Chapter 5

 

Analytical techniques and solutions for linear elastic solids

 

 

 

5.9 Energetics of Dislocations in Elastic Solids

 

Dislocations play a crucial role in determining the response of crystalline materials to stress.  For example, plastic flow in ductile metals occurs as a result of dislocation motion; dislocation emission from a crack tip can determine whether a material is ductile or brittle; and stress induced dislocation nucleation plays a critical role in semiconductor devices.

 

Dislocations tend to move through a crystal in response to stress.  The goal of this and following sections is to derive some results that can be used to predict this motion.

 

 

5.9.1 Classical solution for potential energy of an isolated dislocation loop in an infinite solid

 

In this section, we show that the energy of an isolated dislocation loop with burgers vector b in an infinite solid can be calculated using the following expressions:

V D = E 32π(1+ν) C C 2 R(xξ) x p x p b i b j τ i (x) τ j (ξ)d s x d s ξ + E 16π(1 ν 2 ) C C ikl jmn b k b m 2 R(xξ) x i x j τ l (x) τ n (ξ)d s x d s ξ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamOvamaaCaaaleqabaGaamirai abg6HiLcaakiabg2da9maalaaabaGaamyraaqaaiaaiodacaaIYaGa eqiWdaNaaiikaiaaigdacqGHRaWkcqaH9oGBcaGGPaaaamaapufaba Waa8qvaeaadaWcaaqaaiabgkGi2oaaCaaaleqabaGaaGOmaaaakiaa dkfacaGGOaGaaCiEaiabgkHiTiaah67acaGGPaaabaGaeyOaIyRaam iEamaaBaaaleaacaWGWbaabeaakiabgkGi2kaadIhadaWgaaWcbaGa amiCaaqabaaaaOGaamOyamaaBaaaleaacaWGPbaabeaakiaadkgada WgaaWcbaGaamOAaaqabaGccqaHepaDdaWgaaWcbaGaamyAaaqabaGc caGGOaGaaCiEaiaacMcacqaHepaDdaWgaaWcbaGaamOAaaqabaGcca GGOaGaaCOVdiaacMcacaWGKbGaam4CamaaBaaaleaacaWH4baabeaa aeaacaWGdbaabeqdcqWIr4E0cqGHRiI8aaWcbaGaam4qaaqab0GaeS yeUhTaey4kIipakiaadsgacaWGZbWaaSbaaSqaaiaah67aaeqaaOGa aGPaVlaaykW7aeaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlabgUcaRmaalaaabaGaamyraaqaaiaaigdacaaI2aGaeqiWdaNaai ikaiaaigdacqGHsislcqaH9oGBdaahaaWcbeqaaiaaikdaaaGccaGG PaaaamaapufabaWaa8qvaeaacqGHiiIZdaWgaaWcbaGaamyAaiaadU gacaWGSbaabeaakiabgIGiopaaBaaaleaacaWGQbGaamyBaiaad6ga aeqaaOGaamOyamaaBaaaleaacaWGRbaabeaakiaadkgadaWgaaWcba GaamyBaaqabaGcdaWcaaqaaiabgkGi2oaaCaaaleqabaGaaGOmaaaa kiaadkfacaGGOaGaaCiEaiabgkHiTiaah67acaGGPaaabaGaeyOaIy RaamiEamaaBaaaleaacaWGPbaabeaakiabgkGi2kaadIhadaWgaaWc baGaamOAaaqabaaaaOGaeqiXdq3aaSbaaSqaaiaadYgaaeqaaOGaai ikaiaahIhacaGGPaGaeqiXdq3aaSbaaSqaaiaad6gaaeqaaOGaaiik aiaah67acaGGPaGaamizaiaadohadaWgaaWcbaGaaCiEaaqabaGcca WGKbGaam4CamaaBaaaleaacaWH+oaabeaaaeaacaWGdbaabeqdcqWI r4E0cqGHRiI8aaWcbaGaam4qaaqab0GaeSyeUhTaey4kIipaaaaa@C85A@

Here, R(xξ)= ( x k ξ k )( x k ξ k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkfacaGGOaGaaCiEaiabgkHiTiaah6 7acaGGPaGaeyypa0ZaaOaaaeaadaqadaqaaiaadIhadaWgaaWcbaGa am4AaaqabaGccqGHsislcqaH+oaEdaWgaaWcbaGaam4Aaaqabaaaki aawIcacaGLPaaadaqadaqaaiaadIhadaWgaaWcbaGaam4AaaqabaGc cqGHsislcqaH+oaEdaWgaaWcbaGaam4AaaqabaaakiaawIcacaGLPa aaaSqabaaaaa@4615@  and the integral is taken around the dislocation line twice.  In the first integral, ξ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaah67aaaa@31D2@  is held fixed, and x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahIhaaaa@3189@  varies with position around the dislocation line; then ξ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaah67aaaa@31D2@  is varied for the second line integral.

 

 

Difficulties with evaluating the potential energy in the classical solution: In practice, this is a purely formal result MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  in the classical solution, the energy of a dislocation is always infinite.   You can see this clearly using the solution for a straight dislocation in an infinite solid given in Section 5.3.4.  Recall that the stress state for 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 is given (in polar coordinates) by 

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@

The strain energy density distribution around the dislocation follows as

U= 1+ν 2E { ( σ rr 2 + σ θθ 2 +2 σ rθ 2 )ν ( σ rr + σ θθ ) 2 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwfacqGH9aqpdaWcaaqaaiaaigdacq GHRaWkcqaH9oGBaeaacaaIYaGaamyraaaadaGadaqaamaabmaabaGa eq4Wdm3aa0baaSqaaiaadkhacaWGYbaabaGaaGOmaaaakiabgUcaRi abeo8aZnaaDaaaleaacqaH4oqCcqaH4oqCaeaacaaIYaaaaOGaey4k aSIaaGOmaiabeo8aZnaaDaaaleaacaWGYbGaeqiUdehabaGaaGOmaa aaaOGaayjkaiaawMcaaiabgkHiTiabe27aUnaabmaabaGaeq4Wdm3a aSbaaSqaaiaadkhacaWGYbaabeaakiabgUcaRiabeo8aZnaaBaaale aacqaH4oqCcqaH4oqCaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaa caaIYaaaaaGccaGL7bGaayzFaaaaaa@5CE4@

We can use this to calculate the total strain energy in an annular region around the dislocation, with inner radius a, and outer radius b.   The result is

V= a b 0 2π U(r,θ)rdθdr = E( b 1 2 + b 2 2 ) 8π(1 ν 2 ) log( b a ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAfacqGH9aqpdaWdXbqaamaapehaba GaamyvaiaacIcacaWGYbGaaiilaiabeI7aXjaacMcacaWGYbGaamiz aiabeI7aXjaadsgacaWGYbaaleaacaaIWaaabaGaaGOmaiabec8aWb qdcqGHRiI8aOGaeyypa0ZaaSaaaeaacaWGfbGaaiikaiaadkgadaqh aaWcbaGaaGymaaqaaiaaikdaaaGccqGHRaWkcaWGIbWaa0baaSqaai aaikdaaeaacaaIYaaaaOGaaiykaaqaaiaaiIdacqaHapaCcaGGOaGa aGymaiabgkHiTiabe27aUnaaCaaaleqabaGaaGOmaaaakiaacMcaaa GaciiBaiaac+gacaGGNbWaaeWaaeaadaWcaaqaaiaadkgaaeaacaWG HbaaaaGaayjkaiaawMcaaaWcbaGaamyyaaqaaiaadkgaa0Gaey4kIi paaaa@5EF2@

Taking a0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadggacqGHsgIRcaaIWaaaaa@3415@  gives an infinite energy, because the strain energy density varies as 1/ r 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaigdacaGGVaGaamOCamaaCaaaleqaba GaaGOmaaaaaaa@33D6@  near the dislocation core.

 

Various ways to avoid this problem have been proposed.   The simplest approach is to neglect the strain energy in a tubular region with small radius r 0 | b |/4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkhadaWgaaWcbaGaaGimaaqabaGccq GHijYUdaabdaqaaiaahkgaaiaawEa7caGLiWoacaGGVaGaaGinaaaa @399E@  surrounding the dislocation, on the grounds that the elastic solution does not accurately characterize the atomic-scale deformation near the dislocation core. This works for straight dislocations, but is not easy to apply to 3D dislocation loops.   A more satisfactory approach is described in the next section.

 

Application to a circular prismatic dislocation loop As an example, we attempt to apply the general formula to calculate the energy of a circular dislocation loop, with radius a, which lies in the ( x 1 , x 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacIcacaWG4bWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaadIhadaWgaaWcbaGaaGOmaaqabaGccaGGPaaaaa@366E@  plane, and has a burgers vector b=b e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahkgacqGH9aqpcaWGIbGaaCyzamaaBa aaleaacaaIZaaabeaaaaa@3537@  that is perpendicular to the plane of the loop.  For this case, the contour integral for the potential energy reduces to

V D = Ea b 2 8(1 ν 2 ) 0 2π 1+cosθ 2 2(1cosθ) dθ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAfadaahaaWcbeqaaiaadseacqGHEi sPaaGccqGH9aqpdaWcaaqaaiaadweacaWGHbGaamOyamaaCaaaleqa baGaaGOmaaaaaOqaaiaaiIdacaGGOaGaaGymaiabgkHiTiabe27aUn aaCaaaleqabaGaaGOmaaaakiaacMcaaaWaa8qCaeaadaWcaaqaaiaa igdacqGHRaWkciGGJbGaai4BaiaacohacqaH4oqCaeaacaaIYaWaaO aaaeaacaaIYaGaaiikaiaaigdacqGHsislciGGJbGaai4Baiaacoha cqaH4oqCcaGGPaaaleqaaaaakiaadsgacqaH4oqCaSqaaiaaicdaae aacaaIYaGaeqiWdahaniabgUIiYdaaaa@567A@

(To see this, note that the result of the integral with respect to x must be independent of ξ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaah67aaaa@31D2@  by symmetry). As expected, the integral is divergent.  In the classical theory, the energy of the loop is estimated by truncating the integral around the singularity, so that

V D = Ea b 2 8(1 ν 2 ) r/a 2πr/a 1+cosθ 2 2(1cosθ) dθ Ea b 2 4(1 ν 2 ) { log( 4a r )1 }+Ο (ρ/a) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAfadaahaaWcbeqaaiaadseacqGHEi sPaaGccqGH9aqpdaWcaaqaaiaadweacaWGHbGaamOyamaaCaaaleqa baGaaGOmaaaaaOqaaiaaiIdacaGGOaGaaGymaiabgkHiTiabe27aUn aaCaaaleqabaGaaGOmaaaakiaacMcaaaWaa8qCaeaadaWcaaqaaiaa igdacqGHRaWkciGGJbGaai4BaiaacohacqaH4oqCaeaacaaIYaWaaO aaaeaacaaIYaGaaiikaiaaigdacqGHsislciGGJbGaai4Baiaacoha cqaH4oqCcaGGPaaaleqaaaaakiaadsgacqaH4oqCaSqaaiaadkhaca GGVaGaamyyaaqaaiaaikdacqaHapaCcqGHsislcaWGYbGaai4laiaa dggaa0Gaey4kIipakiabgIKi7oaalaaabaGaamyraiaadggacaWGIb WaaWbaaSqabeaacaaIYaaaaaGcbaGaaGinaiaacIcacaaIXaGaeyOe I0IaeqyVd42aaWbaaSqabeaacaaIYaaaaOGaaiykaaaadaGadaqaai GacYgacaGGVbGaai4zamaabmaabaWaaSaaaeaacaaI0aGaamyyaaqa aiaadkhaaaaacaGLOaGaayzkaaGaeyOeI0IaaGymaaGaay5Eaiaaw2 haaiabgUcaRiabf+5apjaacIcacqaHbpGCcaGGVaGaamyyaiaacMca daahaaWcbeqaaiaaikdaaaaaaa@7A62@

where r is a small cut-off distance.  This is somewhat similar to the core cutoff radius r 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbWaaSbaaSqaaiaaicdaaeqaaa aa@34A3@ , but the relationship between r 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbWaaSbaaSqaaiaaicdaaeqaaa aa@34A3@  and r is not clear.

 

A Circular glide loop, which has burgers vector b (with magnitude b) in the plane of the loop, has energy

V D = Ea b 2 (2ν) 4(1 ν 2 ) { log( 4a r )2 }+Ο (ρ/a) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAfadaahaaWcbeqaaiaadseacqGHEi sPaaGccqGH9aqpcqGHijYUdaWcaaqaaiaadweacaWGHbGaamOyamaa CaaaleqabaGaaGOmaaaakiaacIcacaaIYaGaeyOeI0IaeqyVd4Maai ykaaqaaiaaisdacaGGOaGaaGymaiabgkHiTiabe27aUnaaCaaaleqa baGaaGOmaaaakiaacMcaaaWaaiWaaeaaciGGSbGaai4BaiaacEgada qadaqaamaalaaabaGaaGinaiaadggaaeaacaWGYbaaaaGaayjkaiaa wMcaaiabgkHiTiaaikdaaiaawUhacaGL9baacqGHRaWkcqqHFoWtca GGOaGaeqyWdiNaai4laiaadggacaGGPaWaaWbaaSqabeaacaaIYaaa aaaa@5820@

 

Derivation of the solution for the energy of a 3D dislocation loop

1.      Let [ u i (ξ), ε pq (ξ), σ pq (ξ)] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUfacaWG1bWaaSbaaSqaaiaadMgaae qaaOGaaiikaiaah67acaGGPaGaaiilaiabew7aLnaaBaaaleaacaWG WbGaamyCaaqabaGccaGGOaGaaCOVdiaacMcacaGGSaGaeq4Wdm3aaS baaSqaaiaadchacaWGXbaabeaakiaacIcacaWH+oGaaiykaiaac2fa aaa@455B@  denote the displacement, strain and stress induced by the dislocation loop.   The total potential energy of the solid can be calculated by integrating the strain energy density over the volume of the solid

V D = V 1 2 σ ij ε ij dV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAfadaahaaWcbeqaaiaadseacqGHEi sPaaGccqGH9aqpdaWdrbqaamaalaaabaGaaGymaaqaaiaaikdaaaGa eq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakiabew7aLnaaBaaale aacaWGPbGaamOAaaqabaGccaWGKbGaamOvaaWcbaGaamOvaaqab0Ga ey4kIipaaaa@42DA@

2.      The potential energy can also be expressed in terms of the displacement field in the solid, as

V D = V 1 2 σ ij u i x j dV = 1 2 V ( σ ij u i ) x j u i ( σ ij ) x j dV = 1 2 V ( σ ij u i ) x j dV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAfadaahaaWcbeqaaiaadseacqGHEi sPaaGccqGH9aqpdaWdrbqaamaalaaabaGaaGymaaqaaiaaikdaaaGa eq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakmaalaaabaGaeyOaIy RaamyDamaaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2kaadIhadaWg aaWcbaGaamOAaaqabaaaaOGaamizaiaadAfaaSqaaiaadAfaaeqani abgUIiYdGccqGH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaaamaapefa baWaaSaaaeaacqGHciITcaGGOaGaeq4Wdm3aaSbaaSqaaiaadMgaca WGQbaabeaakiaadwhadaWgaaWcbaGaamyAaaqabaGccaGGPaaabaGa eyOaIyRaamiEamaaBaaaleaacaWGQbaabeaaaaGccqGHsislcaWG1b WaaSbaaSqaaiaadMgaaeqaaOWaaSaaaeaacqGHciITcaGGOaGaeq4W dm3aaSbaaSqaaiaadMgacaWGQbaabeaakiaacMcaaeaacqGHciITca WG4bWaaSbaaSqaaiaadQgaaeqaaaaakiaadsgacaWGwbaaleaacaWG wbaabeqdcqGHRiI8aOGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaa aadaWdrbqaamaalaaabaGaeyOaIyRaaiikaiabeo8aZnaaBaaaleaa caWGPbGaamOAaaqabaGccaWG1bWaaSbaaSqaaiaadMgaaeqaaOGaai ykaaqaaiabgkGi2kaadIhadaWgaaWcbaGaamOAaaqabaaaaOGaamiz aiaadAfaaSqaaiaadAfaaeqaniabgUIiYdaaaa@7B0A@

where we have used the symmetry of σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaaaaa@3454@  and recalled that the stress field satisfies the equilibrium equation σ ij / x j =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkGi2kabeo8aZnaaBaaaleaacaWGPb GaamOAaaqabaGccaGGVaGaeyOaIyRaamiEamaaBaaaleaacaWGQbaa beaakiabg2da9iaaicdaaaa@3BBF@

3.      Applying the divergence theorem, and taking into account the discontinuity in u i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaamyAaaqabaaaaa@329C@  across S,

V D = 1 2 S σ ij ( m j ) u i + dA + 1 2 S σ ij m j u i dA = 1 2 S σ ij m j b i dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAfadaahaaWcbeqaaiaadseacqGHEi sPaaGccqGH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaaamaapefabaGa eq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakiaacIcacqGHsislca WGTbWaaSbaaSqaaiaadQgaaeqaaOGaaiykaiaadwhadaqhaaWcbaGa amyAaaqaaiabgUcaRaaakiaadsgacaWGbbaaleaacaWGtbaabeqdcq GHRiI8aOGaey4kaSYaaSaaaeaacaaIXaaabaGaaGOmaaaadaWdrbqa aiabeo8aZnaaBaaaleaacaWGPbGaamOAaaqabaGccaWGTbWaaSbaaS qaaiaadQgaaeqaaOGaamyDamaaDaaaleaacaWGPbaabaGaeyOeI0ca aOGaamizaiaadgeaaSqaaiaadofaaeqaniabgUIiYdGccqGH9aqpda WcaaqaaiaaigdaaeaacaaIYaaaamaapefabaGaeq4Wdm3aaSbaaSqa aiaadMgacaWGQbaabeaakiaad2gadaWgaaWcbaGaamOAaaqabaGcca WGIbWaaSbaaSqaaiaadMgaaeqaaOGaamizaiaadgeaaSqaaiaadofa aeqaniabgUIiYdaaaa@6603@

4.      Next, we substitute the expression given in Section 5.8.4 for σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaaaaa@3454@  and reverse the order of integration

V D = E 32π(1+ν) C S ( [ imp b m τ q + imq b m τ p ] 3 R(xξ) x i x j x j ) m p b q d A x d s ξ + E 16π(1 ν 2 ) C S ( b m imk τ k [ 3 R(xξ) x i x p x q δ pq 3 R(xξ) x i x j x j ] ) m p b q d A x d s ξ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamOvamaaCaaaleqabaGaamirai abg6HiLcaakiabg2da9maalaaabaGaamyraaqaaiaaiodacaaIYaGa eqiWdaNaaiikaiaaigdacqGHRaWkcqaH9oGBcaGGPaaaamaapefaba Waa8quaeaadaqadaqaamaadmaabaGaeyicI48aaSbaaSqaaiaadMga caWGTbGaamiCaaqabaGccaWGIbWaaSbaaSqaaiaad2gaaeqaaOGaeq iXdq3aaSbaaSqaaiaadghaaeqaaOGaey4kaSIaeyicI48aaSbaaSqa aiaadMgacaWGTbGaamyCaaqabaGccaWGIbWaaSbaaSqaaiaad2gaae qaaOGaeqiXdq3aaSbaaSqaaiaadchaaeqaaaGccaGLBbGaayzxaaWa aSaaaeaacqGHciITdaahaaWcbeqaaiaaiodaaaGccaWGsbGaaiikai aahIhacqGHsislcaWH+oGaaiykaaqaaiabgkGi2kaadIhadaWgaaWc baGaamyAaaqabaGccqGHciITcaWG4bWaaSbaaSqaaiaadQgaaeqaaO GaeyOaIyRaamiEamaaBaaaleaacaWGQbaabeaaaaaakiaawIcacaGL PaaacaWGTbWaaSbaaSqaaiaadchaaeqaaOGaamOyamaaBaaaleaaca WGXbaabeaakiaadsgacaWGbbWaaSbaaSqaaiaahIhaaeqaaaqaaiaa dofaaeqaniabgUIiYdaaleaacaWGdbaabeqdcqGHRiI8aOGaamizai aadohadaWgaaWcbaGaaCOVdaqabaaakeaacaaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8Uaey4kaSIaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8+aaSaaaeaacaWGfbaabaGaaGymaiaaiAdacq aHapaCcaGGOaGaaGymaiabgkHiTiabe27aUnaaCaaaleqabaGaaGOm aaaakiaacMcaaaWaa8quaeaadaWdrbqaamaabmaabaGaamOyamaaBa aaleaacaWGTbaabeaakiabgIGiopaaBaaaleaacaWGPbGaamyBaiaa dUgaaeqaaOGaeqiXdq3aaSbaaSqaaiaadUgaaeqaaOWaamWaaeaada WcaaqaaiabgkGi2oaaCaaaleqabaGaaG4maaaakiaadkfacaGGOaGa aCiEaiabgkHiTiaah67acaGGPaaabaGaeyOaIyRaamiEamaaBaaale aacaWGPbaabeaakiabgkGi2kaadIhadaWgaaWcbaGaamiCaaqabaGc cqGHciITcaWG4bWaaSbaaSqaaiaadghaaeqaaaaakiabgkHiTiabes 7aKnaaBaaaleaacaWGWbGaamyCaaqabaGcdaWcaaqaaiabgkGi2oaa CaaaleqabaGaaG4maaaakiaadkfacaGGOaGaaCiEaiabgkHiTiaah6 7acaGGPaaabaGaeyOaIyRaamiEamaaBaaaleaacaWGPbaabeaakiab gkGi2kaadIhadaWgaaWcbaGaamOAaaqabaGccqGHciITcaWG4bWaaS baaSqaaiaadQgaaeqaaaaaaOGaay5waiaaw2faaaGaayjkaiaawMca aaWcbaGaam4uaaqab0Gaey4kIipaaSqaaiaadoeaaeqaniabgUIiYd GccaWGTbWaaSbaaSqaaiaadchaaeqaaOGaamOyamaaBaaaleaacaWG XbaabeaakiaadsgacaWGbbWaaSbaaSqaaiaahIhaaeqaaOGaamizai aadohadaWgaaWcbaGaaCOVdaqabaaaaaa@0315@

5.      Finally, the surface integrals in this expression can be transformed into a contour integral around C by means of Stoke’s theorem

S npj F j x p m n d A x = C F j τ j d s x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaGaeyicI48aaSbaaSqaaiaad6 gacaWGWbGaamOAaaqabaGcdaWcaaqaaiabgkGi2kaadAeadaWgaaWc baGaamOAaaqabaaakeaacqGHciITcaWG4bWaaSbaaSqaaiaadchaae qaaaaakiaad2gadaWgaaWcbaGaamOBaaqabaGccaWGKbGaamyqamaa BaaaleaacaWH4baabeaakiabg2da9maapefabaGaamOramaaBaaale aacaWGQbaabeaakiabes8a0naaBaaaleaacaWGQbaabeaakiaadsga caWGZbWaaSbaaSqaaiaahIhaaeqaaaqaaiaadoeaaeqaniabgUIiYd aaleaacaWGtbaabeqdcqGHRiI8aaaa@5025@

After some tedious index manipulations, this gives the required result.

 

 

 

5.9.2 Non-singular dislocation theory

 

The infinite potential energy associated with the classical description of a dislocation is unphysical, and highly unsatisfactory.  A straightforward approach to avoiding this difficulty was proposed by Cai et al, Journal of the Mechanics and Physics of Solids, 54, 561-587, (2006).

 

In the classical solution, the dislocation core is localized at a single point in space, which leads to an infinite energy.  In practice, dislocation cores are distributed over a small, but finite, area as indicated in the figure. The TEM micrograph is from Tillmann et al Microsc. Microanal. 10, 185–198, 2004.

 

This effect can be modeled approximately by using the classical solution to construct a distributed dislocation core.  To this end, we suppose that the burgers vector of the dislocation can be represented by a distribution bβ(x) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahkgacqaHYoGycaGGOaGaaCiEaiaacM caaaa@356E@ , which must be chosen to satisfy

b(x)=b V β(ξx)d V ξ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahkgacaGGOaGaaCiEaiaacMcacqGH9a qpcaWHIbWaa8quaeaacqaHYoGycaGGOaGaaCOVdiabgkHiTiaahIha caGGPaGaamizaiaadAfadaWgaaWcbaGaaCOVdaqabaaabaGaamOvaa qab0Gaey4kIipaaaa@4243@

where the volume integral extends over the entire infinite solid.  In principle, β(x) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabek7aIjaacIcacaWH4bGaaiykaaaa@3483@  could be constructed to give an accurate description of the atomic-scale strain field in the immediate neighborhood of the dislocation core, but this is difficult to do, and is not the main intent of the theory.  Instead, β(x) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabek7aIjaacIcacaWH4bGaaiykaaaa@3483@  is selected to make the expressions for the energy and stress field of the dislocation as simple as possible.  It is particularly convenient to choose β(x) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabek7aIjaacIcacaWH4bGaaiykaaaa@3483@  to satisfy

R ρ (xξ)= V V R(yz)β(zξ)β(yx)d V z d V y R(x)= x k x k R ρ (x)= x k x k + ρ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkfadaWgaaWcbaGaeqyWdihabeaaki aacIcacaWH4bGaeyOeI0IaaCOVdiaacMcacqGH9aqpdaWdrbqaamaa pefabaGaamOuaiaacIcacaWH5bGaeyOeI0IaaCOEaiaacMcacqaHYo GycaGGOaGaaCOEaiabgkHiTiaah67acaGGPaGaeqOSdiMaaiikaiaa hMhacqGHsislcaWH4bGaaiykaiaadsgacaWGwbWaaSbaaSqaaiaahQ haaeqaaOGaamizaiaadAfadaWgaaWcbaGaaCyEaaqabaaabaGaamOv aaqab0Gaey4kIipaaSqaaiaadAfaaeqaniabgUIiYdGccaaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaamOuaiaacIcacaWH4bGaaiykaiabg2da9maakaaabaGaamiEam aaBaaaleaacaWGRbaabeaakiaadIhadaWgaaWcbaGaam4Aaaqabaaa beaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaamOuamaaBaaale aacqaHbpGCaeqaaOGaaiikaiaahIhacaGGPaGaeyypa0ZaaOaaaeaa caWG4bWaaSbaaSqaaiaadUgaaeqaaOGaamiEamaaBaaaleaacaWGRb aabeaakiabgUcaRiabeg8aYnaaCaaaleqabaGaaGOmaaaaaeqaaaaa @8424@

where ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYbaa@3248@  is a small characteristic length, comparable to the dimensions of the dislocation core. The required distribution cannot be calculated exactly, but is closely approximated by

β(x) 15 8π { 1m ρ 1 3 ( x k x k / ρ 1 2 +1 ) 7/2 + m ρ 2 3 ( x k x k / ρ 2 2 +1 ) 7/2 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabek7aIjaacIcacaWH4bGaaiykaiabgI Ki7oaalaaabaGaaGymaiaaiwdaaeaacaaI4aGaeqiWdahaamaacmaa baWaaSaaaeaacaaIXaGaeyOeI0IaamyBaaqaaiabeg8aYnaaDaaale aacaaIXaaabaGaaG4maaaakmaabmaabaGaamiEamaaBaaaleaacaWG RbaabeaakiaadIhadaWgaaWcbaGaam4AaaqabaGccaGGVaGaeqyWdi 3aa0baaSqaaiaaigdaaeaacaaIYaaaaOGaey4kaSIaaGymaaGaayjk aiaawMcaamaaCaaaleqabaGaaG4naiaac+cacaaIYaaaaaaakiabgU caRmaalaaabaGaamyBaaqaaiabeg8aYnaaDaaaleaacaaIYaaabaGa aG4maaaakmaabmaabaGaamiEamaaBaaaleaacaWGRbaabeaakiaadI hadaWgaaWcbaGaam4AaaqabaGccaGGVaGaeqyWdi3aa0baaSqaaiaa ikdaaeaacaaIYaaaaOGaey4kaSIaaGymaaGaayjkaiaawMcaamaaCa aaleqabaGaaG4naiaac+cacaaIYaaaaaaaaOGaay5Eaiaaw2haaaaa @63C4@

with ρ 1 =0.9038ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYnaaBaaaleaacaaIXaaabeaaki abg2da9iaaicdacaGGUaGaaGyoaiaaicdacaaIZaGaaGioaiabeg8a Ybaa@3A67@ , ρ 2 =0.5451ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYnaaBaaaleaacaaIYaaabeaaki abg2da9iaaicdacaGGUaGaaGynaiaaisdacaaI1aGaaGymaiabeg8a Ybaa@3A63@ , m=0.6575 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2gacqGH9aqpcaaIWaGaaiOlaiaaiA dacaaI1aGaaG4naiaaiwdaaaa@36EB@ .  The distribution can also be shown to satisfy

V β(xξ)β(ξ)dV = 15 8π ρ 3 ( x k x k + ρ 2 ) 7/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaGaeqOSdiMaaiikaiaahIhacq GHsislcaWH+oGaaiykaiabek7aIjaacIcacaWH+oGaaiykaiaadsga caWGwbaaleaacaWGwbaabeqdcqGHRiI8aOGaeyypa0ZaaSaaaeaaca aIXaGaaGynaaqaaiaaiIdacqaHapaCaaWaaSaaaeaacqaHbpGCdaah aaWcbeqaaiaaiodaaaaakeaadaqadaqaaiaadIhadaWgaaWcbaGaam 4AaaqabaGccaWG4bWaaSbaaSqaaiaadUgaaeqaaOGaey4kaSIaeqyW di3aaWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzkaaWaaWbaaSqabe aacaaI3aGaai4laiaaikdaaaaaaaaa@5385@

 

 Nonsingular energy: The expression for the energy of a dislocation loop then reduces to

V D = E 32π(1+ν) C C 2 R ρ x p x p b i b j τ i (x) τ j (ξ)d s x d s ξ + E 16π(1 ν 2 ) C C ikl jmn b k b m 2 R ρ x i x j τ l (x) τ n (ξ)d s x d s ξ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamOvamaaCaaaleqabaGaamirai abg6HiLcaakiabg2da9maalaaabaGaamyraaqaaiaaiodacaaIYaGa eqiWdaNaaiikaiaaigdacqGHRaWkcqaH9oGBcaGGPaaaamaapufaba Waa8qvaeaadaWcaaqaaiabgkGi2oaaCaaaleqabaGaaGOmaaaakiaa dkfadaWgaaWcbaGaeqyWdihabeaaaOqaaiabgkGi2kaadIhadaWgaa WcbaGaamiCaaqabaGccqGHciITcaWG4bWaaSbaaSqaaiaadchaaeqa aaaakiaadkgadaWgaaWcbaGaamyAaaqabaGccaWGIbWaaSbaaSqaai aadQgaaeqaaOGaeqiXdq3aaSbaaSqaaiaadMgaaeqaaOGaaiikaiaa hIhacaGGPaGaeqiXdq3aaSbaaSqaaiaadQgaaeqaaOGaaiikaiaah6 7acaGGPaGaamizaiaadohadaWgaaWcbaGaaCiEaaqabaaabaGaam4q aaqab0GaeSyeUhTaey4kIipaaSqaaiaadoeaaeqaniablgH7rlabgU IiYdGccaWGKbGaam4CamaaBaaaleaacaWH+oaabeaakiaaykW7caaM c8oabaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqGHRaWk daWcaaqaaiaadweaaeaacaaIXaGaaGOnaiabec8aWjaacIcacaaIXa GaeyOeI0IaeqyVd42aaWbaaSqabeaacaaIYaaaaOGaaiykaaaadaWd vbqaamaapufabaGaeyicI48aaSbaaSqaaiaadMgacaWGRbGaamiBaa qabaGccqGHiiIZdaWgaaWcbaGaamOAaiaad2gacaWGUbaabeaakiaa dkgadaWgaaWcbaGaam4AaaqabaGccaWGIbWaaSbaaSqaaiaad2gaae qaaOWaaSaaaeaacqGHciITdaahaaWcbeqaaiaaikdaaaGccaWGsbWa aSbaaSqaaiabeg8aYbqabaaakeaacqGHciITcaWG4bWaaSbaaSqaai aadMgaaeqaaOGaeyOaIyRaamiEamaaBaaaleaacaWGQbaabeaaaaGc cqaHepaDdaWgaaWcbaGaamiBaaqabaGccaGGOaGaaCiEaiaacMcacq aHepaDdaWgaaWcbaGaamOBaaqabaGccaGGOaGaaCOVdiaacMcacaWG KbGaam4CamaaBaaaleaacaWH4baabeaakiaadsgacaWGZbWaaSbaaS qaaiaah67aaeqaaaqaaiaadoeaaeqaniablgH7rlabgUIiYdaaleaa caWGdbaabeqdcqWIr4E0cqGHRiI8aaaaaa@C325@

This is virtually identical to the classical singular solution, except that the derivatives of R ρ (x)= x k x k + ρ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkfadaWgaaWcbaGaeqyWdihabeaaki aacIcacaWH4bGaaiykaiabg2da9maakaaabaGaamiEamaaBaaaleaa caWGRbaabeaakiaadIhadaWgaaWcbaGaam4AaaqabaGccqGHRaWkcq aHbpGCdaahaaWcbeqaaiaaikdaaaaabeaaaaa@3E96@  are bounded everywhere, so the integral is finite.

 

 Nonsingular Stress: The stress due to the dislocation loop can be expressed in terms of a function Β ρ (x) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfk5acnaaBaaaleaacqaHbpGCaeqaaO GaaiikaiaahIhacaGGPaaaaa@363A@ , defined as

Β ρ (x)= V R(y)β(yx)d V y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfk5acnaaBaaaleaacqaHbpGCaeqaaO GaaiikaiaahIhacaGGPaGaeyypa0Zaa8quaeaacaWGsbGaaiikaiaa hMhacaGGPaGaeqOSdiMaaiikaiaahMhacqGHsislcaWH4bGaaiykai aadsgacaWGwbWaaSbaaSqaaiaahMhaaeqaaaqaaiaadAfaaeqaniab gUIiYdaaaa@4667@

This function cannot be calculated exactly, but can be estimated using the approximation to β(x) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabek7aIjaacIcacaWH4bGaaiykaaaa@3483@  as

Β ρ (x)=(1m) x k x k + ρ 1 2 +m x k x k + ρ 2 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfk5acnaaBaaaleaacqaHbpGCaeqaaO GaaiikaiaahIhacaGGPaGaeyypa0JaaiikaiaaigdacqGHsislcaWG TbGaaiykamaakaaabaGaamiEamaaBaaaleaacaWGRbaabeaakiaadI hadaWgaaWcbaGaam4AaaqabaGccqGHRaWkcqaHbpGCdaqhaaWcbaGa aGymaaqaaiaaikdaaaaabeaakiabgUcaRiaad2gadaGcaaqaaiaadI hadaWgaaWcbaGaam4AaaqabaGccaWG4bWaaSbaaSqaaiaadUgaaeqa aOGaey4kaSIaeqyWdi3aa0baaSqaaiaaikdaaeaacaaIYaaaaaqaba aaaa@4E49@

The stress field then becomes

σ pq (x)= E 16π(1+ν) C ( [ imp b m τ q + imq b m τ p ] 3 Β ρ (xξ) x i x j x j ) d s ξ + E 8π(1 ν 2 ) C ( b m imk τ k [ 3 Β ρ (xξ) x i x p x q δ pq 3 Β ρ (xξ) x i x j x j ] ) d s ξ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaeq4Wdm3aaSbaaSqaaiaadchaca WGXbaabeaakiaacIcacaWH4bGaaiykaiabg2da9maalaaabaGaamyr aaqaaiaaigdacaaI2aGaeqiWdaNaaiikaiaaigdacqGHRaWkcqaH9o GBcaGGPaaaamaapefabaWaaeWaaeaadaWadaqaaiabgIGiopaaBaaa leaacaWGPbGaamyBaiaadchaaeqaaOGaamOyamaaBaaaleaacaWGTb aabeaakiabes8a0naaBaaaleaacaWGXbaabeaakiabgUcaRiabgIGi opaaBaaaleaacaWGPbGaamyBaiaadghaaeqaaOGaamOyamaaBaaale aacaWGTbaabeaakiabes8a0naaBaaaleaacaWGWbaabeaaaOGaay5w aiaaw2faamaalaaabaGaeyOaIy7aaWbaaSqabeaacaaIZaaaaOGaeu OKdi0aaSbaaSqaaiabeg8aYbqabaGccaGGOaGaaCiEaiabgkHiTiaa h67acaGGPaaabaGaeyOaIyRaamiEamaaBaaaleaacaWGPbaabeaaki abgkGi2kaadIhadaWgaaWcbaGaamOAaaqabaGccqGHciITcaWG4bWa aSbaaSqaaiaadQgaaeqaaaaaaOGaayjkaiaawMcaaaWcbaGaam4qaa qab0Gaey4kIipakiaadsgacaWGZbWaaSbaaSqaaiaah67aaeqaaaGc baGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlabgUcaRiaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVpaalaaabaGaam yraaqaaiaaiIdacqaHapaCcaGGOaGaaGymaiabgkHiTiabe27aUnaa CaaaleqabaGaaGOmaaaakiaacMcaaaWaa8quaeaadaqadaqaaiaadk gadaWgaaWcbaGaamyBaaqabaGccqGHiiIZdaWgaaWcbaGaamyAaiaa d2gacaWGRbaabeaakiabes8a0naaBaaaleaacaWGRbaabeaakmaadm aabaWaaSaaaeaacqGHciITdaahaaWcbeqaaiaaiodaaaGccqqHsoGq daWgaaWcbaGaeqyWdihabeaakiaacIcacaWH4bGaeyOeI0IaaCOVdi aacMcaaeaacqGHciITcaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaeyOa IyRaamiEamaaBaaaleaacaWGWbaabeaakiabgkGi2kaadIhadaWgaa WcbaGaamyCaaqabaaaaOGaeyOeI0IaeqiTdq2aaSbaaSqaaiaadcha caWGXbaabeaakmaalaaabaGaeyOaIy7aaWbaaSqabeaacaaIZaaaaO GaeuOKdi0aaSbaaSqaaiabeg8aYbqabaGccaGGOaGaaCiEaiabgkHi Tiaah67acaGGPaaabaGaeyOaIyRaamiEamaaBaaaleaacaWGPbaabe aakiabgkGi2kaadIhadaWgaaWcbaGaamOAaaqabaGccqGHciITcaWG 4bWaaSbaaSqaaiaadQgaaeqaaaaaaOGaay5waiaaw2faaaGaayjkai aawMcaaaWcbaGaam4qaaqab0Gaey4kIipakiaadsgacaWGZbWaaSba aSqaaiaah67aaeqaaaaaaa@F878@

Alternatively, one may calculate exactly a modified stress measure, defined as

σ pq (ρ) (x)= V σ pq (ξ)β(ξx)d V ξ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaDaaaleaacaWGWbGaamyCaa qaaiaacIcacqaHbpGCcaGGPaaaaOGaaiikaiaahIhacaGGPaGaeyyp a0Zaa8quaeaacqaHdpWCdaWgaaWcbaGaamiCaiaadghaaeqaaOGaai ikaiaah67acaGGPaGaeqOSdiMaaiikaiaah67acqGHsislcaWH4bGa aiykaiaadsgacaWGwbWaaSbaaSqaaiaah67aaeqaaaqaaiaadAfaae qaniabgUIiYdaaaa@4DF3@

This stress measure is particularly convenient for calculating the force tending to make a dislocation move, as shown in a subsequent section.  In addition, σ pq (ρ) (x) σ pq (x) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaDaaaleaacaWGWbGaamyCaa qaaiaacIcacqaHbpGCcaGGPaaaaOGaaiikaiaahIhacaGGPaGaeyis ISRaeq4Wdm3aaSbaaSqaaiaadchacaWGXbaabeaakiaacIcacaWH4b Gaaiykaaaa@41CF@  except very close to the core of a dislocation.   It is straightforward to show that

σ pq (ρ) (x)= E 16π(1+ν) C ( [ imp b m τ q + imq b m τ p ] 3 R ρ (xξ) x i x j x j ) d s ξ + E 8π(1 ν 2 ) C ( b m imk τ k [ 3 R ρ (xξ) x i x p x q δ pq 3 R ρ (xξ) x i x j x j ] ) d s ξ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaeq4Wdm3aa0baaSqaaiaadchaca WGXbaabaGaaiikaiabeg8aYjaacMcaaaGccaGGOaGaaCiEaiaacMca cqGH9aqpdaWcaaqaaiaadweaaeaacaaIXaGaaGOnaiabec8aWjaacI cacaaIXaGaey4kaSIaeqyVd4MaaiykaaaadaWdrbqaamaabmaabaWa amWaaeaacqGHiiIZdaWgaaWcbaGaamyAaiaad2gacaWGWbaabeaaki aadkgadaWgaaWcbaGaamyBaaqabaGccqaHepaDdaWgaaWcbaGaamyC aaqabaGccqGHRaWkcqGHiiIZdaWgaaWcbaGaamyAaiaad2gacaWGXb aabeaakiaadkgadaWgaaWcbaGaamyBaaqabaGccqaHepaDdaWgaaWc baGaamiCaaqabaaakiaawUfacaGLDbaadaWcaaqaaiabgkGi2oaaCa aaleqabaGaaG4maaaakiaadkfadaWgaaWcbaGaeqyWdihabeaakiaa cIcacaWH4bGaeyOeI0IaaCOVdiaacMcaaeaacqGHciITcaWG4bWaaS baaSqaaiaadMgaaeqaaOGaeyOaIyRaamiEamaaBaaaleaacaWGQbaa beaakiabgkGi2kaadIhadaWgaaWcbaGaamOAaaqabaaaaaGccaGLOa GaayzkaaaaleaacaWGdbaabeqdcqGHRiI8aOGaamizaiaadohadaWg aaWcbaGaaCOVdaqabaaakeaacaaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaey4k aSIaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8+aaSaaaeaacaWGfbaabaGaaGioaiabec8aWjaacIcacaaI XaGaeyOeI0IaeqyVd42aaWbaaSqabeaacaaIYaaaaOGaaiykaaaada WdrbqaamaabmaabaGaamOyamaaBaaaleaacaWGTbaabeaakiabgIGi opaaBaaaleaacaWGPbGaamyBaiaadUgaaeqaaOGaeqiXdq3aaSbaaS qaaiaadUgaaeqaaOWaamWaaeaadaWcaaqaaiabgkGi2oaaCaaaleqa baGaaG4maaaakiaadkfadaWgaaWcbaGaeqyWdihabeaakiaacIcaca WH4bGaeyOeI0IaaCOVdiaacMcaaeaacqGHciITcaWG4bWaaSbaaSqa aiaadMgaaeqaaOGaeyOaIyRaamiEamaaBaaaleaacaWGWbaabeaaki abgkGi2kaadIhadaWgaaWcbaGaamyCaaqabaaaaOGaeyOeI0IaeqiT dq2aaSbaaSqaaiaadchacaWGXbaabeaakmaalaaabaGaeyOaIy7aaW baaSqabeaacaaIZaaaaOGaamOuamaaBaaaleaacqaHbpGCaeqaaOGa aiikaiaahIhacqGHsislcaWH+oGaaiykaaqaaiabgkGi2kaadIhada WgaaWcbaGaamyAaaqabaGccqGHciITcaWG4bWaaSbaaSqaaiaadQga aeqaaOGaeyOaIyRaamiEamaaBaaaleaacaWGQbaabeaaaaaakiaawU facaGLDbaaaiaawIcacaGLPaaaaSqaaiaadoeaaeqaniabgUIiYdGc caWGKbGaam4CamaaBaaaleaacaWH+oaabeaaaaaa@F9F1@

where R ρ (x)= x k x k + ρ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkfadaWgaaWcbaGaeqyWdihabeaaki aacIcacaWH4bGaaiykaiabg2da9maakaaabaGaamiEamaaBaaaleaa caWGRbaabeaakiaadIhadaWgaaWcbaGaam4AaaqabaGccqGHRaWkcq aHbpGCdaahaaWcbeqaaiaaikdaaaaabeaaaaa@3E96@ , as before. 

 

Nonsingular energy of circular dislocation loops.  It is straightforward to calculate the energy of a circular dislocation loop.  Cai et al Journal of the Mechanics and Physics of Solids, 54, 561-587, (2006) give:

 Prismatic Loop: (b perpendicular to loop) V D = Ea b 2 4(1 ν 2 ) { log( 8a ρ )1 }+Ο (ρ/a) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAfadaahaaWcbeqaaiaadseacqGHEi sPaaGccqGH9aqpdaWcaaqaaiaadweacaWGHbGaamOyamaaCaaaleqa baGaaGOmaaaaaOqaaiaaisdacaGGOaGaaGymaiabgkHiTiabe27aUn aaCaaaleqabaGaaGOmaaaakiaacMcaaaWaaiWaaeaaciGGSbGaai4B aiaacEgadaqadaqaamaalaaabaGaaGioaiaadggaaeaacqaHbpGCaa aacaGLOaGaayzkaaGaeyOeI0IaaGymaaGaay5Eaiaaw2haaiabgUca Riabf+5apjaacIcacqaHbpGCcaGGVaGaamyyaiaacMcadaahaaWcbe qaaiaaikdaaaaaaa@5281@

 Glide Loop: (b in the plane of the loop): V D Ea b 2 8(1+ν) { 2ν 1ν [ log( 8a r )2 ] 1 2 }+Ο (ρ/a) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAfadaahaaWcbeqaaiaadseacqGHEi sPaaGccqGHijYUdaWcaaqaaiaadweacaWGHbGaamOyamaaCaaaleqa baGaaGOmaaaaaOqaaiaaiIdacaGGOaGaaGymaiabgUcaRiabe27aUj aacMcaaaWaaiWaaeaadaWcaaqaaiaaikdacqGHsislcqaH9oGBaeaa caaIXaGaeyOeI0IaeqyVd4gaamaadmaabaGaciiBaiaac+gacaGGNb WaaeWaaeaadaWcaaqaaiaaiIdacaWGHbaabaGaamOCaaaaaiaawIca caGLPaaacqGHsislcaaIYaaacaGLBbGaayzxaaGaeyOeI0YaaSaaae aacaaIXaaabaGaaGOmaaaaaiaawUhacaGL9baacqGHRaWkcqqHFoWt caGGOaGaeqyWdiNaai4laiaadggacaGGPaWaaWbaaSqabeaacaaIYa aaaaaa@5CA1@

 

 

 

 

5.9.3 Energy of a dislocation loop in a stressed, finite elastic solid

 

The figure shows a dislocation loop in an elastic solid.  Assume that:

1.      The solid is an isotropic, homogeneous, linear elastic material with Young’s modulus E and Poisson’s ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH9oGBaaa@347E@

2.      The solid contains a dislocation, which is characterized by the loop C, and the burgers vector b for the dislocation, following the conventions described in the preceding section.  As before, we can imagine creating the dislocation loop by cutting the crystal over some surface S, and displacing the two material surfaces adjacent to the cut by the burgers vector.  The figure shows the dislocation loop to be completely contained within the solid, but this is not necessary MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  the surface S could intersect the exterior boundary of the solid, in which case the dislocation line C would start and end on the solid’s surface.

3.      Part of the boundary of the solid 1 R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITdaWgaaWcbaGaaGymaaqaba GccaWGsbaaaa@35F4@  is subjected to a prescribed displacement, while the remainder 2 R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITdaWgaaWcbaGaaGOmaaqaba GccaWGsbaaaa@35F5@  is subjected to a prescribed traction.  Note that there is some ambiguity in specifying the prescribed displacement.  In some problems, the solid contains a dislocation before it is loaded: if so, displacements are measured relative to the solid with traction free boundary, but containing a dislocation.  In other problems, the dislocation may be nucleated during deformation.  In this case, displacements are measured with respect to the initial, stress free and undislocated solid.  In the discussion to follow, we consider only the latter case.

 

 

To express the potential energy in a useful form, it is helpful to define several measures of stress and strain in the solid, as follows:

1.      The actual fields in the loaded solid containing the dislocation will be denoted by [ u i , ε ij , σ ij ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGaamyDamaaBaaaleaacaWGPb aabeaakiaacYcacqaH1oqzdaWgaaWcbaGaamyAaiaadQgaaeqaaOGa aiilaiabeo8aZnaaBaaaleaacaWGPbGaamOAaaqabaGccaGGDbaaaa@3F94@ .  Note that u i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaadMgaaeqaaa aa@34DA@  is measured with respect to a stress-free solid, which contains no dislocations.  The displacement is discontinuous across S.

2.      The fields induced by the applied loading in an un-dislocated solid will be denoted by [ u i * , ε ij * , σ ij * ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGaamyDamaaDaaaleaacaWGPb aabaGaaiOkaaaakiaacYcacqaH1oqzdaqhaaWcbaGaamyAaiaadQga aeaacaGGQaaaaOGaaiilaiabeo8aZnaaDaaaleaacaWGPbGaamOAaa qaaiaacQcaaaGccaGGDbaaaa@41A1@ .  The displacement field u i * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaa0baaSqaaiaadMgaaeaaca GGQaaaaaaa@3589@  is continuous.

3.      The fields in a solid containing a dislocation, but with 2 R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITdaWgaaWcbaGaaGOmaaqaba GccaWGsbaaaa@35F5@  traction free, and with zero displacement on 1 R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITdaWgaaWcbaGaaGymaaqaba GccaWGsbaaaa@35F4@  will be denoted by [ u i D , ε ij D , σ ij D ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGaamyDamaaDaaaleaacaWGPb aabaGaamiraaaakiaacYcacqaH1oqzdaqhaaWcbaGaamyAaiaadQga aeaacaWGebaaaOGaaiilaiabeo8aZnaaDaaaleaacaWGPbGaamOAaa qaaiaadseaaaGccaGGDbaaaa@41F2@

4.      The fields in an infinite solid containing a dislocation with line C and burgers vector b will be denoted by [ u i D , ε ij D , σ ij D ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGaamyDamaaDaaaleaacaWGPb aabaGaamiraiabg6HiLcaakiaacYcacqaH1oqzdaqhaaWcbaGaamyA aiaadQgaaeaacaWGebGaeyOhIukaaOGaaiilaiabeo8aZnaaDaaale aacaWGPbGaamOAaaqaaiaadseacqGHEisPaaGccaGGDbaaaa@4645@ .  If the dislocation line terminates on the solid’s surface, any convenient procedure can be used to close the loop when deriving the infinte solid solution, but the fields will depend on this choice.

5.      The difference between the fields for a dislocation in a bounded solid and the solution for a dislocation in an infinite solid will be denoted by [ u i I = u i D u i D , ε ij I = ε ij D ε ij D , σ ij I = σ ij D σ ij D ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGaamyDamaaDaaaleaacaWGPb aabaGaamysaaaakiabg2da9iaadwhadaqhaaWcbaGaamyAaaqaaiaa dseaaaGccqGHsislcaWG1bWaa0baaSqaaiaadMgaaeaacaWGebGaey OhIukaaOGaaiilaiabew7aLnaaDaaaleaacaWGPbGaamOAaaqaaiaa dMeaaaGccqGH9aqpcqaH1oqzdaqhaaWcbaGaamyAaiaadQgaaeaaca WGebaaaOGaeyOeI0IaeqyTdu2aa0baaSqaaiaadMgacaWGQbaabaGa amiraiabg6HiLcaakiaacYcacqaHdpWCdaqhaaWcbaGaamyAaiaadQ gaaeaacaWGjbaaaOGaeyypa0Jaeq4Wdm3aa0baaSqaaiaadMgacaWG QbaabaGaamiraaaakiabgkHiTiabeo8aZnaaDaaaleaacaWGPbGaam OAaaqaaiaadseacqGHEisPaaGccaGGDbaaaa@6445@

 

The potential energy of the solid can be expressed as

V= V D + V D* + V * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGwbGaeyypa0JaamOvamaaCaaale qabaGaamiraaaakiabgUcaRiaadAfadaahaaWcbeqaaiaadseacaGG QaaaaOGaey4kaSIaamOvamaaCaaaleqabaGaaiOkaaaaaaa@3C85@

where

  V D = S 1 2 σ ij D b i m j dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGwbWaaWbaaSqabeaacaWGebaaaO Gaeyypa0Zaa8quaeaadaWcaaqaaiaaigdaaeaacaaIYaaaaiabeo8a ZnaaDaaaleaacaWGPbGaamOAaaqaaiaadseaaaGccaWGIbWaaSbaaS qaaiaadMgaaeqaaOGaamyBamaaBaaaleaacaWGQbaabeaakiaadsga caWGbbaaleaacaWGtbaabeqdcqGHRiI8aaaa@44C1@  is the strain energy of the dislocation itself

  V D* = S σ ij * b i m j dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGwbWaaWbaaSqabeaacaWGebGaai Okaaaakiabg2da9maapefabaGaeq4Wdm3aa0baaSqaaiaadMgacaWG QbaabaGaaiOkaaaakiaadkgadaWgaaWcbaGaamyAaaqabaGccaWGTb WaaSbaaSqaaiaadQgaaeqaaOGaamizaiaadgeaaSqaaiaadofaaeqa niabgUIiYdaaaa@43CD@  is the work done to introduce the dislocation into the externally applied stress

  V * = R 1 2 σ ij * ε ij * dV 2R t i u i * dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGwbWaaWbaaSqabeaacaGGQaaaaO Gaeyypa0Zaa8quaeaadaWcaaqaaiaaigdaaeaacaaIYaaaaiabeo8a ZnaaDaaaleaacaWGPbGaamOAaaqaaiaacQcaaaGccqaH1oqzdaqhaa WcbaGaamyAaiaadQgaaeaacaGGQaaaaOGaamizaiaadAfaaSqaaiaa dkfaaeqaniabgUIiYdGccqGHsisldaWdrbqaaiaadshadaWgaaWcba GaamyAaaqabaGccaWG1bWaa0baaSqaaiaadMgaaeaacaGGQaaaaOGa amizaiaadgeaaSqaaiabgkGi2kaaikdacaWGsbaabeqdcqGHRiI8aa aa@51B9@  is the potential energy of the applied loads

 

The strain energy of the dislocation can also be expressed as a sum of two terms:

V D = V D + V I MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAfadaahaaWcbeqaaiaadseaaaGccq GH9aqpcaWGwbWaaWbaaSqabeaacaWGebGaeyOhIukaaOGaey4kaSIa amOvamaaCaaaleqabaGaamysaaaaaaa@396D@

where

  V D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAfadaahaaWcbeqaaiaadseacqGHEi sPaaaaaa@33CA@  is the energy of a dislocation with line C in an infinite solid, which can be calculated using the expressions in 5.9.2.

   V I = S σ ij I b i m j dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGwbWaaWbaaSqabeaacaWGjbaaaO Gaeyypa0Zaa8quaeaacqaHdpWCdaqhaaWcbaGaamyAaiaadQgaaeaa caWGjbaaaOGaamOyamaaBaaaleaacaWGPbaabeaakiaad2gadaWgaa WcbaGaamOAaaqabaGccaWGKbGaamyqaaWcbaGaam4uaaqab0Gaey4k Iipaaaa@4344@  is the change in potential energy due to the presence of boundaries in the solid. 

 

If the classical approach is used to represent the burgers vector of the dislocation, the energy is infinite, because V D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAfadaahaaWcbeqaaiaadseacqGHEi sPaaaaaa@33CA@  contains a contribution from the singular dislocation core.  The remaining terms are all bounded.  The simplest way to avoid this unsatisfactory behavior is to estimate V D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAfadaahaaWcbeqaaiaadseacqGHEi sPaaaaaa@33CA@  using the non-singular dislocation theory presented in 5.9.2, but use the classical expressions for all the remaining terms.   This is not completely consistent, because in a rigorous non-singular dislocation theory all the terms should be computed by taking a convolution integral with the burgers vector distribution.   However, provided the solid is large compared with the dislocation core, the error in the approximate result is negligible.

 

Derivation

  1. The potential energy of the solid is given by the usual expression V= R 1 2 σ ij ε ij dV 2R t i u i dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGwbGaeyypa0Zaa8quaeaadaWcaa qaaiaaigdaaeaacaaIYaaaaiabeo8aZnaaBaaaleaacaWGPbGaamOA aaqabaGccqaH1oqzdaWgaaWcbaGaamyAaiaadQgaaeqaaOGaamizai aadAfaaSqaaiaadkfaaeqaniabgUIiYdGccqGHsisldaWdrbqaaiaa dshadaWgaaWcbaGaamyAaaqabaGccaWG1bWaaSbaaSqaaiaadMgaae qaaOGaamizaiaadgeaaSqaaiabgkGi2kaaikdacaWGsbaabeqdcqGH RiI8aaaa@4EC7@
  2. The total stress consists of the dislocation fields, together with the externally applied fields, so that

V= R 1 2 [ σ ij D + σ ij * ][ ε ij D + ε ij * ]dV 2R t i ( u i D + u i * )dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGwbGaeyypa0Zaa8quaeaadaWcaa qaaiaaigdaaeaacaaIYaaaamaadmaabaGaeq4Wdm3aa0baaSqaaiaa dMgacaWGQbaabaGaamiraaaakiabgUcaRiabeo8aZnaaDaaaleaaca WGPbGaamOAaaqaaiaacQcaaaaakiaawUfacaGLDbaadaWadaqaaiab ew7aLnaaDaaaleaacaWGPbGaamOAaaqaaiaadseaaaGccqGHRaWkcq aH1oqzdaqhaaWcbaGaamyAaiaadQgaaeaacaGGQaaaaaGccaGLBbGa ayzxaaGaamizaiaadAfaaSqaaiaadkfaaeqaniabgUIiYdGccqGHsi sldaWdrbqaaiaadshadaWgaaWcbaGaamyAaaqabaGcdaqadaqaaiaa dwhadaqhaaWcbaGaamyAaaqaaiaadseaaaGccqGHRaWkcaWG1bWaa0 baaSqaaiaadMgaaeaacaGGQaaaaaGccaGLOaGaayzkaaGaamizaiaa dgeaaSqaaiabgkGi2kaaikdacaWGsbaabeqdcqGHRiI8aaaa@64F3@

  1. This expression can be re-written as

V= R ( 1 2 σ ij D u i D x j + σ ij * u i D x j + 1 2 σ ij * ε ij * )dV 2R t i ( u i D + u i * )dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGwbGaeyypa0Zaa8quaeaadaqada qaamaalaaabaGaaGymaaqaaiaaikdaaaGaeq4Wdm3aa0baaSqaaiaa dMgacaWGQbaabaGaamiraaaakmaalaaabaGaeyOaIyRaamyDamaaDa aaleaacaWGPbaabaGaamiraaaaaOqaaiabgkGi2kaadIhadaWgaaWc baGaamOAaaqabaaaaOGaey4kaSIaeq4Wdm3aa0baaSqaaiaadMgaca WGQbaabaGaaiOkaaaakmaalaaabaGaeyOaIyRaamyDamaaDaaaleaa caWGPbaabaGaamiraaaaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaam OAaaqabaaaaOGaey4kaSYaaSaaaeaacaaIXaaabaGaaGOmaaaacqaH dpWCdaqhaaWcbaGaamyAaiaadQgaaeaacaGGQaaaaOGaeqyTdu2aa0 baaSqaaiaadMgacaWGQbaabaGaaiOkaaaaaOGaayjkaiaawMcaaiaa dsgacaWGwbaaleaacaWGsbaabeqdcqGHRiI8aOGaeyOeI0Yaa8quae aacaWG0bWaaSbaaSqaaiaadMgaaeqaaOWaaeWaaeaacaWG1bWaa0ba aSqaaiaadMgaaeaacaWGebaaaOGaey4kaSIaamyDamaaDaaaleaaca WGPbaabaGaaiOkaaaaaOGaayjkaiaawMcaaiaadsgacaWGbbaaleaa cqGHciITcaaIYaGaamOuaaqab0Gaey4kIipaaaa@73EC@

To see this, note that σ ij D ε ij D = σ ij D ( u i D / x j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaqhaaWcbaGaamyAaiaadQ gaaeaacaWGebaaaOGaeqyTdu2aa0baaSqaaiaadMgacaWGQbaabaGa amiraaaakiabg2da9iabeo8aZnaaDaaaleaacaWGPbGaamOAaaqaai aadseaaaGccaGGOaGaeyOaIyRaamyDamaaDaaaleaacaWGPbaabaGa amiraaaakiaac+cacqGHciITcaWG4bWaaSbaaSqaaiaadQgaaeqaaO Gaaiykaaaa@4B72@  from the symmetry of σ ij D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaqhaaWcbaGaamyAaiaadQ gaaeaacaWGebaaaaaa@375C@  and the strain-displacement relations, and that σ ij D ε ij * = C ijkl ε kl D ε ij * = σ ij * ε ij D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaqhaaWcbaGaamyAaiaadQ gaaeaacaWGebaaaOGaeqyTdu2aa0baaSqaaiaadMgacaWGQbaabaGa aiOkaaaakiabg2da9iaadoeadaWgaaWcbaGaamyAaiaadQgacaWGRb GaamiBaaqabaGccqaH1oqzdaqhaaWcbaGaam4AaiaadYgaaeaacaWG ebaaaOGaeqyTdu2aa0baaSqaaiaadMgacaWGQbaabaGaaiOkaaaaki abg2da9iabeo8aZnaaDaaaleaacaWGPbGaamOAaaqaaiaacQcaaaGc cqaH1oqzdaqhaaWcbaGaamyAaiaadQgaaeaacaWGebaaaaaa@5487@  because of the symmetry of the elasticity tensor C ijkl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGdbWaaSbaaSqaaiaadMgacaWGQb Gaam4AaiaadYgaaeqaaaaa@3778@ .

  1. The terms involving ( u i D / x j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaeyOaIyRaamyDamaaDaaale aacaWGPbaabaGaamiraaaakiaac+cacqGHciITcaWG4bWaaSbaaSqa aiaadQgaaeqaaOGaaiykaaaa@3CA8@  can now be integrated by parts, by writing, for example

σ ij D ( u i D / x j )=( σ ij D u i D )/ x j u i ( σ ij D / x j )=( σ ij D u i D )/ x j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaqhaaWcbaGaamyAaiaadQ gaaeaacaWGebaaaOGaaiikaiabgkGi2kaadwhadaqhaaWcbaGaamyA aaqaaiaadseaaaGccaGGVaGaeyOaIyRaamiEamaaBaaaleaacaWGQb aabeaakiaacMcacqGH9aqpcqGHciITcaGGOaGaeq4Wdm3aa0baaSqa aiaadMgacaWGQbaabaGaamiraaaakiaadwhadaqhaaWcbaGaamyAaa qaaiaadseaaaGccaGGPaGaai4laiabgkGi2kaadIhadaWgaaWcbaGa amOAaaqabaGccqGHsislcaWG1bWaaSbaaSqaaiaadMgaaeqaaOGaai ikaiabgkGi2kabeo8aZnaaDaaaleaacaWGPbGaamOAaaqaaiaadsea aaGccaGGVaGaeyOaIyRaamiEamaaBaaaleaacaWGQbaabeaakiaacM cacqGH9aqpcqGHciITcaGGOaGaeq4Wdm3aa0baaSqaaiaadMgacaWG QbaabaGaamiraaaakiaadwhadaqhaaWcbaGaamyAaaqaaiaadseaaa GccaGGPaGaai4laiabgkGi2kaadIhadaWgaaWcbaGaamOAaaqabaaa aa@6EF3@

because σ ij D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaqhaaWcbaGaamyAaiaadQ gaaeaacaWGebaaaaaa@375C@  is an equilibrium stress field.  Using this result, applying the divergence theorem, and taking into account the discontinuity in the displacement field across S gives

R 1 2 σ ij D u i D x j dV= S+ 1 2 σ ij D u i D+ ( m j )dA + S 1 2 σ ij D u i D m j dA + R 1 2 σ ij D u i D n j dA = S 1 2 σ ij D b i m j dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWdrbqaamaalaaabaGaaGymaaqaai aaikdaaaGaeq4Wdm3aa0baaSqaaiaadMgacaWGQbaabaGaamiraaaa kmaalaaabaGaeyOaIyRaamyDamaaDaaaleaacaWGPbaabaGaamiraa aaaOqaaiabgkGi2kaadIhadaWgaaWcbaGaamOAaaqabaaaaOGaamiz aiaadAfacqGH9aqpdaWdrbqaamaalaaabaGaaGymaaqaaiaaikdaaa Gaeq4Wdm3aa0baaSqaaiaadMgacaWGQbaabaGaamiraaaakiaadwha daqhaaWcbaGaamyAaaqaaiaadseacqGHRaWkaaGccaGGOaGaeyOeI0 IaamyBamaaBaaaleaacaWGQbaabeaakiaacMcacaWGKbGaamyqaaWc baGaam4uaiabgUcaRaqab0Gaey4kIipakiabgUcaRmaapefabaWaaS aaaeaacaaIXaaabaGaaGOmaaaacqaHdpWCdaqhaaWcbaGaamyAaiaa dQgaaeaacaWGebaaaOGaamyDamaaDaaaleaacaWGPbaabaGaamirai abgkHiTaaakiaad2gadaWgaaWcbaGaamOAaaqabaGccaWGKbGaamyq aaWcbaGaam4uaiabgkHiTaqab0Gaey4kIipakiabgUcaRmaapefaba WaaSaaaeaacaaIXaaabaGaaGOmaaaacqaHdpWCdaqhaaWcbaGaamyA aiaadQgaaeaacaWGebaaaOGaamyDamaaDaaaleaacaWGPbaabaGaam iraaaakiaad6gadaWgaaWcbaGaamOAaaqabaGccaWGKbGaamyqaaWc baGaeyOaIyRaamOuaaqab0Gaey4kIipaaSqaaiaadkfaaeqaniabgU IiYdGccqGH9aqpdaWdrbqaamaalaaabaGaaGymaaqaaiaaikdaaaGa eq4Wdm3aa0baaSqaaiaadMgacaWGQbaabaGaamiraaaakiaadkgada WgaaWcbaGaamyAaaqabaGccaWGTbWaaSbaaSqaaiaadQgaaeqaaOGa amizaiaadgeaaSqaaiaadofaaeqaniabgUIiYdaaaa@8FEF@

where we have noted that σ ij D n j u i D =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaqhaaWcbaGaamyAaiaadQ gaaeaacaWGebaaaOGaamOBamaaBaaaleaacaWGQbaabeaakiaadwha daqhaaWcbaGaamyAaaqaaiaadseaaaGccqGH9aqpcaaIWaaaaa@3E26@  on the exterior boundary of the solid, and that u i u i + = b i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaqhaaWcbaGaamyAaaqaaiabgk HiTaaakiabgkHiTiaadwhadaqhaaWcbaGaamyAaaqaaiabgUcaRaaa kiabg2da9iaadkgadaWgaaWcbaGaamyAaaqabaaaaa@3A89@ .  A similar procedure gives

R σ ij * u i D x j dV= S σ ij * b i m j dA + R σ ij * n j u i D dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWdrbqaaiabeo8aZnaaDaaaleaaca WGPbGaamOAaaqaaiaacQcaaaGcdaWcaaqaaiabgkGi2kaadwhadaqh aaWcbaGaamyAaaqaaiaadseaaaaakeaacqGHciITcaWG4bWaaSbaaS qaaiaadQgaaeqaaaaakiaadsgacaWGwbGaeyypa0Zaa8quaeaacqaH dpWCdaqhaaWcbaGaamyAaiaadQgaaeaacaGGQaaaaOGaamOyamaaBa aaleaacaWGPbaabeaakiaad2gadaWgaaWcbaGaamOAaaqabaGccaWG KbGaamyqaaWcbaGaam4uaaqab0Gaey4kIipakiabgUcaRmaapefaba Gaeq4Wdm3aa0baaSqaaiaadMgacaWGQbaabaGaaiOkaaaakiaad6ga daWgaaWcbaGaamOAaaqabaGccaWG1bWaa0baaSqaaiaadMgaaeaaca WGebaaaOGaamizaiaadgeaaSqaaiabgkGi2kaadkfaaeqaniabgUIi YdaaleaacaWGsbaabeqdcqGHRiI8aaaa@633B@

Finally, substituting this result back into the expression for V and noting σ ij * n j = t i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaqhaaWcbaGaamyAaiaadQ gaaeaacaGGQaaaaOGaamOBamaaBaaaleaacaWGQbaabeaakiabg2da 9iaadshadaWgaaWcbaGaamyAaaqabaaaaa@3C7C@  on 2 R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITdaWgaaWcbaGaaGOmaaqaba GccaWGsbaaaa@35F5@  and u i D =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaa0baaSqaaiaadMgaaeaaca WGebaaaOGaeyypa0JaaGimaaaa@376E@  on 1 R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITdaWgaaWcbaGaaGymaaqaba GccaWGsbaaaa@35F4@  gives the required result.

 

 

 

 

5.9.4 Energy of two interacting dislocation loops

 

Consider two dislocation loops in an infinite elastic solid, as shown in the figure.  Assume that

  1. The solid is an isotropic, homogeneous, linear elastic material with Young’s modulus E and Poisson’s ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH9oGBaaa@347E@
  2. The dislocations can be characterized by surfaces, contours and burger’s vectors [ S 1 , C 1 , b (1) ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGaam4uamaaBaaaleaacaaIXa aabeaakiaacYcacaWGdbWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaa hkgadaahaaWcbeqaaiaacIcacaaIXaGaaiykaaaakiaac2faaaa@3C9E@  and [ S 2 , C 2 , b (2) ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGaam4uamaaBaaaleaacaaIYa aabeaakiaacYcacaWGdbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaa hkgadaahaaWcbeqaaiaacIcacaaIYaGaaiykaaaakiaac2faaaa@3CA1@ .

 

The total potential energy of the solid can be calculated from the following expressions

V= V D1 + V D1D2 + V D2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGwbGaeyypa0JaamOvamaaCaaale qabaGaamiraiaaigdaaaGccqGHRaWkcaWGwbWaaWbaaSqabeaacaWG ebGaaGymaiaadseacaaIYaaaaOGaey4kaSIaamOvamaaCaaaleqaba Gaamiraiaaikdaaaaaaa@3FA9@

where

  V D1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGwbWaaWbaaSqabeaacaWGebGaaG ymaaaaaaa@3552@  and V D2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGwbWaaWbaaSqabeaacaWGebGaaG Omaaaaaaa@3553@  are the energies of the two dislocation loops in isolation, which can be computed from the formulas in 5.9.1 (or 5.9.2 if you need a non-singular expression)

  V D1D2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGwbWaaWbaaSqabeaacaWGebGaaG ymaiaadseacaaIYaaaaaaa@36D7@  is an `interaction energy,’ which can be thought of as the work done to introduce dislocation 2 into the stress field associated with dislocation 1 (or vice-versa).  The interaction energy is given by

V D1D2 = E 16π(1+ν) C2 C1 2 R x p x p b i (1) b j (2) τ i (1) (x) τ j (2) (ξ)d s x d s ξ E 16π(1+ν) C2 C1 ijq mnq b i (1) b j (2) 2 R x k x k τ m (1) (x) τ n (2) (ξ)d s x d s ξ + E 8π(1 ν 2 ) C2 C1 ikl jmn b k (1) b m (2) 2 R x i x j τ l (1) (x) τ n (2) (ξ)d s x d s ξ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaadAfadaahaaWcbeqaaiaads eacaaIXaGaamiraiaaikdaaaGccqGH9aqpdaWcaaqaaiaadweaaeaa caaIXaGaaGOnaiabec8aWjaacIcacaaIXaGaey4kaSIaeqyVd4Maai ykaaaadaWdvbqaamaapufabaWaaSaaaeaacqGHciITdaahaaWcbeqa aiaaikdaaaGccaWGsbaabaGaeyOaIyRaamiEamaaBaaaleaacaWGWb aabeaakiabgkGi2kaadIhadaWgaaWcbaGaamiCaaqabaaaaOGaamOy amaaDaaaleaacaWGPbaabaGaaiikaiaaigdacaGGPaaaaOGaamOyam aaDaaaleaacaWGQbaabaGaaiikaiaaikdacaGGPaaaaOGaeqiXdq3a a0baaSqaaiaadMgaaeaacaGGOaGaaGymaiaacMcaaaGccaGGOaGaaC iEaiaacMcacqaHepaDdaqhaaWcbaGaamOAaaqaaiaacIcacaaIYaGa aiykaaaakiaacIcacaWH+oGaaiykaiaadsgacaWGZbWaaSbaaSqaai aahIhaaeqaaaqaaiaadoeacaaIXaaabeqdcqWIr4E0cqGHRiI8aaWc baGaam4qaiaaikdaaeqaniablgH7rlabgUIiYdGccaWGKbGaam4Cam aaBaaaleaacaWH+oaabeaakiaaykW7caaMc8oabaGaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqGHsisl daWcaaqaaiaadweaaeaacaaIXaGaaGOnaiabec8aWjaacIcacaaIXa Gaey4kaSIaeqyVd4MaaiykaaaadaWdvbqaamaapufabaGaeyicI48a aSbaaSqaaiaadMgacaWGQbGaamyCaaqabaGccqGHiiIZdaWgaaWcba GaamyBaiaad6gacaWGXbaabeaakiaadkgadaqhaaWcbaGaamyAaaqa aiaacIcacaaIXaGaaiykaaaakiaadkgadaqhaaWcbaGaamOAaaqaai aacIcacaaIYaGaaiykaaaakmaalaaabaGaeyOaIy7aaWbaaSqabeaa caaIYaaaaOGaamOuaaqaaiabgkGi2kaadIhadaWgaaWcbaGaam4Aaa qabaGccqGHciITcaWG4bWaaSbaaSqaaiaadUgaaeqaaaaakiabes8a 0naaDaaaleaacaWGTbaabaGaaiikaiaaigdacaGGPaaaaOGaaiikai aahIhacaGGPaGaeqiXdq3aa0baaSqaaiaad6gaaeaacaGGOaGaaGOm aiaacMcaaaGccaGGOaGaaCOVdiaacMcacaWGKbGaam4CamaaBaaale aacaWH4baabeaakiaadsgacaWGZbWaaSbaaSqaaiaah67aaeqaaaqa aiaadoeacaaIXaaabeqdcqWIr4E0cqGHRiI8aaWcbaGaam4qaiaaik daaeqaniablgH7rlabgUIiYdaakeaacaaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlabgUcaRmaalaaabaGaamyraaqaaiaaiIdacqaH apaCcaGGOaGaaGymaiabgkHiTiabe27aUnaaCaaaleqabaGaaGOmaa aakiaacMcaaaWaa8qvaeaadaWdvbqaaiabgIGiopaaBaaaleaacaWG PbGaam4AaiaadYgaaeqaaOGaeyicI48aaSbaaSqaaiaadQgacaWGTb GaamOBaaqabaGccaWGIbWaa0baaSqaaiaadUgaaeaacaGGOaGaaGym aiaacMcaaaGccaWGIbWaa0baaSqaaiaad2gaaeaacaGGOaGaaGOmai aacMcaaaGcdaWcaaqaaiabgkGi2oaaCaaaleqabaGaaGOmaaaakiaa dkfaaeaacqGHciITcaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaeyOaIy RaamiEamaaBaaaleaacaWGQbaabeaaaaGccqaHepaDdaqhaaWcbaGa amiBaaqaaiaacIcacaaIXaGaaiykaaaakiaacIcacaWH4bGaaiykai abes8a0naaDaaaleaacaWGUbaabaGaaiikaiaaikdacaGGPaaaaOGa aiikaiaah67acaGGPaGaamizaiaadohadaWgaaWcbaGaaCiEaaqaba GccaWGKbGaam4CamaaBaaaleaacaWH+oaabeaaaeaacaWGdbGaaGym aaqab0GaeSyeUhTaey4kIipaaSqaaiaadoeacaaIYaaabeqdcqWIr4 E0cqGHRiI8aaaaaa@369D@

Although this integral is bounded (provided the dislocation lines only meet at discrete points), it is sometimes convenient to replace R(x)= x k x k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGsbGaaiikaiaahIhacaGGPaGaey ypa0ZaaOaaaeaacaWG4bWaaSbaaSqaaiaadUgaaeqaaOGaamiEamaa BaaaleaacaWGRbaabeaaaeqaaaaa@3B49@  by R ρ (x)= x k x k + ρ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGsbWaaSbaaSqaaiabeg8aYbqaba GccaGGOaGaaCiEaiaacMcacqGH9aqpdaGcaaqaaiaadIhadaWgaaWc baGaam4AaaqabaGccaWG4bWaaSbaaSqaaiaadUgaaeqaaOGaey4kaS IaeqyWdi3aaWbaaSqabeaacaaIYaaaaaqabaaaaa@40D4@  for a non-singular treatment of dislocations.

 

HEALTH WARNING: Notice that the expression for the interaction energy is very similar to the formula for the self-energy of a dislocation loop, except that (i) it contains an extra term (which vanishes if b (1) = b (2) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGIbWaaWbaaSqabeaacaGGOaGaaG ymaiaacMcaaaGccqGH9aqpcaWGIbWaaWbaaSqabeaacaGGOaGaaGOm aiaacMcaaaaaaa@3A27@  ), and (ii) the integrals in the interaction energy are twice those in the self-energy.  The latter is an endless source of confusion.

 

Derivation: We can regard the two interacting dislocations as a special case of a dislocation loop subjected to an applied stress:  one dislocation generates the `applied stress,’ which influences the second dislocation.  The total potential energy follows as V= V D + V D* + V * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGwbGaeyypa0JaamOvamaaCaaale qabaGaamiraaaakiabgUcaRiaadAfadaahaaWcbeqaaiaadseacaGG QaaaaOGaey4kaSIaamOvamaaCaaaleqabaGaaiOkaaaaaaa@3C85@ , where V D = V D1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGwbWaaWbaaSqabeaacaWGebaaaO Gaeyypa0JaamOvamaaCaaaleqabaGaamiraiaaigdaaaaaaa@3833@ , V * = V D2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGwbWaaWbaaSqabeaacaGGQaaaaO Gaeyypa0JaamOvamaaCaaaleqabaGaamiraiaaikdaaaaaaa@3819@  are the potential energies of the two isolated dislocations, and V D* = V D1D2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGwbWaaWbaaSqabeaacaWGebGaai Okaaaakiabg2da9iaadAfadaahaaWcbeqaaiaadseacaaIXaGaamir aiaaikdaaaaaaa@3A66@  is the interaction energy.  We have that

V D1D2 = S1 σ ij D2 b i (1) m j (1) dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGwbWaaWbaaSqabeaacaWGebGaaG ymaiaadseacaaIYaaaaOGaeyypa0Zaa8quaeaacqaHdpWCdaqhaaWc baGaamyAaiaadQgaaeaacaWGebGaaGOmaaaakiaadkgadaqhaaWcba GaamyAaaqaaiaacIcacaaIXaGaaiykaaaakiaad2gadaqhaaWcbaGa amOAaaqaaiaacIcacaaIXaGaaiykaaaakiaadsgacaWGbbaaleaaca WGtbGaaGymaaqab0Gaey4kIipaaaa@4B1B@

where σ ij D2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaqhaaWcbaGaamyAaiaadQ gaaeaacaWGebGaaGOmaaaaaaa@3818@  is the stress induced by dislocation 2.  We can express this stress in terms of a line integral around dislocation 2.  Finally, the surface integral over S1 can reduced to a contour integral around dislocation 1 by applying Stokes theorem.

 

 

 

5.9.5 Driving force for dislocation motion MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E7@  The Peach-Koehler formula

 

If a dislocation is subjected to stress, it tends to move through the crystal.  This motion is the mechanism for plastic flow in a crystalline solid, as discussed in Section 3.7.12.

 

The tendency of a dislocation to move can be quantified by a force.  This force needs to be interpreted carefully: it is not a mechanical force (in the sense of Newtonian mechancics) that induces motion of a material particle, but rather a generalized force (in the sense of Lagrangean mechanics) that causes a rearrangement of atoms around the dislocation core.   It is sometimes known as a `configurational force’

 

The generalized force for dislocation motion is defined as follows.

  1. Consider an elastic solid, which contains a dislocation loop. The loop is characterized by a curve C, the tangent vector τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHepaaaa@3416@ , and the burgers vector b. As usual, we can imagine creating the dislocation loop by cutting the crystal over some surface S that is bounded by C, and displacing the two material surfaces adjacent to the cut by the burgers vector. 
  2. Suppose that the dislocation moves, so that a point at x(s) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH4bGaaiikaiaadohacaGGPaaaaa@3617@  on C advances to a new position x(s)+δa(s)n(s) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH4bGaaiikaiaadohacaGGPaGaey 4kaSIaeqiTdqMaamyyaiaacIcacaWGZbGaaiykaiaah6gacaGGOaGa am4CaiaacMcaaaa@3F1E@ , where n(s) is a unit vector normal to C, as shown in the figure (in the figure, the dislocation moves ina single plane, but this is not necessary).
  3. As the dislocation moves, the potential energy of the solid changes by an amount δV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazcaWGwbaaaa@3546@ .  This change of energy provides the driving force for dislocation motion.
  4. The driving force is defined as a vector function of arc-length around the dislocation F(s) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHgbGaaiikaiaadohacaGGPaaaaa@35E6@ , whose direction is perpendicular to C, and which satisfies

δV= C F(s)n(s)δa(s)ds MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHsislcqaH0oazcaWGwbGaeyypa0 Zaa8quaeaacaWHgbGaaiikaiaadohacaGGPaGaeyyXICTaaCOBaiaa cIcacaWGZbGaaiykaiabes7aKjaadggacaGGOaGaam4CaiaacMcaca WGKbGaam4CaaWcbaGaam4qaaqab0Gaey4kIipaaaa@49BA@

for all possible choices of n(s) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHUbGaaiikaiaadohacaGGPaaaaa@360E@  and δa(s) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazcaWGHbGaaiikaiaadohaca GGPaaaaa@37A2@  (the change in energy is negative because the displacement is in the same direction as the force).

 

The Peach-Koehler formula states that the driving force for dislocation motion can be computed from the following formula

F i (s)= ijk σ jm (s) b m τ k (s) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGgbWaaSbaaSqaaiaadMgaaeqaaO GaaiikaiaadohacaGGPaGaeyypa0JaeyicI48aaSbaaSqaaiaadMga caWGQbGaam4AaaqabaGccqaHdpWCdaWgaaWcbaGaamOAaiaad2gaae qaaOGaaiikaiaadohacaGGPaGaamOyamaaBaaaleaacaWGTbaabeaa kiabes8a0naaBaaaleaacaWGRbaabeaakiaacIcacaWGZbGaaiykaa aa@4A09@

where σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaaaa@3692@  is the total stress acting on the dislocation at a point s along the curve C (the stress includes contributions from the dislocation itself, as well as stresses generated by external loading on the solid).

 

The Peach-Koehler equation is meaningless without further discussion, because the classical solution predicts that the stress acting on the dislocation line is infinite.   To avoid this, we need to partition the stress according to its various origins, as described in Section 5.9.3.

 

 

 

 

  1. We assume that the dislocation loop lies within an elastic solid, which is subjected to some external loading.  The external fields subject part of the boundary of the solid 1 R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITdaWgaaWcbaGaaGymaaqaba GccaWGsbaaaa@35F4@  to a prescribed displacement; and the remainder of the boundary 2 R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITdaWgaaWcbaGaaGOmaaqaba GccaWGsbaaaa@35F5@  to a prescribed traction.

 

  1. The actual fields in the loaded solid containing the dislocation will be denoted by [ u i , ε ij , σ ij ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGaamyDamaaBaaaleaacaWGPb aabeaakiaacYcacqaH1oqzdaWgaaWcbaGaamyAaiaadQgaaeqaaOGa aiilaiabeo8aZnaaBaaaleaacaWGPbGaamOAaaqabaGccaGGDbaaaa@3F94@ . 

 

  1. The fields induced by the applied loading in an un-dislocated solid will be denoted by [ u i * , ε ij * , σ ij * ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGaamyDamaaDaaaleaacaWGPb aabaGaaiOkaaaakiaacYcacqaH1oqzdaqhaaWcbaGaamyAaiaadQga aeaacaGGQaaaaOGaaiilaiabeo8aZnaaDaaaleaacaWGPbGaamOAaa qaaiaacQcaaaGccaGGDbaaaa@41A1@ .

 

  1. The fields in a solid containing a dislocation, but with 2 R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITdaWgaaWcbaGaaGOmaaqaba GccaWGsbaaaa@35F5@  traction free, and with zero displacement on 1 R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHciITdaWgaaWcbaGaaGymaaqaba GccaWGsbaaaa@35F4@  will be denoted by [ u i D , ε ij D , σ ij D ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGaamyDamaaDaaaleaacaWGPb aabaGaamiraaaakiaacYcacqaH1oqzdaqhaaWcbaGaamyAaiaadQga aeaacaWGebaaaOGaaiilaiabeo8aZnaaDaaaleaacaWGPbGaamOAaa qaaiaadseaaaGccaGGDbaaaa@41F2@

 

  1. The fields in an infinite solid containing a dislocation with line C and burgers vector b will be denoted by [ u i D , ε ij D , σ ij D ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGaamyDamaaDaaaleaacaWGPb aabaGaamiraiabg6HiLcaakiaacYcacqaH1oqzdaqhaaWcbaGaamyA aiaadQgaaeaacaWGebGaeyOhIukaaOGaaiilaiabeo8aZnaaDaaale aacaWGPbGaamOAaaqaaiaadseacqGHEisPaaGccaGGDbaaaa@4645@ .  If the dislocation line terminates on the solid’s surface, any convenient procedure can be used to close the loop when deriving the infinte solid solution, but the fields will depend on this choice.

 

  1. The difference between the fields for a dislocation in a bounded solid and the solution for a dislocation in an infinite solid will be denoted by [ u i I = u i D u i D , ε ij I = ε ij D ε ij D , σ ij I = σ ij D σ ij D ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGaamyDamaaDaaaleaacaWGPb aabaGaamysaaaakiabg2da9iaadwhadaqhaaWcbaGaamyAaaqaaiaa dseaaaGccqGHsislcaWG1bWaa0baaSqaaiaadMgaaeaacaWGebGaey OhIukaaOGaaiilaiabew7aLnaaDaaaleaacaWGPbGaamOAaaqaaiaa dMeaaaGccqGH9aqpcqaH1oqzdaqhaaWcbaGaamyAaiaadQgaaeaaca WGebaaaOGaeyOeI0IaeqyTdu2aa0baaSqaaiaadMgacaWGQbaabaGa amiraiabg6HiLcaakiaacYcacqaHdpWCdaqhaaWcbaGaamyAaiaadQ gaaeaacaWGjbaaaOGaeyypa0Jaeq4Wdm3aa0baaSqaaiaadMgacaWG QbaabaGaamiraaaakiabgkHiTiabeo8aZnaaDaaaleaacaWGPbGaam OAaaqaaiaadseacqGHEisPaaGccaGGDbaaaa@6445@

 

The Peach-Koehler force can then be divided into contributions from three sources:

F i (s)= F i D (s)+ F i I (s)+ F i * (s) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGgbWaaSbaaSqaaiaadMgaaeqaaO GaaiikaiaadohacaGGPaGaeyypa0JaamOramaaDaaaleaacaWGPbaa baGaamiraiabg6HiLcaakiaacIcacaWGZbGaaiykaiabgUcaRiaadA eadaqhaaWcbaGaamyAaaqaaiaadMeaaaGccaGGOaGaam4CaiaacMca cqGHRaWkcaWGgbWaa0baaSqaaiaadMgaaeaacaGGQaaaaOGaaiikai aadohacaGGPaaaaa@4A49@

where

  1. F i D (s)= ijk σ jm D (s) b m τ k (s) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGgbWaa0baaSqaaiaadMgaaeaaca WGebGaeyOhIukaaOGaaiikaiaadohacaGGPaGaeyypa0JaeyicI48a aSbaaSqaaiaadMgacaWGQbGaam4AaaqabaGccqaHdpWCdaqhaaWcba GaamOAaiaad2gaaeaacaWGebGaeyOhIukaaOGaaiikaiaadohacaGG PaGaamOyamaaBaaaleaacaWGTbaabeaakiabes8a0naaBaaaleaaca WGRbaabeaakiaacIcacaWGZbGaaiykaaaa@4E7F@  is the `self-force’ of the dislocation, i.e. the force exerted by the stresses generated by the dislocation itself.   This force always acts so as to reduce the length of the dislocation line.  In the classical solution, this force is infinite.  The procedure described in Section  5.9.2 can be used to remove the singularity MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  in this case the stress in the Peach-Koehler formula should be calculated using the expression 

σ pq D (x)= E 16π(1+ν) C ( [ imp b m τ q + imq b m τ p ] 3 R ρ (xξ) x i x j x j ) d s ξ + E 8π(1 ν 2 ) C ( b m imk τ k [ 3 R ρ (xξ) x i x p x q δ pq 3 R ρ (xξ) x i x j x j ] ) d s ξ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaeq4Wdm3aa0baaSqaaiaadchaca WGXbaabaGaamiraiabg6HiLcaakiaacIcacaWH4bGaaiykaiabg2da 9maalaaabaGaamyraaqaaiaaigdacaaI2aGaeqiWdaNaaiikaiaaig dacqGHRaWkcqaH9oGBcaGGPaaaamaapefabaWaaeWaaeaadaWadaqa aiabgIGiopaaBaaaleaacaWGPbGaamyBaiaadchaaeqaaOGaamOyam aaBaaaleaacaWGTbaabeaakiabes8a0naaBaaaleaacaWGXbaabeaa kiabgUcaRiabgIGiopaaBaaaleaacaWGPbGaamyBaiaadghaaeqaaO GaamOyamaaBaaaleaacaWGTbaabeaakiabes8a0naaBaaaleaacaWG WbaabeaaaOGaay5waiaaw2faamaalaaabaGaeyOaIy7aaWbaaSqabe aacaaIZaaaaOGaamOuamaaBaaaleaacqaHbpGCaeqaaOGaaiikaiaa hIhacqGHsislcaWH+oGaaiykaaqaaiabgkGi2kaadIhadaWgaaWcba GaamyAaaqabaGccqGHciITcaWG4bWaaSbaaSqaaiaadQgaaeqaaOGa eyOaIyRaamiEamaaBaaaleaacaWGQbaabeaaaaaakiaawIcacaGLPa aaaSqaaiaadoeaaeqaniabgUIiYdGccaWGKbGaam4CamaaBaaaleaa caWH+oaabeaaaOqaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqGHRaWkcaaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7daWcaaqaaiaadweaaeaacaaI4aGaeqiWdaNaaiikaiaaigdacqGH sislcqaH9oGBdaahaaWcbeqaaiaaikdaaaGccaGGPaaaamaapefaba WaaeWaaeaacaWGIbWaaSbaaSqaaiaad2gaaeqaaOGaeyicI48aaSba aSqaaiaadMgacaWGTbGaam4AaaqabaGccqaHepaDdaWgaaWcbaGaam 4AaaqabaGcdaWadaqaamaalaaabaGaeyOaIy7aaWbaaSqabeaacaaI ZaaaaOGaamOuamaaBaaaleaacqaHbpGCaeqaaOGaaiikaiaahIhacq GHsislcaWH+oGaaiykaaqaaiabgkGi2kaadIhadaWgaaWcbaGaamyA aaqabaGccqGHciITcaWG4bWaaSbaaSqaaiaadchaaeqaaOGaeyOaIy RaamiEamaaBaaaleaacaWGXbaabeaaaaGccqGHsislcqaH0oazdaWg aaWcbaGaamiCaiaadghaaeqaaOWaaSaaaeaacqGHciITdaahaaWcbe qaaiaaiodaaaGccaWGsbWaaSbaaSqaaiabeg8aYbqabaGccaGGOaGa aCiEaiabgkHiTiaah67acaGGPaaabaGaeyOaIyRaamiEamaaBaaale aacaWGPbaabeaakiabgkGi2kaadIhadaWgaaWcbaGaamOAaaqabaGc cqGHciITcaWG4bWaaSbaaSqaaiaadQgaaeqaaaaaaOGaay5waiaaw2 faaaGaayjkaiaawMcaaaWcbaGaam4qaaqab0Gaey4kIipakiaadsga caWGZbWaaSbaaSqaaiaah67aaeqaaaaaaa@F912@

where R ρ (x)= x k x k + ρ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkfadaWgaaWcbaGaeqyWdihabeaaki aacIcacaWH4bGaaiykaiabg2da9maakaaabaGaamiEamaaBaaaleaa caWGRbaabeaakiaadIhadaWgaaWcbaGaam4AaaqabaGccqGHRaWkcq aHbpGCdaahaaWcbeqaaiaaikdaaaaabeaaaaa@3E96@ .   Note that, if the dislocation remains straight, the total length of the dislocation line does not change as the dislocation moves.  In this case, the self-force is zero.  In 2D descriptions of dislocation motion, therefore, the core singularity has no effect MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  this is why it has been possible to live with the classical dislocation fields for so long.

  1. F i I (s)= ijk σ jm I (s) b m τ k (s) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGgbWaa0baaSqaaiaadMgaaeaaca WGjbaaaOGaaiikaiaadohacaGGPaGaeyypa0JaeyicI48aaSbaaSqa aiaadMgacaWGQbGaam4AaaqabaGccqaHdpWCdaqhaaWcbaGaamOAai aad2gaaeaacaWGjbaaaOGaaiikaiaadohacaGGPaGaamOyamaaBaaa leaacaWGTbaabeaakiabes8a0naaBaaaleaacaWGRbaabeaakiaacI cacaWGZbGaaiykaaaa@4BA7@  is a force generated by stress associated with the solid’s boundaries.  These are generally non-singular.  This force is often referred to as the `image force’
  2. F i * (s)= ijk σ jm * (s) b m τ k (s) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGgbWaa0baaSqaaiaadMgaaeaaca GGQaaaaOGaaiikaiaadohacaGGPaGaeyypa0JaeyicI48aaSbaaSqa aiaadMgacaWGQbGaam4AaaqabaGccqaHdpWCdaqhaaWcbaGaamOAai aad2gaaeaacaGGQaaaaOGaaiikaiaadohacaGGPaGaamOyamaaBaaa leaacaWGTbaabeaakiabes8a0naaBaaaleaacaWGRbaabeaakiaacI cacaWGZbGaaiykaaaa@4B67@  is the force exerted on the dislocation by externally applied loading.  This, too, is generally nonsingular.

 

 

Derivation:  The following expression for the total energy of a dislocation in an elastic solid was derived in Section 5.9.3.

V= V D + V D* + V * MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGwbGaeyypa0JaamOvamaaCaaale qabaGaamiraaaakiabgUcaRiaadAfadaahaaWcbeqaaiaadseacaGG QaaaaOGaey4kaSIaamOvamaaCaaaleqabaGaaiOkaaaaaaa@3C85@

where

  V D = S 1 2 σ ij D b i m j dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGwbWaaWbaaSqabeaacaWGebaaaO Gaeyypa0Zaa8quaeaadaWcaaqaaiaaigdaaeaacaaIYaaaaiabeo8a ZnaaDaaaleaacaWGPbGaamOAaaqaaiaadseaaaGccaWGIbWaaSbaaS qaaiaadMgaaeqaaOGaamyBamaaBaaaleaacaWGQbaabeaakiaadsga caWGbbaaleaacaWGtbaabeqdcqGHRiI8aaaa@44C1@  is the strain energy of the dislocation itself

  V D* = S σ ij * b i m j dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGwbWaaWbaaSqabeaacaWGebGaai Okaaaakiabg2da9maapefabaGaeq4Wdm3aa0baaSqaaiaadMgacaWG QbaabaGaaiOkaaaakiaadkgadaWgaaWcbaGaamyAaaqabaGccaWGTb WaaSbaaSqaaiaadQgaaeqaaOGaamizaiaadgeaaSqaaiaadofaaeqa niabgUIiYdaaaa@43CD@  is the work done to introduce the dislocation into the externally applied stress

  V * = R 1 2 σ ij * ε ij * dV 2R t i u i * dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGwbWaaWbaaSqabeaacaGGQaaaaO Gaeyypa0Zaa8quaeaadaWcaaqaaiaaigdaaeaacaaIYaaaaiabeo8a ZnaaDaaaleaacaWGPbGaamOAaaqaaiaacQcaaaGccqaH1oqzdaqhaa WcbaGaamyAaiaadQgaaeaacaGGQaaaaOGaamizaiaadAfaaSqaaiaa dkfaaeqaniabgUIiYdGccqGHsisldaWdrbqaaiaadshadaWgaaWcba GaamyAaaqabaGccaWG1bWaa0baaSqaaiaadMgaaeaacaGGQaaaaOGa amizaiaadgeaaSqaaiabgkGi2kaaikdacaWGsbaabeqdcqGHRiI8aa aa@51B9@  is the potential energy of the applied loads

We wish to calculate the change in potential energy resulting from a small change in area δS MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazcaWGtbaaaa@3543@  as the dislocation line advances by a small distance δa(s)n(s) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazcaWGHbGaaiikaiaadohaca GGPaGaaCOBaiaacIcacaWGZbGaaiykaaaa@3AEA@ .  We consider each term in the potential energy

  1. The last term is independent of S, and therefore δ V * =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazcaWGwbWaaWbaaSqabeaaca GGQaaaaOGaeyypa0JaaGimaaaa@37EB@ .

  1. The change in V D* MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGwbWaaWbaaSqabeaacaWGebGaai Okaaaaaaa@3545@  follows as  δ V D* = δS σ ij * b i m j dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazcaWGwbWaaWbaaSqabeaaca WGebGaaiOkaaaakiabg2da9maapefabaGaeq4Wdm3aa0baaSqaaiaa dMgacaWGQbaabaGaaiOkaaaakiaadkgadaWgaaWcbaGaamyAaaqaba GccaWGTbWaaSbaaSqaaiaadQgaaeqaaOGaamizaiaadgeaaSqaaiab es7aKjaadofaaeqaniabgUIiYdaaaa@4717@ , where δS MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazcaWGtbaaaa@3543@  is the increment in area swept by the dislocation.  Note that an area element swept by the advancing dislocation line can be expressed as mdA=δan×τds MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHTbGaamizaiaadgeacqGH9aqpcq aH0oazcaWGHbGaaGPaVlaaykW7caWHUbGaey41aqRaaCiXdiaadsga caWGZbaaaa@4251@ , so we can write

δ V D* = C σ ij * b i jkl n k τ l δa(s)ds MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazcaWGwbWaaWbaaSqabeaaca WGebGaaiOkaaaakiabg2da9maapefabaGaeq4Wdm3aa0baaSqaaiaa dMgacaWGQbaabaGaaiOkaaaakiaadkgadaWgaaWcbaGaamyAaaqaba GccqGHiiIZdaWgaaWcbaGaamOAaiaadUgacaWGSbaabeaakiaad6ga daWgaaWcbaGaam4AaaqabaGccqaHepaDdaWgaaWcbaGaamiBaaqaba GccqaH0oazcaWGHbGaaiikaiaadohacaGGPaGaamizaiaadohaaSqa aiaadoeaaeqaniabgUIiYdaaaa@51E8@

  1. The change in V D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGwbWaaWbaaSqabeaacaWGebaaaa aa@3497@  can be written as

δ V D = S 1 2 δ σ ij D b i m j dA + δS 1 2 σ ij D b i m j dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazcaWGwbWaaWbaaSqabeaaca WGebaaaOGaeyypa0Zaa8quaeaadaWcaaqaaiaaigdaaeaacaaIYaaa aiabes7aKjabeo8aZnaaDaaaleaacaWGPbGaamOAaaqaaiaadseaaa GccaWGIbWaaSbaaSqaaiaadMgaaeqaaOGaamyBamaaBaaaleaacaWG QbaabeaakiaadsgacaWGbbaaleaacaWGtbaabeqdcqGHRiI8aOGaey 4kaSYaa8quaeaadaWcaaqaaiaaigdaaeaacaaIYaaaaiabeo8aZnaa DaaaleaacaWGPbGaamOAaaqaaiaadseaaaGccaWGIbWaaSbaaSqaai aadMgaaeqaaOGaamyBamaaBaaaleaacaWGQbaabeaakiaadsgacaWG bbaaleaacqaH0oazcaWGtbaabeqdcqGHRiI8aaaa@59B6@

To calculate the change in stress δ σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazcqaHdpWCdaWgaaWcbaGaam yAaiaadQgaaeqaaaaa@3837@  arising from the motion of the dislocation line, recall that the displacement and stress due to the dislocation loop can be calculated from the expression

u k (x)= S m i Σ ij (k) (xξ) b j d A ξ σ pq D = C pqkl u k x l = C pqkl S m i x l Σ ij (k) (xξ) b j d A ξ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaam4AaaqabaGcca GGOaGaaCiEaiaacMcacqGH9aqpdaWdrbqaaiaad2gadaWgaaWcbaGa amyAaaqabaGccqqHJoWudaqhaaWcbaGaamyAaiaadQgaaeaacaGGOa Gaam4AaiaacMcaaaGccaGGOaGaaCiEaiabgkHiTiaah67acaGGPaGa amOyamaaBaaaleaacaWGQbaabeaakiaadsgacaWGbbWaaSbaaSqaai aah67aaeqaaaqaaiaadofaaeqaniabgUIiYdGccaaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlabeo8aZnaaDaaaleaacaWGWbGaamyCaaqaaiaadseaaa GccqGH9aqpcaWGdbWaaSbaaSqaaiaadchacaWGXbGaam4AaiaadYga aeqaaOWaaSaaaeaacqGHciITcaWG1bWaaSbaaSqaaiaadUgaaeqaaa GcbaGaeyOaIyRaamiEamaaBaaaleaacaWGSbaabeaaaaGccqGH9aqp caWGdbWaaSbaaSqaaiaadchacaWGXbGaam4AaiaadYgaaeqaaOWaa8 quaeaacaWGTbWaaSbaaSqaaiaadMgaaeqaaOWaaSaaaeaacqGHciIT aeaacqGHciITcaWG4bWaaSbaaSqaaiaadYgaaeqaaaaakiabfo6atn aaDaaaleaacaWGPbGaamOAaaqaaiaacIcacaWGRbGaaiykaaaakiaa cIcacaWH4bGaeyOeI0IaaCOVdiaacMcacaWGIbWaaSbaaSqaaiaadQ gaaeqaaOGaamizaiaadgeadaWgaaWcbaGaaCOVdaqabaaabaGaam4u aaqab0Gaey4kIipaaaa@9AFB@

where Σ ij (k) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHJoWudaqhaaWcbaGaamyAaiaadQ gaaeaacaGGOaGaam4AaiaacMcaaaaaaa@389D@  is the stress due to a point force in the (bounded) elastic solid.  The change in stress therefore follows as

δ σ pq = C pqkl C x l Σ ij (k) (xξ) b j jkl n k τ l δa(s)d s ξ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=xi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes7aKjabeo8aZnaaBaaaleaacaWGWb GaamyCaaqabaGccqGH9aqpcaWGdbWaaSbaaSqaaiaadchacaWGXbGa am4AaiaadYgaaeqaaOWaa8quaeaadaWcaaqaaiabgkGi2cqaaiabgk Gi2kaadIhadaWgaaWcbaGaamiBaaqabaaaaOGaeu4Odm1aa0baaSqa aiaadMgacaWGQbaabaGaaiikaiaadUgacaGGPaaaaOGaaiikaiaahI hacqGHsislcaWH+oGaaiykaiaadkgadaWgaaWcbaGaamOAaaqabaGc cqGHiiIZdaWgaaWcbaGaamOAaiaadUgacaWGSbaabeaakiaad6gada WgaaWcbaGaam4AaaqabaGccqaHepaDdaWgaaWcbaGaamiBaaqabaGc cqaH0oazcaWGHbGaaiikaiaadohacaGGPaGaamizaiaadohadaWgaa WcbaGaaCOVdaqabaaabaGaam4qaaqab0Gaey4kIipaaaa@6227@

This shows that

δ V D = 1 2 S C pqkl C x l Σ ij (k) (xξ) b j jkl n k τ l δa(s)d s ξ dA + δS 1 2 σ ij D b i m j dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazcaWGwbWaaWbaaSqabeaaca WGebaaaOGaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaadaWdrbqa aiaadoeadaWgaaWcbaGaamiCaiaadghacaWGRbGaamiBaaqabaGcda WdrbqaamaalaaabaGaeyOaIylabaGaeyOaIyRaamiEamaaBaaaleaa caWGSbaabeaaaaGccqqHJoWudaqhaaWcbaGaamyAaiaadQgaaeaaca GGOaGaam4AaiaacMcaaaGccaGGOaGaaCiEaiabgkHiTiaah67acaGG PaGaamOyamaaBaaaleaacaWGQbaabeaakiabgIGiopaaBaaaleaaca WGQbGaam4AaiaadYgaaeqaaOGaamOBamaaBaaaleaacaWGRbaabeaa kiabes8a0naaBaaaleaacaWGSbaabeaakiabes7aKjaadggacaGGOa Gaam4CaiaacMcacaWGKbGaam4CamaaBaaaleaacaWH+oaabeaaaeaa caWGdbaabeqdcqGHRiI8aOGaaGPaVlaaykW7caWGKbGaamyqaaWcba Gaam4uaaqab0Gaey4kIipakiabgUcaRmaapefabaWaaSaaaeaacaaI XaaabaGaaGOmaaaacqaHdpWCdaqhaaWcbaGaamyAaiaadQgaaeaaca WGebaaaOGaamOyamaaBaaaleaacaWGPbaabeaakiaad2gadaWgaaWc baGaamOAaaqabaGccaWGKbGaamyqaaWcbaGaeqiTdqMaam4uaaqab0 Gaey4kIipaaaa@7D80@

Reversing the order of integration in the first integral and using the expression for σ pq MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamiCaiaadg haaeqaaaaa@36A0@  then gives

δ V D = δS σ ij D b i m j dA = C σ ij D b i jkl n k τ l δa(s)ds MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazcaWGwbWaaWbaaSqabeaaca WGebaaaOGaeyypa0Zaa8quaeaacqaHdpWCdaqhaaWcbaGaamyAaiaa dQgaaeaacaWGebaaaOGaamOyamaaBaaaleaacaWGPbaabeaakiaad2 gadaWgaaWcbaGaamOAaaqabaGccaWGKbGaamyqaaWcbaGaeqiTdqMa am4uaaqab0Gaey4kIipakiabg2da9maapefabaGaeq4Wdm3aa0baaS qaaiaadMgacaWGQbaabaGaamiraaaakiaadkgadaWgaaWcbaGaamyA aaqabaGccqGHiiIZdaWgaaWcbaGaamOAaiaadUgacaWGSbaabeaaki aad6gadaWgaaWcbaGaam4AaaqabaGccqaHepaDdaWgaaWcbaGaamiB aaqabaGccqaH0oazcaWGHbGaaiikaiaadohacaGGPaGaamizaiaado haaSqaaiaadoeaaeqaniabgUIiYdaaaa@619D@

  1. Finally, combining the results of (3) and (4) and noting that ijk = jik MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHiiIZdaWgaaWcbaGaamyAaiaadQ gacaWGRbaabeaakiabg2da9iabgkHiTiabgIGiopaaBaaaleaacaWG QbGaamyAaiaadUgaaeqaaaaa@3DBD@  then gives

δV= C kjl [ σ ij D + σ ij * ] b i τ l n k δa(s)ds = C F(s)n(s)δa(s)ds MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqGHsislcqaH0oazcaWGwbGaeyypa0 Zaa8quaeaacqGHiiIZdaWgaaWcbaGaam4AaiaadQgacaWGSbaabeaa kmaadmaabaGaeq4Wdm3aa0baaSqaaiaadMgacaWGQbaabaGaamiraa aakiabgUcaRiabeo8aZnaaDaaaleaacaWGPbGaamOAaaqaaiaacQca aaaakiaawUfacaGLDbaacaWGIbWaaSbaaSqaaiaadMgaaeqaaOGaeq iXdq3aaSbaaSqaaiaadYgaaeqaaOGaamOBamaaBaaaleaacaWGRbaa beaakiabes7aKjaadggacaGGOaGaam4CaiaacMcacaWGKbGaam4Caa WcbaGaam4qaaqab0Gaey4kIipakiabg2da9maapefabaGaaCOraiaa cIcacaWGZbGaaiykaiabgwSixlaah6gacaGGOaGaam4CaiaacMcacq aH0oazcaWGHbGaaiikaiaadohacaGGPaGaamizaiaadohaaSqaaiaa doeaaeqaniabgUIiYdaaaa@6C2C@

This has to hold for all possible δa(s) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH0oazcaWGHbGaaiikaiaadohaca GGPaaaaa@37A2@ , which shows that F i (s)= ijk σ jm (s) b m τ k (s) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGgbWaaSbaaSqaaiaadMgaaeqaaO GaaiikaiaadohacaGGPaGaeyypa0JaeyicI48aaSbaaSqaaiaadMga caWGQbGaam4AaaqabaGccqaHdpWCdaWgaaWcbaGaamOAaiaad2gaae qaaOGaaiikaiaadohacaGGPaGaamOyamaaBaaaleaacaWGTbaabeaa kiabes8a0naaBaaaleaacaWGRbaabeaakiaacIcacaWGZbGaaiykaa aa@4A09@  as required.