---
title: Fractional Maxwell Models in Viscoelasticity
url: https://www.emergentmind.com/topics/fractional-maxwell-models
type: topic
---

# Fractional Maxwell Models in Viscoelasticity

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 $\Gamma(\cdot)$ is the gamma function. Using
\[
\mu := b_1/(2a_1), \qquad \tau^\alpha := b_1/(2\mu),
\]
the Laplace-domain complex shear modulus is
\[
\widetilde G(s) = \frac{2\mu}{s} \frac{(s\tau)^{\alpha}}{1 + (s\tau)^{\alpha}}.
\]
The classical case is recovered for $\alpha = 1$ [1110.3400].

## 2. Linear Rheology: Creep, Relaxation, and Viscosity

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

**Creep Compliance:**
\[
J(t) = \frac{1}{2\mu} \left[ 1 + \frac{(t/\tau)^{\alpha}}{\Gamma(1+\alpha)} \right]
\]
shows sub-linear power-law creep for $0<\alpha<1$. As $t \to 0^+$, $J(t) \to 1/(2\mu)$; as $t \to \infty$, $J(t) \sim t^{\alpha}/(2\mu \Gamma(1+\alpha)\tau^{\alpha})$ [1110.3400].

**Relaxation Modulus:**
\[
G(t) = 2\mu\, E_{\alpha}\left(-\left(\frac{t}{\tau}\right)^{\alpha}\right),
\]
where $E_{\alpha}(z)$ is the one-parameter Mittag–Leffler function. For $t \to 0^+$, $G(0^+)=2\mu$; for $t \to \infty$, $G(t) \sim t^{-\alpha}$ exhibits slow, power-law relaxation [1110.3400].

**Effective Viscosity:**
\[
\eta_{\text{eff}}(t) = \frac{\Gamma(1+\alpha)}{\alpha} \mu \left(\frac{t}{\tau}\right)^{1-\alpha}
\]
shows that for $\alpha<1$, viscosity grows with time (“solidification”), while for $\alpha \to 1$ (classical limit) it becomes constant [1110.3400].

## 3. Extensions and Generalizations

### 3.1. Multi-Parameter Fractional Maxwell and Generalized Models

**Two spring-pots (Scott-Blair elements) in series:**
\[
\sigma(t) + \lambda_r\, D^{r} \sigma(t) = K_{\beta}\, D^{\beta}\varepsilon(t), \quad r = \beta-\alpha, \quad 0 < \alpha < \beta < 2,
\]
covers the full spectrum from purely elastic to viscous-inertial, with relaxation modulus [2002.04581, 2311.13173]:
\[
G(t) = K_\alpha\, t^{-\alpha} E_{\beta-\alpha, 1-\alpha}\left( -\frac{K_\alpha}{K_\beta} t^{\beta-\alpha} \right),
\]
with $E_{a,b}(z)$ the two-parameter Mittag–Leffler function.

**Distributed-order models:**
\[
\int_{0}^{1} c(\alpha)\, D_t^\alpha[\sigma(t)]\, d\alpha = E\int_{0}^1 c(\alpha) D_t^\alpha[\varepsilon(t)]\, d\alpha,
\]
where $c(\alpha)$ is a weighting function, realizes continuous spectra of memory kernels suited to complex hierarchical materials [2202.05878].

**Variable-order and Prabhakar models:**
Replacement of the constant order $\alpha$ by $\alpha(t)$ (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 [1705.09246, 2311.16305].

### 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 [1701.06350].

**Maxwell models with Hadamard-type derivatives**: Inclusion of a logarithmic kernel encodes both memory and time-dependent viscosity, leading to ultra-slow, logarithmic relaxation [2204.02783].

## 4. Wave Propagation, Oscillations, and Physical Interpretation

### 4.1. Viscoelastic Wave Propagation

The fractional Maxwell model in wave propagation leads to:
\[
\Sigma(s) = \frac{s^{\alpha}}{1+\tau_M s^{\alpha}} E(s),
\]
so frequency-dependent complex moduli, phase velocities, attenuation, and quality factors exhibit anomalous dispersion and attenuation directly governed by $\alpha$ [1802.00825, 2506.04257]. Fractional models are essential in geophysical applications, such as deformation and stress relaxation in the lithosphere [2506.04257].

### 4.2. Oscillatory and Nonlinear Regimes

Oscillatory excitation of fractional Maxwell elements yields broadened damping windows and nonclassical critical damping criteria [1701.02155]. 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 [2402.04969].

## 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 [2311.13173, 2210.01308]. Time-discrete schemes (e.g., L1 approximation) enable efficient implementation within visco-elasto-plastic return-mapping solvers [2210.01308].

## 6. Physical Interpretation and Parameter Selection

The fractional order $\alpha$ governs the material memory:

- $\alpha \to 0$ approaches purely elastic solid (no permanent flow).
- $\alpha \to 1$ reproduces the classical Maxwell model (exponential relaxation, steady viscous flow).
- $0<\alpha<1$ exhibits power-law creep and relaxation, empirically matching many polymers, biomaterials, and rocks [1110.3400, 2003.07834].

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 [1110.3400, 1602.03541, 1701.06350, 2506.04257].

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.

Source: https://www.emergentmind.com/topics/fractional-maxwell-models