Shear Transformation Zone (STZ) Models

This section documents the Shear Transformation Zone (STZ) theory for amorphous solids—a microscopic framework for plasticity based on localized structural rearrangements.

Glass Transition Physics

Common Physical Framework

Models in this category describe materials near or below the glass transition—where thermal fluctuations become insufficient for structural relaxation on experimental timescales. These materials exhibit:

Characteristic Signatures:

  • Cage effect: Particles trapped by neighbors, requiring cooperative rearrangements

  • Aging: Properties evolve with waiting time (time since preparation)

  • Yield stress: Finite stress required for macroscopic flow

  • Power-law rheology: \(G'(\omega) \sim G''(\omega) \sim \omega^n\) with weak frequency dependence

  • Structural relaxation: \(\alpha\)-relaxation timescale diverges at glass transition

Key Control Parameters:

Model

Parameter

Physical meaning

SGR

\(x\) (noise temperature)

Ratio of activation energy to trap depth

ITT-MCT

\(\varepsilon\) (separation parameter)

Distance from ideal glass transition

STZ

\(\chi\) (effective temperature)

Configurational disorder

EPM

\(\sigma/\sigma_y\) (stress ratio)

Proximity to yield

Glass Transition Regimes:

  • Liquid regime (above \(T_g\) or critical point): Equilibrium relaxation, aging absent

  • Glass regime (below \(T_g\)): Frozen structure, aging, yield stress emerges

  • Critical point: Power-law divergences, scale-free avalanches

Related Concepts:

Quick Reference

Model

Parameters

Use Case

Shear Transformation Zone (STZ)

5-7 (G, \(\sigma_y\), \(\chi\), \(\tau_0\), \(\varepsilon_0\), …)

Amorphous solids, metallic glasses, granular materials

Overview

The Shear Transformation Zone (STZ) theory, developed by Falk and Langer, provides a microscopic statistical mechanics framework for plasticity in amorphous materials. Unlike crystalline solids where plasticity occurs via dislocations, amorphous materials deform through localized clusters of atoms—Shear Transformation Zones—that rearrange cooperatively under stress.

Key physics:

  • Localized rearrangements: Plasticity occurs in discrete STZ regions (~10-100 atoms)

  • Two-state model: STZs exist in (+) and (-) orientations relative to shear

  • Effective temperature: Configurational disorder tracked by \(\chi\) (chi)

  • Rate-dependent: Thermal activation + mechanical driving

  • Disorder dynamics: \(\chi\) evolves with plastic strain and aging

Materials described by STZ:

  • Metallic glasses (bulk metallic glasses, thin films)

  • Polymer glasses (PMMA, PS below Tg)

  • Colloidal glasses

  • Granular materials (athermal limit)

  • Amorphous silicon, silica glasses

Physical Framework

Two-State STZ Model:

STZs are modeled as bistable units that can flip between (+) and (-) orientations:

(+) state                     (-) state
   ●●●         ←→                ●●●
  ● ● ●     shear flip          ● ● ●
   ●●●                           ●●●

Favors γ > 0              Favors γ < 0

The net plastic strain rate depends on the population imbalance:

\[\dot{\gamma}^{pl} = \varepsilon_0 \Gamma(\sigma, \chi) (n_+ - n_-)\]

where \(\Gamma\) is the transition rate and \(\varepsilon_0\) is strain per STZ flip.

Effective Temperature \(\chi\) :

The configurational disorder is characterized by an effective temperature \(\chi\) that:

  • Increases under plastic deformation (disorder created by rearrangements)

  • Decreases during aging (structural relaxation toward equilibrium)

  • Governs STZ density: More STZs at higher \(\chi\) (more disordered states)

The evolution of \(\chi\) is governed by:

\[\dot{\chi} = \frac{\chi_{ss}(\dot{\gamma}) - \chi}{\tau_\chi} + \frac{\text{energy from STZ flips}}{\text{specific heat}}\]

Steady-State Flow:

At steady state, the STZ model predicts:

  • Yield stress: \(\sigma_y\) emerges from the competition between creation and annihilation of STZs

  • Rate dependence: Logarithmic or power-law depending on regime

  • Temperature sensitivity: Arrhenius activation for thermal STZ flips

Key Parameters

Parameter

Symbol

Units

Physical Meaning

Shear modulus

G0

Pa

Elastic stiffness

Yield stress

\(\sigma_y\)

Pa

Threshold for plastic flow

Effective temp.

\(\chi\)

—

Configurational disorder (0 = ordered)

Attempt time

tau0

s

Microscopic attempt frequency

STZ strain

epsilon0

—

Strain per STZ flip (~0.1-1)

Steady-state \(\chi\)

\(\chi_{ss}\)

—

Disorder level under flow

Model Predictions

Flow Curve:

The STZ model predicts rate-dependent yield stress behavior:

  • Low rates: Yield stress \(\sigma_y\) (athermal limit)

  • Intermediate rates: Logarithmic strengthening \(\sigma \sim \sigma_y + A \cdot \ln(\dot{\gamma})\)

  • High rates: Power-law or saturation

Transient Response:

  • Stress overshoot: Peak stress during startup (\(\chi\) evolution)

  • Strain softening: Post-yield stress reduction as disorder increases

  • Strain hardening: At very high strains, disorder saturates

Shear Banding:

The STZ model naturally predicts shear band formation when:

  • Strain softening is strong (large \(\chi\) increase per strain)

  • Thermal diffusion is weak compared to mechanical driving

  • Material has positive feedback between disorder and flow rate

Quick Start

STZ Conventional model:

from rheojax.models import STZConventional
import numpy as np

# Create model
model = STZConventional()

# Set parameters for a metallic glass
model.parameters.set_value('G0', 40e9)        # Pa (metallic glass)
model.parameters.set_value('sigma_y', 1e9)   # Pa
model.parameters.set_value('chi_inf', 0.1)   # Steady-state disorder
model.parameters.set_value('tau0', 1e-12)    # s (atomic timescale)

# Fit to flow curve
gamma_dot = np.logspace(-4, 2, 50)
model.fit(gamma_dot, stress_data, test_mode='flow_curve')

Startup flow prediction:

# Fit startup data with stress overshoot
t = np.linspace(0, 10, 1000)
model.fit(t, stress_data, test_mode='startup', gamma_dot=1.0)

# Predict startup stress
stress = model.predict(t)

# Find stress overshoot
stress_peak = np.max(stress)
strain_peak = t[np.argmax(stress)] * 1.0

Bayesian inference:

# Bayesian with NLSQ warm-start
result = model.fit_bayesian(
    gamma_dot, stress_data,
    test_mode='flow_curve',
    num_warmup=1000,
    num_samples=2000,
    num_chains=4,
    seed=42
)

# Parameter correlations
import arviz as az
az.plot_pair(result.inference_data, var_names=['sigma_y', 'chi_inf'])

Model Documentation

See Also

References

  1. Falk, M. L. & Langer, J. S. (1998). “Dynamics of viscoplastic deformation in amorphous solids.” Phys. Rev. E, 57, 7192–7205. DOI: 10.1103/PhysRevE.57.7192 PDF

  2. Langer, J. S. & Pechenik, L. (2003). “Dynamics of shear-transformation zones in amorphous plasticity: Energetic constraints in a minimal theory.” Phys. Rev. E, 68, 061507. DOI: 10.1103/PhysRevE.68.061507 PDF

  3. Langer, J. S. (2008). “Shear-transformation-zone theory of plastic deformation near the glass transition.” Phys. Rev. E, 77, 021502. DOI: 10.1103/PhysRevE.77.021502 PDF

  4. Falk, M. L. & Langer, J. S. (2011). “Deformation and failure of amorphous, solidlike materials.” Annu. Rev. Condens. Matter Phys., 2, 353–373. DOI: 10.1146/annurev-conmatphys-062910-140452

  5. Manning, M. L., Langer, J. S., & Carlson, J. M. (2007). “Strain localization in a shear transformation zone model for amorphous solids.” Phys. Rev. E, 76, 056106. DOI: 10.1103/PhysRevE.76.056106

  6. Shi, Y. & Falk, M. L. (2005). “Strain localization and percolation of stable structure in amorphous solids.” Phys. Rev. Lett., 95, 095502. DOI: 10.1103/PhysRevLett.95.095502

  7. Johnson, W. L. & Samwer, K. (2005). “A universal criterion for plastic yielding of metallic glasses with a (T/Tg)^(2/3) temperature dependence.” Phys. Rev. Lett., 95, 195501. DOI: 10.1103/PhysRevLett.95.195501

  8. Schuh, C. A., Hufnagel, T. C., & Ramamurty, U. (2007). “Mechanical behavior of amorphous alloys.” Acta Mater., 55, 4067–4109. DOI: 10.1016/j.actamat.2007.01.052