Fractional Viscoelastic Models

This section documents the fractional calculus-based viscoelastic models that capture power-law relaxation and broad spectral behavior.

Quick Reference

Model	Parameters	Use Case
Fractional Maxwell Gel (Fractional)	3	Gels with terminal flow, SpringPot + dashpot
Fractional Maxwell Liquid (Fractional)	3	Viscoelastic liquids, spring + SpringPot
Generalized Fractional Maxwell (Two-Order)	4	Two-order generalized, hierarchical relaxation
Fractional Kelvin-Voigt (Fractional)	3	Solids with bounded creep, spring \(\parallel\) SpringPot
Fractional Zener Solid-Solid (Fractional)	4	Solid-Solid Zener, low-frequency plateau
Fractional Zener Solid-Liquid (Fractional)	4	Solid-Liquid Zener, terminal flow
Fractional Zener Liquid-Liquid (Fractional)	4	Liquid-Liquid Zener, double flow
Fractional Kelvin-Voigt-Zener (Fractional)	5	Complex retardation, bounded creep + plateau
Fractional Jeffreys Model (Fractional)	5	Polymer solutions, two relaxation modes
Fractional Burgers Model (Fractional)	6	Primary creep, four-element
Fractional Poynting-Thomson (Fractional)	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: \(G(t) \sim t^{-\alpha}\) for broad spectra

• Parsimonious fitting: Fewer parameters than multi-mode models

• Physical insight: Fractional order \(\alpha\) relates to structural heterogeneity

The SpringPot element is the fundamental building block, interpolating between ideal spring (\(\alpha\) = 0) and dashpot (\(\alpha\) = 1) behavior.

Fractional Calculus Fundamentals

This model uses fractional calculus for power-law viscoelastic behavior. For mathematical foundations—SpringPot element, Mittag-Leffler functions, physical meaning of fractional order \(\alpha\), and derivation from molecular theory—see:

/user_guide/fractional_viscoelasticity_reference

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

Material Type	Recommended Model	Alternatives	Key Indicator
Gel (terminal flow)	FMG	FZSL	\(G'' > G'\) at low \(\omega\)
Polymer melt	FML	FMG, FZSL	\(G''\) crosses \(G'\) once
Crosslinked gel	FKV, FZSS	—	\(G'\) 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 (\(\alpha\) ≈ 0.5)	\(\tan\delta \approx\) const

Decision Flowchart:

1. Does material flow at long times (\(G'' > G'\) as \(\omega \to 0\))? - Yes \(\to\) Maxwell family (FMG, FML, FZSL, FZLL) - No \(\to\) Kelvin-Voigt family or FZSS

2. Is there a high-frequency plateau in \(G'\)? - Yes \(\to\) Models with spring in series (FML, FZSS, FZSL) - No \(\to\) Models starting with SpringPot (FMG, FKV)

3. Are two power-law regimes visible? - Yes \(\to\) Two-Order FM or FBurgers - No \(\to\) Single-order models

Key Parameters

Parameter	Symbol	Units	Physical Meaning
Fractional order	\(\alpha\)	—	0 = solid, 1 = liquid, 0.5 = critical gel
SpringPot constant	\(c_\alpha\)	Pa·s\(^{\alpha}\)	Sets magnitude (unusual units)
Shear modulus	\(G\)	Pa	Elastic plateau stiffness
Viscosity	\(\eta\)	Pa·s	Terminal viscosity (when present)
Relaxation time	\(\tau\)	s	Crossover frequency \(\omega \approx 1/\tau\)

Physical interpretation of \(\alpha\):

• \(\alpha \to 0\): Nearly elastic, broad relaxation spectrum

• \(\alpha \to 0.3\) –0.5: Typical for soft solids, gels

• \(\alpha \to 0.5\): Critical gel, self-similar structure

• \(\alpha \to 0.7\) –0.9: Approaching Newtonian behavior

• \(\alpha \to 1\): Classical dashpot (Newtonian)

Quick Start

Fractional Maxwell Gel (soft gels):

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):

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:

# 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:

• Fractional Maxwell Gel (Fractional)
• Fractional Maxwell Liquid (Fractional)
• Generalized Fractional Maxwell (Two-Order)

Kelvin-Voigt Family:

• Fractional Kelvin-Voigt (Fractional)
• Fractional Kelvin-Voigt-Zener (Fractional)

Zener Family:

• Fractional Zener Solid-Solid (Fractional)
• Fractional Zener Solid-Liquid (Fractional)
• Fractional Zener Liquid-Liquid (Fractional)

Extended Models:

• Fractional Jeffreys Model (Fractional)
• Fractional Burgers Model (Fractional)
• Fractional Poynting-Thomson (Fractional)

See Also

• Classical Viscoelastic Models — Integer-order building blocks

• /user_guide/fractional_viscoelasticity_reference — Mathematical foundations

• Soft Glassy Rheology (SGR) Models — Power-law from disordered structure (SGR approach)

• Mastercurve (Time-Temperature Superposition) — Time-temperature superposition

• /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