Scaling of Dislocation Strengthening by Multiple Obstacle Types
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THE plastic behavior of metal alloys is controlled by dislocation glide through a field of dispersed obstacles or precipitates, forest dislocations, or solute atoms introduced during alloying processing. The existence of several types of obstacles operating together is common. Second-phase obstacles can be introduced or modified by alloy chemistry and aging time. Such obstacles may have different chemical composition, crystallographic structure, or particle size, leading to varying resistance to the movement of dislocations.[1,2] Precipitates of bimodal size distribution can be produced in some binary and multisolute alloys by double-aging treatment.[3] Precipitate strengthening is often accompanied by some solid solution strengthening.[4,5] Solute atoms are also added intentionally in order to raise the total critical resolved shear stress (CRSS) in many systems, and both solute and forest strengthening arise in many systems. The strengthening mechanisms associated with a single type of obstacle have been widely investigated theoretically. Recent atomistic and discrete dislocation models have investigated the interactions of a dislocation with single precipitates,[6–8] forest dislocations,[9,10] irradiation defects,[7,11,12] and solutes.[13,14] Interactions of dislocations with a statistical field of obstacles have been performed to a lesser degree using these detailed Y. DONG, Graduate Student, T. NOGARET, Postdoctoral Research Associate, and W.A. CURTIN, Elisha Benjamin Andraws Professor, are with the Division of Engineering, Brown University, Providence, RI 02912. Contact e-mail: [email protected] Manuscript submitted October 8, 2009. Article published online June 8, 2010 1954—VOLUME 41A, AUGUST 2010
approaches. Most studies of interactions with a field of obstacles have been performed using a line-tension model and the point-obstacle model.[15–21] In this framework, the analytic Friedel model is appropriate[22] and is in good agreement with numerical simulations.[18] The strengthening associated with simultaneous operation of multiple obstacle types is less clear. Precipitate strengthening accompanied by solid solute strengthening leads to some total CRSS st, but the individual contributions due to the precipitates, sp, and due to the solid solution matrix, ss, are difficult to identify. Only a few experiments addressing the problem of a mixture of different obstacles have been carried out.[3,23] Based on these limited experiments and early computer simulations,[24] a widely accepted ad hoc model is sat ¼ sa1 þ sa2
½1
where st is the total CRSS, and s1 and s2 are the individual strengths due to two different mechanisms (e.g., two different obstacles). Here, a is an exponent whose value varies between 1 and 2 depending on details of the mechanisms involved. Based on Friedel’s model, Kocks and co-workers[25] first supposed that the critical spacing of different obstacles is independent and hence suggested a = 1, which was supported by some experiments.[4] a = 1 is often assumed for interpreting operating mechanisms from
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