Fractional Maxwell Models in Viscoelasticity

• Fractional Maxwell models are generalized viscoelastic frameworks that use fractional derivatives to capture power-law memory and anomalous relaxation effects.
• They provide closed-form solutions for creep, relaxation, and effective viscosity, enhancing the modeling of polymers, biological tissues, and geomaterials.
• Extensions like variable-order, distributed-order, and infinite-order models broaden their applicability in simulating complex mechanical responses and multi-physics coupling.

A fractional Maxwell model is a generalization of the classical Maxwell viscoelastic model, formulated by replacing integer-order time derivatives in the constitutive stress–strain law with derivatives of fractional order. This approach captures the power-law memory effects and anomalous relaxation dynamics observed in complex materials such as polymers, biological tissues, geomaterials, and soft matter. Fractional Maxwell models have become a fundamental analytical tool for rheology, time-domain and frequency-domain behavior of viscoelastic media, wave propagation, micromechanics, and multi-physics coupling.

1. Mathematical Formulation

Let $\sigma(t)$ denote the stress and $\varepsilon(t)$ the strain. The classical Maxwell model is given by: $$\sigma(t) + \lambda \frac{d\sigma(t)}{dt} = E \frac{d\varepsilon(t)}{dt},$$ where $E$ is the spring modulus, $\lambda$ is a relaxation time, and the dashpot viscosity is $\eta = E\lambda$.

The fractional Maxwell model replaces the first-order derivatives with Caputo (or Riemann–Liouville) fractional derivatives of order $\alpha \in (0,1]$: $$\sigma(t) + a_1\, {}^C D^{\alpha} \sigma(t) = b_1\, {}^C D^{\alpha} \varepsilon(t),$$ with

$${}^C D^{\alpha} f(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t (t-\tau)^{-\alpha} f'(\tau)\, d\tau,$$

where $a_1, b_1 > 0$ are material constants, and $\varepsilon(t)$0 is the gamma function. Using

$\varepsilon(t)$1

the Laplace-domain complex shear modulus is

$\varepsilon(t)$2

The classical case is recovered for $\varepsilon(t)$3 (Mainardi et al., 2011).

2. Linear Rheology: Creep, Relaxation, and Viscosity

The fractional Maxwell model admits closed-form expressions for the principal rheological functions.

Creep Compliance:

$\varepsilon(t)$4

shows sub-linear power-law creep for $\varepsilon(t)$5. As $\varepsilon(t)$6, $\varepsilon(t)$7; as $\varepsilon(t)$8, $\varepsilon(t)$9 (Mainardi et al., 2011).

Relaxation Modulus:

$$\sigma(t) + \lambda \frac{d\sigma(t)}{dt} = E \frac{d\varepsilon(t)}{dt},$$0

where $$\sigma(t) + \lambda \frac{d\sigma(t)}{dt} = E \frac{d\varepsilon(t)}{dt},$$1 is the one-parameter Mittag–Leffler function. For $$\sigma(t) + \lambda \frac{d\sigma(t)}{dt} = E \frac{d\varepsilon(t)}{dt},$$2, $$\sigma(t) + \lambda \frac{d\sigma(t)}{dt} = E \frac{d\varepsilon(t)}{dt},$$3; for $$\sigma(t) + \lambda \frac{d\sigma(t)}{dt} = E \frac{d\varepsilon(t)}{dt},$$4, $$\sigma(t) + \lambda \frac{d\sigma(t)}{dt} = E \frac{d\varepsilon(t)}{dt},$$5 exhibits slow, power-law relaxation (Mainardi et al., 2011).

Effective Viscosity:

$$\sigma(t) + \lambda \frac{d\sigma(t)}{dt} = E \frac{d\varepsilon(t)}{dt},$$6

shows that for $$\sigma(t) + \lambda \frac{d\sigma(t)}{dt} = E \frac{d\varepsilon(t)}{dt},$$7, viscosity grows with time (“solidification”), while for $$\sigma(t) + \lambda \frac{d\sigma(t)}{dt} = E \frac{d\varepsilon(t)}{dt},$$8 (classical limit) it becomes constant (Mainardi et al., 2011).

3. Extensions and Generalizations

3.1. Multi-Parameter Fractional Maxwell and Generalized Models

Two spring-pots (Scott-Blair elements) in series:

$$\sigma(t) + \lambda \frac{d\sigma(t)}{dt} = E \frac{d\varepsilon(t)}{dt},$$9

covers the full spectrum from purely elastic to viscous-inertial, with relaxation modulus (2002.04581, Santelli et al., 2023): $E$0 with $E$1 the two-parameter Mittag–Leffler function.

Distributed-order models:

$E$2

where $E$3 is a weighting function, realizes continuous spectra of memory kernels suited to complex hierarchical materials (Ferrás et al., 2022).

Variable-order and Prabhakar models:

Replacement of the constant order $E$4 by $E$5 (Scarpi operator) or by Prabhakar fractional derivatives further generalizes the kernel, enabling time-evolving memory effects, multimodal response, and interpolation between exponential and power-law relaxation (Giusti et al., 2017, Giusti et al., 2023).

3.2. Modified and Infinite-Order Models

Bessel-based infinite-order models: Constitutive laws written as infinite sums of higher-order (fractional and integer) derivatives, e.g., via Bessel function generating series, with fractional Maxwell scaling at intermediate times (Giusti, 2017).

Maxwell models with Hadamard-type derivatives: Inclusion of a logarithmic kernel encodes both memory and time-dependent viscosity, leading to ultra-slow, logarithmic relaxation (Garra et al., 2022).

4. Wave Propagation, Oscillations, and Physical Interpretation

4.1. Viscoelastic Wave Propagation

The fractional Maxwell model in wave propagation leads to: $E$6 so frequency-dependent complex moduli, phase velocities, attenuation, and quality factors exhibit anomalous dispersion and attenuation directly governed by $E$7 (Brown et al., 2018, Mahato et al., 2 Jun 2025). Fractional models are essential in geophysical applications, such as deformation and stress relaxation in the lithosphere (Mahato et al., 2 Jun 2025).

4.2. Oscillatory and Nonlinear Regimes

Oscillatory excitation of fractional Maxwell elements yields broadened damping windows and nonclassical critical damping criteria (Pritchard et al., 2017). The power-law relaxation and creep impart non-exponential, scale-free behavior. Coupling to nonlinear thermodynamic frameworks (e.g., RET) enables embedding power-law anomalous relaxation in thermodynamically sound, nonlinear viscoelasticity (Giusti et al., 2024).

5. Multi-Mode, Numerical, and Computational Approaches

Practical simulations employ multi-mode fractional Maxwell models, involving either additive branches in parallel/series or distributed-order formulations to fit experimental data over decades of timescales. Direct Lagrangian integration using Smoothed Particle Hydrodynamics (SPH), history buffer schemes, and convolution quadrature are state-of-the-art for evaluating weakly-singular memory integrals in complex geometries and loading histories (Santelli et al., 2023, Suzuki et al., 2022). Time-discrete schemes (e.g., L1 approximation) enable efficient implementation within visco-elasto-plastic return-mapping solvers (Suzuki et al., 2022).

6. Physical Interpretation and Parameter Selection

The fractional order $E$8 governs the material memory:

• $E$9 approaches purely elastic solid (no permanent flow).
• $\lambda$0 reproduces the classical Maxwell model (exponential relaxation, steady viscous flow).
• $\lambda$1 exhibits power-law creep and relaxation, empirically matching many polymers, biomaterials, and rocks (Mainardi et al., 2011, Bonfanti et al., 2020).

Distributed and variable-order extensions give additional flexibility, critical for fitting materials with broad relaxation spectra or structural evolution.

7. Applications and Significance

Fractional Maxwell models underpin the characterization of viscoelastic power-law rheology in soft condensed matter, complex fluids, geomechanics, and engineered meta-materials. They are essential for realistic modeling of time- and frequency-dependent mechanical responses, attenuation and dispersion in wave propagation, nonlinear relaxation phenomena, as well as circuit analogs and electromagnetic field memory (Mainardi et al., 2011, Moreles et al., 2016, Giusti, 2017, Mahato et al., 2 Jun 2025).

Their analytical tractability, closed-form solutions, and ability to capture continuous spectral relaxation make them indispensable for contemporary rheological modeling well beyond the limits of classical integer-order theories.