A composite model for the grain-size dependence of yield stress of nanograined materials
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NED materials are typically characterized by an average grain size that is less than several tens of nanometers. This class of materials behaves quite differently from their coarse-grained counterparts.[1,2] Two of the most widely reported differences are the departure from the Hall–Petch[3,4] relation of the yield stress and the increased ductility (superplasticity). The increased ductility has been observed in both metals and ceramics. For instance, a nanocrystalline copper—with a purity better than 99.993 at pct and an average grain size of 28 nm— can deform up to 5100 pct at room temperature,[5] and a composite ceramic consisting of 40 vol pct ZrO2-30 vol pct spinel-30 vol pct Al2O3, with an average grain size of 210 nm, can deform up to 1050 pct at 1650°C without failure.[6] For the yield stress, the Hall–Petch relation is known to govern the grain-size dependence for coarsegrained materials, but for nanograined materials, recent experiments on copper and palladium have shown a departure, and a decrease in some cases in the nanorange, in the Hall–Petch plot.[7,8,9] These markedly different characteristics are primarily caused by the increased role played by the grain boundaries in a nanograined material. For the traditional coarse-grained materials, the grain boundary can generally be regarded as a region of almost zero thickness whose deformation contributes little to the overall plastic strain, but when the grain size reduces to the nanometer scale, a sigB. JIANG, Postdoctoral Fellow, and G.J. WENG, Professor, are with the Department of Mechanical and Aerospace Engineering, Rutgers University, New Brunswick, NJ 08903. Contact e-mail: [email protected] Manuscript submitted May 23, 2002. METALLURGICAL AND MATERIALS TRANSACTIONS A
nificant portion of atoms exists at the grain boundary and its deformation can have a dominant effect on the overall plastic strain of the material. In this article, a micromechanics-based composite model with a finite grain-boundary volume fraction will be developed to study the plasticity and the departure of the Hall–Petch plot of the yield stress during the coarse to nanograin transition. We shall leave the issue of superplasticity to a later study where deformation up to several hundreds or thousands of percents requires a large-strain formulation. Within the small-strain framework, our primary objective is to determine how the overall elastoplastic behavior of the material and its yield strength change as the grain size decreases from the coarse-grain to the nanograin range. II. RATIONALE OF THE COMPOSITE MODEL There have been some molecular dynamic simulations of atomic structures for nanograined materials. A representative one from Schiotz et al.[10] for a nanocrystalline copper using 100,000 atoms is reproduced in Figure 1(a). The microstructure is marked by a distinct layer of grain boundary whose volume fraction is not negligible. For continuum modeling with a composite approach, such a microstructure can be represented by the schematic diagram in Figure 1(b), where each
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