Fractional Burgers Model (Fractional)

Quick Reference

  • Use when: Complex creep with glassy compliance, fractional retardation, and viscous flow

  • Parameters: 5 (\(J_g, J_k, \alpha, \tau_k, \eta_1\))

  • Key equation: \(J(t) = J_g + \frac{t^{\alpha}}{\eta_1\Gamma(1+\alpha)} + J_k[1 - E_{\alpha}(-(t/\tau_k)^{\alpha})]\)

  • Test modes: Relaxation, creep, oscillation

  • Material examples: Polymer composites, asphalt binders, bituminous materials, viscoelastic solids under load

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

\(J_g\)

1/Pa

Glassy compliance (instantaneous elastic response)

\(\eta_1\)

Pa·s

Viscosity of Maxwell dashpot (controls terminal flow)

\(J_k\)

1/Pa

Kelvin compliance magnitude (retardation amplitude)

\(\alpha\)

dimensionless

Fractional order (0 < \(\alpha\) < 1, controls power-law character)

\(\tau_k\)

s

Retardation time (characteristic Kelvin timescale)

\(E_{\alpha}(z)\)

dimensionless

One-parameter Mittag-Leffler function

\(\Gamma(z)\)

dimensionless

Gamma function

Overview

The Fractional Burgers Model combines a Maxwell element in series with a Fractional Kelvin-Voigt element, creating a five-parameter model that captures glassy compliance, viscous flow, and fractional retardation in a single compact framework. This model extends the classical four-element Burgers model by replacing the Kelvin-Voigt dashpot with a SpringPot, enabling power-law retardation instead of exponential relaxation.

The Fractional Burgers model is particularly effective for materials exhibiting complex creep behavior with both instantaneous elastic response, delayed fractional retardation, and long-term viscous flow. Common applications include polymer composites under load, asphalt binders, bituminous materials, and viscoelastic solids undergoing time-dependent deformation.

Mechanical Analogue:

[Maxwell Arm: Spring Gg + Dashpot η1] ---- series ---- [Fractional KV: Spring + SpringPot (Jk, α, τk)]

Physical Foundations

The Fractional Burgers model combines three distinct mechanical responses:

  1. Instantaneous elastic response (glassy compliance \(J_g\))

  2. Fractional retardation (SpringPot in Kelvin arm with time constant \(\tau_k\))

  3. Long-term viscous flow (dashpot viscosity \(\eta_1\))

Microstructural Interpretation:

  • \(J_g\): Instantaneous bond stretching, glassy modulus

  • Fractional KV arm: Distributed retardation from hierarchical polymer network rearrangements

  • Maxwell dashpot: Irreversible chain flow, reptation, or permanent deformation

Governing Equations

Time Domain (Creep Compliance):

\[J(t) = J_g + \frac{t^{\alpha}}{\eta_1\,\Gamma(1+\alpha)} + J_k\left[1 - E_{\alpha}\!\left(-\left(\frac{t}{\tau_k}\right)^{\alpha}\right)\right]\]

where \(E_{\alpha}(z)\) is the one-parameter Mittag-Leffler function:

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

Frequency Domain (Complex Compliance):

\[J^{*}(\omega) = J_g + \frac{(i\omega)^{-\alpha}}{\eta_1\,\Gamma(1-\alpha)} + \frac{J_k}{1 + (i\omega\tau_k)^{\alpha}}\]

Complex Modulus:

\[G^{*}(\omega) = \frac{1}{J^{*}(\omega)}\]

Note: The inversion \(G^* = 1/J^*\) is exact for linear viscoelastic materials.

Parameters

Parameters

Name

Symbol

Units

Bounds

Notes

Jg

\(J_g\)

1/Pa

[1e-9, 1e3]

Glassy compliance (instantaneous response)

eta1

\(\eta_1\)

Pa·s

[1e-6, 1e12]

Viscosity (Maxwell arm, controls terminal flow)

Jk

\(J_k\)

1/Pa

[1e-9, 1e3]

Kelvin compliance (retardation magnitude)

alpha

\(\alpha\)

dimensionless

[0.05, 0.95]

Fractional order (0.2-0.7 typical for polymers)

tau_k

\(\tau_k\)

s

[1e-6, 1e6]

Retardation time (characteristic Kelvin timescale)

Regimes and Behavior

Short Time (\(t \ll \tau_k\)):

\[J(t) \approx J_g + \frac{t^{\alpha}}{\eta_1\,\Gamma(1+\alpha)}\]

Instantaneous glassy compliance plus early-time fractional flow from Maxwell arm.

Intermediate Time (\(t \sim \tau_k\)):

\[J(t) \approx J_g + J_k\left[1 - E_{\alpha}\!\left(-\left(\frac{t}{\tau_k}\right)^{\alpha}\right)\right]\]

Fractional retardation dominated by Kelvin arm with power-law approach to equilibrium.

Long Time (\(t \gg \tau_k\)):

\[J(t) \approx J_g + J_k + \frac{t}{\eta_1}\]

Unbounded creep (liquid-like) with constant compliance offset from glassy and Kelvin contributions.

Validity and Assumptions

  • Linear viscoelasticity: Strain amplitudes remain small (< 5-10% typically)

  • Isothermal conditions: Temperature constant throughout measurement

  • Time-invariant material: No aging, degradation, or structural evolution

  • Supported test modes: Creep (primary), oscillation

  • Fractional order bounds: 0.05 < \(\alpha\) < 0.95 for numerical stability

  • Liquid-like behavior: Unbounded creep at long times (\(\eta_1\) finite)

What You Can Learn

This section explains how to translate fitted Fractional Burgers parameters into material insights and actionable knowledge.

Parameter Interpretation

Glassy Compliance ( \(J_g\) ):

The instantaneous elastic response upon stress application.

  • For graduate students: \(J_g\) reflects short-range bond stretching and angle deformation in the glassy state. For polymers, \(J_g \approx 1/G_\infty\) where \(G_\infty\) is the glassy modulus (~1 GPa for many polymers).

  • For practitioners: \(J_g\) sets the immediate strain upon loading. Critical for impact resistance and short-time deformation.

Kelvin Compliance ( \(J_k\) ):

Controls the magnitude of delayed (retarded) elastic deformation.

  • Retardation magnitude: \(\Delta J = J_k\)

  • For polymers, relates to chain rearrangements in constrained environments

  • Typical values: \(10^{-6}\) to \(10^{-2}\) Pa\(^{-1}\)

Fractional Order ( \(\alpha\) ):

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

  • \(\alpha \approx 0.2\)–0.3: Very broad spectrum, highly heterogeneous (filled systems)

  • \(\alpha \approx 0.4\)–0.5: Moderate breadth, typical for polymer composites

  • \(\alpha \approx 0.6\)–0.7: Narrower spectrum, more uniform structure

  • \(\alpha \to 1\): Exponential retardation (classical Burgers)

Physical interpretation: Lower \(\alpha\) indicates greater polydispersity in relaxation times arising from structural heterogeneity, filler distribution, or molecular weight distribution.

Viscosity ( \(\eta_1\) ):

Controls the rate of unbounded creep at long times.

  • Slope of J(t) at long times: dJ/dt = 1/\(\eta_1\)

  • For polymers, relates to molecular weight via \(\eta_1 \sim M_w^{3.4}\) (reptation)

  • Determines processability and long-term dimensional stability

Retardation Time ( \(\tau_k\) ):

Characteristic timescale for the fractional Kelvin-Voigt relaxation.

  • Marks the transition from glassy to retardation-dominated regime

  • Temperature-dependent: follows WLF or Arrhenius behavior

Material Classification

Burgers Behavior Classification

Parameter Pattern

Material Type

Examples

Key Characteristics

High \(J_k/J_g\) (> 10)

Soft viscoelastic solid

Polymer composites, filled elastomers

Large delayed compliance

Low \(\alpha\) (< 0.3)

Highly heterogeneous

Asphalt, bitumen, nanocomposites

Very broad spectrum

High \(\eta_1 (> 10^6\) Pa·s)

High MW polymer

Melts, concentrated solutions

Slow terminal flow

Low \(\eta_1 (< 10^3\) Pa·s)

Low MW or diluted

Modified bitumen, soft materials

Rapid creep

Diagnostic Indicators

  • \(J_g \approx 0\) or poorly constrained: Insufficient early-time data; use faster sampling or estimate from high-frequency \(G'\)

  • Linear \(J(t)\) at all times: No retardation; use simple Maxwell liquid instead

  • \(\alpha\) near bounds (0.05 or 0.95): Data may not support fractional behavior; try classical Burgers (\(\alpha\) = 1)

  • Strong \(J_k\) - \(\tau_k\) correlation: Need better data coverage in intermediate regime

Application Examples

Asphalt Pavement Design:

Use Burgers model to predict rutting under sustained traffic load. The terminal flow (\(\eta_1\)) determines permanent deformation rate, while \(J_k\) and \(\alpha\) control elastic recovery.

Polymer Composite Selection:

Compare \(J_k\) values between formulations. Lower \(J_k\) means better dimensional stability under load. Monitor \(\alpha\) for filler dispersion quality.

Food Texture Analysis:

Fit creep data from cheese or dough. High \(J_k\) indicates soft, easily deformable texture. Use \(\alpha\) to quantify structural heterogeneity.

Fitting Guidance

Recommended Data Collection:

  1. Creep test (primary): 4-5 decades in time (e.g., 0.1 s - \(10^4\) s)

  2. Sampling: Log-spaced, minimum 50 points per decade

  3. Stress level: Within LVR, verify with amplitude sweep

  4. Temperature control: ±0.1°C for polymers, ±0.5°C for bitumen

Initialization Strategy:

# From creep data J(t)
Jg_init = J(t_min)  # Instantaneous compliance
eta1_init = t / (J(t) - J(t_min)) at long time  # Terminal slope
Jk_init = (J(t_mid) - Jg_init) * 0.5  # Mid-range magnitude
tau_k_init = t where retardation is 50% complete
alpha_init = 0.5  # Default starting point

Optimization Tips:

  • Fit in log(compliance) space for better conditioning

  • Use weighted least squares with log-spaced weights

  • Constrain \(J_g < J_k\) (glassy stiffer than Kelvin)

  • Verify residuals are random, not systematic

Common Pitfalls:

  • Overfitting: Don’t fit Burgers if classical 4-element model suffices

  • Underfitting: If residuals show curvature, may need additional Kelvin element

  • Wrong regime: Ensure data captures all three regimes (glassy, retardation, flow)

Usage

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

# Create model
model = FractionalBurgersModel()

# Fit to experimental creep data
t_exp = np.logspace(-1, 4, 100)  # 0.1 s to 10,000 s
J_exp = load_creep_data()  # Load your data

# Automatic fit
model.fit(t_exp, J_exp, test_mode='creep')

# Inspect fitted parameters
print(f"Jg = {model.parameters.get_value('Jg'):.2e} Pa⁻¹")
print(f"Jk = {model.parameters.get_value('Jk'):.2e} Pa⁻¹")
print(f"α = {model.parameters.get_value('alpha'):.3f}")
print(f"τk = {model.parameters.get_value('tau_k'):.2e} s")
print(f"η₁ = {model.parameters.get_value('eta1'):.2e} Pa·s")

# Predict creep at new times
t_new = np.logspace(-2, 5, 200)
data = RheoData(x=t_new, y=np.zeros_like(t_new), domain='time')
data.metadata['test_mode'] = 'creep'
J_pred = model.predict(data)

# Bayesian uncertainty quantification
result = model.fit_bayesian(
    t_exp, J_exp,
    num_warmup=1000,
    num_samples=2000,
    test_mode='creep'
)
intervals = model.get_credible_intervals(result.posterior_samples, credibility=0.95)

See Also

Material Examples

Polymer Composites (\(J_g \approx 10^{-6}-10^{-5}\) 1/Pa, \(\alpha \approx 0.3-0.5\)):

  • Filled elastomers (carbon black, silica fillers)

  • Fiber-reinforced polymers under sustained load

  • Polymer nanocomposites (clay, CNT fillers)

Asphalt and Bitumen (\(\eta_1 \approx 10^4-10^7\) Pa·s, \(\alpha \approx 0.4-0.6\)):

  • Asphalt concrete (temperature-dependent)

  • Bituminous binders for road pavements

  • Roofing materials

Food Materials (\(J_k \approx 10^{-4}-10^{-2}\), \(\alpha \approx 0.2-0.5\)):

  • Cheese (long-term creep)

  • Dough (wheat flour, viscoelastic retardation)

  • Semi-solid fats (margarine, butter)

Biological Tissues (\(\alpha \approx 0.2-0.4\)):

  • Ligaments and tendons under sustained stress

  • Intervertebral discs (viscoelastic creep)

Experimental Design

Creep Test (Primary Application):

  1. Step stress: Apply constant stress \(\sigma_0\) within LVR

  2. Time span: Cover 4-5 decades (e.g., 0.1 s - \(10^4\) s)

  3. Sampling: Log-spaced to capture all three regimes

  4. Analysis: Fit \(J(t)\) to identify \(J_g\) (instantaneous), \(J_k\) (retardation), \(\eta_1\) (slope at long time)

Frequency Sweep (Oscillatory):

  1. Frequency range: 0.001-100 rad/s (wide span critical)

  2. Strain amplitude: Within LVR (0.5-5%)

  3. Analysis: Fit \(G'(\omega)\), \(G''(\omega)\) simultaneously

  4. Verification: Check terminal flow region (\(G'' \sim \omega\), \(G' \sim \omega^2\))

Fitting Strategies

Initialization from Creep Data:

  1. \(J_g\): Extrapolate \(J(t \to 0)\) (instantaneous compliance)

  2. \(\eta_1\): Slope of \(J(t)\) at long time \(\to 1/\eta_1\)

  3. \(J_k\): Mid-time plateau height minus \(J_g\)

  4. \(\tau_k\): Time where retardation is half-complete

  5. \(\alpha\): Curvature of retardation region in log-log plot

Optimization:

  • Use weighted least squares (log-spaced weights)

  • Constrain \(J_g < J_k\) (glassy stiffer than Kelvin)

  • Fit in compliance space for creep, modulus space for oscillation

  • Verify residuals random across all time/frequency decades

Usage Example

from rheojax.models import FractionalBurgersModel
import numpy as np

# Create model
model = FractionalBurgersModel()

# Set typical parameters for polymer composite
model.parameters.set_value('Jg', 1e-6)       # 1/Pa
model.parameters.set_value('eta1', 1e5)      # Pa·s
model.parameters.set_value('Jk', 5e-6)       # 1/Pa
model.parameters.set_value('alpha', 0.4)     # dimensionless
model.parameters.set_value('tau_k', 10.0)    # s

# Predict creep compliance
t = np.logspace(-1, 4, 100)
J_t = model.predict(t, test_mode='creep')

# Fit to experimental creep data
# t_exp, J_exp = load_creep_data()
# model.fit(t_exp, J_exp, test_mode='creep')

Limiting Behavior

  • \(\alpha \to 1\): Classical Burgers with exponential Kelvin retardation

  • \(J_k \to 0\): Maxwell + fractional flow only (no retardation)

  • \(\eta_1 \to \infty\): Fractional Kelvin-Voigt (bounded creep, no flow)

  • \(\tau_k \to 0\): Instantaneous Kelvin response, \(J(t) = J_g + J_k + t/\eta_1\)

  • \(\tau_k \to \infty\): Kelvin arm inactive, simple Maxwell

Model Comparison

Burgers vs Fractional Burgers:

  • Classical Burgers: Exponential retardation (\(\alpha = 1\))

  • Fractional Burgers: Power-law retardation (0 < \(\alpha\) < 1)

  • Use Fractional when creep shows curved transition in log-log plots

Burgers vs Fractional Maxwell Gel:

  • Burgers: 5 parameters, includes delayed elasticity (Kelvin arm)

  • FMG: 3 parameters, single relaxation mode

  • Use Burgers for complex creep with multiple timescales

Troubleshooting

Issue: Cannot identify \(J_g\) from data

  • Cause: Insufficient early-time resolution

  • Solution: Use faster sampling or estimate from high-frequency modulus

Issue: Oscillatory fit poor at low frequencies

  • Cause: Terminal flow region not captured

  • Solution: Extend frequency sweep to lower \(\omega\) (< 0.01 rad/s)

Issue: Parameter correlation (\(J_k\) and \(\tau_k\))

  • Cause: Insufficient data in retardation regime

  • Solution: Focus measurements on intermediate timescale (\(t \sim \tau_k\))

Tips & Best Practices

  1. Fit creep first: Compliance space more natural for Burgers model

  2. Verify terminal flow: Confirm linear \(J(t)\) vs \(t\) at long time

  3. Check bounds: Ensure \(J_g < J_k\) (physically meaningful)

  4. Use transforms: Apply FFTAnalysis to convert creep → oscillation

  5. Log-log plots: Visualize all three regimes clearly

References

See Also