hw4sol - 1/8 Anderson G4:Users:peterand:Central...

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Unformatted text preview: 1/8 Anderson G4:Users:peterand:Central files:Courses:MSE 765:2005:hw4sol.doc Materials Science and Engineering 765 mechanical behavior of materials Spring 2005 Homework 4 (corrected 5.16.05) with solutions Due: Thursday, May 19, by 5pm. Show all work necessary to arrive at your answers. 45 points max on homework 4 1. Hall-Petch relation for yield strength as a function of grain size In class, we derived a relation for the force per unit length acting on an obstacle such as a grain boundary, in terms of the uniform applied shear stress τ nb appl on the active slip plane, the critical local shear stress τ nb s needed to operate a source in the center of the grain, grain size D , and isotropic elastic properties μ and ν : In the first step, we obtained the applied stress needed to operate the source in terms of the parameters mentioned, the number N of dislocations in the pile-up, dislocation line energy e , and the source length scale s . σ nb applto operatesource = 2 e bs + μ b π 1 − ν ( ) D N In the second part, we obtained the force exerted on the obstacle in terms of N , the applied stress, and the grain size D : F obs L = σ nb appl − μ Nb 4 π 1 − ν ( ) D Nb a. 4 points: Use these two expressions to solve for the critical applied stress σ nb crit to push the pile-up past the obstacle. Assume that the critical force to overcome the 2/8...
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hw4sol - 1/8 Anderson G4:Users:peterand:Central...

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