Deformation and Coble Creep of Nanocrystalline Materials
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Deformation and Coble Creep of Nanocrystalline Materials
C.S. Pande and R. A. Masumura Materials Science and Technology Division, Naval Research Laboratory, Washington, DC 20375-5343, USA.
ABSTRACT
Modeling of strengthening by nanocrystalline materials need consideration of dislocation interactions and sliding due to Coble creep, both of which may be acting simultaneously. Such a mechanism is considered in this paper. It is shown that a model based on using Coble creep (with a threshold stress) for finer grains and conventional Hall-Petch strengthening for larger grains, appears to be most successful in explaining experimental results provided care is taken to incorporate into the analysis the effect of grain size distribution occurring in most specimens. A generalized expression relating yield stress to grain size is also proposed.
INTRODUCTION
For relatively large grain sizes (for grains larger than about a micron) relationship between yield stress τ and grain size d, is described very well by the classic Hall-Petch relationship [1,2] viz.,
τ = τ o + k d −1/ 2
(1)
where τo is the friction stress, and k is a constant often referred to as the Hall-Petch slope which varies from material to material. Masumura et al. [3] have plotted some of the available data in a Hall-Petch plot. They find that the yield stress-grain size exponent for relatively large grains appears to be very close to -1/2 and generally this trend continues until the very fine grain regime
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(~100 nm) is reached. However in nanocrystalline materials whose grain sizes are of nanometer (nm) dimensions, the applicability and validity of Eq. (1) has been questioned [4,5]. A close analysis of experimental Hall-Petch data in a variety of materials shows that even though the plot of τ vs. d
-1/2
forms a continuous curve, but below a critical grain size Hall-Petch slope is nearly
zero (with no increase in strength on decreasing grain size) or in some cases at least where the strength actually decreases with decreasing grain size ("Inverse Hall Petch") [5]. In this paper we are mostly interested in the mechanism or mechanisms responsible for yield stress applicable to whole range of grain sizes but especially for the lowest grain sizes.
DISLOCATION MODELS
For large grain sizes most of the models use a mechanism based on dislocations. They account very well for the grain size dependence of the stress, τ,in Eq. (1); most of these can be rationalized in terms of a dislocation pile up model [6]. In deriving the Hall-Petch relation, the role of grain boundaries as a barrier to dislocation model is considered in various models. In one type of model [7,8,9], the grain boundary acts as a barrier to pile up of dislocations, causing stresses to concentrate and activating dislocation sources in the neighboring grains, thus initiating slip from grain to grain. In the other type of models [10,11] the grain boundaries are regarded as dislocations barriers limiting the mean free path of the dislocations, thereby increasing strain hardening, re
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