Gradient Effects at Macro, Micro, and Nano Scales

The effects of gradients on the localization and patterning of deformation at various scales ranging from the millimeter (macroscale) down to the nanometer (nanoscale) are discussed. At the nanoscale, a dominant mechanism of deformation is the rearrangement of free nano volume and exchange of momentum between bulk and grain boundary space. At the microscale, a most common mechanism of deformation is dislocation motion. At the macroscale, deformation patterning occurs by the transport of strain from one region of the deforming material to another. In each case, higher order gradients of the respective nano, micro or macro variables are essential to be included in the constitutive equations. This leads to various classes of material behavior including a gradient theory of elasticity, a gradient theory of defect dynamics, and a gradient theory of macroscopic plasticity. Within such "internal-length" theories it is possible to discuss the occurrence of deformation patterns at various scales, the determination of width, spacing and velocity of deformation bands, as well as the structure of the crack tip and the removal of associated strain or stress singularities.