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MIT OpenCourseWare http://ocw.mit.edu For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms3.22 Mechanical Properties of MaterialsSpring 2008.Problem Set #4 Due: Tuesday, April 15 by 5:00 PM 1. A non-planar dislocation loop is shown in the diagram below. Each of the dislocation seg-ments lies parallel to one of the coordinate axes. The segment AB is a left-handed screw dislocation. ABCDEFGHzxy(a) Identify the character (positive edge, negative edge, right-handed screw (RHS), left-handed screw (LHS)) of each segment of the dislocation loop. (b) A positive shear stress τzx = τxz is applied. Indicate the direction of the Peach-Koehler force for each segment, and list those segments for which the force is zero. 2. Solute strengthening includes a point defect-based obstacle to dislocation motion, where this obstacle is an impurity or solute atom in the ”host” or ”matrix” crystal lattice. Those taking 3.21 in parallel have already considered the equilibrium situation of a solute atom interacting with an edge dislocation self-stress field (Balluffi, Allen & Carter, Kinetics of Materials; Section 3.5.2). Here, we will continue the kinetic consequences of this analysis. (a) Assume a point defect to be well-represented by an elastic sphere of effective radius r(1 + δ), where r is the radius of the host lattice (or matrix) atom and δ is the posi-tive/negative difference between the solute and matrix atomic radii. Note that δ is pos-itive if the solute atom exceeds the host atom in size; and negative if the solute atom is smaller. Assuming the atomic size mismatch is sufficiently small, let the solute atom be a substitutional impurity in the lattice, and express the change in volume at that lattice site ΔV due to replacing a lattice vacancy (the volume of the missing atom in the host lattice) with a solute atom, in terms of the solute atom radius and induced strain �. (b) Cottrell considered the equilibrium position or “atmosphere” of the solute atom due to the self-stress of a dislocation, as a function of distance from the dislocation core. In general, the elastic strain energy Ue of the crystal increases as a pΔV term, where p is equal to the hydrostatic stress acting on that point-defected volume ΔV . Using the elastic solutions to dislocation self-stress and the definition of hydrostatic stress, graph the hydrostatic isostress contours as a function of (x, y) or (r, θ) distance around both an edge dislocation core and around a screw dislocation core. (Mathematica, Matlab, etc. are handy rendering programs for analytical equations.) Here, you can use the required 3.22 Mechanical Behavior of Materials 1 Prof. K.J. Van Vlietstructural and elastic constants for bcc (also called α-phase) Fe. Cottrell combined all these structural/elastic terms into a constant, such that Ue was reduced to a material-dependent constant modifying a dependence on just θ! and r. (c) From the above graphs, show the most energetically stable location of a smaller solute atom like C in terms of θ, the angular position defined by your coordinate system at the edge and screw dislocation cores. (Aside: Note that the above analysis was for a substitutional atom, but C is in fact an interstitial defect in Fe; this point is usually neglected in analysis of point-line defect interactions, but does affect the number of lattice points available to both species.) (d) The equilibrium or steady-state concentration of solute atoms as a function of distance from the dislocation core can be predicted from thermodynamics, based on an Arrhenius dependence on the core-distance-dependent energy barrier Ue. Predict this concentra-tion at a location (0, -b) from the core of a positive edge dislocation in Fe at room temperature (300 K), assuming that the concentration of C solute far from the disloca-tion is 0.1% and that the binding energy of C to the edge dislocation core is 0.5 eV. Explain implications in terms of distribution of the C solute within the Fe lattice. (e) Cottrell considered these interactions even before dislocations were observed, so he analyzed the effect of any applied pressure on the solute atom. Use Peach-Koehler to take the opposite view: What is the driving force for glide and/or climb due to the hydrostatic pressure created by the solute atom that is located at (0, -b) from a positive edge dislocation core and from a positive screw dislocation core? (f) Now as the dislocation glides under applied shear stress on its slip plane, the role of C in solute strengthening depends on a competition between solute diffusion (keep-ing up with the core) and maintained dislocation velocity. For example, if diffusion times/distances are comparable to glide times/distances, the C will keep up with the core and continue to impede glide. This is called dynamic strain aging, and leads to serrated yielding in unaxial tension (stress above yield point dropping as dislocations break free of solute; and increasing as solute diffuses back to the core. How fast/far is that? Here, we make analogy to Cottrell’s classic strain aging argument, which showed scaling between the time required for a moving solute atom to diffuse into a stationary dislocation core (opposite case to our dynamic strain aging), show how the concentra-tion or number of solute atom obstacles at the bcc Fe dislocation core depends on the diffusion time t of the C solute (see BAC’s KoM). Then compare this time for a solute atom to diffuse a distance |b| with the time it takes for a dislocation to move a distance |b| for a typical metal (see your Orowan strain rate solution from PS3). 3. We discussed how two edge dislocations could intersect, resulting in an edge-type jog in one of them. Show whether it is possible for pure (edge or screw) dislocations to intersect in such a way that would result in formation of a sessile kink in either of the two dislocations. These could be two edges, two screws, or one edge/one screw. 4. You are given an Al- 2 at.% Cu alloy where the Cu is in solution with the Al. Your boss then asks you to make the material as strong as possible. (a) Before you do anything to the material, first estimate the amount of solute (in at.%) necessary to put one solute atom at each site


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MIT 3 22 - Problem Set 4

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