Problems for Chapter 5

 

Analytical Techniques and Solutions for Linear Elastic Solids

 

 

 

5.3.  Complex Variable Solutions to Static Linear Elasticity Problems

 

 

5.3.1.      A long cylinder is made from an isotropic, linear elastic solid with shear modulus μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH8oqBaaa@347C@ .  The solid is loaded so

 (i) the resultant forces and moments acting on the ends of the cylinder are zero;

 (ii) the body force b=b( x 1 , x 2 ) e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHIbGaeyypa0JaamOyaiaacIcaca WG4bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadIhadaWgaaWcbaGa aGOmaaqabaGccaGGPaGaaCyzamaaBaaaleaacaaIZaaabeaaaaa@3D5B@  in the interior of the solid acts parallel to the axis of the cylinder; and (iii) Any tractions t=t( x 1 , x 2 ) e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH0bGaeyypa0JaamiDaiaacIcaca WG4bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadIhadaWgaaWcbaGa aGOmaaqabaGccaGGPaGaaCyzamaaBaaaleaacaaIZaaabeaaaaa@3D7F@  or displacements u * = u * ( x 1 , x 2 ) e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bWaaWbaaSqabeaacaGGQaaaaO Gaeyypa0JaamyDamaaCaaaleqabaGaaiOkaaaakiaacIcacaWG4bWa aSbaaSqaaiaaigdaaeqaaOGaaiilaiaadIhadaWgaaWcbaGaaGOmaa qabaGccaGGPaGaaCyzamaaBaaaleaacaaIZaaabeaaaaa@3F4B@  imposed on the sides of the cylinder are parallel to the axis of the cylinder.

Under these conditions, the displacement field at a point far from the ends of the cylinder has the form u=u( x 1 , x 2 ) e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWH1bGaeyypa0JaamyDaiaacIcaca WG4bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadIhadaWgaaWcbaGa aGOmaaqabaGccaGGPaGaaCyzamaaBaaaleaacaaIZaaabeaaaaa@3D81@ , and the solid is said to deform in a state of anti-plane shear.

5.3.1.1.            Calculate the strain field in the solid in terms of u.

5.3.1.2.            Find an expression for the nonzero stress components in the solid, in terms of u and material properties.

5.3.1.3.            Find the equations of equilibrium for the nonzero stress components.

5.3.1.4.            Write down boundary conditions for stress and displacement on the side of the cylinder

5.3.1.5.            Hence, show that the governing equations for u reduce to

2 u x 1 2 + 2 u x 2 2 +b=0{ u(x)= u * (x)x S 1 }{ μ( u(x) x 1 n 1 (x)+ u(x) x 2 n 2 (x) )=t(x)x S 1 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2oaaCaaaleqaba GaaGOmaaaakiaadwhaaeaacqGHciITcaWG4bWaa0baaSqaaiaaigda aeaacaaIYaaaaaaakiabgUcaRmaalaaabaGaeyOaIy7aaWbaaSqabe aacaaIYaaaaOGaamyDaaqaaiabgkGi2kaadIhadaqhaaWcbaGaaGOm aaqaaiaaikdaaaaaaOGaey4kaSIaamOyaiabg2da9iaaicdacaaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8+aaiWaaeaacaWG1bGaaiikaiaahI hacaGGPaGaeyypa0JaamyDamaaCaaaleqabaGaaiOkaaaakiaacIca caWH4bGaaiykaiaaykW7caaMc8UaaGPaVlaaykW7caWH4bGaeyicI4 Saam4uamaaBaaaleaacaaIXaaabeaaaOGaay5Eaiaaw2haaiaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8+aaiWaae aacqaH8oqBdaqadaqaamaalaaabaGaeyOaIyRaamyDaiaacIcacaWH 4bGaaiykaaqaaiabgkGi2kaadIhadaWgaaWcbaGaaGymaaqabaaaaO GaamOBamaaBaaaleaacaaIXaaabeaakiaacIcacaWH4bGaaiykaiab gUcaRmaalaaabaGaeyOaIyRaamyDaiaacIcacaWH4bGaaiykaaqaai abgkGi2kaadIhadaWgaaWcbaGaaGOmaaqabaaaaOGaamOBamaaBaaa leaacaaIYaaabeaakiaacIcacaWH4bGaaiykaaGaayjkaiaawMcaai abg2da9iaadshacaGGOaGaaCiEaiaacMcacaaMc8UaaGPaVlaaykW7 caaMc8UaaCiEaiabgIGiolaadofadaWgaaWcbaGaaGymaaqabaaaki aawUhacaGL9baacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 aaa@B2E0@

 

 

 

 

5.3.2.      Let Θ(z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHyoqucaGGOaGaamOEaiaacMcaaa a@3695@  be an analytic function of a complex number z= x 1 +i x 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG6bGaeyypa0JaamiEamaaBaaale aacaaIXaaabeaakiabgUcaRiaadMgacaWG4bWaaSbaaSqaaiaaikda aeqaaaaa@3A6E@ .  Let v( x 1 , x 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG2bGaaiikaiaadIhadaWgaaWcba GaaGymaaqabaGccaGGSaGaamiEamaaBaaaleaacaaIYaaabeaakiaa cMcaaaa@39A7@ , w( x 1 , x 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG3bGaaiikaiaadIhadaWgaaWcba GaaGymaaqabaGccaGGSaGaamiEamaaBaaaleaacaaIYaaabeaakiaa cMcaaaa@39A8@  denote the real and imaginary parts of Θ(z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHyoqucaGGOaGaamOEaiaacMcaaa a@3695@ .

5.3.2.1.            Since Θ(z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHyoqucaGGOaGaamOEaiaacMcaaa a@3695@  is analytic, the real and imaginary parts must satisfy the Cauchy Riemann conditions

v x 1 = w x 2 w x 1 = v x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2kaadAhaaeaacq GHciITcaWG4bWaaSbaaSqaaiaaigdaaeqaaaaakiabg2da9maalaaa baGaeyOaIyRaam4DaaqaaiabgkGi2kaadIhadaWgaaWcbaGaaGOmaa qabaaaaOqbaeqabeWaaaqaaaqaaaqaaaaadaWcaaqaaiabgkGi2kaa dEhaaeaacqGHciITcaWG4bWaaSbaaSqaaiaaigdaaeqaaaaakiabg2 da9iabgkHiTmaalaaabaGaeyOaIyRaamODaaqaaiabgkGi2kaadIha daWgaaWcbaGaaGOmaaqabaaaaaaa@4CDB@

Show that

2 v x 1 2 + 2 v x 2 2 =0 2 w x 1 2 + 2 w x 2 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaWcaaqaaiabgkGi2oaaCaaaleqaba GaaGOmaaaakiaadAhaaeaacqGHciITcaWG4bWaa0baaSqaaiaaigda aeaacaaIYaaaaaaakiabgUcaRmaalaaabaGaeyOaIy7aaWbaaSqabe aacaaIYaaaaOGaamODaaqaaiabgkGi2kaadIhadaqhaaWcbaGaaGOm aaqaaiaaikdaaaaaaOGaeyypa0JaaGimaiaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVpaalaaaba GaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaam4DaaqaaiabgkGi2kaa dIhadaqhaaWcbaGaaGymaaqaaiaaikdaaaaaaOGaey4kaSYaaSaaae aacqGHciITdaahaaWcbeqaaiaaikdaaaGccaWG3baabaGaeyOaIyRa amiEamaaDaaaleaacaaIYaaabaGaaGOmaaaaaaGccqGH9aqpcaaIWa aaaa@71A7@

5.3.2.2.            Deduce that the displacement and stress in a solid that is free of body force, and loaded on its boundary so as to induce a state anti-plane shear (see problem 1) can be derived an analytic function Θ(z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHyoqucaGGOaGaamOEaiaacMcaaa a@3695@ , using the representation

2μu( x 1 , x 2 )=Θ(z)+ Θ(z) ¯ σ 31 i σ 32 =Θ'(z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaaikdacqaH8oqBcaWG1bGaai ikaiaadIhadaWgaaWcbaGaaGymaaqabaGccaGGSaGaamiEamaaBaaa leaacaaIYaaabeaakiaacMcacqGH9aqpcqqHyoqucaGGOaGaamOEai aacMcacqGHRaWkdaqdaaqaaiabfI5arjaacIcacaWG6bGaaiykaaaa aeaacqaHdpWCdaWgaaWcbaGaaG4maiaaigdaaeqaaOGaeyOeI0Iaam yAaiabeo8aZnaaBaaaleaacaaIZaGaaGOmaaqabaGccqGH9aqpcqqH yoqucaGGNaGaaiikaiaadQhacaGGPaaaaaa@53F4@

 

 

 

 

5.3.3.      Calculate the displacements and stresses generated when the complex potential

Θ(z)= F 2π log(z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHyoqucaGGOaGaamOEaiaacMcacq GH9aqpcqGHsisldaWcaaqaaiaadAeaaeaacaaIYaGaeqiWdahaaiGa cYgacaGGVbGaai4zaiaacIcacaWG6bGaaiykaaaa@4104@

is substituted into the representation described in Problem 2. Show that the solution represents the displacement and stress in an infinite solid due to a line force acting in the e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaiodaaeqaaa aa@349D@  direction at the origin.

 

 

 

5.3.4.      Calculate the displacements and stresses generated when the complex potential

Θ(z)= iμb 2π log(z) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHyoqucaGGOaGaamOEaiaacMcacq GH9aqpcqGHsisldaWcaaqaaiaadMgacqaH8oqBcaWGIbaabaGaaGOm aiabec8aWbaaciGGSbGaai4BaiaacEgacaGGOaGaamOEaiaacMcaaa a@43C4@

is substituted into the representation described in Problem 2. Show that the solution represents the displacement and stress due to a screw dislocation in an infinite solid, with burgers vector and line direction parallel to e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHLbWaaSbaaSqaaiaaiodaaeqaaa aa@349D@ . (To do this, you need to show that (a) the displacement field has the correct character; and (b) the resultant force acting on a circular arc surrounding the dislocation is zero)

 

 

 

5.3.5.      Calculate the displacements and stresses generated when the complex potential

Θ(z)= τ 0 ( z a 2 z ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHyoqucaGGOaGaamOEaiaacMcacq GH9aqpcqaHepaDdaWgaaWcbaGaaGimaaqabaGcdaqadaqaaiaadQha cqGHsisldaWcaaqaaiaadggadaahaaWcbeqaaiaaikdaaaaakeaaca WG6baaaaGaayjkaiaawMcaaaaa@40AD@

is substituted into the representation described in Problem 2. Show that the solution represents the displacement and stress in an infinite solid, which contains a hole with radius a at the origin, and is subjected to anti-plane shear at infinity.

 

 

 

5.3.6.      Calculate the displacements and stresses generated when the complex potential

Θ(z)=i τ 0 z 2 a 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHyoqucaGGOaGaamOEaiaacMcacq GH9aqpcaWGPbGaeqiXdq3aaSbaaSqaaiaaicdaaeqaaOWaaOaaaeaa caWG6bWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaamyyamaaCaaale qabaGaaGOmaaaaaeqaaaaa@3FFC@

is substituted into the representation described in Problem 2. Show that the solution represents the displacement and stress in an infinite solid, which contains a crack with length a at the origin, and is subjected to a prescribed anti-plane shear stress at infinity.  Use the procedure given in Section 5.3.6 to calculate z 2 a 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaadaGcaaqaaiaadQhadaahaaWcbeqaai aaikdaaaGccqGHsislcaWGHbWaaWbaaSqabeaacaaIYaaaaaqabaaa aa@3784@

 

 

 

5.3.7.      Consider complex potentials Ω(z)=az+b,ω(z)=cz+d MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHPoWvcaGGOaGaamOEaiaacMcacq GH9aqpcaWGHbGaamOEaiabgUcaRiaadkgacaGGSaGaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7cqaHjpWDcaGGOaGaamOEaiaacMcacqGH9a qpcaWGJbGaamOEaiabgUcaRiaadsgaaaa@4CA4@ , where a, b, c, d are complex numbers.  Let

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

σ 11 + σ 22 =2( Ω (z)+ Ω (z) ¯ ) σ 11 σ 22 +2i σ 12 =2( z Ω (z) ¯ + ω (z) ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiqacaWaaaGcbaGaeq4Wdm3aaSbaaSqaaiaaigdacaaIXa aabeaakiabgUcaRiabeo8aZnaaBaaaleaacaaIYaGaaGOmaaqabaGc cqGH9aqpcaaIYaWaaeWaaeaacuqHPoWvgaqbaiaacIcacaWG6bGaai ykaiabgUcaRmaanaaabaGafuyQdCLbauaacaGGOaGaamOEaiaacMca aaaacaGLOaGaayzkaaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlabeo8aZnaaBaaaleaacaaIXaGaaGym aaqabaGccqGHsislcqaHdpWCdaWgaaWcbaGaaGOmaiaaikdaaeqaaO Gaey4kaSIaaGOmaiaadMgacqaHdpWCdaWgaaWcbaGaaGymaiaaikda aeqaaOGaeyypa0JaeyOeI0IaaGOmamaabmaabaGaamOEaiaaykW7da qdaaqaaiqbfM6axzaagaGaaiikaiaadQhacaGGPaaaaiabgUcaRmaa naaabaGafqyYdCNbauaacaGGOaGaamOEaiaacMcaaaaacaGLOaGaay zkaaaaaa@88F8@

be a displacement and stress field derived from these potentials.

5.3.7.1.            Find values of a,b,c,d that represent a rigid displacement u 1 = w 1 +α x 2 , u 2 = w 2 α x 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwhadaWgaaWcbaGaaGymaaqabaGccq GH9aqpcaWG3bWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaeqySdeMa amiEamaaBaaaleaacaaIYaaabeaakiaacYcacaaMc8UaaGPaVlaayk W7caWG1bWaaSbaaSqaaiaaikdaaeqaaOGaeyypa0Jaam4DamaaBaaa leaacaaIYaaabeaakiabgkHiTiabeg7aHjaadIhadaWgaaWcbaGaaG ymaaqabaaaaa@4857@  where w 1 , w 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEhadaWgaaWcbaGaaGymaaqabaGcca GGSaGaam4DamaaBaaaleaacaaIYaaabeaaaaa@34E9@  are (real) constants representing a translation, and α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg7aHbaa@3207@  is a real constant representing an infinitesimal rotation.

5.3.7.2.            Find values of a,b,c,d that correspond to a state of uniform stress

Note that the solutions to 7.1 and 7.2 are not unique.

 

 

 

 

5.3.8.      Show that the complex potentials

Ω(z)= p a a 2 p b b 2 4( b 2 a 2 ) z ω(z)= ( p a p b ) 2( b 2 a 2 )z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiqacaWaaaGcbaGaeuyQdCLaaiikaiaadQhacaGGPaGaey ypa0ZaaSaaaeaacaWGWbWaaSbaaSqaaiaadggaaeqaaOGaamyyamaa CaaaleqabaGaaGOmaaaakiabgkHiTiaadchadaWgaaWcbaGaamOyaa qabaGccaWGIbWaaWbaaSqabeaacaaIYaaaaaGcbaGaaGinaiaacIca caWGIbWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaamyyamaaCaaale qabaGaaGOmaaaakiaacMcaaaGaamOEauaabeqabqaaaaqaaaqaaaqa aaqaaaaacqaHjpWDcaGGOaGaamOEaiaacMcacqGH9aqpcqGHsislda WcaaqaaiaacIcacaWGWbWaaSbaaSqaaiaadggaaeqaaOGaeyOeI0Ia amiCamaaBaaaleaacaWGIbaabeaakiaacMcaaeaacaaIYaGaaiikai aadkgadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWGHbWaaWbaaSqa beaacaaIYaaaaOGaaiykaiaadQhaaaaaaa@5A99@

give the stress and displacement field in a pressurized circular cylinder which deforms in plane strain (it is best to solve this problem using polar coordinates)

 

 

 

 

 

 

 

5.3.9.      The complex potentials

Ω(z)=i E( b 1 +i b 2 ) 8π(1 ν 2 ) log(z) ω(z)=i E( b 1 i b 2 ) 8π(1 ν 2 ) log(z) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8XjY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiqacaWaaaGcbaGaeuyQdCLaaiikaiaadQhacaGGPaGaey ypa0JaeyOeI0IaamyAamaalaaabaGaamyramaabmaabaGaamOyamaa BaaaleaacaaIXaaabeaakiabgUcaRiaadMgacaWGIbWaaSbaaSqaai aaikdaaeqaaaGccaGLOaGaayzkaaaabaGaaGioaiabec8aWjaacIca caaIXaGaeyOeI0IaeqyVd42aaWbaaSqabeaacaaIYaaaaOGaaiykaa aaciGGSbGaai4BaiaacEgacaGGOaGaamOEaiaacMcafaqabeqaeaaa aeaaaeaaaeaaaeaaaaGaeqyYdCNaaiikaiaadQhacaGGPaGaeyypa0 JaamyAamaalaaabaGaamyraiaacIcacaWGIbWaaSbaaSqaaiaaigda aeqaaOGaeyOeI0IaamyAaiaadkgadaWgaaWcbaGaaGOmaaqabaGcca GGPaaabaGaaGioaiabec8aWjaacIcacaaIXaGaeyOeI0IaeqyVd42a aWbaaSqabeaacaaIYaaaaOGaaiykaaaaciGGSbGaai4BaiaacEgaca GGOaGaamOEaiaacMcaaaa@6806@

generate the plane strain solution to an edge dislocation at the origin of an infinite solid.  Work through the algebra necessary to determine the stresses (you can check your answer using the solution given in Section 5.3.4).   Verify that the resultant force exerted by the internal tractions on a circular surface surrounding the dislocation is zero.

 

 

 

 

5.3.10.  When a stress field acts on a dislocation, the dislocation tends to move through the solid.  Formulas for these forces are derived in Section 5.9.5.   For the particular case of a straight edge dislocation, with burgers vector b i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkgadaWgaaWcbaGaamyAaaqabaaaaa@3269@ , the force can be calculated as follows:

*      Let σ ij D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaDaaaleaacaWGPbGaamOAaa qaaiaadseacqGHEisPaaaaaa@366F@  denote the stress field in an infinite solid containing the dislocation (calculated using the formulas in Section 5.3.4

*      Let σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaaaaa@3434@  denote the actual stress field in the solid (including the effects of the dislocation itself, as well as corrections due to boundaries in the solid, or externally applied fields)

*      Define Δ σ ij = σ ij σ ij D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejabeo8aZnaaBaaaleaacaWGPb GaamOAaaqabaGccqGH9aqpcqaHdpWCdaWgaaWcbaGaamyAaiaadQga aeqaaOGaeyOeI0Iaeq4Wdm3aa0baaSqaaiaadMgacaWGQbaabaGaam iraiabg6HiLcaaaaa@4174@  denote the difference between these quantities.

The force can then be calculated as F i = ij3 Δ σ jk b k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAeadaWgaaWcbaGaamyAaaqabaGccq GH9aqpcqGHiiIZdaWgaaWcbaGaamyAaiaadQgacaaIZaaabeaakiab fs5aejabeo8aZnaaBaaaleaacaWGQbGaam4AaaqabaGccaWGIbWaaS baaSqaaiaadUgaaeqaaaaa@3EF2@ .

Consider two edge dislocations in an infinite solid, each with burgers vector b 1 =b, b 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkgadaWgaaWcbaGaaGymaaqabaGccq GH9aqpcaWGIbGaaiilaiaaykW7caaMc8UaaGPaVlaadkgadaWgaaWc baGaaGOmaaqabaGccqGH9aqpcaaIWaaaaa@3D17@ .  One dislocation is located at the origin, the other is at position ( x 1 , x 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacIcacaWG4bWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaadIhadaWgaaWcbaGaaGOmaaqabaGccaGGPaaaaa@364E@ .  Plot contours of the horizontal component of force acting on the second dislocation due to the stress field of the dislocation at the origin. (normalize the force as 8π(1 ν 2 ) F 1 ( x 1 2 + x 2 2 )/Eb MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaiIdacqaHapaCcaGGOaGaaGymaiabgk HiTiabe27aUnaaCaaaleqabaGaaGOmaaaakiaacMcacaWGgbWaaSba aSqaaiaaigdaaeqaaOWaaeWaaeaadaGcaaqaaiaadIhadaqhaaWcba GaaGymaaqaaiaaikdaaaGccqGHRaWkcaWG4bWaa0baaSqaaiaaikda aeaacaaIYaaaaaqabaaakiaawIcacaGLPaaacaGGVaGaamyraiaadk gaaaa@4485@  ). 

 

 

 

 

 

5.3.11.  The figure shows an edge dislocation below the surface of an elastic solid.  Use the solution given in Section 5.3.12, together with the formula in Problem 5.3.9 to calculate an expression for the force acting on the dislocation.

 

 

 

 

 

5.3.12.  The figure shows an edge dislocation with burgers vector b 1 =b, b 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkgadaWgaaWcbaGaaGymaaqabaGccq GH9aqpcqGHsislcaWGIbGaaiilaiaadkgadaWgaaWcbaGaaGOmaaqa baGccqGH9aqpcaaIWaaaaa@3963@  that lies in a strained elastic film with thickness h.   The film and substrate have the same elastic moduli. The stress in the film consists of the stress due to the dislocation, together with a tensile stress σ 11 = σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIXaGaaGymaa qabaGccqGH9aqpcqaHdpWCdaWgaaWcbaGaaGimaaqabaaaaa@3786@ .  Calculate the force acting on the dislocation, and hence find the film thickness for which the dislocation will be attracted to the free surface and escape from the film.  You will need to use the formula given in problem 5.3.10 to calculate the force on the dislocation.

 

 

 

 

5.3.13.  The figure shows a dislocation in an elastic solid with Young’s modulus E and Poisson’s ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe27aUbaa@3220@ , which is bonded to a rigid solid.   The solution can be generated from complex potentials

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

where

Ω 0 (z)=i E( b 1 +i b 2 ) 8π(1 ν 2 ) log(zih) ω 0 (z)=i E( b 1 i b 2 ) 8π(1 ν 2 ) log(zih)+ E( b 1 +i b 2 ) 8π(1 ν 2 ) h zih MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa aiaabeqaaiqacaWaaaGceaqabeaacqqHPoWvdaWgaaWcbaGaaGimaa qabaGccaGGOaGaamOEaiaacMcacqGH9aqpcqGHsislcaWGPbWaaSaa aeaacaWGfbWaaeWaaeaacaWGIbWaaSbaaSqaaiaaigdaaeqaaOGaey 4kaSIaamyAaiaadkgadaWgaaWcbaGaaGOmaaqabaaakiaawIcacaGL PaaaaeaacaaI4aGaeqiWdaNaaiikaiaaigdacqGHsislcqaH9oGBda ahaaWcbeqaaiaaikdaaaGccaGGPaaaaiGacYgacaGGVbGaai4zaiaa cIcacaWG6bGaeyOeI0IaamyAaiaadIgacaGGPaaabaGaeqyYdC3aaS baaSqaaiaaicdaaeqaaOGaaiikaiaadQhacaGGPaGaeyypa0JaamyA amaalaaabaGaamyraiaacIcacaWGIbWaaSbaaSqaaiaaigdaaeqaaO GaeyOeI0IaamyAaiaadkgadaWgaaWcbaGaaGOmaaqabaGccaGGPaaa baGaaGioaiabec8aWjaacIcacaaIXaGaeyOeI0IaeqyVd42aaWbaaS qabeaacaaIYaaaaOGaaiykaaaaciGGSbGaai4BaiaacEgacaGGOaGa amOEaiabgkHiTiaadMgacaWGObGaaiykaiabgUcaRmaalaaabaGaam yramaabmaabaGaamOyamaaBaaaleaacaaIXaaabeaakiabgUcaRiaa dMgacaWGIbWaaSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaaaba GaaGioaiabec8aWjaacIcacaaIXaGaeyOeI0IaeqyVd42aaWbaaSqa beaacaaIYaaaaOGaaiykaaaadaWcaaqaaiaadIgaaeaacaWG6bGaey OeI0IaamyAaiaadIgaaaaaaaa@8523@

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

Ω 1 (z)=( z Ω 0 ( z ¯ ) ¯ + ω 0 ( z ¯ ) ¯ )/( 34ν ) ω 1 (z)=(34ν) Ω 0 ( z ¯ ) ¯ z( Ω 0 ( z ¯ ) ¯ +z Ω 0 ( z ¯ ) ¯ + ω 0 ( z ¯ ) ¯ )/(34ν) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaeuyQdC1aaSbaaSqaaiaaigdaae qaaOGaaiikaiaadQhacaGGPaGaeyypa0ZaaeWaaeaacaWG6bWaa0aa aeaacuqHPoWvgaqbamaaBaaaleaacaaIWaaabeaakiaacIcaceWG6b GbaebacaGGPaaaaiabgUcaRmaanaaabaGaeqyYdC3aaSbaaSqaaiaa icdaaeqaaOGaaiikaiqadQhagaqeaiaacMcaaaaacaGLOaGaayzkaa Gaai4lamaabmaabaGaaG4maiabgkHiTiaaisdacqaH9oGBaiaawIca caGLPaaaaeaacqaHjpWDdaWgaaWcbaGaaGymaaqabaGccaGGOaGaam OEaiaacMcacqGH9aqpcaGGOaGaaG4maiabgkHiTiaaisdacqaH9oGB caGGPaWaa0aaaeaacqqHPoWvdaWgaaWcbaGaaGimaaqabaGccaGGOa GabmOEayaaraGaaiykaaaacqGHsislcaWG6bWaaeWaaeaadaqdaaqa aiqbfM6axzaafaWaaSbaaSqaaiaaicdaaeqaaOGaaiikaiqadQhaga qeaiaacMcaaaGaey4kaSIaamOEamaanaaabaGafuyQdCLbayaadaWg aaWcbaGaaGimaaqabaGccaGGOaGabmOEayaaraGaaiykaaaacqGHRa WkdaqdaaqaaiqbeM8a3zaafaWaaSbaaSqaaiaaicdaaeqaaOGaaiik aiqadQhagaqeaiaacMcaaaaacaGLOaGaayzkaaGaai4laiaacIcaca aIZaGaeyOeI0IaaGinaiabe27aUjaacMcaaaaa@76B4@

corrects the solution to satisfy the zero displacement boundary condition at the interface.

5.3.13.1.        Show that the solution satisfies the zero displacement boundary condition

5.3.13.2.        Calculate the force acting on the dislocation, in terms of h and relevant material properties.  You will need to use the formula from 5.3.10 to calculate the force on the dislocation.

5.3.13.3.        Calculate the distribution of stress along the interface between the elastic and rigid solids, in terms of h and x 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaaGymaaqabaaaaa@324C@

 

 

 

 

 

 

5.3.14.  The figure shows a rigid cylindrical inclusion with radius a embedded in an isotropic elastic matrix.  The solid is subjected to a uniform uniaxial stress σ 11 = σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIXaGaaGymaa qabaGccqGH9aqpcqaHdpWCdaWgaaWcbaGaaGimaaqabaaaaa@3786@  at infinity.  The goal of this problem is to calculate the stress fields in the matrix.

5.3.14.1.        Write down the boundary conditions on the displacement field at r=a

5.3.14.2.        Show that the boundary conditions can be satisfied by complex potentials of the form

Ω(z)= σ 0 4 ( z+ α a 2 z )ω(z)= σ 0 2 ( z+ β a 2 z + λ a 4 z 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfM6axjaacIcacaWG6bGaaiykaiabg2 da9maalaaabaGaeq4Wdm3aaSbaaSqaaiaaicdaaeqaaaGcbaGaaGin aaaadaqadaqaaiaadQhacqGHRaWkdaWcaaqaaiabeg7aHjaadggada ahaaWcbeqaaiaaikdaaaaakeaacaWG6baaaaGaayjkaiaawMcaaiaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaeqyYdCNaaiikaiaadQhacaGGPaGaeyypa0JaeyOeI0YaaSaaae aacqaHdpWCdaWgaaWcbaGaaGimaaqabaaakeaacaaIYaaaamaabmaa baGaamOEaiabgUcaRmaalaaabaGaeqOSdiMaamyyamaaCaaaleqaba GaaGOmaaaaaOqaaiaadQhaaaGaey4kaSYaaSaaaeaacqaH7oaBcaWG HbWaaWbaaSqabeaacaaI0aaaaaGcbaGaamOEamaaCaaaleqabaGaaG 4maaaaaaaakiaawIcacaGLPaaaaaa@72FF@

where α,β,λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg7aHjaacYcacqaHYoGycaGGSaGaeq 4UdWgaaa@36BC@  are three real valued coefficients whose values you will need to determine.  The algebra in this problem can be simplified by noting that z ¯ = a 2 /z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqadQhagaqeaiabg2da9iaadggadaahaa WcbeqaaiaaikdaaaGccaGGVaGaamOEaaaa@3610@  on the boundary of the inclusion.

5.3.14.3.        Find an expression for the stresses acting at the inclusion/matrix boundary

5.3.14.4.        The interface between inclusion and matrix fails when the normal stress acting on the interface reaches a critical stress σ crit MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGJbGaamOCai aadMgacaWG0baabeaaaaa@361D@ .  Find an expression for the maximum tensile stress that can be applied to the material without causing failure.

 

 

 

 

5.3.15.  The figure shows a cylindrical inclusion with radius a and Young’s modulus and Poisson’s ratio E p , ν p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweadaWgaaWcbaGaamiCaaqabaGcca GGSaGaeqyVd42aaSbaaSqaaiaadchaaeqaaaaa@35E6@   embedded in an isotropic elastic matrix with elastic constants E,ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweacaGGSaGaeqyVd4gaaa@339A@ .  The solid is subjected to a uniform uniaxial stress σ 11 = σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIXaGaaGymaa qabaGccqGH9aqpcqaHdpWCdaWgaaWcbaGaaGimaaqabaaaaa@3786@  at infinity.  The goal of this problem is to calculate the stress fields in both the particle and the matrix.  The analylsis can be simplified greatly by assuming a priori that the stress in the particle is uniform (this is not obvious, but can be checked after the full solution has been obtained).   Assume, therefore, that the stress in the inclusion is σ 11 = p 11 , σ 22 = p 22 s 12 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIXaGaaGymaa qabaGccqGH9aqpcaWGWbWaaSbaaSqaaiaaigdacaaIXaaabeaakiaa cYcacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabeo8aZnaaBaaale aacaaIYaGaaGOmaaqabaGccqGH9aqpcaWGWbWaaSbaaSqaaiaaikda caaIYaaabeaakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caWGZbWaaSbaaSqaaiaaigdacaaIYaaabeaakiabg2da9iaa icdaaaa@5631@ , where p 11 , p 22 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchadaWgaaWcbaGaaGymaiaaigdaae qaaOGaaiilaiaadchadaWgaaWcbaGaaGOmaiaaikdaaeqaaaaa@3652@  are to be determined.   The solution inside the particle can therefore be derived from complex potentials

Ω p (z)= p 11 + p 22 4 z ω p (z)= p 22 p 11 2 z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfM6axnaaBaaaleaacaWGWbaabeaaki aacIcacaWG6bGaaiykaiabg2da9maalaaabaGaamiCamaaBaaaleaa caaIXaGaaGymaaqabaGccqGHRaWkcaWGWbWaaSbaaSqaaiaaikdaca aIYaaabeaaaOqaaiaaisdaaaGaamOEaiaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqaHjpWDda WgaaWcbaGaamiCaaqabaGccaGGOaGaamOEaiaacMcacqGH9aqpdaWc aaqaaiaadchadaWgaaWcbaGaaGOmaiaaikdaaeqaaOGaeyOeI0Iaam iCamaaBaaaleaacaaIXaGaaGymaaqabaaakeaacaaIYaaaaiaadQha aaa@5C32@

In addition, assume that the solution in the matrix can be derived from complex potentials of the form

Ω(z)= σ 0 4 ( z+ α a 2 z )ω(z)= σ 0 2 ( z+ β a 2 z + λ a 4 z 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfM6axjaacIcacaWG6bGaaiykaiabg2 da9maalaaabaGaeq4Wdm3aaSbaaSqaaiaaicdaaeqaaaGcbaGaaGin aaaadaqadaqaaiaadQhacqGHRaWkdaWcaaqaaiabeg7aHjaadggada ahaaWcbeqaaiaaikdaaaaakeaacaWG6baaaaGaayjkaiaawMcaaiaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaeqyYdCNaaiikaiaadQhacaGGPaGaeyypa0JaeyOeI0YaaSaaae aacqaHdpWCdaWgaaWcbaGaaGimaaqabaaakeaacaaIYaaaamaabmaa baGaamOEaiabgUcaRmaalaaabaGaeqOSdiMaamyyamaaCaaaleqaba GaaGOmaaaaaOqaaiaadQhaaaGaey4kaSYaaSaaaeaacqaH7oaBcaWG HbWaaWbaaSqabeaacaaI0aaaaaGcbaGaamOEamaaCaaaleqabaGaaG 4maaaaaaaakiaawIcacaGLPaaaaaa@72FF@

where α,β,λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg7aHjaacYcacqaHYoGycaGGSaGaeq 4UdWgaaa@36BC@  are three real valued coefficients whose values you will need to determine.

5.3.15.1.        Write down the boundary conditions on the displacement field at r=a.  Express this boundary condition as an equation relating Ω p , ω p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfM6axnaaBaaaleaacaWGWbaabeaaki aacYcacqaHjpWDdaWgaaWcbaGaamiCaaqabaaaaa@36BF@  to Ω,ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfM6axjaacYcacqaHjpWDaaa@3473@  in terms of material properties.

5.3.15.2.        Write down boundary conditions on the stress components σ rr , σ rθ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGYbGaamOCaa qabaGccaGGSaGaeq4Wdm3aaSbaaSqaaiaadkhacqaH4oqCaeqaaaaa @399B@  at r=a. Express this boundary condition as an equation relating Ω p , ω p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfM6axnaaBaaaleaacaWGWbaabeaaki aacYcacqaHjpWDdaWgaaWcbaGaamiCaaqabaaaaa@36BF@  to Ω,ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfM6axjaacYcacqaHjpWDaaa@3473@ .

5.3.15.3.        Calculate expressions for p 11 , p 22 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchadaWgaaWcbaGaaGymaiaaigdaae qaaOGaaiilaiaadchadaWgaaWcbaGaaGOmaiaaikdaaeqaaaaa@3652@ , α,β,λ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg7aHjaacYcacqaHYoGycaGGSaGaeq 4UdWgaaa@36BC@  in terms of σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIWaaabeaaaa a@3311@  and material properties.

5.3.15.4.        The inclusion fractures when the maximum principal stress acting in the particle reaches a critical stress σ crit MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGJbGaamOCai aadMgacaWG0baabeaaaaa@361D@ .  Find an expression for the maximum tensile stress that can be applied to the material without causing failure.

 

 

 

5.3.16.  The figure shows a slit crack in an infinite solid.   Using the solution given in Section 5.3.6, calculate the stress field very near the right hand crack tip (i.e. find the stresses in the limit as r 1 /a0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkhadaWgaaWcbaGaaGymaaqabaGcca GGVaGaamyyaiabgkziUkaaicdaaaa@3690@  ).  Show that the results are consistent with the asymptotic crack tip field given in Section 5.2.9, and deduce an expression for the crack tip stress intensity factors in terms of σ 22 , σ 12 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8sk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaqhaaWcbaGaaGOmaiaaik daaeaacqGHEisPaaGccaGGSaGaeq4Wdm3aa0baaSqaaiaaigdacaaI YaaabaGaeyOhIukaaaaa@3D41@  and a.

 

 

 

 

 

5.3.17.  Two identical cylindrical roller bearings with radius 1cm are pressed into contact by a force P per unit out of plane length as indicated in the figure.   The bearings are made from 52100 steel with a uniaxial tensile yield stress of 2.8GPa.   Calculate the force (per unit length) that will just initiate yield in the bearings, and calculate the width of the contact strip between the bearings at this load.

 

 

 

 

5.3.18.  The figure shows a pair of identical involute spur gears.  The contact between the two gears can be idealized as a line contact between two cylindrical surfaces.  The goal of this problem is to find an expression for the maximum torque Q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGrbaaaa@339C@  that can be transmitted through the gears.  The gears can be idealized as isotropic, linear elastic solids with Young’s modulus E and Poisson’s ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaH9oGBaaa@347E@ .

 

As a representative configuration, consider the instant when a single pair of gear teeth make contact exactly at the pitch point.  At this time, the geometry can be idealized as contact between two cylinders, with radius r= R p sinϕ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGYbGaeyypa0JaamOuamaaBaaale aacaWGWbaabeaakiGacohacaGGPbGaaiOBaiabew9aMbaa@3B65@ , where R p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGsbWaaSbaaSqaaiaadchaaeqaaa aa@34BE@  is the pitch circle radius of the gears and ϕ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHvpGzaaa@348E@  is the pressure angle.  The cylinders are pressed into contact by a force P=Q/( R p cosϕ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGqbGaeyypa0Jaamyuaiaac+caca GGOaGaamOuamaaBaaaleaacaWGWbaabeaakiGacogacaGGVbGaai4C aiabew9aMjaacMcaaaa@3E20@ .

5.3.18.1.        Find a formula for the area of contact between the two gear teeth, in terms of Q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGrbaaaa@339C@ , R p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGsbWaaSbaaSqaaiaadchaaeqaaa aa@34BE@ , ϕ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHvpGzaaa@348E@ , b and representative material properties.

5.3.18.2.        Find a formula for the maximum contact pressure acting on the contact area, in terms of Q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGrbaaaa@339C@ , R p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGsbWaaSbaaSqaaiaadchaaeqaaa aa@34BE@ , ϕ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHvpGzaaa@348E@ , b and representative material properties.

5.3.18.3.        Suppose that the gears have uniaxial tensile yield stress Y.  Find a formula for the critical  value of Q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGrbaaaa@339C@  required to initiate yield in the gears.