Prediction of ideal orientations in velocity gradient-driven processes for large plastic deformations of crystals

TL;DR

The paper tackles predicting crystal texture evolution under large, velocity-gradient-driven deformations by identifying stable lattice orientation attractors through a linear stability analysis of a rigid-visco-plastic crystal plasticity model in an Eulerian framework. It develops a 2-D three-slip-system reduction to obtain analytical slip-rate expressions and tractable stability results, while outlining the challenges of a full 3-D treatment. The authors characterize stationary orientations and their basins, validate predictions against polycrystal and mono-crystal problems, and demonstrate alignment between attractor-based predictions and high-resolution CP simulations, including scenarios with void growth and slip-band formation. The work provides a principled approach to predict final textures in manufacturing processes involving large strains and localized deformation, enabling texture control and material-property optimization.

Abstract

We focus on the crystal lattice ideal orientations, also referred to as preferred or attractor orientations, in crystalline materials, and how they can be used to predict the final texture of polycrystals after manufacturing processes. The simplified crystal plasticity (CP) models used here capture the main features of microstructural evolution in monocrystalline and polycrystalline materials undergoing velocity-gradient-driven processes, without considering hardening or softening effects. The evolution of the lattice orientation is described by a nonlinear ordinary differential equation, and a linear stability analysis is performed to identify the permanent orientations that act as attractors (i.e., the ideal or preferred orientations). Although our linear stability analysis is generally applicable, it is detailed using a simplified two-dimensional model with three slip systems. This approach successfully predicts lattice orientation attractors for large strains by analyzing the interplay between deformation and rotation, initial orientation, and the interaction between different slip systems under applied loads. Three fundamental problems in CP illustrate the effectiveness of the theory: polycrystal deformation under homogeneous velocity gradient loading, void evolution under radial loading, and slip band formation in a monocrystal. High-resolution CP numerical simulations, enhanced using re-meshing techniques, provide further validation of our findings concerning the impact of initial crystallographic orientations, deformation mechanisms, and loading conditions on the evolution of orientation attractors and the final crystal texture.

Figures (15)

• Figure 1: (a) Two-dimensional model with three slip systems. (b) Definition of angles: $\psi$, the principal strain rate direction, and $\theta$, the lattice orientation. (c) Reference slip rates $\dot{\gamma}_1^{\text{ref}}(\theta)$, $\dot{\gamma}_2^{\text{ref}}(\theta)$, and $\dot{\gamma}_3^{\text{ref}}(\theta)$ as functions of $\theta$ for an FCC crystal (Schmidt model). (d) Reference slip rate $\dot{\gamma}_1^{\text{ref}}(\theta)$ as a function of $\theta$ in the Perzyna model, for different values of the viscosity parameter: $\eta = 0.5\eta^{\text{ref}}$ (magenta), $\eta = 0.4\eta^{\text{ref}}$ (green), $\eta = 0.33\eta^{\text{ref}}$ (blue), $\eta = 0.30\eta^{\text{ref}}$ (red), and $\eta = 0$ (i.e., the Schmidt model, yellow), with $\eta^{\text{ref}} = \tau_c / \dot{\Gamma}^{\text{ref}}$. (e) The function $\theta \mapsto \mathcal{G}(\theta)$ for a FCC crystal (green, $\phi = 54.7^\circ$) and a hexagonal crystal (blue, $\phi = 60^\circ$).
• Figure 2: The function $\theta \mapsto \mathcal{G}(\theta)$ in the case of in-plane deformation of an FCC crystal. Its intersection with the line $y = 2\tilde{\omega}^*/d^*$ defines the stationary lattice orientations $\tilde{\theta}^{\text{st}}$. The attractors $\tilde{\theta}^{\text{att}}$ are shown as solid colored discs (orange, blue, and green), and the colored segments (matching the color of each attractor) represent their respective basins of attraction.
• Figure 3: Numerical simulation of 200 time/strain trajectories $t \to (t,\cos(\theta(t)), \sin(\theta(t)))$ of the crystal orientation for different choices of the initial orientation $\theta_0$ and $\psi^*=0$. Left: case i) ($\vert \omega^*\vert < d^*$) with 3 attractors. Middle: case ii) ($-\omega^* = d^*$) with 3 half-attractors. Right: case iii) ($\vert \omega^*\vert > d^*$) with no attractors.
• Figure 4: Left: Schematic representation of a polycrystal subjected to a homogeneous velocity gradient loading. Right: Spatial distribution $(x_1, x_2) \mapsto \theta^0(x_1, x_2)$ of the initial lattice orientation, shown in a color scale ranging from $-30^\circ$ to $30^\circ$.
• Figure 5: Top: Computed Eulerian distribution $(x_1, x_2) \mapsto \theta(t, x_1, x_2)$ of the lattice orientation, shown in a color scale ranging from $-30^\circ$ to $30^\circ$. Middle: Expected Eulerian distribution $(x_1, x_2) \mapsto \theta^{\text{att}}(t, x_1, x_2)$ of the orientation attractor, in the same color scale $(-30^\circ, 30^\circ)$. Bottom: Eulerian distribution $(x_1, x_2) \mapsto \vert \theta(t, x_1, x_2) - \theta^{\text{att}}(x_1, x_2) \vert$ representing the difference between the lattice orientation and the estimated attractor, shown in a color scale from $0^\circ$ to $30^\circ$. The distributions are shown for four time points: $t = 0$ ($\epsilon^{\text{eng}} = 0$), $t = T/3$ ($\epsilon^{\text{eng}} = 0.166$), $t = 2T/3$ ($\epsilon^{\text{eng}} = 0.33$), and $t = T$ ($\epsilon^{\text{eng}} = 0.5$).