Finite compressibility and strain hardening in elasto-plastic models of amorphous matter

1. Problem Statement

The plastic yielding and ductility of amorphous solids (e.g., metallic glasses, colloidal glasses, emulsions) are governed by local shear transformations. A critical, yet often overlooked, factor in theoretical models is finite compressibility, characterized by the ratio of the bulk modulus ($K$) to the shear modulus ($\mu$).

• The Gap: While the Poisson ratio ($\nu$) or the ratio $K/\mu$ is empirically linked to the ductility of metallic glasses (high $K/\mu$ correlates with higher ductility), most recent mesoscopic elasto-plastic models treat materials as incompressible or ignore the quantitative dependence of stress redistribution on compressibility.
• The Question: How does varying the dimensionless compressibility ($K/\mu$) affect the macroscopic yielding behavior, specifically the transition from elastic to plastic regimes in cyclic shear and the steady-state flow stress in forward shear?

2. Methodology

The authors employ a mesoscopic elasto-plastic automaton model based on a space-filling tiling of identical potential shear transformation sites.

• Model Construction:
  • The system is a 2D square grid ($L \times L$) where each site has a piecewise-quadratic strain-energy function.
  • Local yielding occurs when the local elastic strain exceeds a threshold (set to unity). Upon yielding, the site undergoes a discrete plastic strain increment (eigenstrain).
  • Staggered Grid Discretization: To solve the linear elasticity equations, the authors use a staggered grid scheme (common in seismology) to ensure kinematic compatibility. This avoids pathological behaviors found in standard centered-difference schemes regarding off-diagonal stress components.
• Eshelby Inclusion Problem:
  • The model incorporates the Eshelby back stress ($\sigma_0$). When a site yields, it induces a stress field in the surrounding matrix.
  • The magnitude of this stress redistribution depends on the dimensionless parameter $\alpha = \frac{2K}{K+\mu}$.
  • $\alpha$ ranges from 0 (maximally compressible) to 2 (incompressible).
• Simulation Protocols:
  1. Cyclic Shear: The system is subjected to oscillatory shear with amplitude $\Gamma$. The study focuses on the terminal limit cycles.
  2. Forward Steady Shear: The system is sheared monotonically until it reaches a steady-state flow.

3. Key Contributions

1. Quantitative Link between Compressibility and Yielding: The paper establishes that the Eshelby back stress ($\sigma_0$) is the primary mechanism linking compressibility ($K/\mu$) to macroscopic yielding. As $K/\mu$ increases (material becomes less compressible), $\sigma_0$ becomes more negative.
2. Emergent Hardening without Ad-hoc Rules: The model demonstrates complex hardening behaviors (isotropic and kinematic) emerging purely from elastic interactions and the Eshelby back stress, without introducing microscopic hardening parameters or rules.
3. Unified Scaling Laws: The authors derive scaling relations for the critical cyclic amplitude ($\Gamma_0$) and the steady-state flow stress ($\sigma_y$) in terms of $\sigma_0$ and $\alpha$, showing how these macroscopic quantities depend on the microscopic elastic moduli.

4. Key Results

A. Cyclic Shear Behavior

The system exhibits three regimes as the strain amplitude $\Gamma$ increases:

1. Elastic Regime (ER): $\Gamma < \Gamma_0$. No plastic rearrangement; the system returns to its initial state.
2. Reversible Plastic Regime (RPR): $\Gamma_0 < \Gamma < \Gamma_y$. The system enters limit cycles with plastic rearrangements that are microscopically reversible (returning to previous configurations).
3. Diffusive Regime (DR): $\Gamma > \Gamma_y$. Irreversible, diffusive motion.

Findings on $\Gamma_0$ (Hysteresis Transition):

• Dependence on Compressibility: The transition amplitude $\Gamma_0$ increases as $K/\mu$ decreases (i.e., more compressible systems yield earlier). Conversely, less compressible (higher $K/\mu$) systems are effectively "harder," exhibiting a larger purely elastic regime.
• The Relation: $\Gamma_0$ is determined by the back stress:
  $$\Gamma_0 = \frac{1 + \sigma_0}{2}$$
  Since $\sigma_0$ becomes more negative as $K/\mu$ increases, $\Gamma_0$ increases.
• Plastic Onset: For $\Gamma > \Gamma_0$, plasticity initiates at a strain $\gamma^* = \Gamma_0 - (\Gamma - \Gamma_0)$. The plastic strain rate jumps sharply to $\approx 0.2$ at $\gamma^*$.
• Stress Distribution: The probability distribution of local stresses, $P(\sigma)$, develops a characteristic "shoulder" at $\sigma_0$. This feature dictates the onset of plasticity.

B. Forward Steady Shear Behavior

• Flow Stress ($\sigma_y$): The steady-state flow stress decreases as $K/\mu$ increases.
• Scaling: Similar to cyclic shear, the flow stress is governed by the back stress:
  $$\sigma_y = \sigma'_y + \sigma_0(1 - \sigma'_y)$$
  where $\sigma'_y \approx 0.54$ is a universal constant derived from the rescaled stress distribution.
• Data Collapse: By rescaling stress and strain using $\sigma_0$ and $\alpha$, the loading curves for different compressibilities collapse onto master curves, confirming the universality of the mechanism.

C. Emergent Strain Hardening

The model naturally reproduces two distinct hardening regimes observed in continuum plasticity:

1. Isotropic Hardening ($\Gamma < \Gamma_0$): The elastic range expands symmetrically as the system is cycled.
2. Kinematic Hardening ($\Gamma > \Gamma_0$): The elastic range translates in stress space without changing size. The transition between these two regimes occurs precisely at the critical amplitude $\Gamma_0$.

• Mechanism: This behavior arises solely from the geometry of the stress redistribution kernel (the Eshelby solution) and the threshold condition, without any explicit hardening laws.

5. Significance and Implications

• Reconciling Theory and Experiment: The results provide a quantitative explanation for why metallic glasses with high $K/\mu$ (low Poisson ratio) are more ductile. In the model, higher $K/\mu$ leads to a larger negative back stress, which suppresses yielding in cyclic loading (increasing $\Gamma_0$) and alters the flow stress in forward shear.
• Role of Compressibility: The study highlights that finite compressibility is not a minor correction but a fundamental driver of plasticity onset and hardening. It challenges mean-field theories that often treat stress redistribution amplitude as an ad-hoc parameter, showing instead that it is physically determined by elastic moduli.
• Broad Applicability: While the model is simple, the findings suggest that the dependence of yielding on $K/\mu$ should be observable in a wide range of amorphous systems, including metallic alloys, colloidal glasses, and emulsions, where $K/\mu$ can be tuned via composition, age, or processing history.
• Methodological Advance: The use of the staggered grid discretization provides a robust numerical framework for solving Eshelby problems in periodic lattices, correcting previous numerical artifacts.

In summary, the paper demonstrates that the Poisson ratio (via $K/\mu$) controls the amplitude of the Eshelby back stress, which in turn dictates the macroscopic yield limits, flow stress, and the nature of strain hardening in amorphous materials.