Coarse-graining amorphous plasticity: impact of rejuvenation and disorder

Technical Summary: Coarse-graining Amorphous Plasticity: Impact of Rejuvenation and Disorder

Problem Statement
The mechanical behavior of amorphous solids, characterized by phenomena such as aging, rejuvenation, shear banding, and the Bauschinger effect, arises from localized structural rearrangements (Shear Transformations) interacting through an elastic medium. While atomistic simulations capture these features, they are computationally expensive. Conversely, mesoscopic Elasto-Plastic Models (EPMs) offer a computationally efficient framework but often rely on oversimplified assumptions that may fail to quantitatively reproduce atomistic results. The central problem addressed is the "coarse-graining" challenge: determining how to transfer parameters from atomistic scales to mesoscopic lattice models to reproduce the physics of amorphous plasticity, specifically focusing on the role of the underlying disordered potential energy landscape (PEL), the evolution of disorder upon loading, and the rejuvenation of the glassy state.

Methodology
The study employs a comparative approach between Athermal Quasistatic (AQS) atomistic simulations and a simplified scalar mesoscopic EPM.

1. Atomistic Simulations:

   • Three distinct 2D binary Lennard-Jones glass samples were prepared with contrasting thermal histories:
     • MCT: Instantly quenched from a temperature near the Mode Coupling Transition ($T_p \approx 0.38$).
     • ESL: Instantly quenched from a higher temperature ($T_p = 0.351$), representing a poorly annealed (soft) glass.
     • GQ: Gradually quenched from $T_p = 0.351$, representing a well-annealed (hard) glass.
   • Samples were subjected to AQS shear loading up to a global strain of $\gamma = 5$.
   • Local properties were characterized using the "frozen matrix method," which isolates circular regions to measure local yield stresses ($\sigma_c$), internal stress fluctuations ($\delta\sigma$), and stress drops ($\Delta\sigma$) across 18 shearing directions.

2. Mesoscopic Model:

   • A scalar lattice model was implemented on a square grid with bi-periodic boundary conditions.
   • Dynamics: The model features local threshold dynamics where a cell yields if local stress exceeds a threshold $\sigma_c$. Plastic events trigger long-range elastic stress redistribution (Eshelby kernel) and a renewal of the local threshold.
   • Key Parameters: The study focuses on two dimensionless parameters:
     • Disorder Strength ($s$): The ratio of yield stress fluctuations to the stress drop induced by a plastic event.
     • Age/Rejuvenation Parameter ($r$): Defined as $r = (\sigma_{c}^{initial} - \sigma_{c}^{renewed}) / \sigma_{c}^{renewed}$. It quantifies the difference between the initial glass state and the state of the material after a plastic rearrangement.
   • Up-scaling Strategy: Parameters for the mesoscopic model (distributions of yield stresses, slip increments, and renewal distributions) were directly extracted from the atomistic data. The study tested two specific hypotheses:
     • Disordered Scenario: The renewal distribution of yield stresses is identical to the initial distribution ($r=0$).
     • Rejuvenation Scenario: The renewal distribution corresponds to the MCT glass (a "fresher" state), implying $r > 0$.
     • Correlation Hypothesis: A direct correlation was introduced between the local yield stress and the magnitude of the subsequent slip increment ($\varepsilon_0 \propto \sigma_c^2$), reflecting the hierarchical structure of the PEL.

Key Contributions and Results

1. Stress-Strain Behavior and Transients:

   • Simple models without correlation between yield stress and slip increments failed to reproduce the transient stress peak observed in atomistic simulations, producing only monotonic curves.
   • Introducing a correlation between the local yield stress and the slip increment amplitude allowed the mesoscopic model to reproduce the full stress-strain curve, including the transient peak and the steady-state flow stress, for both soft (ESL) and hard (GQ) glasses.
   • The steady-state flow stress was found to be controlled primarily by the disorder strength parameter $s$ (linked to the mean slip amplitude), rather than the specific details of the rejuvenation scenario.

2. Rejuvenation and Local Yield Stress Evolution:

   • Soft Glass (ESL): The atomistic simulations showed a slight hardening (increase in mean local yield stress) followed by a plateau. The mesoscopic model with rejuvenation ($r \approx 0.14$) achieved quasi-quantitative agreement with this evolution. Without rejuvenation, the model overestimated the yield stress.
   • Hard Glass (GQ): Atomistic simulations showed an initial hardening followed by a softening regime associated with shear banding. The mesoscopic model without rejuvenation could not reproduce the softening phase, showing only continuous hardening. The inclusion of the rejuvenation scenario ($r \approx 0.65$) was crucial to qualitatively recover the decrease in mean yield stress, although quantitative agreement remained imperfect.

3. Localization and Shear Banding:

   • For the hard glass (GQ), the atomistic simulations exhibited clear shear banding.
   • The mesoscopic model with rejuvenation and correlation successfully reproduced the localization index evolution: a rapid increase during band nucleation followed by a gradual decrease as the band broadened.
   • Models without rejuvenation showed significant localization but underestimated the intensity and persistence of the shear band.

4. Strain and Stress Fluctuations:

   • While the mesoscopic model reproduced the trend of strain variance evolution, it systematically overestimated the magnitude of fluctuations by a factor of ~1.5 compared to atomistic results.
   • The internal stress field fluctuations were overestimated even more significantly (by a factor of 2–3). The authors attribute this discrepancy to the scalar nature of the model (ignoring orientational disorder of slip directions) and the neglect of elastic heterogeneity and non-affine effects in the mesoscopic formulation.

Significance and Claims
The paper claims that a coarse-grained scalar model can reproduce the salient features of amorphous plasticity, provided that specific structural information from the atomistic scale is incorporated. The primary significance lies in identifying two critical ingredients for quantitative agreement:

1. Correlation: The necessity of correlating local stress drops with local yield stresses, which reflects the hierarchical nature of the Potential Energy Landscape (deeper wells correspond to larger slips).
2. Rejuvenation: The validation of the rejuvenation scenario, where plastic rearrangements transform the local glassy structure into a state resembling a glass quenched near the Mode Coupling Transition. This mechanism is essential for reproducing the softening behavior and shear banding observed in well-annealed glasses.

The authors modestly acknowledge that the model is "too simple and caricatural" for complete quantitative agreement, particularly regarding fluctuation amplitudes. They conclude that while the scalar model captures the essential physics of flow stress and localization, future work must address elastic heterogeneity, tensorial disorder, and memory effects to fully resolve the discrepancies in fluctuation statistics.