGccqGH9aqpdaGcaaqaaiaadkhadaWgaaWcbaGaaGym aaqabaaabeaakiaadwgadaahaaWcbeqaaiabgkHiTiaadMgacqaH4o qCdaWgaaadbaGaaGymaaqabaWccaGGVaGaaGOmaaaakmaakaaabaGa amOCamaaBaaaleaacaaIYaaabeaaaeqaaOGaamyzamaaCaaaleqaba GaeyOeI0IaamyAaiabeI7aXnaaBaaameaacaaIYaaabeaaliaac+ca caaIYaaaaaaaaa@98E1@

where the angles and distances r 1 , θ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbWaaSbaaSqaaiaaigdaaeqaaO GaaiilaiabeI7aXnaaBaaaleaacaaIXaaabeaaaaa@37FB@  and r 2 , θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbWaaSbaaSqaaiaaikdaaeqaaO GaaiilaiabeI7aXnaaBaaaleaacaaIYaaabeaaaaa@37FD@  are shown in the figure, and θ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH4oqCdaWgaaWcbaGaaGymaaqaba aaaa@3563@  and θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH4oqCdaWgaaWcbaGaaGOmaaqaba aaaa@3564@  must lie in the ranges 0 θ 1 π0 θ 2 π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgs MiJkabeI7aXnaaBaaaleaacaaIXaaabeaakiabgsMiJkabec8aWjaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGimaiabgsMiJkabeI7aXnaaBaaaleaacaaI YaaabeaakiabgsMiJkabec8aWbaa@5907@ .

 

The full displacement and stress fields can be determined without difficulty, but are too lengthy to write out in full.  However, important features of the solution can be extracted.  In particular:

 

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

p( x 1 )= σ 22 ( x 1 , x 2 =0)= F 2 π a 2 x 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbGaaiikaiaadIhadaWgaaWcba GaaGymaaqabaGccaGGPaGaeyypa0JaeyOeI0Iaeq4Wdm3aaSbaaSqa aiaaikdacaaIYaaabeaakiaacIcacaWG4bWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaadIhadaWgaaWcbaGaaGOmaaqabaGccqGH9aqpcaaI WaGaaiykaiabg2da9maalaaabaGaamOramaaBaaaleaacaaIYaaabe aaaOqaaiabec8aWnaakaaabaGaamyyamaaCaaaleqabaGaaGOmaaaa kiabgkHiTiaadIhadaqhaaWcbaGaaGymaaqaaiaaikdaaaaabeaaaa aaaa@4E13@

 

2.      Surface displacement: The displacement of the surface is

u 2 ={ F 2 (1ν) πμ log( | x 1 |+ x 1 2 a 2 )+ d 2 | x 1 |>a F 2 (1ν) πμ log(a)+ d 2 | x 1 |<a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaaikdaaeqaaO Gaeyypa0ZaaiqaaeaafaqabeGabaaabaGaeyOeI0YaaSaaaeaacaWG gbWaaSbaaSqaaiaaikdaaeqaaOGaaiikaiaaigdacqGHsislcqaH9o GBcaGGPaaabaGaeqiWdaNaeqiVd0gaaiGacYgacaGGVbGaai4zamaa bmaabaWaaqWaaeaacaWG4bWaaSbaaSqaaiaaigdaaeqaaaGccaGLhW UaayjcSdGaey4kaSYaaOaaaeaacaWG4bWaa0baaSqaaiaaigdaaeaa caaIYaaaaOGaeyOeI0IaamyyamaaCaaaleqabaGaaGOmaaaaaeqaaa GccaGLOaGaayzkaaGaey4kaSIaamizamaaBaaaleaacaaIYaaabeaa kiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7daabdaqaaiaadIhadaWgaaWcbaGaaGymaa qabaaakiaawEa7caGLiWoacqGH+aGpcaWGHbaabaGaeyOeI0YaaSaa aeaacaWGgbWaaSbaaSqaaiaaikdaaeqaaOGaaiikaiaaigdacqGHsi slcqaH9oGBcaGGPaaabaGaeqiWdaNaeqiVd0gaaiGacYgacaGGVbGa ai4zaiaacIcacaWGHbGaaiykaiaaykW7cqGHRaWkcaWGKbWaaSbaaS qaaiaaikdaaeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7daabdaqa aiaadIhadaWgaaWcbaGaaGymaaqabaaakiaawEa7caGLiWoacqGH8a apcaWGHbaaaaGaay5Eaaaaaa@E59B@

Note that there is no unambiguous way to determine the value of d 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbWaaSbaaSqaaiaaikdaaeqaaa aa@3497@ .  It is tempting, for example, to attempt to calculate d 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbWaaSbaaSqaaiaaikdaaeqaaa aa@3497@  by assuming that the surface remains fixed at some point far from the indenter.  However, in this case d 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbWaaSbaaSqaaiaaikdaaeqaaa aa@3497@  increases without limit as the distance of the fixed point from the indenter increases. 

 

3.      Contact stiffness: the stiffness of a contact is defined as the ratio of the force acting on the indenter to its displacement k c = F 2 / u 2 (z=0) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGRbWaaSbaaSqaaiaadogaaeqaaO Gaeyypa0JaamOramaaBaaaleaacaaIYaaabeaakiaac+cacaWG1bWa aSbaaSqaaiaaikdaaeqaaOGaaiikaiaadQhacqGH9aqpcaaIWaGaai ykaaaa@3E4E@ , and is of considerable interest in practical applications.  Unfortunately, the solution for an infinite solid cannot be used to estimate the stiffness of a 2D contact (the stiffness depends on d 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbWaaSbaaSqaaiaaikdaaeqaaa aa@3497@  ).  Of course, the stiffness of a contact between two finite sized elastic solids is well defined MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  but the stiffness depends on the overall geometry of the two contacting solids, and varies as k c =μ/[(1ν)log(R/a)] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGRbWaaSbaaSqaaiaadogaaeqaaO Gaeyypa0JaeqiVd0Maai4laiaacUfacaGGOaGaaGymaiabgkHiTiab e27aUjaacMcaciGGSbGaai4BaiaacEgacaGGOaGaamOuaiaac+caca WGHbGaaiykaiaac2faaaa@4555@ , where R is a characteristic length comparable to the specimen size, and a is the contact width.

 

 

 

5.3.9 Frictionless parabolic (cylindrical) indenter in contact with a half-space

 

A rigid, parabolic punch with profile

f(r)= r 2 /(2R) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGMbGaaiikaiaadkhacaGGPaGaey ypa0JaamOCamaaCaaaleqabaGaaGOmaaaakiaac+cacaGGOaGaaGOm aiaadkfacaGGPaaaaa@3C90@

(and infinite length perpendicular to the plane of the figure) is pushed into an elastic half-space by a force F 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGgbWaaSbaaSqaaiaaikdaaeqaaa aa@3489@  (this profile is often used to approximate a cylinder with radius R).  The interface between the two solids is frictionless, and cannot withstand any tensile stress.    The indenter sinks into the elastic solid, so that the two solids make contact over a finite region a< x 1 <a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHsislcaWGHbGaeyipaWJaamiEam aaBaaaleaacaaIXaaabeaakiabgYda8iaadggaaaa@3975@ , where

a= 4R F 2 /π E * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabg2 da9maakaaabaGaaGinaiaadkfacaWGgbWaaSbaaSqaaiaaikdaaeqa aOGaai4laiabec8aWjaadweadaahaaWcbeqaaiaacQcaaaaabeaaaa a@4061@               E * =E/(1 ν 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweadaahaaWcbeqaaiaacQcaaaGccq GH9aqpcaWGfbGaai4laiaacIcacaaIXaGaeyOeI0IaeqyVd42aaWba aSqabeaacaaIYaaaaOGaaiykaaaa@3A46@

The solution is generated from the following complex potentials

Ω(z)= i F 2 2π a 2 { z z 2 a 2 z 2 a 2 log( z+ z 2 a 2 ) }+ Ei d 2 4(1 ν 2 ) ω(z)= Ω( z ¯ ) ¯ z Ω (z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHPoWvcaGGOaGaamOEaiaacMcacq GH9aqpdaWcaaqaaiaadMgacaWGgbWaaSbaaSqaaiaaikdaaeqaaaGc baGaaGOmaiabec8aWjaadggadaahaaWcbeqaaiaaikdaaaaaaOWaai WaaeaacaWG6bWaaOaaaeaacaWG6bWaaWbaaSqabeaacaaIYaaaaOGa eyOeI0IaamyyamaaCaaaleqabaGaaGOmaaaaaeqaaOGaeyOeI0Iaam OEamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaadggadaahaaWcbeqa aiaaikdaaaGcciGGSbGaai4BaiaacEgadaqadaqaaiaadQhacqGHRa WkdaGcaaqaaiaadQhadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWG HbWaaWbaaSqabeaacaaIYaaaaaqabaaakiaawIcacaGLPaaaaiaawU hacaGL9baacaaMc8Uaey4kaSYaaSaaaeaacaWGfbGaamyAaiaadsga daWgaaWcbaGaaGOmaaqabaaakeaacaaI0aGaaiikaiaaigdacqGHsi slcqaH9oGBdaahaaWcbeqaaiaaikdaaaGccaGGPaaaaiaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaeqyY dCNaaiikaiaadQhacaGGPaGaeyypa0JaeyOeI0Yaa0aaaeaacqqHPo WvcaGGOaGabmOEayaaraGaaiykaaaacqGHsislcaWG6bGafuyQdCLb auaacaGGOaGaamOEaiaacMcaaaa@8E18@

where d 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbWaaSbaaSqaaiaaikdaaeqaaa aa@3497@  is an arbitrary constant, representing an unknown rigid displacement. Note that the solution is valid only for Im(z)>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaciGGjbGaaiyBaiaacIcacaWG6bGaai ykaiabg6da+iaaicdaaaa@38A0@ . You can use the formulas given at the end of Section 5.3.1 to determine displacements and stress directly from the Ω(z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHPoWvcaGGOaGaamOEaiaacMcaaa a@36AC@ .  In addition, the formulas in 5.3.7 should be used to determine correct sign for the square root.

 

Important features of the solution are:

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

p( x 1 )= σ 22 ( x 1 , x 2 =0)= 2 F 2 π a 2 a 2 x 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbGaaiikaiaadIhadaWgaaWcba GaaGymaaqabaGccaGGPaGaeyypa0JaeyOeI0Iaeq4Wdm3aaSbaaSqa aiaaikdacaaIYaaabeaakiaacIcacaWG4bWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaadIhadaWgaaWcbaGaaGOmaaqabaGccqGH9aqpcaaI WaGaaiykaiabg2da9maalaaabaGaaGOmaiaadAeadaWgaaWcbaGaaG OmaaqabaaakeaacqaHapaCcaWGHbWaaWbaaSqabeaacaaIYaaaaaaa kmaakaaabaGaamyyamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaadI hadaqhaaWcbaGaaGymaaqaaiaaikdaaaaabeaaaaa@50A8@

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

u 2 ={ 2 F 2 π E * a 2 { x 1 x 1 2 a 2 x 1 2 a 2 log( x 1 + x 1 2 a 2 ) }+ d 2 | x 1 |>a 2 F 2 π E * a 2 ( log(a)+ x 1 2 )+ d 2 | x 1 |<a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaaikdaaeqaaO Gaeyypa0ZaaiqaaeaafaqabeGabaaabaWaaSaaaeaacaaIYaGaamOr amaaBaaaleaacaaIYaaabeaaaOqaaiabec8aWjaadweadaahaaWcbe qaaiaacQcaaaGccaWGHbWaaWbaaSqabeaacaaIYaaaaaaakmaacmaa baGaamiEamaaBaaaleaacaaIXaaabeaakmaakaaabaGaamiEamaaDa aaleaacaaIXaaabaGaaGOmaaaakiabgkHiTiaadggadaahaaWcbeqa aiaaikdaaaaabeaakiabgkHiTiaadIhadaqhaaWcbaGaaGymaaqaai aaikdaaaGccqGHsislcaWGHbWaaWbaaSqabeaacaaIYaaaaOGaciiB aiaac+gacaGGNbWaaeWaaeaacaWG4bWaaSbaaSqaaiaaigdaaeqaaO Gaey4kaSYaaOaaaeaacaWG4bWaa0baaSqaaiaaigdaaeaacaaIYaaa aOGaeyOeI0IaamyyamaaCaaaleqabaGaaGOmaaaaaeqaaaGccaGLOa GaayzkaaaacaGL7bGaayzFaaGaey4kaSIaamizamaaBaaaleaacaaI YaaabeaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7daabdaqaaiaadIhadaWgaaWcba GaaGymaaqabaaakiaawEa7caGLiWoacqGH+aGpcaWGHbaabaGaeyOe I0YaaSaaaeaacaaIYaGaamOramaaBaaaleaacaaIYaaabeaaaOqaai abec8aWjaadweadaahaaWcbeqaaiaacQcaaaGccaWGHbWaaWbaaSqa beaacaaIYaaaaaaakmaabmaabaGaciiBaiaac+gacaGGNbGaaiikai aadggacaGGPaGaey4kaSIaamiEamaaDaaaleaacaaIXaaabaGaaGOm aaaaaOGaayjkaiaawMcaaiaaykW7cqGHRaWkcaWGKbWaaSbaaSqaai aaikdaaeqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8+a aqWaaeaacaWG4bWaaSbaaSqaaiaaigdaaeqaaaGccaGLhWUaayjcSd GaeyipaWJaamyyaaaaaiaawUhaaaaa@26C0@

As discussed in 5.3.8, d 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbWaaSbaaSqaaiaaikdaaeqaaa aa@3497@  or the contact stiffness cannot be determined uniquely.

3.      Stress field

σ 11 = 2 F 2 π a 2 ( m( 1+  ( x 2 2 + n 2 ) ( m 2 + n 2 ) )2 x 2 ) σ 22 =  2 F 2 π a 2 m( 1 ( x 2 2 + n 2 ) ( m 2 + n 2 ) ) σ 12 = 2 F 2 π a 2 n ( m 2 x 2 2 ) ( m 2 + n 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIXaGaaGymaa qabaGccqGH9aqpcqGHsisldaWcaaqaaiaaikdacaWGgbWaaSbaaSqa aiaaikdaaeqaaaGcbaGaeqiWdaNaamyyamaaCaaaleqabaGaaGOmaa aaaaGcdaqadaqaaiaad2gadaqadaqaaiaaigdacqGHRaWkcaqGGaWa aSaaaeaacaGGOaGaamiEamaaDaaaleaacaaIYaaabaGaaGOmaaaaki abgUcaRiaad6gadaahaaWcbeqaaiaaikdaaaGccaGGPaaabaGaaiik aiaad2gadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWGUbWaaWbaaS qabeaacaaIYaaaaOGaaiykaaaaaiaawIcacaGLPaaacqGHsislcaaI YaGaamiEamaaBaaaleaacaaIYaaabeaaaOGaayjkaiaawMcaaiaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaeq4W dm3aaSbaaSqaaiaaikdacaaIYaaabeaakiabg2da9iaabccacqGHsi sldaWcaaqaaiaaikdacaWGgbWaaSbaaSqaaiaaikdaaeqaaaGcbaGa eqiWdaNaamyyamaaCaaaleqabaGaaGOmaaaaaaGccaWGTbWaaeWaae aacaaIXaGaeyOeI0YaaSaaaeaacaGGOaGaamiEamaaDaaaleaacaaI YaaabaGaaGOmaaaakiabgUcaRiaad6gadaahaaWcbeqaaiaaikdaaa GccaGGPaaabaGaaiikaiaad2gadaahaaWcbeqaaiaaikdaaaGccqGH RaWkcaWGUbWaaWbaaSqabeaacaaIYaaaaOGaaiykaaaaaiaawIcaca GLPaaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua eq4Wdm3aaSbaaSqaaiaaigdacaaIYaaabeaakiabg2da9iabgkHiTm aalaaabaGaaGOmaiaadAeadaWgaaWcbaGaaGOmaaqabaaakeaacqaH apaCcaWGHbWaaWbaaSqabeaacaaIYaaaaaaakiaad6gadaWcaaqaai aacIcacaWGTbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaamiEamaa DaaaleaacaaIYaaabaGaaGOmaaaakiaacMcaaeaacaGGOaGaamyBam aaCaaaleqabaGaaGOmaaaakiabgUcaRiaad6gadaahaaWcbeqaaiaa ikdaaaGccaGGPaaaaaaa@A05B@

m= ( c 1 + c 2 )/2 n= x 1 ( c 1 c 2 )/(2 x 1 2 ) c 1 =1 x 1 2 x 2 2 c 2 = c 1 2 +4 x 1 2 x 2 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2gacqGH9aqpdaGcaaqaamaabmaaba Gaam4yamaaBaaaleaacaaIXaaabeaakiabgUcaRiaadogadaWgaaWc baGaaGOmaaqabaaakiaawIcacaGLPaaacaGGVaGaaGOmaaWcbeaaki aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caWGUbGaeyypa0JaamiEamaaBaaaleaacaaIXaaabe aakmaakaaabaWaaeWaaeaacaWGJbWaaSbaaSqaaiaaigdaaeqaaOGa eyOeI0Iaam4yamaaBaaaleaacaaIYaaabeaaaOGaayjkaiaawMcaai aac+cacaGGOaGaaGOmaiaadIhadaqhaaWcbaGaaGymaaqaaiaaikda aaGccaGGPaaaleqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGJbWaaSbaaSqa aiaaigdaaeqaaOGaeyypa0JaaGymaiabgkHiTiaadIhadaqhaaWcba GaaGymaaqaaiaaikdaaaGccqGHsislcaWG4bWaa0baaSqaaiaaikda aeaacaaIYaaaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caWGJbWaaSbaaSqaaiaaikdaaeqaaOGaeyypa0Za aOaaaeaacaWGJbWaa0baaSqaaiaaigdaaeaacaaIYaaaaOGaey4kaS IaaGinaiaadIhadaqhaaWcbaGaaGymaaqaaiaaikdaaaGccaWG4bWa a0baaSqaaiaaikdaaeaacaaIYaaaaaqabaaaaa@8DB7@

 

4.      Critical load required to cause yield.  The elastic limit is best calculated using the Tresca yield criterion, which gives

F 2 /a=2.616Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGgbWaaSbaaSqaaiaaikdaaeqaaO Gaai4laiaadggacqGH9aqpcaaIYaGaaiOlaiaaiAdacaaIXaGaaGOn aiaadMfaaaa@3BA9@

where Y is the tensile yield stress of the solid.  To derive this result, note that the stresses are proportional to F 2 /a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGgbWaaSbaaSqaaiaaikdaaeqaaO Gaai4laiaadggaaaa@361C@ .  This means we can write

σ ij =( F 2 /a) σ ^ ij ( x i /a) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaeyypa0JaaiikaiaadAeadaWgaaWcbaGaaGOmaaqabaGc caGGVaGaamyyaiaacMcacuaHdpWCgaqcamaaBaaaleaacaWGPbGaam OAaaqabaGccaGGOaGaamiEamaaBaaaleaacaWGPbaabeaakiaac+ca caWGHbGaaiykaaaa@454A@

where σ ^ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaHdpWCgaqcamaaBaaaleaacaWGPb GaamOAaaqabaaaaa@36A2@  is the stress induced at x i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG4bWaaSbaaSqaaiaadMgaaeqaaa aa@34DD@  for a contact with a=1 subjected to load F 2 =1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGgbWaaSbaaSqaaiaaikdaaeqaaO Gaeyypa0JaaGymaaaa@3644@ .  The yield criterion can therefore be expressed as

F 2 a max (x1,x2) { ( σ ^ 11 σ ^ 22 ) 2 +4 σ ^ 12 2 }=Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiaadAeadaWgaaWcbaGaaG OmaaqabaaakeaacaWGHbaaaiGac2gacaGGHbGaaiiEamaaBaaaleaa caGGOaGaamiEaiaaigdacaGGSaGaamiEaiaaikdacaGGPaaabeaakm aacmaabaWaaOaaaeaacaGGOaGafq4WdmNbaKaadaWgaaWcbaGaaGym aiaaigdaaeqaaOGaeyOeI0Iafq4WdmNbaKaadaWgaaWcbaGaaGOmai aaikdaaeqaaOGaaiykamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaa isdacuaHdpWCgaqcamaaDaaaleaacaaIXaGaaGOmaaqaaiaaikdaaa aabeaaaOGaay5Eaiaaw2haaiabg2da9iaadMfaaaa@5238@

where max (x1,x2) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaciGGTbGaaiyyaiaacIhadaWgaaWcba GaaiikaiaadIhacaaIXaGaaiilaiaadIhacaaIYaGaaiykaaqabaaa aa@3B40@  denotes maximizing with respect to position in the solid.   The figure shows contours of ( σ ^ 11 σ ^ 22 ) 2 +4 σ ^ 12 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGcaaqaaiaacIcacuaHdpWCgaqcam aaBaaaleaacaaIXaGaaGymaaqabaGccqGHsislcuaHdpWCgaqcamaa BaaaleaacaaIYaGaaGOmaaqabaGccaGGPaWaaWbaaSqabeaacaaIYa aaaOGaey4kaSIaaGinaiqbeo8aZzaajaWaa0baaSqaaiaaigdacaaI YaaabaGaaGOmaaaaaeqaaaaa@42E2@ : the maximum value is approximately 0.3823, and occurs on the symmetry axis at a depth of about 0.78a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaaIWaGaaiOlaiaaiEdacaaI4aGaam yyaaaa@369B@ .  Substituting this value back into the yield criterion gives the result.

 

 

 

5.3.10 Line contact between two non-conformal frictionless elastic solids

 

The solution in the preceding section can be generalized to find stress and displacement caused by contact between two elastic solids.  The solution assumes:

1.      The two contacting solids initially meet at along a line perpendicular to the plane of the figure (the line of initial contact lies on the line connecting the centers of curvature of the two solids)

2.      The two contacting solids have radii of curvature R 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkfadaWgaaWcbaGaaGymaaqabaaaaa@3226@  and R 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkfadaWgaaWcbaGaaGOmaaqabaaaaa@3227@  at the point of initial contact.  A convex surface has a positive radius of curvature; a concave surface (like the internal surface of a hole) has a negative radius of curvature

3.      The two solids have Young’s modulus and Poissons ratio E 1 , ν 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweadaWgaaWcbaGaaGymaaqabaGcca GGSaGaeqyVd42aaSbaaSqaaiaaigdaaeqaaaaa@3572@  and E 2 , ν 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweadaWgaaWcbaGaaGOmaaqabaGcca GGSaGaeqyVd42aaSbaaSqaaiaaikdaaeqaaaaa@3574@ .

4.      The two solids are pushed into contact by a force F 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAeadaWgaaWcbaGaaGOmaaqabaaaaa@321B@

 

The solution is expressed in terms of an effective contact radius and an effective modulus, defined as

R= R 1 R 2 R 1 + R 2 E * = E 1 E 2 (1 ν 1 2 ) E 2 +(1 ν 2 2 ) E 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkfacqGH9aqpdaWcaaqaaiaadkfada WgaaWcbaGaaGymaaqabaGccaWGsbWaaSbaaSqaaiaaikdaaeqaaaGc baGaamOuamaaBaaaleaacaaIXaaabeaakiabgUcaRiaadkfadaWgaa WcbaGaaGOmaaqabaaaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caWGfbWaaWbaaSqabeaacaGGQaaa aOGaeyypa0ZaaSaaaeaacaWGfbWaaSbaaSqaaiaaigdaaeqaaOGaam yramaaBaaaleaacaaIYaaabeaaaOqaaiaacIcacaaIXaGaeyOeI0Ia eqyVd42aa0baaSqaaiaaigdaaeaacaaIYaaaaOGaaiykaiaadweada WgaaWcbaGaaGOmaaqabaGccqGHRaWkcaGGOaGaaGymaiabgkHiTiab e27aUnaaDaaaleaacaaIYaaabaGaaGOmaaaakiaacMcacaWGfbWaaS baaSqaaiaaigdaaeqaaaaaaaa@6BEE@

The contact width and contact pressure can be determined by substituting these values into the formulas given in the preceding section.   The full stress and displacement field in each solid can be calculated from the potential given in the preceding section, by adopting a coordinate system that points into the solid of interest.

 

 

 

5.3.11 Sliding contact between two rough elastic cylinders

 

Two elastic cylinders with elastic constants E 1 , ν 1 , E 2 , ν 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweadaWgaaWcbaGaaGymaaqabaGcca GGSaGaeqyVd42aaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadweadaWg aaWcbaGaaGOmaaqabaGccaGGSaGaeqyVd42aaSbaaSqaaiaaikdaae qaaaaa@3B38@ , radii R 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkfadaWgaaWcbaGaaGymaaqabaaaaa@3226@ , R 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkfadaWgaaWcbaGaaGOmaaqabaaaaa@3227@  and infinite length perpendicular to the plane of the figure, are pushed into contact by a forces F 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGgbWaaSbaaSqaaiaaikdaaeqaaa aa@3489@  acting perpendicular to the line of contact, and  F 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGgbWaaSbaaSqaaiaaigdaaeqaaa aa@3488@  acting parallel to the tangent plane.   The interface between the two solids has a coefficient of friction f MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAgaaaa@3153@ , and cannot withstand any tensile stress.  The tangential force is sufficient to cause the two solids to slide against each other, so that F 1 =f F 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGgbWaaSbaaSqaaiaaigdaaeqaaO Gaeyypa0JaamOzaiaadAeadaWgaaWcbaGaaGOmaaqabaaaaa@3836@ . We give the solution for solid (1) only: the solution for the second solid can be found by exchanging the moduli appropriately.

 

The coordinate system has origin at the initial point of contact between the two solids. The two solids make contact over a finite region a< x 1 <b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHsislcaWGHbGaeyipaWJaamiEam aaBaaaleaacaaIXaaabeaakiabgYda8iaadkgaaaa@3976@ , where

a= 4R F 2 (1+2γ)/π(12γ) E * b= 4R F 2 (12γ)/(1+2γ)π E * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabg2 da9maakaaabaGaaGinaiaadkfacaWGgbWaaSbaaSqaaiaaikdaaeqa aOGaaiikaiaaigdacqGHRaWkcaaIYaGaeq4SdCMaaiykaiaac+cacq aHapaCcaGGOaGaaGymaiabgkHiTiaaikdacqaHZoWzcaGGPaGaamyr amaaCaaaleqabaGaaiOkaaaaaeqaaOGaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaamOyaiabg2da9maakaaabaGaaGinaiaadkfacaWGgbWaaSbaaS qaaiaaikdaaeqaaOGaaiikaiaaigdacqGHsislcaaIYaGaeq4SdCMa aiykaiaac+cacaGGOaGaaGymaiabgUcaRiaaikdacqaHZoWzcaGGPa GaeqiWdaNaamyramaaCaaaleqabaGaaiOkaaaaaeqaaaaa@71CD@

and

R= R 1 R 2 R 1 + R 2 E * = E 1 E 2 (1 ν 1 2 ) E 2 +(1 ν 2 2 ) E 1 β= ( 12 ν 2 )/ μ 2 ( 12 ν 1 )/ μ 1 2( 1 ν 2 )/ μ 2 +2( 1 ν 1 )/ μ 1 γ= 1 π tan 1 (βf) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkfacqGH9aqpdaWcaaqaaiaadkfada WgaaWcbaGaaGymaaqabaGccaWGsbWaaSbaaSqaaiaaikdaaeqaaaGc baGaamOuamaaBaaaleaacaaIXaaabeaakiabgUcaRiaadkfadaWgaa WcbaGaaGOmaaqabaaaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaadwea daahaaWcbeqaaiaacQcaaaGccqGH9aqpdaWcaaqaaiaadweadaWgaa WcbaGaaGymaaqabaGccaWGfbWaaSbaaSqaaiaaikdaaeqaaaGcbaGa aiikaiaaigdacqGHsislcqaH9oGBdaqhaaWcbaGaaGymaaqaaiaaik daaaGccaGGPaGaamyramaaBaaaleaacaaIYaaabeaakiabgUcaRiaa cIcacaaIXaGaeyOeI0IaeqyVd42aa0baaSqaaiaaikdaaeaacaaIYa aaaOGaaiykaiaadweadaWgaaWcbaGaaGymaaqabaaaaOGaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaeqOSdiMaeyypa0ZaaSaaae aadaqadaqaaiaaigdacqGHsislcaaIYaGaeqyVd42aaSbaaSqaaiaa ikdaaeqaaaGccaGLOaGaayzkaaGaai4laiabeY7aTnaaBaaaleaaca aIYaaabeaakiabgkHiTmaabmaabaGaaGymaiabgkHiTiaaikdacqaH 9oGBdaWgaaWcbaGaaGymaaqabaaakiaawIcacaGLPaaacaGGVaGaeq iVd02aaSbaaSqaaiaaigdaaeqaaaGcbaGaaGOmamaabmaabaGaaGym aiabgkHiTiabe27aUnaaBaaaleaacaaIYaaabeaaaOGaayjkaiaawM caaiaac+cacqaH8oqBdaWgaaWcbaGaaGOmaaqabaGccqGHRaWkcaaI YaWaaeWaaeaacaaIXaGaeyOeI0IaeqyVd42aaSbaaSqaaiaaigdaae qaaaGccaGLOaGaayzkaaGaai4laiabeY7aTnaaBaaaleaacaaIXaaa beaaaaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 Uaeq4SdCMaeyypa0JaeyOeI0YaaSaaaeaacaaIXaaabaGaeqiWdaha aiGacshacaGGHbGaaiOBamaaCaaaleqabaGaeyOeI0IaaGymaaaaki aacIcacqaHYoGycaWGMbGaaiykaiaaykW7aaa@A884@

 

Only the derivatives of the complex potentials for this solution can be found analytically: they are

Ω'(z)= (f+i) F 2 πab { z ( z+a ) 1/2γ ( zb ) 1/2+γ } ω (z)= Ω ( z ¯ ) ¯ z Ω (z) Ω (z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHPoWvcaGGNaGaaiikaiaadQhaca GGPaGaeyypa0JaeyOeI0YaaSaaaeaacaGGOaGaamOzaiabgUcaRiaa dMgacaGGPaGaamOramaaBaaaleaacaaIYaaabeaaaOqaaiabec8aWj aadggacaWGIbaaamaacmaabaGaamOEaiabgkHiTmaabmaabaGaamOE aiabgUcaRiaadggaaiaawIcacaGLPaaadaahaaWcbeqaaiaaigdaca GGVaGaaGOmaiabgkHiTiabeo7aNbaakmaabmaabaGaamOEaiabgkHi TiaadkgaaiaawIcacaGLPaaadaahaaWcbeqaaiaaigdacaGGVaGaaG OmaiabgUcaRiabeo7aNbaaaOGaay5Eaiaaw2haaiaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7cuaHjpWDgaqbaiaacIcacaWG6bGaai ykaiabg2da9iabgkHiTmaanaaabaGafuyQdCLbauaacaGGOaGabmOE ayaaraGaaiykaaaacqGHsislcaWG6bGafuyQdCLbauGbauaacaGGOa GaamOEaiaacMcacqGHsislcuqHPoWvgaqbaiaacIcacaWG6bGaaiyk aaaa@9056@

 

 

Note that the solution is valid only for Im(z)>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaciGGjbGaaiyBaiaacIcacaWG6bGaai ykaiabg6da+iaaicdaaaa@38A0@ . You can use the formulas given at the end of Section 5.3.1 to determine stresses directly from Ω (z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuqHPoWvgaqbaiaacIcacaWG6bGaai ykaaaa@36B8@ .  In addition, the branch of (z+a) γ1/2 (zb) γ1/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaacIcacaWG6b Gaey4kaSIaamyyaiaacMcadaahaaWcbeqaaiabeo7aNjabgkHiTiaa igdacaGGVaGaaGOmaaaakiaacIcacaWG6bGaeyOeI0IaamOyaiaacM cadaahaaWcbeqaaiabgkHiTiabeo7aNjabgkHiTiaaigdacaGGVaGa aGOmaaaaaaa@4952@  must be selected so that

` ( zb ) 1/2+γ ( z+a ) 1/2γ = r 1 1/2γ e i(1/2γ) θ 1 r 2 1/2+γ e i(1/2+γ) θ 2 ( zb ) 1/2+γ ( z+a ) 1/2γ ¯ = r 1 1/2γ e i(1/2γ) θ 1 r 2 1/2+γ e i(1/2+γ) θ 2 ( z ¯ b ) 1/2+γ ( z ¯ +a ) 1/2γ = r 1 1/2γ e i(1/2γ) θ 1 r 2 1/2+γ e i(1/2+γ) θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaWaaeWaae aacaWG6bGaeyOeI0IaamOyaaGaayjkaiaawMcaamaaCaaaleqabaGa aGymaiaac+cacaaIYaGaey4kaSIaeq4SdCgaaOWaaeWaaeaacaWG6b Gaey4kaSIaamyyaaGaayjkaiaawMcaamaaCaaaleqabaGaaGymaiaa c+cacaaIYaGaeyOeI0Iaeq4SdCgaaOGaeyypa0JaamOCamaaDaaale aacaaIXaaabaGaaGymaiaac+cacaaIYaGaeyOeI0Iaeq4SdCgaaOGa amyzamaaCaaaleqabaGaamyAaiaacIcacaaIXaGaai4laiaaikdacq GHsislcqaHZoWzcaGGPaGaeqiUde3aaSbaaWqaaiaaigdaaeqaaaaa kiaadkhadaqhaaWcbaGaaGOmaaqaaiaaigdacaGGVaGaaGOmaiabgU caRiabeo7aNbaakiaadwgadaahaaWcbeqaaiaadMgacaGGOaGaaGym aiaac+cacaaIYaGaey4kaSIaeq4SdCMaaiykaiabeI7aXnaaBaaame aacaaIYaaabeaaaaaakeaadaqdaaqaamaabmaabaGaamOEaiabgkHi TiaadkgaaiaawIcacaGLPaaadaahaaWcbeqaaiaaigdacaGGVaGaaG OmaiabgUcaRiabeo7aNbaakmaabmaabaGaamOEaiabgUcaRiaadgga aiaawIcacaGLPaaadaahaaWcbeqaaiaaigdacaGGVaGaaGOmaiabgk HiTiabeo7aNbaaaaGccqGH9aqpcaWGYbWaa0baaSqaaiaaigdaaeaa caaIXaGaai4laiaaikdacqGHsislcqaHZoWzaaGccaWGLbWaaWbaaS qabeaacqGHsislcaWGPbGaaiikaiaaigdacaGGVaGaaGOmaiabgkHi Tiabeo7aNjaacMcacqaH4oqCdaWgaaadbaGaaGymaaqabaaaaOGaam OCamaaDaaaleaacaaIYaaabaGaaGymaiaac+cacaaIYaGaey4kaSIa eq4SdCgaaOGaamyzamaaCaaaleqabaGaeyOeI0IaamyAaiaacIcaca aIXaGaai4laiaaikdacqGHRaWkcqaHZoWzcaGGPaGaeqiUde3aaSba aWqaaiaaikdaaeqaaaaaaOqaamaabmaabaGabmOEayaaraGaeyOeI0 IaamOyaaGaayjkaiaawMcaamaaCaaaleqabaGaaGymaiaac+cacaaI YaGaey4kaSIaeq4SdCgaaOWaaeWaaeaaceWG6bGbaebacqGHRaWkca WGHbaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIXaGaai4laiaaikda cqGHsislcqaHZoWzaaGccqGH9aqpcaWGYbWaa0baaSqaaiaaigdaae aacaaIXaGaai4laiaaikdacqGHsislcqaHZoWzaaGccaWGLbWaaWba aSqabeaacqGHsislcaWGPbGaaiikaiaaigdacaGGVaGaaGOmaiabgk HiTiabeo7aNjaacMcacqaH4oqCdaWgaaadbaGaaGymaaqabaaaaOGa amOCamaaDaaaleaacaaIYaaabaGaaGymaiaac+cacaaIYaGaey4kaS Iaeq4SdCgaaOGaamyzamaaCaaaleqabaGaeyOeI0IaamyAaiaacIca caaIXaGaai4laiaaikdacqGHRaWkcqaHZoWzcaGGPaGaeqiUde3aaS baaWqaaiaaikdaaeqaaaaaaaaa@DD25@

where the angles and distances r 1 , θ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbWaaSbaaSqaaiaaigdaaeqaaO GaaiilaiabeI7aXnaaBaaaleaacaaIXaaabeaaaaa@37FB@  and r 2 , θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbWaaSbaaSqaaiaaikdaaeqaaO GaaiilaiabeI7aXnaaBaaaleaacaaIYaaabeaaaaa@37FD@  are shown in the figure, and θ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH4oqCdaWgaaWcbaGaaGymaaqaba aaaa@3563@  and θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH4oqCdaWgaaWcbaGaaGOmaaqaba aaaa@3564@  must lie in the ranges 0 θ 1 π0 θ 2 π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKc9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgs MiJkabeI7aXnaaBaaaleaacaaIXaaabeaakiabgsMiJkabec8aWjaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGimaiabgsMiJkabeI7aXnaaBaaaleaacaaI YaaabeaakiabgsMiJkabec8aWbaa@5907@ .

 

Important features of the solution are:

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

p( x 1 )= σ 22 ( x 1 , x 2 =0)= 2 F 2 πab 1+ β 2 f 2 a+ x 1 x 1 b ( x 1 +a x 1 b ) γ q( x 1 )= σ 12 =fp( x 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGWbGaaiikaiaadIhadaWgaaWcba GaaGymaaqabaGccaGGPaGaeyypa0JaeyOeI0Iaeq4Wdm3aaSbaaSqa aiaaikdacaaIYaaabeaakiaacIcacaWG4bWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaadIhadaWgaaWcbaGaaGOmaaqabaGccqGH9aqpcaaI WaGaaiykaiabg2da9maalaaabaGaaGOmaiaadAeadaWgaaWcbaGaaG OmaaqabaaakeaacqaHapaCcaWGHbGaamOyamaakaaabaGaaGymaiab gUcaRiabek7aInaaCaaaleqabaGaaGOmaaaakiaadAgadaahaaWcbe qaaiaaikdaaaaabeaaaaGcdaGcaaqaaiaadggacqGHRaWkcaWG4bWa aSbaaSqaaiaaigdaaeqaaaqabaGcdaGcaaqaaiaadIhadaWgaaWcba GaaGymaaqabaGccqGHsislcaWGIbaaleqaaOWaaeWaaeaadaWcaaqa aiaadIhadaWgaaWcbaGaaGymaaqabaGccqGHRaWkcaWGHbaabaGaam iEamaaBaaaleaacaaIXaaabeaakiabgkHiTiaadkgaaaaacaGLOaGa ayzkaaWaaWbaaSqabeaacqaHZoWzaaGccaaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caWGXbGaaiikaiaadIhadaWgaaWcbaGaaGym aaqabaGccaGGPaGaeyypa0Jaeq4Wdm3aaSbaaSqaaiaaigdacaaIYa aabeaakiabg2da9iaadAgacaWGWbGaaiikaiaadIhadaWgaaWcbaGa aGymaaqabaGccaGGPaaaaa@7C02@

In practice, the value of γ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo7aNbaa@320F@  is very small (generally less than 0.05), and you can approximate the solution by assuming that γ=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo7aNjabg2da9iaaicdaaaa@33CF@  without significant error.

2.      Approximate expressions for stresses.  For γ=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo7aNjabg2da9iaaicdaaaa@33CF@ , the stresses can be written in a simple form.  The stresses due to the vertical force are given in Section 5.3.8.  The stresses due to the friction force are

σ 11 = 2f F 2 π a 2 ( n( 2 ( x 2 2 m 2 ) ( m 2 + n 2 ) )2 x 1 ) σ 22 = 2f F 2 π a 2 n ( m 2 x 2 2 ) ( m 2 + n 2 ) σ 12 = 2f F 2 π a 2 m( 1 ( x 2 2 + n 2 ) ( m 2 + n 2 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIXaGaaGymaa qabaGccqGH9aqpcqGHsisldaWcaaqaaiaaikdacaWGMbGaamOramaa BaaaleaacaaIYaaabeaaaOqaaiabec8aWjaadggadaahaaWcbeqaai aaikdaaaaaaOWaaeWaaeaacaWGUbWaaeWaaeaacaaIYaGaeyOeI0Ya aSaaaeaacaGGOaGaamiEamaaDaaaleaacaaIYaaabaGaaGOmaaaaki abgkHiTiaad2gadaahaaWcbeqaaiaaikdaaaGccaGGPaaabaGaaiik aiaad2gadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWGUbWaaWbaaS qabeaacaaIYaaaaOGaaiykaaaaaiaawIcacaGLPaaacqGHsislcaaI YaGaamiEamaaBaaaleaacaaIXaaabeaaaOGaayjkaiaawMcaaiaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqaHdpWCdaWg aaWcbaGaaGOmaiaaikdaaeqaaOGaeyypa0JaeyOeI0YaaSaaaeaaca aIYaGaamOzaiaadAeadaWgaaWcbaGaaGOmaaqabaaakeaacqaHapaC caWGHbWaaWbaaSqabeaacaaIYaaaaaaakiaad6gadaWcaaqaaiaacI cacaWGTbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaamiEamaaDaaa leaacaaIYaaabaGaaGOmaaaakiaacMcaaeaacaGGOaGaamyBamaaCa aaleqabaGaaGOmaaaakiabgUcaRiaad6gadaahaaWcbeqaaiaaikda aaGccaGGPaaaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl abeo8aZnaaBaaaleaacaaIXaGaaGOmaaqabaGccqGH9aqpcqGHsisl daWcaaqaaiaaikdacaWGMbGaamOramaaBaaaleaacaaIYaaabeaaaO qaaiabec8aWjaadggadaahaaWcbeqaaiaaikdaaaaaaOGaamyBamaa bmaabaGaaGymaiabgkHiTmaalaaabaGaaiikaiaadIhadaqhaaWcba GaaGOmaaqaaiaaikdaaaGccqGHRaWkcaWGUbWaaWbaaSqabeaacaaI YaaaaOGaaiykaaqaaiaacIcacaWGTbWaaWbaaSqabeaacaaIYaaaaO Gaey4kaSIaamOBamaaCaaaleqabaGaaGOmaaaakiaacMcaaaaacaGL OaGaayzkaaGaaGPaVlaaykW7caaMc8oaaa@A377@

m= ( c 1 + c 2 )/2 n= x 1 ( c 1 c 2 )/(2 x 1 2 ) c 1 =1 x 1 2 x 2 2 c 2 = c 1 2 +4 x 1 2 x 2 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2gacqGH9aqpdaGcaaqaamaabmaaba Gaam4yamaaBaaaleaacaaIXaaabeaakiabgUcaRiaadogadaWgaaWc baGaaGOmaaqabaaakiaawIcacaGLPaaacaGGVaGaaGOmaaWcbeaaki aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caWGUbGaeyypa0JaamiEamaaBaaaleaacaaIXaaabe aakmaakaaabaWaaeWaaeaacaWGJbWaaSbaaSqaaiaaigdaaeqaaOGa eyOeI0Iaam4yamaaBaaaleaacaaIYaaabeaaaOGaayjkaiaawMcaai aac+cacaGGOaGaaGOmaiaadIhadaqhaaWcbaGaaGymaaqaaiaaikda aaGccaGGPaaaleqaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGJbWaaSbaaSqa aiaaigdaaeqaaOGaeyypa0JaaGymaiabgkHiTiaadIhadaqhaaWcba GaaGymaaqaaiaaikdaaaGccqGHsislcaWG4bWaa0baaSqaaiaaikda aeaacaaIYaaaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caWGJbWaaSbaaSqaaiaaikdaaeqaaOGaeyypa0Za aOaaaeaacaWGJbWaa0baaSqaaiaaigdaaeaacaaIYaaaaOGaey4kaS IaaGinaiaadIhadaqhaaWcbaGaaGymaaqaaiaaikdaaaGccaWG4bWa a0baaSqaaiaaikdaaeaacaaIYaaaaaqabaaaaa@8DB7@

 

 

 

5.3.12 Dislocation near the surface of a half-space

 

The figure shows a dislocation with burgers vector b= b 1 e 1 + b 2 e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahkgacqGH9aqpcaWGIbWaaSbaaSqaai aaigdaaeqaaOGaaCyzamaaBaaaleaacaaIXaaabeaakiabgUcaRiaa dkgadaWgaaWcbaGaaGOmaaqabaGccaWHLbWaaSbaaSqaaiaaikdaae qaaaaa@3AA1@  located at a depth h below the surface of an isotropic linear elastic half-space, with Young’s modulus E MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweaaaa@3130@  and Poisson’s ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe27aUbaa@321E@ .  The surface of the half-space is traction free.

 

The solution is given by the sum of two potentials:

Ω(z)= Ω 0 (z)+ Ω 1 (z)ω(z)= ω 0 (z)+ ω 1 (z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfM6axjaacIcacaWG6bGaaiykaiabg2 da9iabfM6axnaaBaaaleaacaaIWaaabeaakiaacIcacaWG6bGaaiyk aiabgUcaRiabfM6axnaaBaaaleaacaaIXaaabeaakiaacIcacaWG6b GaaiykaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaeqyYdCNaaiikaiaadQhacaGGPa Gaeyypa0JaeqyYdC3aaSbaaSqaaiaaicdaaeqaaOGaaiikaiaadQha caGGPaGaey4kaSIaeqyYdC3aaSbaaSqaaiaaigdaaeqaaOGaaiikai aadQhacaGGPaaaaa@6114@

where

Ω 0 (z)=i E( b 1 +i b 2 ) 8π(1 ν 2 ) log(zih) ω 0 (z)=i E( b 1 i b 2 ) 8π(1 ν 2 ) log(zih)+ E( b 1 +i b 2 ) 8π(1 ν 2 ) h zih MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiqacaWaaaGcbaGaeuyQdC1aaSbaaSqaaiaaicdaaeqaaO GaaiikaiaadQhacaGGPaGaeyypa0JaeyOeI0IaamyAamaalaaabaGa amyramaabmaabaGaamOyamaaBaaaleaacaaIXaaabeaakiabgUcaRi aadMgacaWGIbWaaSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaaa baGaaGioaiabec8aWjaacIcacaaIXaGaeyOeI0IaeqyVd42aaWbaaS qabeaacaaIYaaaaOGaaiykaaaaciGGSbGaai4BaiaacEgacaGGOaGa amOEaiabgkHiTiaadMgacaWGObGaaiykaiaaykW6caaMcSUaaGPaRl aaykW6caaMcSUaaGPaRlaaykW6caaMcSUaeqyYdC3aaSbaaSqaaiaa icdaaeqaaOGaaiikaiaadQhacaGGPaGaeyypa0JaamyAamaalaaaba GaamyraiaacIcacaWGIbWaaSbaaSqaaiaaigdaaeqaaOGaeyOeI0Ia amyAaiaadkgadaWgaaWcbaGaaGOmaaqabaGccaGGPaaabaGaaGioai abec8aWjaacIcacaaIXaGaeyOeI0IaeqyVd42aaWbaaSqabeaacaaI YaaaaOGaaiykaaaaciGGSbGaai4BaiaacEgacaGGOaGaamOEaiabgk HiTiaadMgacaWGObGaaiykaiabgUcaRmaalaaabaGaamyramaabmaa baGaamOyamaaBaaaleaacaaIXaaabeaakiabgUcaRiaadMgacaWGIb WaaSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaaabaGaaGioaiab ec8aWjaacIcacaaIXaGaeyOeI0IaeqyVd42aaWbaaSqabeaacaaIYa aaaOGaaiykaaaadaWcaaqaaiaadIgaaeaacaWG6bGaeyOeI0IaamyA aiaadIgaaaaaaa@9154@

is the solution for a dislocation at position z 0 =ih MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadQhadaWgaaWcbaGaaGimaaqabaGccq GH9aqpcaWGPbGaamiAaaaa@3536@  in an infinite solid, and

Ω 1 (z)=z Ω 0 ( z ¯ ) ¯ ω 0 ( z ¯ ) ¯ ω 1 (z)=z ω 0 ( z ¯ ) ¯ Ω 0 ( z ¯ ) ¯ +z Ω 0 ( z ¯ ) ¯ + z 2 Ω 0 ( z ¯ ) ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaeuyQdC1aaSbaaSqaaiaaigdaae qaaOGaaiikaiaadQhacaGGPaGaeyypa0JaeyOeI0IaamOEamaanaaa baGafuyQdCLbauaadaWgaaWcbaGaaGimaaqabaGccaGGOaGabmOEay aaraGaaiykaaaacqGHsisldaqdaaqaaiabeM8a3naaBaaaleaacaaI WaaabeaakiaacIcaceWG6bGbaebacaGGPaaaaaqaaiabeM8a3naaBa aaleaacaaIXaaabeaakiaacIcacaWG6bGaaiykaiabg2da9iaadQha daqdaaqaaiqbeM8a3zaafaWaaSbaaSqaaiaaicdaaeqaaOGaaiikai qadQhagaqeaiaacMcaaaGaeyOeI0Yaa0aaaeaacqqHPoWvdaWgaaWc baGaaGimaaqabaGccaGGOaGabmOEayaaraGaaiykaaaacqGHRaWkca WG6bWaa0aaaeaacuqHPoWvgaqbamaaBaaaleaacaaIWaaabeaakiaa cIcaceWG6bGbaebacaGGPaaaaiabgUcaRiaadQhadaahaaWcbeqaai aaikdaaaGcdaqdaaqaaiqbfM6axzaagaWaaSbaaSqaaiaaicdaaeqa aOGaaiikaiqadQhagaqeaiaacMcaaaaaaaa@648B@

corrects the solution to satisfy the traction free boundary condition at the surface.

 

The displacement and stress fields can be computed by substituting Ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfM6axbaa@31F4@  and ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeM8a3baa@3233@  into the standard formulas given in Sect 5.3.1 (do not use the half-space representation).  A symbolic manipulation program makes the calculation painless. Most symbolic manipulation programs will not be able to differentiate the complex conjugate of a function, so the derivatives of Ω 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfM6axnaaBaaaleaacaaIXaaabeaaaa a@32DD@  and ω 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeM8a3naaBaaaleaacaaIXaaabeaaaa a@331C@  should be calculated by substituting appropriate derivatives of Ω 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfM6axnaaBaaaleaacaaIWaaabeaaaa a@32DC@  and ω 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeM8a3naaBaaaleaacaaIWaaabeaaaa a@331B@  into the following formulas

Ω 1 (z)=z Ω 0 ( z ¯ ) ¯ Ω 0 ( z ¯ ) ¯ ω 0 ( z ¯ ) ¯ Ω 1 (z)=z Ω 0 ( z ¯ ) ¯ 2 Ω 0 ( z ¯ ) ¯ ω 0 ( z ¯ ) ¯ ω 1 (z)=z ω 0 ( z ¯ ) ¯ + ω 0 ( z ¯ ) ¯ +3z Ω 0 ( z ¯ ) ¯ + z 2 Ω 0 ( z ¯ ) ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGafuyQdCLbauaadaWgaaWcbaGaaG ymaaqabaGccaGGOaGaamOEaiaacMcacqGH9aqpcqGHsislcaWG6bWa a0aaaeaacuqHPoWvgaqbgaqbamaaBaaaleaacaaIWaaabeaakiaacI caceWG6bGbaebacaGGPaaaaiabgkHiTmaanaaabaGafuyQdCLbauaa daWgaaWcbaGaaGimaaqabaGccaGGOaGabmOEayaaraGaaiykaaaacq GHsisldaqdaaqaaiqbeM8a3zaafaWaaSbaaSqaaiaaicdaaeqaaOGa aiikaiqadQhagaqeaiaacMcaaaaabaGafuyQdCLbauGbauaadaWgaa WcbaGaaGymaaqabaGccaGGOaGaamOEaiaacMcacqGH9aqpcqGHsisl caWG6bWaa0aaaeaacuqHPoWvgaqbgaqbgaqbamaaBaaaleaacaaIWa aabeaakiaaykW7caaMc8UaaiikaiqadQhagaqeaiaacMcaaaGaeyOe I0IaaGOmamaanaaabaGafuyQdCLbauGbauaadaWgaaWcbaGaaGimaa qabaGccaGGOaGabmOEayaaraGaaiykaaaacqGHsisldaqdaaqaaiqb eM8a3zaafyaafaWaaSbaaSqaaiaaicdaaeqaaOGaaiikaiqadQhaga qeaiaacMcaaaaabaGafqyYdCNbauaadaWgaaWcbaGaaGymaaqabaGc caGGOaGaamOEaiaacMcacqGH9aqpcaWG6bWaa0aaaeaacuaHjpWDga qbgaqbamaaBaaaleaacaaIWaaabeaakiaacIcaceWG6bGbaebacaGG PaaaaiabgUcaRmaanaaabaGafqyYdCNbauaadaWgaaWcbaGaaGimaa qabaGccaGGOaGabmOEayaaraGaaiykaaaacqGHRaWkcaaIZaGaamOE amaanaaabaGafuyQdCLbauGbauaadaWgaaWcbaGaaGimaaqabaGcca GGOaGabmOEayaaraGaaiykaaaacqGHRaWkcaWG6bWaaWbaaSqabeaa caaIYaaaaOWaa0aaaeaacuqHPoWvgaqbgaqbgaqbamaaBaaaleaaca aIWaaabeaakiaaykW7caaMc8UaaiikaiqadQhagaqeaiaacMcaaaaa aaa@8C09@

As an example, the variation of stress along the line x 1 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaaGymaaqabaGccq GH9aqpcaaIWaaaaa@3416@  is given by

σ 22 = E b 1 π(1 ν 2 ) 2h x 2 2 ( x 2 +h) 3 ( x 2 h) σ 11 = E b 1 π(1 ν 2 ) 2 h 2 x 2 ( x 2 +h) 3 ( x 2 h) σ 12 = E b 2 π(1 ν 2 ) 2 h 2 x 2 ( x 2 +h) 3 ( x 2 h) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIYaGaaGOmaa qabaGccqGH9aqpdaWcaaqaaiaadweacaWGIbWaaSbaaSqaaiaaigda aeqaaaGcbaGaeqiWdaNaaiikaiaaigdacqGHsislcqaH9oGBdaahaa WcbeqaaiaaikdaaaGccaGGPaaaamaalaaabaGaaGOmaiaadIgacaWG 4bWaa0baaSqaaiaaikdaaeaacaaIYaaaaaGcbaGaaiikaiaadIhada WgaaWcbaGaaGOmaaqabaGccqGHRaWkcaWGObGaaiykamaaCaaaleqa baGaaG4maaaakiaacIcacaWG4bWaaSbaaSqaaiaaikdaaeqaaOGaey OeI0IaamiAaiaacMcaaaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlabeo8aZnaaBaaaleaacaaIXaGaaGymaaqabaGccq GH9aqpdaWcaaqaaiaadweacaWGIbWaaSbaaSqaaiaaigdaaeqaaaGc baGaeqiWdaNaaiikaiaaigdacqGHsislcqaH9oGBdaahaaWcbeqaai aaikdaaaGccaGGPaaaamaalaaabaGaaGOmaiaadIgadaahaaWcbeqa aiaaikdaaaGccaWG4bWaaSbaaSqaaiaaikdaaeqaaaGcbaGaaiikai aadIhadaWgaaWcbaGaaGOmaaqabaGccqGHRaWkcaWGObGaaiykamaa CaaaleqabaGaaG4maaaakiaacIcacaWG4bWaaSbaaSqaaiaaikdaae qaaOGaeyOeI0IaamiAaiaacMcaaaGaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8Uaeq4Wdm3aaSbaaSqaaiaaigdacaaIYaaabeaaki abg2da9maalaaabaGaamyraiaadkgadaWgaaWcbaGaaGOmaaqabaaa keaacqaHapaCcaGGOaGaaGymaiabgkHiTiabe27aUnaaCaaaleqaba GaaGOmaaaakiaacMcaaaWaaSaaaeaacqGHsislcaaIYaGaamiAamaa CaaaleqabaGaaGOmaaaakiaadIhadaWgaaWcbaGaaGOmaaqabaaake aacaGGOaGaamiEamaaBaaaleaacaaIYaaabeaakiabgUcaRiaadIga caGGPaWaaWbaaSqabeaacaaIZaaaaOGaaiikaiaadIhadaWgaaWcba GaaGOmaaqabaGccqGHsislcaWGObGaaiykaaaaaaa@A03A@