Dislocation Pile-Ups, Material Strength Levels, and Thermal Activation
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INTRODUCTION
AMONG many research topics in micromechanics investigated by James C.M. Li, with colleagues and students, are elaborations of (1) dislocation pile-ups and (2) thermally activated dislocation dynamics.[1,2] Li’s reports on both topics have contributed much in establishment of a fundamental basis for understanding the microstructural parameters determining the strength and fracturing levels of engineering materials—which understanding has been extended in recent years to the important topic of the strength properties of nanopolycrystalline materials[3]—and for understanding the influence of temperature and strain rate on those strength properties. The following account provides documentation of the efforts as related to broader research developments on the several subjects of (1) dislocation pile-up theory and its extensions; (2) pile-up applications to Hall–Petch (H–P) type demonstrations of grain size influence on material stress–strain behaviors;[4] and (3) the thermal activation-strain rate analysis (TASRA) for description of strain rate (and temperature) characterizations of material strength levels.[5] II.
DISLOCATION PILE-UP ANALYSES
A. Pile-Up Model Descriptions Discrete and continuum mechanics investigations of dislocation pile-ups began in the 1950s.[6,7] Earlier, Zener had pointed both to the nature of a slip band being analogous to a shear crack in determining a reciprocal square root of grain size dependence of yield stress and to coalesced dislocations at a pile-up tip
RONALD W. ARMSTRONG, Professor Emeritus, is with the Center for Engineering Concepts Development, Department of Mechanical Engineering, University of Maryland, College Park, MD 20742. Contact e-mail: [email protected] Manuscript submitted May 25, 2015. METALLURGICAL AND MATERIALS TRANSACTIONS A
providing a physical mechanism for cleavage fracture to occur.[8,9] Figure 1 provides a summary of results obtained from the often-referenced Eshelby, Frank, and Nabarro analysis for specification of dislocation pile-up parameters.[6] In the figure, s is the effective shear stress, s* is the concentrated shear stress acting on the locked dislocation at x = 0, n is the total number of dislocations, G is shear modulus, a is average dislocation character, xn1 is the last dislocation in the pile-up, b is dislocation Burgers vector, and x1 is the separation of the leading dislocation and the locked one. Equation [1] in Figure 1 was obtained by Cottrell from consideration of the virtual work done in displacing the pile-up as compared with displacing the locked dislocation.[10] Equation [2] is obtained by employing the asymptotic large value of n dependence on the product of s and xn1. Equation [3] is obtained from the same Eshelby et al. analysis in terms of an alternative dependence of s on x1.[11] Li and Chou provided a valuable follow-up analysis of numerical and continuum pile-up descriptions involving single-ended, double-ended, and circular geometries that differ from each other only by numerical factors.[1] Figure 2 shows the (s/G) depende
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