Nanocrystalline grain boundaries that slip and separate: A gradient theory that accounts for grain-boundary stress and conditions at a triple-junction

A substantial fraction of the volume in a nanocrystalline material is occupied by intercrystalline grain boundaries; as the grain sizes decrease below ≈30nm these boundaries begin to play a major role in the inelastic deformation of the material. For such grain sizes, and under ambient pressures and moderate strain rates, dislocation-based slip processes in the grain interiors are essentially shut off and inelastic deformation occurs primarily by slip and separation at the grain boundaries. We here develop a continuum mechanical theory of such grain boundaries based on the notion of a cohesive interface across which the displacement suffers a jump discontinuity, an approach that allows us to develop elastic and inelastic descriptions of slip and separation. As a means of capturing the small length scales involved we allow for a microscopic polar stress that expends power over the surface gradient of the inelastic slip rate. Using the principle of virtual power we deduce interfacial force balances for the grain boundaries which when combined with thermodynamically consistent constitutive equations result in viscoinelastic flow rules for the grain boundaries in the form of partial differential equations. A second application of the virtual-power principle yields nonstandard conditions that balance grain-boundary tractions at a triple junction.