Hébraud-Lequeux (HL) Models =========================== This section documents the Hébraud-Lequeux model for soft glassy materials—a mean-field kinetic theory for yield stress fluids with noise-activated plasticity. .. include:: /_includes/glass_transition_physics.rst Quick Reference --------------- .. list-table:: :widths: 25 15 60 :header-rows: 1 * - Model - Parameters - Use Case * - :doc:`hebraud_lequeux` - 3 (:math:`\alpha`, :math:`\sigma_c`, :math:`\tau`) - Mean-field plasticity, noise-activated flow, soft glasses Overview -------- The **Hébraud-Lequeux (HL) model** is a mesoscopic constitutive theory for soft glassy materials that captures the interplay between elastic loading, plastic yielding, and noise-activated structural relaxation. Originally developed to explain the rheology of concentrated emulsions, it provides a physically-motivated framework for yield stress fluids. **Key physics:** - **Mean-field approach**: Material represented as ensemble of mesoscopic elements - **Elastic loading**: Elements store stress until yield threshold - **Plastic yielding**: Stress released when local stress exceeds :math:`\sigma_c` - **Noise activation**: Plastic events occur with rate proportional to noise amplitude - **Mechanical noise**: Yielding events generate noise that activates neighbors **Connection to other models:** - **SGR**: HL can be viewed as a mean-field limit of SGR dynamics - **EPM**: HL lacks spatial resolution but captures similar physics - **Fluidity models**: HL's noise parameter relates to fluidity evolution The HL model bridges the gap between phenomenological yield stress models (Bingham, Herschel-Bulkley) and microscopic theories (mode-coupling), providing mechanistic insight while remaining computationally tractable. Physical Framework ------------------ **Mesoscopic Elements:** The material is coarse-grained into identical mesoscopic elements, each characterized by local stress :math:`\sigma_{el}`. Elements: 1. **Load elastically**: :math:`d\sigma_{el}/dt = G \cdot \dot{\gamma}` under macroscopic shear 2. **Yield plastically**: Reset to :math:`\sigma_{el} = 0` when :math:`|\sigma_{el}| > \sigma_c` 3. **Relax via noise**: Activated hopping with rate :math:`\sim \exp(-U/D)` where :math:`D` is noise **Stress Distribution:** The probability distribution :math:`P(\sigma_{el}, t)` of local stresses evolves according to a Fokker-Planck equation with: - Convective flux from elastic loading - Diffusive spreading from mechanical noise - Boundary conditions from plastic yielding **Macroscopic Stress:** .. math:: \sigma = \int_{-\sigma_c}^{\sigma_c} \sigma_{el} \, P(\sigma_{el}, t) \, d\sigma_{el} Key Parameters -------------- .. list-table:: :widths: 15 10 15 60 :header-rows: 1 * - Parameter - Symbol - Units - Physical Meaning * - Noise coupling - :math:`\alpha` - — - Rate of plastic events generating noise * - Yield threshold - :math:`\sigma_c` - Pa - Local stress for plastic yielding * - Relaxation time - :math:`\tau` - s - Microscopic relaxation timescale Model Predictions ----------------- **Flow Curve:** The HL model predicts a yield stress with continuous transition: .. math:: \sigma(\dot{\gamma}) = \sigma_y + \eta_{eff}\dot{\gamma}^n where :math:`\sigma_y` depends on :math:`\alpha` and :math:`\sigma_c`. **Oscillatory Response:** - **Low frequency**: :math:`G'` plateau, :math:`G''` peak near yield - **High frequency**: Classical Maxwell-like behavior - **Strain amplitude**: Smooth transition from linear to nonlinear **Transient Response:** - **Startup flow**: Stress overshoot for high shear rates - **Creep**: Delayed yielding with characteristic waiting time - **Relaxation**: Non-exponential decay with stretched dynamics Quick Start ----------- **Hébraud-Lequeux model:** .. code-block:: python from rheojax.models import HebraudLequeux import numpy as np # Create model model = HebraudLequeux() # Set parameters model.parameters.set_value('alpha', 0.3) # Noise coupling (< 0.5 = glass) model.parameters.set_value('sigma_c', 50.0) # Pa model.parameters.set_value('tau', 1.0) # s # Fit to flow curve gamma_dot = np.logspace(-2, 1, 30) model.fit(gamma_dot, stress_data, test_mode='steady_shear') # Extract yield stress sigma_y = model.get_yield_stress() print(f"Yield stress: {sigma_y:.1f} Pa") **Bayesian inference:** .. code-block:: python # Bayesian with NLSQ warm-start result = model.fit_bayesian( gamma_dot, stress_data, test_mode='steady_shear', num_warmup=1000, num_samples=2000, num_chains=4, seed=42 ) # Parameter uncertainties intervals = model.get_credible_intervals(result.posterior_samples) print(f"σ_c: [{intervals['sigma_c'][0]:.1f}, {intervals['sigma_c'][1]:.1f}] Pa") Model Documentation ------------------- .. toctree:: :maxdepth: 1 hebraud_lequeux See Also -------- - :doc:`/models/sgr/index` — SGR: trap model approach (HL as mean-field limit) - :doc:`/models/epm/index` — EPM: spatially-resolved plasticity - :doc:`/models/fluidity/index` — Fluidity-based yield stress models - :doc:`/models/flow/herschel_bulkley` — Phenomenological yield stress model - :doc:`/models/stz/index` — STZ: shear transformation zones References ---------- 1. Hébraud, P. & Lequeux, F. (1998). "Mode-coupling theory for the pasty rheology of soft glassy materials." *Phys. Rev. Lett.*, 81, 2934–2937. https://doi.org/10.1103/PhysRevLett.81.2934 2. Hébraud, P., Lequeux, F., Munch, J.P., & Pine, D.J. (1997). "Yielding and rearrangements in disordered emulsions." *Phys. Rev. Lett.*, 78, 4657–4660. https://doi.org/10.1103/PhysRevLett.78.4657 3. Picard, G., Ajdari, A., Lequeux, F., & Bocquet, L. (2005). "Slow flows of yield stress fluids: Complex spatiotemporal behavior within a simple elastoplastic model." *Phys. Rev. E*, 71, 010501. https://doi.org/10.1103/PhysRevE.71.010501 4. Derec, C., Ajdari, A., & Lequeux, F. (2001). "Rheology and aging: A simple approach." *Eur. Phys. J. E*, 4, 355–361. https://doi.org/10.1007/s101890170118 5. Coussot, P., Nguyen, Q.D., Huynh, H.T., & Bonn, D. (2002). "Avalanche behavior in yield stress fluids." *Phys. Rev. Lett.*, 88, 175501. https://doi.org/10.1103/PhysRevLett.88.175501