Gradient Nanomechanics: Applications to Deformation, Fracture, and Diffusion in Nanopolycrystals
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s used as an effective tool to model material behavior and processes across a variety of scales ranging from macroscopic (construction/manufacturing industries) and microscopic (optoelectronics/micro-electro-mechanical systems or MEMS technologies) to planetary (earthquakes/tsunamis) and cosmological (star formation/galaxy clustering) ones. Its applicability to the nanoscale has not been explored systematically since this field has emerged only recently and is usually dominated by computer simulations. Moreover, the key submicroscopic mechanisms that such a continuum description should be based upon are not clear. A first goal of this article is to illustrate how the basic structure of an earlier proposal of the author for media with microstructure can be extended to describe deformation and transport processes at the nanoscale. Continuum models for nanoelasticity, nanoplasticity, and nanodiffusion are derived within such a generalized framework. A second objective is to show the effectiveness of these models in describing nanoscale phenomena observed in benchmark configurations that may not be captured by classical continuum theory. They are concerned with the dependence of the effective elastic modulus on the grain size in elastic bicrystals, the inverse Hall–Petch (H-P) effect in nanopolycrystals, and the curvature of concentration-depth profiles observed E.C. AIFANTIS, Professor, is with the Laboratory of Mechanics and Materials, Polytechnic School, Aristotle University of Thessaloniki, GR-54124 Thessaloniki, Greece. Contact e-mail: mom@mom. gen.auth.gr Manuscript submitted November 8, 2010. Article published online June 4, 2011 METALLURGICAL AND MATERIALS TRANSACTIONS A
in nanophase materials. A third objective is to illustrate with two characteristic examples how dislocation theory and fracture mechanics can be revisited with gradient elasticity to dispense with classical singularities and provide information on dislocation core and crack tip effects, which become important at the nanoscale. A final objective is to show how a simplified approximate treatment of the gradient terms can efficiently capture size-dependent stress-strain curves of deforming nanopolycrystals with varying grain size under different temperature and strain rate conditions. The basic premise that the proposed nanomechanics framework builds upon is the explicit recognition of the critical role of the ‘‘surface to volume’’ ratio in revising the standard continuum mechanics model. This is done by introducing extra terms in the usual balance laws (mass, momentum) or the evolution equations of the basic variables (plastic strain, defect densities) and, then, by making appropriate constitutive assumptions for these extra terms. The constitutive assumptions for the extra terms are motivated directly by the theory of gradient elasticity for the case of elastic deformation (nanoelasticity), gradient plasticity for the case of plastic flow (nanoplasticity), and the theory of double diffusivity for the case of diffusion (nanodiffusion). It turns out that incorporatio
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