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. .. include:: /_includes/glass_transition_physics.rst Quick Reference --------------- .. list-table:: :widths: 25 15 60 :header-rows: 1 * - Model - Parameters - Use Case * - :doc:`stz_conventional` - 5-7 (G, :math:`\sigma_y`, :math:`\chi`, :math:`\tau_0`, :math:`\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 :math:`\chi` (chi) - **Rate-dependent**: Thermal activation + mechanical driving - **Disorder dynamics**: :math:`\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: .. math:: \dot{\gamma}^{pl} = \varepsilon_0 \Gamma(\sigma, \chi) (n_+ - n_-) where :math:`\Gamma` is the transition rate and :math:`\varepsilon_0` is strain per STZ flip. **Effective Temperature** :math:`\chi` **:** The configurational disorder is characterized by an effective temperature :math:`\chi` that: - **Increases** under plastic deformation (disorder created by rearrangements) - **Decreases** during aging (structural relaxation toward equilibrium) - **Governs STZ density**: More STZs at higher :math:`\chi` (more disordered states) The evolution of :math:`\chi` is governed by: .. math:: \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**: :math:`\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 -------------- .. list-table:: :widths: 15 10 15 60 :header-rows: 1 * - Parameter - Symbol - Units - Physical Meaning * - Shear modulus - G0 - Pa - Elastic stiffness * - Yield stress - :math:`\sigma_y` - Pa - Threshold for plastic flow * - Effective temp. - :math:`\chi` - — - Configurational disorder (0 = ordered) * - Attempt time - tau0 - s - Microscopic attempt frequency * - STZ strain - epsilon0 - — - Strain per STZ flip (~0.1-1) * - Steady-state :math:`\chi` - :math:`\chi_{ss}` - — - Disorder level under flow Model Predictions ----------------- **Flow Curve:** The STZ model predicts rate-dependent yield stress behavior: - **Low rates**: Yield stress :math:`\sigma_y` (athermal limit) - **Intermediate rates**: Logarithmic strengthening :math:`\sigma \sim \sigma_y + A \cdot \ln(\dot{\gamma})` - **High rates**: Power-law or saturation **Transient Response:** - **Stress overshoot**: Peak stress during startup (:math:`\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 :math:`\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:** .. code-block:: python 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:** .. code-block:: python # 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:** .. code-block:: python # 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 ------------------- .. toctree:: :maxdepth: 1 stz_conventional See Also -------- - :doc:`/models/epm/index` — EPM: mesoscopic plasticity on lattice - :doc:`/models/sgr/index` — SGR: trap model for soft glasses - :doc:`/models/hl/index` — Hébraud-Lequeux: mean-field plasticity - :doc:`/models/flow/herschel_bulkley` — Phenomenological yield stress - :doc:`/models/dmt/index` — Thixotropic structural kinetics 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 `_ :download:`PDF <../../../reference/falk_langer_1998_stz.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 `_ :download:`PDF <../../../reference/langer_pechenik_2003_stz_minimal.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 `_ :download:`PDF <../../../reference/langer_2008_stz_dynamics.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 `_