Does shape affect shape change at the nanoscale?
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Introduction The fundamental mechanism of shape change, or plasticity, in crystalline metals is the generation, motion, and interaction of dislocations.1 A major triumph of classical crystal plasticity theories is that they predict the mechanical properties of metals based directly on the density of dislocations, ρ, in the material.2 One key assumption of these theories, however, is that the dimensions of the body undergoing deformation are much larger than the characteristic spacing between dislocations, which may be estimated as ρ −1/ 2 . Thus, taking 2 •10 7 − 3•1011 cm/cm3 as the representative range of dislocation densities encountered in well-annealed to heavily cold-worked metals,3 we expect classical crystal plasticity theories to break down for domain sizes below ∼0.2–2 µm. One example of this breakdown is size-dependent mechanical properties, sometimes referred to as the “smaller is stronger” effect,4 observed in particles, wires, and sheets with submicron dimensions.5 Another is the elevation in yield and flow strength of polycrystalline metals that occurs as grain sizes are reduced to ∼100 nm and below.6 In both examples, the dimensions of crystalline domains are so small compared to the characteristic spacing between dislocations, that the frequency of dislocation interactions with free surfaces or solid-state interfaces (i.e., grain and phase boundaries7) equals or exceeds the frequency of dislocation interactions with each other. Thus, the properties of surfaces and interfaces, especially the way
they interact with dislocations,8 are of prime importance for understanding and predicting the mechanical behavior of submicron scale (i.e., nanoscale) metallic objects or aggregates thereof. The paramount importance of surfaces and interfaces for plasticity within nanoscale domains motivates a reevaluation of the role of domain shape in plastic deformation. In continuum crystal plasticity theories, where domain dimensions are substantially larger than the spacing between dislocations, the mechanical properties at any point in the material are determined solely by the local dislocation density. While domain shapes may affect solutions to boundary value problems,9 they have no influence on the constitutive relations governing the mechanical response of individual representative volume elements. Is that still the case at the nanoscale? The shape of nanoscale domains determines the type and arrangement of surfaces and interfaces bounding them (i.e., their area, orientation, curvature, connectivity, and crystallographic character). These, in turn, affect the degree of dislocation confinement along different slip planes and directions. Are the plastic strains (i.e., shape changes) induced by dislocation motion indifferent to these parameters or does shape affect shape change at the nanoscale? An unambiguous demonstration that shape does indeed matter for plasticity at the nanoscale may be obtained by considering examples of materials with extreme differences
Michael J. Demkowicz, Department of Materials Science a
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