Fractional Viscoelastic Models ============================== This section documents the fractional calculus-based viscoelastic models that capture power-law relaxation and broad spectral behavior. Quick Reference --------------- .. list-table:: :widths: 30 15 55 :header-rows: 1 * - Model - Parameters - Use Case * - :doc:`fractional_maxwell_gel` - 3 - Gels with terminal flow, SpringPot + dashpot * - :doc:`fractional_maxwell_liquid` - 3 - Viscoelastic liquids, spring + SpringPot * - :doc:`fractional_maxwell_model` - 4 - Two-order generalized, hierarchical relaxation * - :doc:`fractional_kelvin_voigt` - 3 - Solids with bounded creep, spring :math:`\parallel` SpringPot * - :doc:`fractional_zener_ss` - 4 - Solid-Solid Zener, low-frequency plateau * - :doc:`fractional_zener_sl` - 4 - Solid-Liquid Zener, terminal flow * - :doc:`fractional_zener_ll` - 4 - Liquid-Liquid Zener, double flow * - :doc:`fractional_kv_zener` - 5 - Complex retardation, bounded creep + plateau * - :doc:`fractional_jeffreys` - 5 - Polymer solutions, two relaxation modes * - :doc:`fractional_burgers` - 6 - Primary creep, four-element * - :doc:`fractional_poynting_thomson` - 5 - Solid with multiple timescales Overview -------- Fractional viscoelastic models replace integer-order derivatives in classical constitutive equations with fractional-order derivatives, enabling: - **Power-law relaxation**: :math:`G(t) \sim t^{-\alpha}` for broad spectra - **Parsimonious fitting**: Fewer parameters than multi-mode models - **Physical insight**: Fractional order :math:`\alpha` relates to structural heterogeneity The **SpringPot** element is the fundamental building block, interpolating between ideal spring (:math:`\alpha` = 0) and dashpot (:math:`\alpha` = 1) behavior. .. include:: /_includes/fractional_seealso.rst Model Hierarchy --------------- :: Fractional Models │ ├── Maxwell Family (Series) │ ├── FractionalMaxwellGel (FMG) │ │ └── SpringPot ── Dashpot │ │ └── Gel with terminal flow │ │ │ ├── FractionalMaxwellLiquid (FML) │ │ └── Spring ── SpringPot │ │ └── Viscoelastic liquid │ │ │ └── FractionalMaxwellModel (Two-Order) │ └── SpringPot(α) ── SpringPot(β) │ └── Hierarchical relaxation │ ├── Kelvin-Voigt Family (Parallel) │ ├── FractionalKelvinVoigt │ │ └── Spring \parallel SpringPot │ │ └── Solid with bounded creep │ │ │ └── FractionalKVZener │ └── Spring ── [Spring \parallel SpringPot] │ └── Complex retardation │ ├── Zener Family (Combined) │ ├── FractionalZenerSS │ │ └── Spring ── [SpringPot \parallel Spring] │ │ └── Solid-Solid, plateau at both limits │ │ │ ├── FractionalZenerSL │ │ └── Spring ── [SpringPot \parallel Dashpot] │ │ └── Solid-Liquid, terminal flow │ │ │ └── FractionalZenerLL │ └── Dashpot ── [SpringPot \parallel Dashpot] │ └── Liquid-Liquid, double flow │ └── Extended Models ├── FractionalJeffreys │ └── Two relaxation modes │ ├── FractionalBurgers │ └── Four-element, primary creep │ └── FractionalPoyntingThomson └── Multiple timescales When to Use Which Model ----------------------- .. list-table:: :widths: 25 25 25 25 :header-rows: 1 * - Material Type - Recommended Model - Alternatives - Key Indicator * - Gel (terminal flow) - FMG - FZSL - :math:`G'' > G'` at low :math:`\omega` * - Polymer melt - FML - FMG, FZSL - :math:`G''` crosses :math:`G'` once * - Crosslinked gel - FKV, FZSS - — - :math:`G'` plateau both limits * - Biological tissue - FKV - FZSS - Bounded compliance * - Hierarchical material - Two-Order FM - FBurgers - Two power-law slopes * - Critical gel (gel point) - SpringPot - FMG (:math:`\alpha` ≈ 0.5) - :math:`\tan\delta \approx` const **Decision Flowchart:** 1. Does material flow at long times (:math:`G'' > G'` as :math:`\omega \to 0`)? - **Yes** :math:`\to` Maxwell family (FMG, FML, FZSL, FZLL) - **No** :math:`\to` Kelvin-Voigt family or FZSS 2. Is there a high-frequency plateau in :math:`G'`? - **Yes** :math:`\to` Models with spring in series (FML, FZSS, FZSL) - **No** :math:`\to` Models starting with SpringPot (FMG, FKV) 3. Are two power-law regimes visible? - **Yes** :math:`\to` Two-Order FM or FBurgers - **No** :math:`\to` Single-order models Key Parameters -------------- .. list-table:: :widths: 15 10 15 60 :header-rows: 1 * - Parameter - Symbol - Units - Physical Meaning * - Fractional order - :math:`\alpha` - — - 0 = solid, 1 = liquid, 0.5 = critical gel * - SpringPot constant - :math:`c_\alpha` - Pa·s\ :math:`^{\alpha}` - Sets magnitude (unusual units) * - Shear modulus - :math:`G` - Pa - Elastic plateau stiffness * - Viscosity - :math:`\eta` - Pa·s - Terminal viscosity (when present) * - Relaxation time - :math:`\tau` - s - Crossover frequency :math:`\omega \approx 1/\tau` **Physical interpretation of** :math:`\alpha`: - :math:`\alpha \to 0`: Nearly elastic, broad relaxation spectrum - :math:`\alpha \to 0.3` **–0.5**: Typical for soft solids, gels - :math:`\alpha \to 0.5`: Critical gel, self-similar structure - :math:`\alpha \to 0.7` **–0.9**: Approaching Newtonian behavior - :math:`\alpha \to 1`: Classical dashpot (Newtonian) Quick Start ----------- **Fractional Maxwell Gel (soft gels):** .. code-block:: python from rheojax.models import FractionalMaxwellGel import numpy as np model = FractionalMaxwellGel() omega = np.logspace(-2, 2, 50) # Fit to oscillatory data model.fit(omega, G_star, test_mode='oscillation') # Fractional order indicates spectrum breadth alpha = model.parameters.get_value('alpha') print(f"Fractional order: {alpha:.2f}") **Fractional Kelvin-Voigt (bounded creep):** .. code-block:: python from rheojax.models import FractionalKelvinVoigt model = FractionalKelvinVoigt() model.fit(t, J_t, test_mode='creep') # Equilibrium compliance Ge = model.parameters.get_value('Ge') J_eq = 1 / Ge **Bayesian inference:** .. code-block:: python # Bayesian with warm-start from NLSQ result = model.fit_bayesian( omega, G_star, test_mode='oscillation', num_warmup=1000, num_samples=2000, num_chains=4, seed=42 ) # Credible intervals for fractional order intervals = model.get_credible_intervals(result.posterior_samples) print(f"alpha: [{intervals['alpha'][0]:.2f}, {intervals['alpha'][1]:.2f}]") Model Documentation ------------------- **Maxwell Family:** .. toctree:: :maxdepth: 1 fractional_maxwell_gel fractional_maxwell_liquid fractional_maxwell_model **Kelvin-Voigt Family:** .. toctree:: :maxdepth: 1 fractional_kelvin_voigt fractional_kv_zener **Zener Family:** .. toctree:: :maxdepth: 1 fractional_zener_ss fractional_zener_sl fractional_zener_ll **Extended Models:** .. toctree:: :maxdepth: 1 fractional_jeffreys fractional_burgers fractional_poynting_thomson See Also -------- - :doc:`/models/classical/index` — Integer-order building blocks - :doc:`/user_guide/fractional_viscoelasticity_reference` — Mathematical foundations - :doc:`/models/sgr/index` — Power-law from disordered structure (SGR approach) - :doc:`/transforms/mastercurve` — Time-temperature superposition - :doc:`/examples/advanced/04-fractional-models-deep-dive` — Comparison notebook References ---------- 1. Mainardi, F. (2010). *Fractional Calculus and Waves in Linear Viscoelasticity*. Imperial College Press. https://doi.org/10.1142/p614 2. Schiessel, H., Metzler, R., Blumen, A., & Nonnenmacher, T.F. (1995). "Generalized viscoelastic models: their fractional equations with solutions." *J. Phys. A*, 28, 6567–6584. https://doi.org/10.1088/0305-4470/28/23/012 3. Bagley, R.L. & Torvik, P.J. (1983). "A theoretical basis for the application of fractional calculus to viscoelasticity." *J. Rheol.*, 27, 201–210. https://doi.org/10.1122/1.549724 4. Jaishankar, A. & McKinley, G.H. (2013). "Power-law rheology in the bulk and at the interface." *Proc. R. Soc. A*, 469, 20120284. https://doi.org/10.1098/rspa.2012.0284 5. Friedrich, C. (1991). "Relaxation and retardation functions of the Maxwell model with fractional derivatives." *Rheol. Acta*, 30, 151–158. https://doi.org/10.1007/BF01134604 6. Podlubny, I. (1999). *Fractional Differential Equations*. Academic Press. ISBN: 978-0125588409 7. Gorenflo, R. et al. (2014). *Mittag-Leffler Functions, Related Topics and Applications*. Springer. https://doi.org/10.1007/978-3-662-43930-2