Fractional Kelvin-Voigt (Fractional)

Quick Reference

• Use when: Solid with bounded creep, power-law viscoelastic damping

• Parameters: 2-3 (Ge, \(c_{\alpha, \alpha}\))

• Key equation: \(G^*(\omega) = G_e + c_\alpha (i\omega)^\alpha\)

• Test modes: Oscillation, creep, relaxation

• Material examples: Soft solids, filled polymers, biological tissues, materials with bounded compliance

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

Notation Guide

Symbol	Units	Description
\(G_e\)	Pa	Equilibrium modulus (spring stiffness)
\(c_\alpha\)	Pa·s\(^{\alpha}\)	SpringPot quasi-property (damping coefficient)
\(\alpha\)	dimensionless	Fractional order (0 < \(\alpha\) < 1, controls damping character)
\(\tau_\varepsilon\)	s	Characteristic retardation time, \(\tau_\varepsilon = (c_\alpha/G_e)^{1/\alpha}\)
\(E_\alpha(z)\)	dimensionless	One-parameter Mittag-Leffler function
\(\Gamma(z)\)	dimensionless	Gamma function

Overview

The Fractional Kelvin-Voigt (FKV) model consists of a Hookean spring and a SpringPot element connected in parallel. This configuration describes materials that exhibit solid-like behavior with power-law creep and viscoelastic damping. Unlike the classical Kelvin-Voigt model which combines a spring and dashpot in parallel, the FKV model replaces the dashpot with a SpringPot, introducing fractional-order power-law damping instead of purely viscous dissipation.

The FKV model is particularly effective for characterizing soft solids, filled polymers, biological tissues, and materials that exhibit bounded creep compliance-materials that deform under constant stress but reach an equilibrium strain rather than flowing indefinitely. The fractional order alpha controls the rate and character of this creep process.

Physical Foundations

The FKV model represents the simplest fractional viscoelastic solid, consisting of:

Mechanical Configuration:

[Spring Ge] ---- parallel ---- [SpringPot (c_α, α)]

Microstructural Interpretation:

• Spring (Ge): Permanent network structure (crosslinks, crystalline domains) providing equilibrium elasticity

• SpringPot ( \(c_{\alpha, \alpha}\) ): Distributed viscoelastic damping from hierarchical relaxation processes (chain rearrangements, bond breaking/reformation)

• Solid behavior: Bounded creep to equilibrium compliance J∞ = 1/Ge

The parallel configuration ensures that stress is shared between elastic and viscoelastic components, with the spring providing long-term load-bearing capacity.

What You Can Learn

This section explains how to extract material insights from fitted FKV parameters.

Parameter Interpretation

Equilibrium Modulus (Ge):

The long-time elastic plateau representing permanent network structure.

• For graduate students: Ge relates to crosslink density via rubber elasticity theory: \(G_e \approx \nu k_B T\) where \(\nu\) is network strand density

• For practitioners: Higher Ge means stiffer material; compare to design requirements

SpringPot Constant ( \(c_{\alpha}\) ):

Controls the magnitude of viscoelastic damping.

• High \(c_{\alpha/Ge}\) ratio: Strong damping, slow approach to equilibrium

• Low \(c_{\alpha/Ge}\) ratio: Weak damping, rapid approach to equilibrium

• Units: Pa·s\(^{\alpha}\) (unusual due to fractional calculus)

Fractional Order ( \(\alpha\) ):

Governs the character of power-law damping and spectrum breadth.

• \(\alpha\) → 0: Purely elastic (spring-like), minimal damping

• \(\alpha\) → 0.3-0.5: Typical for soft solids, broad relaxation spectrum

• \(\alpha\) → 0.7-0.9: Approaching classical Kelvin-Voigt (viscous damping)

• \(\alpha\) → 1: Classical Kelvin-Voigt with Newtonian dashpot

Physical meaning: Lower \(\alpha\) indicates broader distribution of relaxation times arising from structural heterogeneity (polydispersity, filler distribution, network inhomogeneity).

Material Classification

FKV Behavior Classification

Parameter Pattern	Material Type	Examples	Key Characteristics
High Ge (> \(10^5 Pa\)), low \(\alpha\)	Stiff crosslinked solid	Thermosets, vulcanized rubber	Minimal creep, strong damping
Moderate Ge (\(10^3-10^5 Pa\)), \(\alpha \sim 0.4\)	Soft viscoelastic solid	Hydrogels, elastomers	Balanced elasticity/damping
Low Ge (< \(10^3\) Pa), high \(\alpha\)	Very soft gel	Weak physical gels	Significant creep, slow recovery

Diagnostic Indicators

• Ge near lower bound: Material may be liquid-like; consider fractional Maxwell gel instead

• \(\alpha\) near 1: Data supports classical Kelvin-Voigt; use simpler model

• Poor fit at long time: Equilibrium not reached; extend measurement time

• \(c_{\alpha and \alpha}\) strongly correlated: Need broader frequency/time coverage

Fitting Guidance

Recommended Data Collection:

1. Creep test: 3-4 decades in time, verify plateau at long times

2. Frequency sweep: 3-4 decades, strain within LVR (< 5%)

3. Temperature control: ±0.1°C for soft materials

Initialization Strategy:

# From creep compliance J(t)
Ge_init = 1 / J(t → ∞)  # Equilibrium compliance
c_alpha_init = Ge_init / (characteristic_time**alpha_init)
alpha_init = 0.5  # Default for soft solids

# From frequency sweep G'(ω), G"(ω)
Ge_init = G'(ω → 0)  # Low-frequency plateau
alpha_init = slope of log(G") vs log(ω) in power-law regime

Optimization Tips:

• Fit in compliance space for creep data (more natural)

• Use frequency-domain fitting for SAOS data

• Constrain 0.05 < \(\alpha\) < 0.95 to avoid numerical issues

• Verify residuals show no systematic trends

See Also

• Fractional Maxwell Gel (Fractional) — provides the series counterpart used for gel-like liquids

• Fractional Kelvin-Voigt-Zener (Fractional) — Kelvin-Voigt element combined with an extra spring for plateau control

• Bingham Plastic — combine bounded creep solids with yield-stress flow models

• MutationNumber — monitor whether the quasi-solid assumption holds during gelation

• ../../examples/advanced/04-fractional-models-deep-dive — notebook comparing Kelvin-Voigt, Maxwell, and Zener fractional families

Governing Equations

The constitutive behavior of the Fractional Kelvin-Voigt model is described by:

Relaxation Modulus:

\[G(t) = G_e + \frac{c_\alpha t^{-\alpha}}{\Gamma(1-\alpha)}\]

where \(G_e\) is the equilibrium modulus, \(c_\alpha\) is the SpringPot constant, and \(\Gamma\) is the gamma function. The relaxation modulus consists of an elastic plateau plus a power-law term that decays in time.

Complex Modulus:

\[G^*(\omega) = G_e + c_\alpha (i\omega)^\alpha\]

This can be decomposed into storage and loss moduli:

\[G'(\omega) = G_e + c_\alpha \omega^\alpha \cos\left(\frac{\alpha\pi}{2}\right)\]

\[G''(\omega) = c_\alpha \omega^\alpha \sin\left(\frac{\alpha\pi}{2}\right)\]

Creep Compliance:

\[J(t) = \frac{1}{G_e}\left[1 - E_\alpha\left(-\left(\frac{t}{\tau_\varepsilon}\right)^\alpha\right)\right]\]

where \(\tau_\varepsilon = (c_\alpha/G_e)^{1/\alpha}\) is the characteristic retardation time and \(E_\alpha(z)\) is the one-parameter Mittag-Leffler function:

\[E_\alpha(z) = \sum_{k=0}^{\infty} \frac{z^k}{\Gamma(\alpha k + 1)}\]

The creep compliance approaches the limiting value \(J_\infty = 1/G_e\) as \(t \to \infty\), confirming solid-like behavior.

Parameters

The Fractional Kelvin-Voigt model has three parameters:

Parameters

Name	Symbol	Units	Bounds	Notes
Ge	\(G_e\)	Pa	[1e-3, 1e9]	Equilibrium modulus
c_alpha	\(c_\alpha\)	Pa·sα	[1e-3, 1e9]	SpringPot constant
alpha	\(\alpha\)	dimensionless	[0, 1]	Fractional order

Validity and Assumptions

• Linear viscoelastic assumption; strain amplitudes remain small.

• Isothermal, time-invariant material parameters throughout the experiment.

• Supported RheoJAX test modes: relaxation, creep, oscillation.

• Fractional orders stay within (0, 1) to keep kernels causal and bounded.

Regimes and Behavior

The Fractional Kelvin-Voigt model exhibits characteristic behavior in different regimes:

Short-Time / High-Frequency Regime (\(t \ll \tau_\varepsilon\) or \(\omega \gg \omega_c\)):

Instantaneous elastic response with additional power-law contribution:

\[G(t) \sim G_e + \frac{c_\alpha t^{-\alpha}}{\Gamma(1-\alpha)}, \quad G^*(\omega) \sim G_e + c_\alpha (i\omega)^\alpha\]

The material behaves as a stiff solid with frequency-dependent damping.

Long-Time / Low-Frequency Regime (\(t \gg \tau_\varepsilon\) or \(\omega \ll \omega_c\)):

Equilibrium elastic plateau:

\[G(t) \to G_e, \quad J(t) \to \frac{1}{G_e}\]

The material reaches a constant equilibrium modulus, confirming solid-like behavior without terminal flow.

Intermediate Regime:

The Mittag-Leffler function in the creep compliance produces a smooth power-law transition from initial response to equilibrium. The characteristic frequency \(\omega_c \sim 1/\tau_\varepsilon\) marks the crossover region where viscoelastic dissipation is most pronounced.

Loss Tangent:

\[\tan\delta = \frac{G''}{G'} = \frac{c_\alpha \omega^\alpha \sin(\alpha\pi/2)}{G_e + c_\alpha \omega^\alpha \cos(\alpha\pi/2)}\]

The loss tangent exhibits a maximum at intermediate frequencies, indicating peak energy dissipation.

Limiting Behavior

The FKV model connects to classical models in limiting cases:

• alpha -> 1: Approaches classical Kelvin-Voigt model with Newtonian damping: \(G^*(\omega) \approx G_e + i\omega c_\alpha\)

• alpha -> 0: Reduces to purely elastic solid: \(G^*(\omega) \to G_e\)

• c:sub:`alpha` -> 0: Pure elastic spring with \(G^*(\omega) = G_e\)

• c:sub:`alpha` -> inf: Diverging damping, non-physical limit

• G:sub:`e` -> inf with fixed c:sub:`alpha`/G:sub:`e`: Infinite stiffness limit

API References

• Module: rheojax.models

• Class: rheojax.models.FractionalKelvinVoigt

Usage

from rheojax.models import FractionalKelvinVoigt
from rheojax.core.data import RheoData
import numpy as np

# Create model instance
model = FractionalKelvinVoigt()

# Set parameters for a filled polymer composite
model.parameters.set_value('Ge', 1e6)         # Pa
model.parameters.set_value('c_alpha', 1e4)    # Pa·s^α
model.parameters.set_value('alpha', 0.5)      # dimensionless

# Predict relaxation modulus
t = np.logspace(-3, 3, 50)
data = RheoData(x=t, y=np.zeros_like(t), domain='time')
data.metadata['test_mode'] = 'relaxation'
G_t = model.predict(data)

# Predict creep compliance showing bounded creep
data_creep = RheoData(x=t, y=np.zeros_like(t), domain='time')
data_creep.metadata['test_mode'] = 'creep'
J_t = model.predict(data_creep)
# J(t->inf) -> 1/Ge (equilibrium compliance)

# Predict complex modulus in frequency domain
omega = np.logspace(-2, 2, 50)
data_freq = RheoData(x=omega, y=np.zeros_like(omega), domain='frequency')
data_freq.metadata['test_mode'] = 'oscillation'
G_star = model.predict(data_freq)

# Extract storage and loss moduli
Gp = G_star.y.real   # G'(omega) includes elastic plateau
Gpp = G_star.y.imag  # G''(omega) shows power-law damping
tan_delta = Gpp / Gp # Peaks at intermediate frequencies

# Fit to experimental oscillatory data
# omega_exp, G_star_exp = load_experimental_data()
# model.fit(omega_exp, G_star_exp, test_mode='oscillation')

For more details on the rheojax.models.FractionalKelvinVoigt class, see the API reference.

See also

• Fractional Maxwell Gel (Fractional) — provides the series counterpart used for gel-like liquids.

• Fractional Kelvin-Voigt-Zener (Fractional) — Kelvin-Voigt element combined with an extra spring for plateau control.

• Bingham Plastic — combine bounded creep solids with yield-stress flow models.

• MutationNumber — monitor whether the quasi-solid assumption holds during gelation.

• ../../examples/advanced/04-fractional-models-deep-dive — notebook comparing Kelvin-Voigt, Maxwell, and Zener fractional families.

References

[1]

Bagley, R. L., and Torvik, P. J. “A theoretical basis for the application of fractional calculus to viscoelasticity.” Journal of Rheology, 27, 201–210 (1983). https://doi.org/10.1122/1.549724

[2]

Makris, N., and Constantinou, M. C. “Fractional-derivative Maxwell model for viscous dampers.” Journal of Structural Engineering, 117, 2708–2724 (1991). https://doi.org/10.1061/%28ASCE%290733-9445%281991%29117:9%282708%29

[3]

Schiessel, H., Metzler, R., Blumen, A., and Nonnenmacher, T. F. “Generalized viscoelastic models: their fractional equations with solutions.” Journal of Physics A, 28, 6567–6584 (1995). https://doi.org/10.1088/0305-4470/28/23/012

[4]

Mainardi, F. Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press (2010). https://doi.org/10.1142/p614

[5]

Friedrich, C. “Relaxation and retardation functions of the Maxwell model with fractional derivatives.” Rheologica Acta, 30, 151–158 (1991). https://doi.org/10.1007/BF01134604

[6]

Metzler, R., Schick, W., Kilian, H.-G., & Nonnenmacher, T. F. “Relaxation in filled polymers: A fractional calculus approach.” Journal of Chemical Physics, 103, 7180-7186 (1995). https://doi.org/10.1063/1.470346

[7]

Friedrich, C. “Relaxation and retardation functions of the Maxwell model with fractional derivatives.” Rheologica Acta, 30, 151-158 (1991). https://doi.org/10.1007/BF01134604

[8]

Heymans, N. & Bauwens, J. C. “Fractal rheological models and fractional differential equations for viscoelastic behavior.” Rheologica Acta, 33, 210-219 (1994). https://doi.org/10.1007/BF00437306

[9]

Nonnenmacher, T. F. & Glöckle, W. G. “A fractional model for mechanical stress relaxation.” Philosophical Magazine Letters, 64, 89-93 (1991). https://doi.org/10.1080/09500839108214672

[10]

Podlubny, I. Fractional Differential Equations. Academic Press (1999). ISBN: 978-0125588409