Fractional Maxwell Model in Viscoelasticity

• Fractional Maxwell model is a viscoelastic framework that replaces integer-order derivatives with fractional ones to capture power-law relaxation and creep behaviors.
• It offers tunable spectra that bridge rate-independent elasticity and rate-dependent viscosity, effectively modeling anomalies like subdiffusion and non-local responses.
• Generalizations include distributed and variable orders, nonclassical kernels, and nonlinear extensions, expanding its application in mechanics, seismology, and electromagnetism.

The fractional Maxwell model generalizes the classical Maxwell viscoelastic model by introducing derivatives of non-integer order, thereby imparting power-law memory effects into the description of stress–strain relations for complex materials or fields. Fractional Maxwell models formalize a continuum between rate-independent elasticity and rate-dependent viscosity, yielding tunable viscoelastic spectra that more accurately fit phenomena such as anomalous stress relaxation, subdiffusion, and non-local electromagnetic or mechanical response. The fractional orders, the structures of the memory kernels, and the resulting constitutive relationships underpin a broad range of theoretical and applied studies in viscoelasticity, continuum mechanics, seismology, soft matter, and fractional electrodynamics.

1. Mathematical Formulation and Model Variants

The canonical (uniaxial) fractional Maxwell model replaces the integer-order time derivatives in the classical Maxwell rheology with Caputo fractional derivatives of order $0 < \alpha < 1$:

$$\sigma(t) + \tau^\alpha D_t^\alpha \sigma(t) = G D_t^\alpha \varepsilon(t)$$

where $D_t^\alpha$ denotes the Caputo fractional derivative:

$$D_t^\alpha f(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t (t-s)^{-\alpha} f'(s) ds$$

with $G$ the elastic modulus, $\tau$ the characteristic time-scale, $\sigma(t)$ the stress, and $\varepsilon(t)$ the strain (Giusti et al., 7 Feb 2024). This model captures stress relaxation and creep behavior through the Mittag–Leffler function:

$$G(t) = G E_{\alpha,1}\left(-\left(\frac{t}{\tau}\right)^{\alpha}\right)$$

which exhibits a transition from early-time power-law decay to late-time asymptotic behaviors (Ferrás et al., 2022).

Generalizations:

• Distributed-order models: Constitutive laws replace single-order derivatives with integrals over orders, parameterized by weighting functions, e.g.,

$$\sigma(t) + \int_0^1 \phi_2(\alpha)\, D_t^\alpha \sigma(t)\, d\alpha = \int_0^1 \phi_1(\alpha)\, D_t^\alpha \varepsilon(t)\, d\alpha$$

where the weights $\phi_{1,2}$ tune the memory spectrum (Ferrás et al., 2022).

• Variable-order models: The order $\alpha$ is replaced by a time-dependent function $\alpha(t)$ governed by a physical relaxation law (e.g., exponential or Mittag-Leffler-type functions), yielding models that interpolate between different regimes during the process (Giusti et al., 2023).
• Fractional Maxwell–like models: One-parameter family models blend features of the fractional Maxwell and Bessel models, e.g.,

$$\left[ 1 + \frac{1}{2(\nu+1)} D_t^{1/2} \right] \sigma(t) = \frac{1}{2(\nu+1)} D_t^{1/2} \varepsilon(t)\,, \quad \nu > -1$$

with analytic forms for creep compliance and relaxation modulus, parameterized by $\nu$ (Colombaro et al., 2016).

• Fractional Maxwell models with non-classical kernels: Utilization of Hadamard- or Prabhakar-type fractional derivatives allows encoding additional physical effects such as time-dependent viscosity or enhanced spectral flexibility (Garra et al., 2022, Giusti et al., 2017).

2. Rheological Interpretation and Physical Parameters

The fractional Maxwell model bridges elastic (spring), viscous (dashpot), and intermediate "spring-pot" components ("Scott–Blair" elements):

• Fractional order $\alpha$: Governs the degree of memory. $\alpha \to 0$ recovers ideal elasticity (Hookean spring), $\alpha \to 1$ yields Newtonian viscosity, and $0 < \alpha < 1$ produces continuous power-law relaxation (Pritchard et al., 2017).
• Viscoelastic spectrum: The model naturally describes materials with a broad or continuous distribution of relaxation times, as seen in polymers, biomaterials, and complex fluids (Ferrás et al., 2022).
• Parameter tuning: In the one-parameter model, $\nu$ modulates the transition between elasticity and fluidity:
  • As $\nu \to -1$, response approaches that of a pure spring.
  • As $\nu \to \infty$, unbounded creep and vanishing stress relaxation emerge (Colombaro et al., 2016).
• Memory kernels: Fractional derivatives introduce nonlocal-in-time kernels; e.g., Caputo or Hadamard kernels, with algebraic or logarithmic singularities, encode the detailed "memory" effect (Garra et al., 2022).

3. Material Functions: Relaxation, Creep, and Moduli

The central material functions (relaxation modulus, creep compliance, and complex moduli) have closed-form analytical expressions involving Mittag–Leffler or generalized Mittag–Leffler (Prabhakar) functions:

Model type	Relaxation modulus $G(t)$	Creep compliance $J(t)$
Fractional (Caputo)	$G(t)=G\,E_{\alpha,1}(-((t/\tau)^\alpha))$	$$J(t)=\displaystyle \frac{1}{G}\left[1 + \frac{(t/\tau)^{\alpha}}{\Gamma(1+\alpha)}\right]$$
One-param ($\nu$)	$G_{as}(t;\nu)=E_{1/2}(-2(\nu+1)t^{1/2})$	$$J_{as}(t;\nu)=1 + \dfrac{4(\nu+1)}{\sqrt{\pi} t^{1/2}}$$
Distributed order	$$G(t)=\int\phi(\alpha)\,t^{-\alpha}/\Gamma(1-\alpha)d\alpha$$	determined by weights $\phi(\alpha)$

• Laplace-domain representation: For constant $\alpha$, $$\widetilde G(s) = b_1\, s^{\alpha-1}/(1+a_1 s^\alpha)$$, with analytic inversion via Mittag–Leffler functions (Giusti et al., 2023).
• Oscillatory and SAOS properties: The storage $G'(\omega)$ and loss $G''(\omega)$ moduli under oscillatory shear are explicit, e.g.:

$$G^*(\omega) = \frac{G (i\omega \tau)^\alpha}{1 + (i\omega\tau)^\alpha}$$

where the phase angle and scaling with frequency directly encode the fractional order (Santelli et al., 2023).

4. Generalizations: Nonlinear, Nonlocal, and Multiphysics Extensions

• Nonlinear viscoelasticity: Embedding the fractional Maxwell model into RET (Rational Extended Thermodynamics) allows construction of nonlinear models whose relaxation solution coincides (for special viscous energies) with the Mittag–Leffler law of linear fractional Maxwell (Giusti et al., 7 Feb 2024).
• Electromagnetic field theory: Fractional Maxwell equations with Caputo (time) and Riesz (space) derivatives generalize electrodynamics, yielding models for anomalous wave propagation and memory in electromagnetic metamaterials. Nonlocality is intrinsic via convolutional field equations in both space and time (Nasrolahpour, 2022, Pérez et al., 2019, Heydeman et al., 2022).
• Geophysical applications: Fractional Maxwell rheology accurately models crustal and lithospheric relaxation after seismic events, with $\alpha$ tuned to fit geodetic and postseismic data, yielding fields with algebraic (power-law) relaxation tails and significantly altering long-term deformation (Mahato et al., 2 Jun 2025).
• Return-mapping and computational frameworks: Fractional viscoelasticity is incorporated within return-mapping algorithms for visco-elasto-plasticity, with efficient time-stepping (L1-schemes) and implicit updates accommodating full power-law memory (Suzuki et al., 2022).

5. Analytical, Numerical, and Well-Posedness Results

• Spectral and Laplace Techniques: Solutions utilize spectral expansions and Laplace inversion, taking advantage of the explicit Laplace images of fractional kernels and inversion via Mittag–Leffler functions (Heibig et al., 2011, Giusti et al., 2023).
• Numerical time-stepping: L1 discretization is standard for Caputo derivatives; such schemes are unconditionally stable and allow for history-efficient updates in time-fractional problems (Suzuki et al., 2022).
• Well-posedness and regularity: Existence and uniqueness of weak solutions for the objective three-dimensional fractional Maxwell fluid model are established under physically natural initial and boundary data, with precise regularity estimates depending on the smoothness of initial data (Heibig et al., 2011).
• Validation and applications: Smoothed Particle Hydrodynamics (SPH) and FDTD/spectral methods are used for direct simulation of fractional viscoelasticity and electrodynamics, capturing nonlocality, memory, and validating models against analytic predictions in both time and frequency domains (Santelli et al., 2023, Nasrolahpour, 2022).

6. Physical Implications, Parameter Estimation, and Domain-Specific Usage

• Biomechanics: Fractional exponents in the range $0.1 \lesssim \alpha \lesssim 0.7$ fit biological tissues exhibiting high-Q, weakly damped oscillations, extended power-law memory, and robustness across scales (Pritchard et al., 2017).
• Glassy and amorphous systems: Power-law and "ultraslow" relaxation captured via Hadamard or distributed kernels explain super-aging kinetics, broad relaxation peaks, and long-time dissipative behavior not accessible to integer-order models (Garra et al., 2022).
• Material parameter identification: Relaxation/creep fits, oscillatory tests, and time/frequency-domain measurements constrain $\alpha$, memory kernels, and moduli—optimally via Laplace/FFT-based fitting to data (Suzuki et al., 2022, Santelli et al., 2023).
• Electromagnetic wave phenomena: Fractional Maxwell equations describe anomalous attenuation, subdiffusive wave fronts, and the emergence of nonlocal flux tubes (Polyakov mechanism) in gauge-theoretic and condensed-matter settings (Heydeman et al., 2022, Nasrolahpour, 2022).

7. Limitations, Open Problems, and Future Developments

• Physical interpretation of fractional order: Unlike the single relaxation time in classical models, the meaning of $\alpha$ is material- and context-dependent, often emergent from fits to dynamics across time scales (Mahato et al., 2 Jun 2025).
• Memory kernel generality: While simple fractional models capture broad spectra, physical memory mechanisms may demand variable order, distributed, or Prabhakar-type kernels for full accuracy (Giusti et al., 2023, Giusti et al., 2017).
• Numerical complexity: History dependence and Mittag–Leffler evaluations incur computational costs; efficient schemes and model reduction remain active areas (Suzuki et al., 2022, Santelli et al., 2023).
• Nonlocality and coupling: Extending scalar viscoelastic models to frame-indifferent, tensorial, or multiphysics settings introduces challenges in objectivity, discretization, and interpretation of fractional operators (Heibig et al., 2011, Pérez et al., 2019).

The fractional Maxwell model and its generalizations provide a rigorous, flexible, and physically grounded rheological paradigm for modeling memory, relaxation, and nonlocal phenomena across materials science, soft matter, biomechanics, seismology, and electromagnetism, supported by both analytic theory and computational approaches (Colombaro et al., 2016, Giusti et al., 7 Feb 2024, Santelli et al., 2023, Mahato et al., 2 Jun 2025).