Plastic deformation as a phase transition: a combinatorial model of plastic flow in copper single crystals

TL;DR

This work reframes plastic deformation in copper single crystals as a phase-transition problem by developing a fully discrete mesoscale model based on combinatorial mean-field theory and discrete exterior calculus. It rewrites a continuum slip model into a cell-complex formulation with energy $\mathcal{H}$ governed by parameters $\alpha$ and $\lambda$, and uses a Metropolis-Hastings sampler to simulate a network of microslips on a 3D tessellation. The simulations reveal a spectrum of deformation phases and both first-order (stress-driven) and second-order (mean-field-driven) transitions, linking microscopic slip configurations to macroscopic localization phenomena. While offering insights beyond continuum theories, the approach also notes limitations such as the absence of strain hardening and full thermodynamics, suggesting avenues for future refinement.

Abstract

Continuum models of plasticity fail to capture the richness of microstructural evolution because the continuum is a homogeneous construction. The present study shows that an alternative way is available at the mesoscale in the form of truly discrete constructions and in the discrete exterior calculus. A pre-existing continuum mean-field model with two parameters is rewritten in the language of the latter to model the properties of a network of plastic slip events in a perfect copper single crystal under uniaxial tension. The behaviour of the system is simulated in a triangular 2D mesh in 3D space employing a Metropolis-Hastings algorithm. Phases of distinct character emerge and both first-order and second-order phase transitions are observed. The phases represent arrangements of the plastic slip network with different combinations of collinear, coplanar, non-collinear and non-coplanar active slip systems. Furthermore, some of these phases can be interpreted as representing crystallographic phenomena like activation of secondary slip systems, strain localisation and fracture or amorphisation. The first-order transitions mostly occur as functions of the applied stress, while the second-order transitions occur exclusively as functions of the mean-field coupling parameter. The former are reminiscent of transitions in other statistical-mechanical models, while the latter find parallels in experimental observations.

Figures (9)

• Figure 1: (a) The computational unit cell for a face-centred cubic (FCC) structure; (b) The $(111)$ slip planes in the computational unit cell, each composed of four 2-cells; (c) A $2 \times 2 \times 2$ construction for the FCC structure, with all slip 2-cells highlighted.
• Figure 2: Values of the self-energy $\alpha$ for different slip systems as the stress tensor of uniaxial tension in the $z$ direction is rotated about the $x$ axis, as shown on the right. The values were obtained from Eq. \ref{['eq: alpha']}. The values are scaled by the magnitude of the applied stress at yield $\sigma_y$ and the cubic root of the volume of the computational UC. The slip systems are labelled using Schmid's and Boas's notation, shown in Table \ref{['table: Schmid&Boas']}.
• Figure 3: One hundred single runs of the Metropolis-Hastings algorithm at different magnitudes of the externally applied stress. (a)--(c) Fraction of microslip events. (d)--(f) Energy surplus, i.e. the cumulative $-\mathcal{H}$\ref{['eq: discrete_lagrangian']}. (g)--(i) Norm of the mean-field vector $\vec{p}$ as in Eq. \ref{['eq: macrodefect_cochain']} at each step of the algorithm. The self-energy parameter $\alpha$ was fixed at 465.15 GPa and the mean-field coupling strength was set to $\lambda =4\: \alpha$.
• Figure 4: The material simulated in Fig. \ref{['fig: individual_simulations']} could have different deformation states which are characterised in part by the mean-field vector $\vec{p}$, as noted in the text after Eq. \ref{['eq: macrodefect_cochain']}. From left to right: $\vec{p}=(0,0,0)$, $\vec{p} \propto (0,0,1)$, $\vec{p} \propto (1,1,0)$, $\vec{p} \propto (1,1,0)$, and $\vec{p} \propto (1,2,1)$. Diagrams are not up to scale.
• Figure 5: (a)--(c) the fraction of microslip events, (d)--(f) the energy surplus and (g)--(i) the mean-field norm of the equilibrium states reached by the simulated system. Each data point seen here is the last data point taken from graphs such as Fig. \ref{['fig: individual_simulations']}. All simulations were run for 1 million steps. The self-energy parameter $\alpha$ was set to 465.15 GPa and the mean-field coupling parameter $\lambda$ was varied according to the captions on each individual plot. The text in each plot refers to the phases identified in Table \ref{['table: phases']} and indicate which phases were observed in the nearby collection of data points. See the main text for more information.