Since the 1950s, a large body of work has been published on connecting the curvature of a crystal lattice to geometrically necessary dislocations densities of a crystal lattice. Studying dislocation transmission through grains and across their boundaries requires the lattice curvature to be preserved. However, traditional crystal plasticity models and their numerical implementations do not formally preserve lattice curvature. In this paper, a continuous crystal lattice orientation finite element method (LOFEM) is proposed to rectify this impediment to the inclusion of dislocation-based constitutive models. The methodology is first presented, and then it is demonstrated for tension and compression deformations of a copper polycrystal. It is shown that under the same deformation histories, the lattice continuity constraint alters the evolving state in comparison to the traditional approach, including retarding the rate at which the crystallographic texture strengthens under monotonic deformation. Taking advantage of the finite element representation of the lattice orientation, the Nye tensor is computed on lattices misoriented by deformation and is subsequently used to compute evolving dislocation density distributions.
Citation: Robert Carson and Paul Dawson 2019 JMPS 126 pg. 1-19
Over the past six decades, work has been published on the connection of the geometrically necessary dislocations (GNDs) densities of a crystal lattice to the distortion of the lattice. The original theory is built on the Nye tensor, α, which connects the two notions of the lattice curvature and GND densities. In the work conducted by Kroener, the Nye tensor for the case of large plastic but small elastic strains required both the lattice orientation and lattice stretch be continuous within a crystal. Subsequently, it has been shown that only the gradient of the lattice orientation is necessary in the calculation of the Nye tensor for the large plastic but small elastic strain case, and as a consequence, only the lattice orientation need be continuous for the curvature of the lattice to be preserved through GNDs.
Continuity of the lattice has implications beyond the computation of the Nye tensor. For example, there continues to be interest in the long-standing issue of the mechanisms by which dislocations cross grain boundaries. Further, because of its relevance to deformation and failure mechanisms, the localization of deformation in bands within the interior of grains continues to be examined in the context of how dislocations traverse discrete volumes within a grain. Continuity of the lattice bears on these issues because of their connections to the movements and distributions of dislocations.
Over the past two decades, there has been increasing interest is using crystal plasticity models in simulation that introduce quantities related to the Nye tensor. While several of these models include dislocation interactions, none of them has directly imposed the restriction of a continuous lattice orientation for each grain. However, strain gradient and dislocation based methods, which depend on quantities derived by the Nye tensor, do need to meet the minimum requirements placed on the lattice by the Nye tensor. While dislocation and strain gradient methods do impose further restrictions on the lattice orientation evolution, neither method strictly enforces continuity of the lattice. Indeed, because models based solely on slip do not require lattice continuity, numerical implementations typically do not impose it. Rather, the evolution equations for lattice orientation have been treated as unconnected sets of ordinary differential equations that are integrated independently. Using standard finite element methodologies it is possible to treat the lattice orientation as a field over a grain and impose a continuity constraint through the choice of interpolation functions, thereby meeting the requirements on lattice curvature imposed on the computation of the Nye tensor. This new method brings the lattice representation further in line with the continuous lattice assumptions made in several electron back-scattering diffraction (EBSD) and high energy x-ray diffraction (HEXD) crystallographic theories. Through this effort, we bridge the gap between the underlying assumptions made by diffraction measurements and dislocation modeling frameworks.