Problems for Chapter 5

 

Analytical Techniques and Solutions for Linear Elastic Solids

 

 

 

5.9.  Energetics of Dislocations

 

 

5.9.1.      Calculate the stress induced by a straight screw dislocation in an infinite solid using the formula in Section 5.8.4.  Compare the solution with the result of the calculation in Problem 5.3.4.

 

 

 

5.9.2.      The figure shows two nearby straight screw dislocations in an infinite solid, with line direction perpendicular to the plane of the figure.  The screw dislocations can be introduced into the solid by cutting the plane between the dislocations and displacing the upper of the surfaces created by the cut ( S MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadofadaahaaWcbeqaaiabgkHiTaaaaa a@325A@  ) by b e 3 /2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkgacaWHLbWaaSbaaSqaaiaaiodaae qaaOGaai4laiaaikdaaaa@349F@ , and the lower ( S + MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadofadaahaaWcbeqaaiabgUcaRaaaaa a@324F@  ) by b e 3 /2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkHiTiaadkgacaWHLbWaaSbaaSqaai aaiodaaeqaaOGaai4laiaaikdaaaa@358C@ , and re-connecting the surfaces.   The solid deforms in anti-plane shear, with a displacement field of the form u=u( x 1 , x 2 ) e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahwhacqGH9aqpcaWG1bGaaiikaiaadI hadaWgaaWcbaGaaGymaaqabaGccaGGSaGaamiEamaaBaaaleaacaaI YaaabeaakiaacMcacaWHLbWaaSbaaSqaaiaaiodaaeqaaaaa@3B23@

5.9.2.1.            Write down nonzero components of stress and strain in the solid

 

5.9.2.2.            Show that the total strain energy of the solid (per unit out of plane distance) can be expressed as

U= 1 2 A σ 3α u x α dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwfacqGH9aqpdaWcaaqaaiaaigdaae aacaaIYaaaamaapefabaGaeq4Wdm3aaSbaaSqaaiaaiodacqaHXoqy aeqaaOWaaSaaaeaacqGHciITcaWG1baabaGaeyOaIyRaamiEamaaBa aaleaacqaHXoqyaeqaaaaakiaadsgacaWGbbaaleaacaWGbbaabeqd cqGHRiI8aaaa@438B@

 

5.9.2.3.            Show that the potential energy can be re-written as

U= 1 2 d/2 d/2 σ 32 ( x 1 )bd x 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwfacqGH9aqpcqGHsisldaWcaaqaai aaigdaaeaacaaIYaaaamaapehabaGaeq4Wdm3aaSbaaSqaaiaaioda caaIYaaabeaakiaacIcacaWG4bWaaSbaaSqaaiaaigdaaeqaaOGaai ykaiaadkgacaWGKbGaamiEamaaBaaaleaacaaIXaaabeaaaeaacqGH sislcaWGKbGaai4laiaaikdaaeaacaWGKbGaai4laiaaikdaa0Gaey 4kIipaaaa@4724@

where the integral is taken along the line x 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaaGOmaaqabaGccq GH9aqpcaaIWaaaaa@3417@ .

 

5.9.2.4.            Use the solution for a screw dislocation given in Problem 5.3.4 (or 5.9.1) to show that the energy can be calculated as

U= μ b 2 8π d/2 d/2 ( 1 x 1 +d/2 + 1 d/2 x 1 )d x 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadwfacqGH9aqpdaWcaaqaaiabeY7aTj aadkgadaahaaWcbeqaaiaaikdaaaaakeaacaaI4aGaeqiWdahaamaa pehabaWaaeWaaeaadaWcaaqaaiaaigdaaeaacaWG4bWaaSbaaSqaai aaigdaaeqaaOGaey4kaSIaamizaiaac+cacaaIYaaaaiabgUcaRmaa laaabaGaaGymaaqaaiaadsgacaGGVaGaaGOmaiabgkHiTiaadIhada WgaaWcbaGaaGymaaqabaaaaaGccaGLOaGaayzkaaGaamizaiaadIha daWgaaWcbaGaaGymaaqabaaabaGaeyOeI0Iaamizaiaac+cacaaIYa aabaGaamizaiaac+cacaaIYaaaniabgUIiYdaaaa@518B@

Note that the integral is unbounded, as expected.  Calculate a bounded expression by truncating the integral at d/2ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaGGVaGaaGOmaiabgkHiTiabeg 8aYbaa@356D@  and d/2+ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkHiTiaadsgacaGGVaGaaGOmaiabgU caRiabeg8aYbaa@364F@

 

5.9.2.5.            Calculate the force exerted on one dislocation by the other by differentiating the expression for the energy.  Is the force attractive or repulsive?

 

5.9.2.6.            Check your answer using the Peach-Koehler formula.

 

 

 

5.9.3.      Calculate the stress induced by an edge dislocation in an infinite solid using the formula in Section 5.8.4.  Compare the solution with the result given in 5.3.4

 

 

 

5.9.4.      Calculate the nonsingular stress σ ij (ρ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaDaaaleaacaWGPbGaamOAaa qaaiaacIcacqaHbpGCcaGGPaaaaaaa@374E@  induced by a screw dislocation in an infinite solid using the formula in Section 5.9.2.  Compare the solution with the result of the calculation in Problem 5.3.4.

 

 

 

5.9.5.      Calculate the nonsingular self- energy per unit length of a straight dislocation, using the approach discussed in Section 5.9.2.  (To do this, you have to calculate the energy of a dislocation segment with finite length, then take the limit of the energy per unit length as the dislocation length goes to infinity).

 

 

 

5.9.6.      Calculate the self-energy of a square prismatic dislocation loop with side length L.  Use nonsingular dislocation theory, and give your answer to zeroth order in the parameter ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYbaa@3228@

 

 

 

 

5.9.7.      Suppose that the dislocation loop described in the preceding problem is subjected to a uniaxial tensile stress σ ij = σ 0 δ i3 δ j3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpcqaHdpWCdaWgaaWcbaGaaGimaaqabaGccqaH0oaz daWgaaWcbaGaamyAaiaaiodaaeqaaOGaeqiTdq2aaSbaaSqaaiaadQ gacaaIZaaabeaaaaa@3EFA@ .  Calculate the total potential energy of the system.   Display your result as a graph of normalized potential energy V D (1 ν 2 )/EL b 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAfadaahaaWcbeqaaiaadseaaaGcca GGOaGaaGymaiabgkHiTiabe27aUnaaCaaaleqabaGaaGOmaaaakiaa cMcacaGGVaGaamyraiaadYeacaWGIbWaaWbaaSqabeaacaaIYaaaaa aa@3C0D@  as a function of L/b, for various values of σ 0 (1 ν 2 )/E MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIWaaabeaaki aacIcacaaIXaGaeyOeI0IaeqyVd42aaWbaaSqabeaacaaIYaaaaOGa aiykaiaac+cacaWGfbaaaa@3A44@ .  Take ρ=b/4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYjabg2da9iaadkgacaGGVaGaaG inaaaa@3586@  as a representative value.  Hence, estimate (i) an expression for the activation energy required for homogeneous nucleation of a prismatic dislocation loop, as a function of σ 0 (1 ν 2 )/E MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIWaaabeaaki aacIcacaaIXaGaeyOeI0IaeqyVd42aaWbaaSqabeaacaaIYaaaaOGa aiykaiaac+cacaWGfbaaaa@3A44@ ; and (ii) the critical size required for a pre-existing dislocation loop to grow, as a function of σ 0 (1 ν 2 )/E MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIWaaabeaaki aacIcacaaIXaGaeyOeI0IaeqyVd42aaWbaaSqabeaacaaIYaaaaOGa aiykaiaac+cacaWGfbaaaa@3A44@ .

 

 

 

 

5.9.8.      Calculate the self-energy of a rectangular glide dislocation loop with burgers vector b=b e 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahkgacqGH9aqpcaWGIbGaaCyzamaaBa aaleaacaaIXaaabeaaaaa@3515@  and side lengths a,d.  Use nonsingular dislocation theory, and give your answer to zeroth order in the parameter ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYbaa@3228@ .

 

 

 

 

 

5.9.9.      A composite material is made by sandwiching thin layers of a ductile metal between layers of a hard ceramic. Both the metal and the ceramic have identical Young’s modulus E MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweaaaa@3132@  and Poisson’s ratio ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe27aUbaa@3220@ . The figure shows one of the metal layers, which contains a glide dislocation loop on an inclined slip-plane.  The solid is subjected to a uniaxial tensile stress σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaaIWaaabeaaaa a@3311@  perpendicular to the layers.

5.9.9.1.            Calculate the total energy of the dislocation loop, in terms of the applied stress and relevant geometric and material parameters.  Use non-singular dislocation theory to calculate the self-energy of the loop.

5.9.9.2.            Suppose that the layer contains a large number of dislocation loops with initial width d 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgadaWgaaWcbaGaaGimaaqabaaaaa@3237@ .  The layer starts to deform plastically if the stress is large enough to cause the loops to expand in the plane of the film (by increasing the loop dimension d).   Calculate the yield stress of the composite.   How does the yield stress scale with film thickness?