Fractional Viscoelasticity: Mathematical Reference ================================================== .. note:: This is the **definitive reference** for fractional calculus concepts in RheoJAX. All fractional model pages link here instead of duplicating this content. Overview -------- Fractional calculus generalizes differentiation and integration to non-integer orders, providing a powerful mathematical framework for describing complex viscoelastic behavior that cannot be captured by classical integer-order models. In rheology, fractional derivatives enable the modeling of **power-law relaxation**, **broad relaxation spectra**, and **self-similar microstructures** using fewer parameters than multi-mode classical models. Why Fractional Calculus in Rheology? ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Most real materials exhibit viscoelastic behavior that deviates from simple exponential relaxation: **Experimental observations:** - **Power-law relaxation** :math:`G(t) \sim t^{-\alpha}` over multiple time decades - **Broad relaxation spectra** arising from structural heterogeneity - **Frequency-dependent moduli** with parallel slopes in log-log plots - **Non-exponential creep** that cannot be fit with single relaxation times **Classical model limitations:** - Single relaxation time :math:`\tau` (Maxwell, Zener) insufficient for complex materials - Multi-mode models require many parameters (5-20+) with limited physical insight - Exponential functions cannot capture power-law dynamics **Fractional model advantages:** - Capture power-law behavior naturally with 3-5 parameters - Fractional order :math:`\alpha` has clear physical meaning (spectrum breadth, microstructure) - Fewer parameters than multi-mode models while maintaining accuracy - Interpolate smoothly between elastic (:math:`\alpha=0`) and viscous (:math:`\alpha=1`) extremes SpringPot Element ----------------- The SpringPot (Scott-Blair element) is the **fundamental building block** of fractional rheology, generalizing both elastic springs and viscous dashpots into a single element. Mathematical Definition ~~~~~~~~~~~~~~~~~~~~~~~ The SpringPot constitutive equation relates stress and strain through a fractional derivative: .. math:: \sigma(t) = E_0 \, D^\alpha \gamma(t) where: - :math:`E_0`: quasi-property with units Pa·s :math:`^\alpha` - :math:`D^\alpha`: fractional derivative of order :math:`\alpha \in [0, 1]` - :math:`\gamma(t)`: strain as a function of time - :math:`\sigma(t)`: stress as a function of time Limiting Cases ~~~~~~~~~~~~~~ The SpringPot smoothly interpolates between classical elements: .. list-table:: SpringPot Limiting Behavior :header-rows: 1 :widths: 15 35 50 * - :math:`\alpha` Value - Element Type - Constitutive Equation * - :math:`\alpha = 0` - Pure elastic spring - :math:`\sigma = E_0 \gamma` (Hooke's law) * - :math:`0 < \alpha < 1` - Fractional viscoelastic - :math:`\sigma = E_0 D^\alpha \gamma` (intermediate behavior) * - :math:`\alpha = 1` - Pure viscous dashpot - :math:`\sigma = E_0 \, d\gamma/dt` (Newton's law) Frequency-Domain Representation ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ In oscillatory shear (frequency domain), the SpringPot impedance is: .. math:: Z(\omega) = E_0 (i\omega)^\alpha = E_0 \omega^\alpha \left[\cos\left(\frac{\alpha\pi}{2}\right) + i\sin\left(\frac{\alpha\pi}{2}\right)\right] This reveals that the SpringPot **simultaneously contributes to both storage and loss moduli** with a constant phase angle: .. math:: \delta = \frac{\alpha\pi}{2} where :math:`\delta` is the loss angle (phase shift between stress and strain). **Physical interpretation:** - :math:`\alpha = 0`: :math:`\delta = 0^\circ` (purely elastic, no phase shift) - :math:`\alpha = 0.5`: :math:`\delta = 45^\circ` (balanced viscoelasticity) - :math:`\alpha = 1`: :math:`\delta = 90^\circ` (purely viscous, maximum phase shift) The storage and loss moduli contributions scale as: .. math:: G'(\omega) &\sim \omega^\alpha \cos(\alpha\pi/2) \\ G''(\omega) &\sim \omega^\alpha \sin(\alpha\pi/2) **Key insight:** Both moduli have **parallel slopes** of :math:`\alpha` in log-log plots, which is the hallmark signature of fractional viscoelasticity. Mittag-Leffler Functions ------------------------- Mittag-Leffler functions play the same role in fractional viscoelasticity as exponential functions do in classical models. They provide the **exact analytical solutions** for fractional differential equations governing viscoelastic constitutive relations. One-Parameter Mittag-Leffler Function ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The one-parameter Mittag-Leffler function is defined by the infinite series: .. math:: E_\alpha(z) = \sum_{k=0}^{\infty} \frac{z^k}{\Gamma(\alpha k + 1)} where :math:`\Gamma` is the gamma function (generalization of factorial to real numbers). **Key Properties:** 1. **Recovers exponential**: :math:`E_1(z) = \exp(z)` (classical limit) 2. **Initial value**: :math:`E_\alpha(0) = 1` for all :math:`\alpha > 0` 3. **Asymptotic behavior**: - Short times: :math:`E_\alpha(-t^\alpha) \approx 1 - t^\alpha/\Gamma(\alpha+1)` - Intermediate times: :math:`E_\alpha(-t^\alpha) \sim t^{-\alpha}` (power-law decay) - Long times: :math:`E_\alpha(-t^\alpha) \sim \exp(-A \, t^{\alpha/(1-\alpha)})` (stretched exponential) 4. **Interpolation**: Smoothly interpolates between exponential (:math:`\alpha=1`) and power-law (:math:`0<\alpha<1`) **Physical Meaning in Relaxation:** The relaxation modulus for fractional models typically has the form: .. math:: G(t) = G_0 \, E_\alpha\left(-\left(\frac{t}{\tau_\alpha}\right)^\alpha\right) This captures: - **Initial plateau**: :math:`G(0) = G_0` (elastic response) - **Power-law relaxation**: :math:`G(t) \sim G_0 (t/\tau_\alpha)^{-\alpha}` at intermediate times - **Broad relaxation spectrum**: Continuous distribution of relaxation times - **Characteristic time** :math:`\tau_\alpha`: Time scale for onset of power-law decay Two-Parameter Mittag-Leffler Function ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The two-parameter generalization adds a second parameter :math:`\beta`: .. math:: E_{\alpha,\beta}(z) = \sum_{k=0}^{\infty} \frac{z^k}{\Gamma(\alpha k + \beta)} **Key Properties:** 1. **Reduces to one-parameter**: :math:`E_{\alpha,1}(z) = E_\alpha(z)` 2. **Initial value**: :math:`E_{\alpha,\beta}(0) = 1/\Gamma(\beta)` 3. **More flexible asymptotics**: Controls short-time behavior via :math:`\beta` **Applications in Fractional Models:** - **Creep compliance**: :math:`J(t)` often involves :math:`E_{\alpha,1+\alpha}(-t^\alpha)` - **Complex constitutive equations**: Fractional Maxwell Liquid uses :math:`E_{1-\alpha,1-\alpha}` - **General viscoelasticity**: Provides exact solutions for arbitrary fractional orders Computational Note ~~~~~~~~~~~~~~~~~~ RheoJAX computes Mittag-Leffler functions using the ``mittag_leffler`` module (:mod:`rheojax.utils.mittag_leffler`), which implements: - **One-parameter**: ``E_alpha(z, alpha)`` via series expansion + asymptotic approximations - **Two-parameter**: ``E_alpha_beta(z, alpha, beta)`` via series expansion These functions are JAX-compatible and GPU-accelerated for fast evaluation in optimization and Bayesian inference. Physical Meaning of Fractional Order α --------------------------------------- The fractional order :math:`\alpha` is **not an arbitrary fitting parameter** -- it has deep physical significance related to material microstructure and relaxation dynamics. 1. Relaxation Spectrum Width ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The fractional order :math:`\alpha` quantifies the **breadth of the relaxation time distribution**: .. list-table:: Spectrum Breadth Interpretation :header-rows: 1 :widths: 15 35 50 * - :math:`\alpha` Value - Spectrum Type - Physical Meaning * - :math:`\alpha = 1` - Narrow (Dirac delta) - Single relaxation time (classical exponential) * - :math:`0.7 < \alpha < 1` - Moderate breadth - Few dominant relaxation processes * - :math:`0.3 < \alpha < 0.7` - Broad distribution - Continuous spectrum over many decades * - :math:`\alpha \to 0` - Very broad (power-law) - Hierarchical or fractal structure, no characteristic time **Mathematical connection:** For fractional models, the relaxation time spectrum :math:`H(\tau)` is approximately: .. math:: H(\tau) \sim \tau^{-(1-\alpha)} \quad \text{for } \tau_{\text{min}} < \tau < \tau_{\text{max}} where: - Narrow spectrum (:math:`\alpha \to 1`): :math:`H(\tau) \to \delta(\tau - \tau_0)` (Dirac delta) - Broad spectrum (:math:`\alpha \approx 0.5`): :math:`H(\tau) \sim \tau^{-0.5}` (power-law distribution) 2. Microstructural Heterogeneity ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Lower :math:`\alpha` values indicate greater **structural heterogeneity** at the molecular/microscopic level: **For cross-linked networks (e.g., elastomers, hydrogels):** - :math:`\alpha < 0.5`: Hierarchical structure with multiple length scales - Broad cross-link density distribution - Polydisperse mesh sizes - Fractal or self-similar network architecture - :math:`\alpha \approx 0.5`: Critical gel-like behavior - Sol-gel transition point - Percolation threshold - Maximum structural disorder - :math:`\alpha > 0.5`: More homogeneous networks - Narrow cross-link density distribution - Approaching regular lattice structure **For polymer melts:** - :math:`\alpha < 0.5`: Broad molecular weight distribution (polydispersity) - Significant chain length heterogeneity - Branched or star polymers - Complex intermolecular interactions - :math:`\alpha \approx 0.7\text{--}0.9`: Relatively monodisperse linear polymers - Narrow molecular weight distribution - Simple chain dynamics (reptation) 3. Material Character (Solid vs. Liquid vs. Gel) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The fractional order :math:`\alpha` influences the **dominant viscoelastic character**: .. list-table:: Material Character Classification :header-rows: 1 :widths: 20 30 50 * - :math:`\alpha` Range - Dominant Character - Typical Materials * - :math:`\alpha < 0.3` - Strong solid-like - Stiff gels, covalently cross-linked elastomers, biological tissues * - :math:`0.3 < \alpha < 0.5` - Solid-like viscoelastic - Soft gels, filled polymers, weak networks * - :math:`\alpha \approx 0.5` - Critical gel (balanced) - Gel point, percolation threshold, :math:`G' \approx G''` across all :math:`\omega` * - :math:`0.5 < \alpha < 0.7` - Liquid-like viscoelastic - Concentrated polymer solutions, weak gels * - :math:`\alpha > 0.7` - Strong liquid-like - Polymer melts, dilute solutions, approaching classical Maxwell **Oscillatory shear signature:** - :math:`\alpha < 0.5`: :math:`G'(\omega) > G''(\omega)` at low frequencies (elastic dominance) - :math:`\alpha \approx 0.5`: :math:`G'(\omega) \approx G''(\omega)` across all frequencies (critical gel) - :math:`\alpha > 0.5`: :math:`G''(\omega) > G'(\omega)` at low frequencies (viscous dominance) 4. Typical α Ranges by Material Class ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Extensive experimental studies have established typical fractional order ranges for common materials: .. list-table:: Fractional Order Ranges by Material :header-rows: 1 :widths: 30 20 50 * - Material Class - Typical :math:`\alpha` - Notes * - **Cross-linked polymer networks** - 0.3 - 0.6 - Natural rubber, synthetic elastomers, cured epoxies * - **Filled elastomers** - 0.2 - 0.5 - Carbon black or silica-filled rubber; lower :math:`\alpha` due to filler-polymer interactions * - **Hydrogels (chemical)** - 0.4 - 0.7 - Covalently cross-linked PVA, alginate, PAA * - **Hydrogels (physical)** - 0.3 - 0.5 - Non-covalent cross-links, weaker structure * - **Biological tissues (soft)** - 0.1 - 0.4 - Skin, tendons, cartilage; very broad spectra from hierarchical collagen/elastin * - **Biological tissues (stiff)** - 0.3 - 0.5 - Bone, dentin, cornea * - **Semi-crystalline polymers** - 0.3 - 0.5 - Polyethylene, polypropylene; crystalline vs amorphous phase relaxation * - **Polymer melts (linear)** - 0.7 - 0.9 - Linear homopolymers; approaching classical Maxwell behavior * - **Polymer melts (branched)** - 0.5 - 0.7 - Long-chain branched polymers, star polymers * - **Concentrated polymer solutions** - 0.5 - 0.8 - Above overlap concentration :math:`c^*` * - **Emulsions** - 0.4 - 0.7 - Droplet size polydispersity and interfacial dynamics * - **Colloidal gels** - 0.2 - 0.4 - Particle network with weak attractive interactions * - **Critical gels** - 0.45 - 0.55 - Sol-gel transition, gelation point Physical Interpretation Summary ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ **Key takeaway:** The fractional order :math:`\alpha` is a **structural fingerprint** that encodes: 1. **How broad the relaxation spectrum is** (spectrum width) 2. **How heterogeneous the microstructure is** (structural disorder) 3. **Whether the material is solid-like or liquid-like** (material character) 4. **What physical processes dominate relaxation** (molecular vs network dynamics) Lower :math:`\alpha` values indicate: - Broader relaxation spectra - More heterogeneous microstructure - More solid-like character - Hierarchical or fractal organization Higher :math:`\alpha` values indicate: - Narrower relaxation spectra - More homogeneous microstructure - More liquid-like character - Approaching classical exponential behavior Fractional Models in RheoJAX ----------------------------- RheoJAX implements **11 fractional models** organized into families based on their mechanical analogs: **Fractional Maxwell Family (4 models):** - :doc:`/models/fractional/fractional_maxwell_gel` — Gel-like with elastic component - :doc:`/models/fractional/fractional_maxwell_liquid` — Liquid-like with memory - :doc:`/models/fractional/fractional_maxwell_model` — Dual SpringPot series (general) - :doc:`/models/fractional/fractional_kelvin_voigt` — Solid-like with slow relaxation **Fractional Zener Family (4 models):** - :doc:`/models/fractional/fractional_zener_ss` — **Most common**: Dual elastic plateaus - :doc:`/models/fractional/fractional_zener_sl` — Solid + fractional liquid - :doc:`/models/fractional/fractional_zener_ll` — Fractional liquid-liquid - :doc:`/models/fractional/fractional_kv_zener` — FKV + series spring **Advanced Fractional Models (3 models):** - :doc:`/models/fractional/fractional_burgers` — Maxwell + FKV (creep + relaxation) - :doc:`/models/fractional/fractional_jeffreys` — Two dashpots + SpringPot - :doc:`/models/fractional/fractional_poynting_thomson` — Multi-plateau solid See :doc:`/models/index` for detailed model documentation. Key References -------------- **Foundational Theory:** - **Mainardi, F. (2010)**. *Fractional Calculus and Waves in Linear Viscoelasticity*. Imperial College Press. ISBN: 978-1-84816-329-4 *The definitive reference on fractional calculus in viscoelasticity.* - **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 *Original derivation of fractional viscoelastic models.* - **Gorenflo, R., Kilbas, A.A., Mainardi, F., Rogosin, S.V. (2014)**. *Mittag-Leffler Functions, Related Topics and Applications*. Springer. https://doi.org/10.1007/978-3-662-43930-2 *Comprehensive treatment of Mittag-Leffler functions.* **Physical Interpretation:** - **Mainardi, F., Spada, G. (2011)**. "Creep, Relaxation and Viscosity Properties for Basic Fractional Models in Rheology." *European Physical Journal Special Topics*, 193, 133-160. https://doi.org/10.1140/epjst/e2011-01387-1 *Physical meaning of fractional parameters in rheology.* - **Friedrich, C., Braun, H. (1992)**. "Generalized Cole-Cole Behavior and its Rheological Relevance." *Rheologica Acta*, 31, 309-322. https://doi.org/10.1007/BF00418328 *Connection between fractional order and relaxation spectrum width.* **Applications:** - **Koeller, R.C. (1984)**. "Applications of fractional calculus to the theory of viscoelasticity." *J. Appl. Mech.* 51, 299–307. https://doi.org/10.1115/1.3167616 *Early application of fractional calculus to viscoelasticity.* - **Metzler, R., Klafter, J. (2000)**. "The Random Walk's Guide to Anomalous Diffusion: A Fractional Dynamics Approach." *Physics Reports*, 339(1), 1-77. https://doi.org/10.1016/S0370-1573(00)00070-3 *Broader context: fractional dynamics in physics.* Further Reading --------------- **Within RheoJAX Documentation:** - :doc:`/user_guide/model_selection` — Decision flowcharts for choosing fractional vs classical models - :doc:`/developer/architecture` — Template Method pattern for smart initialization - :doc:`/examples/advanced/04-fractional-models-deep-dive` — Jupyter notebook with case studies **External Resources:** - **Podlubny, I. (1999)**. *Fractional Differential Equations*. Academic Press. ISBN: 978-0-12-558840-9 - **Hilfer, R. (Ed.) (2000)**. *Applications of Fractional Calculus in Physics*. World Scientific. ISBN: 978-981-02-3457-7 - **Tarasov, V.E. (2010)**. *Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media*. Springer. https://doi.org/10.1007/978-3-642-14003-7 See Also -------- - :doc:`/models/index` — Complete model catalog with governing equations - :doc:`/user_guide/core_concepts` — RheoData, parameters, and test modes - :doc:`/user_guide/modular_api` — Direct model API usage - :doc:`/user_guide/bayesian_inference` — Bayesian inference for fractional models