Elasto-Plastic Models (EPM)

This section documents the Elasto-Plastic Model (EPM) family for modeling spatially-resolved plasticity in amorphous solids.

Quick Reference

Model

Import

Use Case

LatticeEPM

from rheojax.models import LatticeEPM

Scalar stress (\(\sigma_{xy}\)), fast avalanche dynamics

TensorialEPM

from rheojax.models import TensorialEPM

Full tensor, normal stresses (\(N_1\), \(N_2\)), anisotropic yielding

Overview

Elasto-Plastic Models (EPMs) provide a mesoscopic lattice-based framework for modeling the rheology of amorphous solids—glasses, gels, pastes, and dense suspensions. Unlike mean-field approaches (SGR, Hébraud-Lequeux), EPMs explicitly resolve:

  • Spatial heterogeneity via a discrete lattice of mesoscopic blocks

  • Plastic avalanches from stress redistribution cascades

  • Non-local stress redistribution via the Eshelby propagator (quadrupolar symmetry)

  • Shear banding from localized yielding

The implementation leverages JAX for FFT-accelerated simulations on GPU/TPU, achieving O(L^2 log L) complexity for stress redistribution instead of O(L^4) direct summation.

Documentation Highlights:

  • Boxed equations for all key mathematical relations

  • Protocol-specific governing equations with boundary conditions

  • Complete JAX implementation utilities for custom simulations

  • Physical foundations connecting EPM to SGR, fluidity, and STZ models

Model Hierarchy

EPM Family
│
├── LatticeEPM (Scalar)
│   └── Tracks σ_xy only
│   └── O(L^2 log L) FFT acceleration
│   └── 6 parameters: μ, σ_c_mean, σ_c_std, τ_pl, L, dt
│
└── TensorialEPM (Full Tensor)
    │
    ├── von Mises (isotropic)
    │   └── Tracks [σ_xx, σ_yy, σ_xy] + σ_zz
    │   └── N_1, N_2 predictions
    │   └── 9 parameters
    │
    └── Hill (anisotropic)
        └── Directional yield resistance
        └── Fiber suspensions, liquid crystals
        └── Additional: hill_H, hill_N

When to Use Which Model

Feature / Use Case

LatticeEPM (Scalar)

TensorialEPM

Flow curve fitting

✓ Fast (3-5x faster)

✓ Use if \(N_1\) data available

Yield stress determination

✓ Sufficient

✓ More accurate for anisotropic

Stress overshoot

✓ Qualitative

✓ Quantitative

Normal stress differences

✗ Cannot capture

✓ \(N_1\), \(N_2\) predictions

Shear banding analysis

~ Qualitative (\(\sigma_{xy}\) gradients)

✓ Quantitative (\(N_1\) gradients)

Anisotropic materials

✗

✓ Hill criterion

Rod climbing / die swell

✗

✓ Required

Computational cost

1× (baseline)

3-5× slower

Memory usage

1×

3× (tensor storage)

Decision Guide:

  • Start with LatticeEPM for exploratory analysis and flow curve fitting

  • Use TensorialEPM when normal stress data is available, material is anisotropic, or analyzing flow instabilities (edge fracture, shear banding)

Supported Protocols

Both EPM variants support standard rheological protocols. See Elasto-Plastic Models (EPM) for complete mathematical details with boxed governing equations.

Protocol

Description

Key Observable

Math Details

flow_curve

Constant shear rate, steady state

\(\sigma(\dot{\gamma})\), yield stress \(\sigma_y\)

Flow Curve (Rotation / Steady Shear)

startup

Step shear rate from rest

Stress overshoot, peak strain

Start-up of Steady Shear

relaxation

Step strain, stress decay

G(t), relaxation spectrum

Stress Relaxation (Step Strain)

creep

Constant stress (PID controlled)

\(\gamma(t)\), viscosity bifurcation

Creep (Step Stress)

oscillation

Sinusoidal shear (SAOS/LAOS)

\(G'\), \(G''\), Lissajous figures

Oscillation (SAOS and LAOS)

Tip

The Elasto-Plastic Models (EPM) documentation now includes a complete JAX Implementation Utilities section with production-ready code for EPM simulations, including time-stepping kernels, creep controllers, and avalanche relaxation.

Physical Context

EPMs operate at the mesoscopic length scale \(\xi\) (correlation length of plastic events, typically 10-100 particle diameters). At this scale:

  • Material is coarse-grained into discrete blocks with local stress \(\sigma_{ij}\)

  • Plastic yielding is localized and stochastic (quenched disorder)

  • Stress redistribution follows long-range Eshelby coupling (quadrupolar, \(\sim 1/r^2\))

  • Avalanches emerge from cascading plastic events

Ideal materials:

  • Metallic glasses

  • Dense colloidal gels

  • Pastes and foams

  • Granular suspensions

  • Emulsions near jamming

Athermal limit: EPMs assume yielding is purely stress-driven (\(T \to 0\)). For thermal activation, consider SGR models instead.

Key Parameters

Parameter

Symbol

Typical Range

Physical Meaning

Shear modulus

\(\mu\)

10-10,000 Pa

Elastic stiffness of matrix

Mean yield stress

\(\sigma_{c,\text{mean}}\)

0.5-2× \(\sigma_y\)

Local threshold for plastic events

Disorder strength

\(\sigma_{c,\text{std}}\)

0.1-0.5× \(\sigma_{c,\text{mean}}\)

Heterogeneity; controls avalanche statistics

Plastic time

\(\tau_{pl}\)

0.01-10 s

Relaxation after yielding

Lattice size

L

8-128

Fitting: 8-16, Production: 32-128

Time step

dt

0.001-0.05

Must resolve \(\tau_{pl}\) (dt < \(\tau_{pl}\)/10)

TensorialEPM additional parameters:

Parameter

Symbol

Typical Range

Physical Meaning

Poisson ratio

\(\nu\)

0.40-0.49

Plane strain coupling; \(N_1\) magnitude

Normal relax. time

\(\tau_{pl,normal}\)

0.1-10× \(\tau_{pl,shear}\)

Can differ for anisotropic materials

\(N_1\) weight

\(w_{N_1}\)

0.1-10

Prioritize \(N_1\) in combined fitting

Hill H

H

0.5-2.0

Normal stress anisotropy

Hill N

N

1.5-5.0

Shear amplification (\(H=1, N=3\) recovers von Mises)

Quick Start

LatticeEPM (fast fitting):

from rheojax.models import LatticeEPM
import numpy as np

# Small lattice for parameter estimation
model = LatticeEPM(L=16, dt=0.01)

# Fit to flow curve
gamma_dot = np.logspace(-2, 1, 20)
stress = np.array([...])  # experimental data
model.fit(gamma_dot, stress, test_mode='flow_curve')

# Bayesian inference with 4 chains
result = model.fit_bayesian(
    gamma_dot, stress,
    test_mode='flow_curve',
    num_warmup=500,
    num_samples=1000,
    num_chains=4,
    seed=42
)

TensorialEPM (normal stresses):

from rheojax.models import TensorialEPM
import numpy as np

# Initialize with normal stress weight
model = TensorialEPM(L=32, dt=0.01, w_N1=2.0)

# Fit to shear data
model.fit(gamma_dot, stress_exp, test_mode='flow_curve')

# Get N₁ predictions
result = model.predict(gamma_dot, test_mode='flow_curve')
N1 = result.metadata["N1"]

Model Documentation

References

  1. Picard, G., Ajdari, A., Lequeux, F., and Bocquet, L. (2004). “Elastic consequences of a single plastic event: A step towards the microscopic modeling of the flow of yield stress fluids.” European Physical Journal E, 15, 371-381. https://doi.org/10.1140/epje/i2004-10054-8

  2. Nicolas, A., Ferrero, E. E., Martens, K., and Barrat, J.-L. (2018). “Deformation and flow of amorphous solids: Insights from elastoplastic models.” Reviews of Modern Physics, 90, 045006. https://doi.org/10.1103/RevModPhys.90.045006

  3. Eshelby, J. D. (1957). “The determination of the elastic field of an ellipsoidal inclusion, and related problems.” Proceedings of the Royal Society A, 241, 376-396. https://doi.org/10.1098/rspa.1957.0133

  4. Lin, J., Lerner, E., Rosso, A., and Wyart, M. (2014). “Scaling description of the yielding transition in soft amorphous solids at zero temperature.” PNAS, 111, 14382-14387. https://doi.org/10.1073/pnas.1406391111

  5. Budrikis, Z., Castellanos, D. F., Sandfeld, S., Zaiser, M., and Zapperi, S. (2017). “Universal features of amorphous plasticity.” Nature Communications, 8, 15928. https://doi.org/10.1038/ncomms15928

See Also