Consistent determination of geometrically necessary dislocation density from simulations and experiments

Abstract

The use of Nye's dislocation tensor for calculating the density of geometrically necessary dislocations (GND) is widely adopted in the study of plastically deformed materials. The "curl" operation involved in finding the Nye tensor, while conceptually straightforward has been marred with inconsistencies and several different definitions are in use. For the three most common definitions, we show that their consistent application leads to the same result. To eliminate frequently encountered confusion, a summary of expressions for Nye's tensor in terms of elastic and plastic deformation gradient, and for both small and large deformations, is presented. A further question when estimating GND density concerns the optimization technique used to solve the under-determined set of equations linking Nye's tensor and GND density. A systematic comparison of the densities obtained by two widely used techniques, L 1 and L 2 minimisation, shows that both methods yield remarkably similar total GND densities. Thus the mathematically simpler, L 2 , may be preferred over L 1 except when information about the distribution of densities on specific slip systems is required. To illustrate this, we compare experimentally measured lattice distortions beneath nano-indents in pure tungsten, probed using 3D-resolved synchrotron X-ray micro-diffraction, with those predicted by 3D straingradient crystal plasticity finite element calculations. The results are in good agreement and show that the volumetric component of the elastic strain field has a surprisingly small effect on the determined Nye tensor. This is important for experimental techniques, such as micro-beam Laue measurements and HR-EBSD, where only the deviatoric strain component is measured.

Orientation-dependent indentation response of helium-implanted tungsten

“…A model based on this hypothesis was implemented in a CPFE user material subroutine (UMAT) for Abaqus where strain softening was applied to the helium-implanted layer. The UMAT is based on a user-element developed by Dunne et al [29] and is founded on the theory of multiplicative decomposition of the deformation gradient into elastic and plastic components [30][31]. Briefly, the CPFE formulation constrains slip to applicable slip-systems (assumed to be the 12 {110} slip planes with a/2<111> slip vector directions [32]).…”

Hardening and Strain Localisation in Helium-Ion-Implanted Tungsten

“…We assume, consistent with the CPFE calculations, that deformation in bcc tungsten is accommodated by dislocations with a/2<111> Burgers' vector gliding on {110} planes 49,50 (list of Burgers' vector and line directions considered can be found elsewhere 34 ). Furthermore, we assume dislocations to be of either pure edge (with <112> line directions) or pure screw type (with <111> line directions), resulting in 16 dislocation types in total 34 . Figure 6 shows the total GND density computed from HR-EBSD, Laue diffraction and CPFE datasets, plotted on the YZ, XZ and the XY cross sections at the indent centre.…”

“…The indentation experiments were simulated using a strain-gradient CPFE model based on the model proposed by Dunne et al 72,73 where plastic deformation is constrained to occur only in directions consistent with crystallographic slip. Recently we successfully demonstrated the use of strain-gradient CPFE for performing 3D simulations of nano-indentation in pure tungsten 34 . Here, we use a modified version of this UMAT to simulate nano-indentation in helium-implanted tungsten.…”

Dislocation density distribution at slip band-grain boundary intersections

“…The method for GND density calculation follows that developed for HR-EBSD [20,38,39], i.e. solving the Nye-Kröner-Bilby (NKB) equation (equations 1 and 2) but extended to incorporate the gradients in the third dimension, therefore accounting for all nine components of the elastic displacement gradient tensor [6,40]:…”