Acta mater. 49 (2001) 2567–2582 www.elsevier.com/locate/actamat ANALYTICAL AND COMPUTATIONAL DESCRIPTION OF EFFECT OF GRAIN SIZE ON YIELD STRESS OF METALS H. -H. FU, D. J. BENSON and M. A. MEYERS† Dept. of Mechanical and Aerospace Engineering, University of California, San Diego, Mail Code 0411, 9500 Gilman Drive, La Jolla, CA 92093, USA ( Received 28 July 2000; received in revised form 22 January 2001; accepted 22 January 2001 ) Abstract—Four principal factors contribute to grain-boundary strengthening: (a) the grain boundaries act as barriers to plastic flow; (b) the grain boundaries act as dislocation sources; (c) elastic anisotropy causes additional stresses in grain-boundary surroundings; (d) multislip is activated in the grain-boundary regions, whereas grain interiors are initially dominated by single slip, if properly oriented. As a result, the regions adjoining grain boundaries harden at a rate much higher than grain interiors. A phenomenological constitutive equation predicting the effect of grain size on the yield stress of metals is discussed and extended to the nanocrystalline regime. At large grain sizes, it has the Hall–Petch form, and in the nanocrystalline domain the slope gradually decreases until it asymptotically approaches the flow stress of the grain boundaries. The material is envisaged as a composite, comprised of the grain interior, with flow stress s fG , and grain boundary work-hardened layer, with flow stress s fGB . The predictions of this model are compared with experimental measurements over the mono, micro, and nanocrystalline domains. Computational predictions are made of plastic flow as a function of grain size incorporating differences of dislocation accumulation rate in grain- boundary regions and grain interiors. The material is modeled as a monocrystalline core surrounded by a mantle (grain-boundary region) with a high work hardening rate response. This is the first computational plasticity calculation that accounts for grain size effects in a physically-based manner. A discussion of statisti- cally stored and geometrically necessary dislocations in the framework of strain-gradient plasticity is intro- duced to describe these effects. Grain-boundary sliding in the nanocrystalline regime is predicted from calcu- lations using the Raj–Ashby model and incorporated into the computations; it is shown to predispose the material to shear localization.  2001 Published by Elsevier Science Ltd on behalf of Acta Materialia Inc. Keywords: Nanocrystalline materials; Grain size; Hall–Petch 1. INTRODUCTION The grain-size dependence of yield stress in metals has been represented as a D -1/2 relationship since the pioneering work of Hall [1] and Petch [2]. The term Hall–Petch was introduced by Conrad and Schoeck [3] as a tribute to these researchers. The original explanation for this effect, envisaged by Hall and Petch, was that pile-ups formed at grain boundaries, and required a critical stress to break through them. This was followed by Cottrell [4], who suggested a more realistic scenario; that the stress concentration due to pile-ups activated sources in the surrounding grains. Armstrong [5] presents a critical overview of the effects. Departures of this pile-up scenario were proposed by Li [6] and Conrad [7] who invoked grain-boundary dislocation sources and a grain-size † To whom all correspondence should be addressed. Tel.: +1-858-534-4719; fax: +1-858-534-5698 E-mail address: mameyers@mae.ucsd.edu (M. A. Meyers) 1359-6454/01/$20.00  2001 Published by Elsevier Science Ltd on behalf of Acta Materialia Inc. PII:S1359-6454(01)00062-3 dependence of dislocation density, respectively. The important contributions by Ashby [8], Hirth [9], and Thompson [10] further strengthened the argument that causes other than pile-ups were responsible for the grain size effects. Clear evidence for the forma- tion of a layer of high dislocation density in the direct vicinity of the grain boundaries, starting at an applied stress below the global yield stress, is the trans- mission electron microscopy by Murr and Hecker [11] (especially, Fig. 2). In similar experiments in Fe– 3%Si, Suits and Chalmers [12], and Worthington and Smith [13] incontrovertibly demonstrated that stresses are higher in the grain-boundary region than in the grain interiors. This was corroborated by Margolin and Stanescu [14] for β titanium. Meyers and Ash- worth [15] proposed a mechanism based on elastic anisotropy of the grains. Nevertheless, pile-ups are still widely recognized as the dominating effect. The Hall–Petch relationship has recently come under close scrutiny in the context of nanocrystalline materials pioneered by Gleiter and coworkers [16, 17]. Weertman and coworkers [18–21] have investi-