Comparative Analysis of Plasticity-based GND Density Estimation Methods in Crystal Plasticity Finite Element Models

TL;DR

This work probes three approaches to quantify GND densities from the Nye tensor in crystal plasticity finite element models: (i) projection onto all dislocation systems with an L2 pseudoinverse, (ii) projection restricted to active slip systems, and (iii) a CPFE-based shear-gradient method. While all methods reproduce the expected trends—GNDs increasing with strain and decreasing grain size—projection onto all systems yields systematically lower densities than the shear/CPFE approach; this mismatch is largely eliminated when restricting to active slip systems. Across single-crystal and polycrystal tests, the shear-based method and the active-set projection align more closely with analytical predictions, while all approaches capture GND accumulation near grain boundaries and multigrain junctions and reflect Schmid-factor related effects. The study highlights the importance of method choice and mesh sensitivity for quantitatively accurate GND predictions in polycrystal CPFE simulations, with practical implications for size- and gradient-dependent hardening modeling.

Abstract

In crystal plasticity finite element (CPFE) simulations, accurately quantifying geometrically necessary dislocations (GNDs) is critical for capturing strain gradients in polycrystals. We compare different methods for quantifying GNDs, all of which originate from the Nye tensor, which is computed as the curl of the plastic deformation gradient. The projection technique directly decomposes the Nye tensor onto individual screw and edge dislocation components to compute GNDs. This approach requires converting a nine-component Nye tensor into densities for a larger number of dislocation systems, a fundamentally underdetermined (non-unique) process, which is resolved using $L2$ minimization. In contrast, when employing CPFE analysis, one could directly compute dislocation densities on each slip system using shear gradients. Projection and slip gradient methods are compared with respect to their prediction of GNDs with changing grain size, strain, and grain neighborhoods, including multigrain junctions. Although these techniques match analytical GND densities for single slip, single crystal deformation, and are consistent with anticipated overall GND trends, we find that the GND densities from projection techniques are significantly lower than those predicted from CPFE-based slip gradients in polycrystals. A suggested improvement of only using the active dislocation systems in the projection technique almost entirely resolved this mismatch.

Figures (16)

• Figure 1: Deformation gradient decomposition.
• Figure 2: Single crystal beam shear loading and dimensioning adapted from Demir2024 (left) and mesh in the undeformed and deformed configurations (right).
• Figure 3: Sample microstructure and loading direction used for GND analysis.
• Figure 4: Median GND density in the symmetric tension vs non-symmetric shear case. The inset shows an adjusted GND density axis for clarity with the symmetric case.
• Figure 5: GND density distributions for the sheared single crystal beam using (a) the dislocation system approach and (b) the shear approach.