In Encyclopedia of Nanotechnology, B. Bhushan, Editor. 2012, Springer

Publication year: 2012

Plasticity theory is the mathematical formalism that describes the constitutive model of a material undergoing permanent deformation upon loading. For polycrystalline metals at low temperature and strain rate, the *J*_{2 }theory is the simplest adequate model. Classic plasticity theory does not include any explicit length scale, and as a result, the constitutive behavior is independent of the sample dimensions. As the characteristic length of a sample is reduced to the micro (and nano) scale, careful experimental observations clearly reveal the presence of a size effect that is not accounted for by the classical theory. Strain gradient plasticity is a formalism devised to extend plasticity theory to these smaller scales. For most metals, strain gradient plasticity is intended to apply to objects in the range from roughly 100 nm to 100 μm. Above 100 μm, the theory converges with the classical theory and below 100 nm surface and grain boundary effects not accounted for in the theory begin to dominate the behavior. By assuming that the plastic work (or in some theories, the yield strength) depends not only on strain but also on strain gradients (a hypothesis physically grounded in dislocation theory and, in particular, in the notion of geometrically necessary dislocations (GND) associated with incompatibility due to strain gradients), an intrinsic length scale is naturally introduced, allowing the theory to capture size effects. According to most theories, the intrinsic length scale is of the order of the distance between dislocation-clipping obstacles or cellular dislocation structures (typically, submicron to tens of microns). This continuum theory is appropriate for length scales that remain large relative to the distance between dislocations. As the sample length scale is dropped below this level, dislocations must be modeled individually, and discrete dislocations simulations (DSS) are the preferred approach. At even smaller scales, molecular dynamics (MD) becomes the applicable tool. This article presents a brief overview of one of the simplest continuum strain gradient plasticity theories that reduces to the classical *J*_{2} theory when the scale of the deformation becomes large compared to the material length scale. This simple theory captures the essence of the experimental trends observed to date regarding size effects in submicron to micron scale plasticity.

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