Four-Parameter Fractional Maxwell Model

• The four-parameter Fractional Maxwell Model is a viscoelastic framework that replaces integer-order operators with fractional ones to capture complex power-law behaviors.
• It unifies various parametrizations—including series Scott–Blair elements, two-springpot forms, fractional Zener, and Hadamard-type models—highlighting its flexible operator structure.
• The model underpins practical applications such as numerical fitting of hydrogels, wave propagation analysis, and stress relaxation studies in advanced materials.

The four-parameter Fractional Maxwell Model (FMM) is a class of Maxwell-type linear viscoelastic constitutive models in which one replaces the integer-order operators of classical spring–dashpot rheology by fractional operators, thereby obtaining relaxation, creep, and oscillatory responses governed by power laws or Mittag–Leffler functions rather than single exponentials. In the cited literature, the label refers to several closely related parametrizations: a series combination of two Scott–Blair elements with parameters $(\mathbb{E}_1,\mathbb{E}_2,\beta_1,\beta_2)$ (Suzuki et al., 2022), the equivalent two-springpot form $(\mathbb{V},\mathbb{G},\alpha,\beta)$ (Ferrás et al., 2022), a fractional Zener realization that is rheologically a fractional Maxwell branch in parallel with a spring (Nasholm et al., 2012), and modified Maxwell models in which the Caputo derivative is replaced by Prabhakar or Hadamard-type operators (Giusti et al., 2017, Garra et al., 2022).

1. Canonical meanings of “four-parameter FMM”

A standard and explicit four-parameter FMM is the series combination of two Scott–Blair elements. In the formulation used for fractional visco-elasto-plasticity, the constitutive equation is

$\sigma(t) + \frac{\mathbb{E}_2}{\mathbb{E}_1}\, \prescript{C}{0}{}\mathcal{D}_t^{\beta_2-\beta_1} \sigma(t) = \mathbb{E}_2\,\prescript{C}{0}{}\mathcal{D}_t^{\beta_2}\varepsilon(t), \qquad 0<\beta_1<\beta_2<1,$

so the four parameters are $\mathbb{E}_1,\mathbb{E}_2,\beta_1,\beta_2$ (Suzuki et al., 2022). In the distributed-order Maxwell literature, the same structure is written as

$$\sigma(t)+\frac{\mathbb{V}}{\mathbb{G}}\frac{d^{\alpha-\beta}\sigma(t)}{dt^{\alpha-\beta}} = \mathbb{V}\,\frac{d^{\alpha}\gamma(t)}{dt^{\alpha}}, \qquad 0<\beta\le \alpha<1,$$

with four parameters $(\mathbb{V},\mathbb{G},\alpha,\beta)$; the classical Maxwell model is recovered at $\alpha=1,\beta=0$ (Ferrás et al., 2022).

A second widespread meaning is the physically admissible four-parameter fractional Zener model,

$$\sigma(t) + \tau_{\epsilon}^{\alpha}\frac{\partial^{\alpha}\sigma(t)}{\partial t^{\alpha}} = E_0\left[\epsilon(t)+\tau_{\sigma}^{\alpha}\frac{\partial^{\alpha}\epsilon(t)}{\partial t^{\alpha}}\right],$$

with independent parameters $E_0,\tau_\sigma,\tau_\epsilon,\alpha$ when $\alpha=\beta$ (Nasholm et al., 2012). In that representation the model is a fractional Maxwell element in parallel with an additional spring, and it is therefore a four-parameter fractional Maxwell-type structure in the spring–dashpot sense (Nasholm et al., 2012).

A third four-parameter interpretation appears in modified fractional Maxwell models with nonstandard kernels. The Hadamard-type model has constitutive law

$(\mathbb{V},\mathbb{G},\alpha,\beta)$0

and the physically meaningful parameter set is $(\mathbb{V},\mathbb{G},\alpha,\beta)$1, where $(\mathbb{V},\mathbb{G},\alpha,\beta)$2 is the elastic modulus, $(\mathbb{V},\mathbb{G},\alpha,\beta)$3 the initial viscosity, $(\mathbb{V},\mathbb{G},\alpha,\beta)$4 the strain-hardening coefficient, and $(\mathbb{V},\mathbb{G},\alpha,\beta)$5 the fractional order (Garra et al., 2022).

Realization	Four parameters	Representative constitutive form
Two Scott–Blair elements in series	$(\mathbb{V},\mathbb{G},\alpha,\beta)$6	$(\mathbb{V},\mathbb{G},\alpha,\beta)$7
Two-springpot Maxwell form	$(\mathbb{V},\mathbb{G},\alpha,\beta)$8	$(\mathbb{V},\mathbb{G},\alpha,\beta)$9
Fractional Zener / Maxwell + spring	$\sigma(t) + \frac{\mathbb{E}_2}{\mathbb{E}_1}\, \prescript{C}{0}{}\mathcal{D}_t^{\beta_2-\beta_1} \sigma(t) = \mathbb{E}_2\,\prescript{C}{0}{}\mathcal{D}_t^{\beta_2}\varepsilon(t), \qquad 0<\beta_1<\beta_2<1,$0	$\sigma(t) + \frac{\mathbb{E}_2}{\mathbb{E}_1}\, \prescript{C}{0}{}\mathcal{D}_t^{\beta_2-\beta_1} \sigma(t) = \mathbb{E}_2\,\prescript{C}{0}{}\mathcal{D}_t^{\beta_2}\varepsilon(t), \qquad 0<\beta_1<\beta_2<1,$1
Modified Hadamard-type Maxwell	$\sigma(t) + \frac{\mathbb{E}_2}{\mathbb{E}_1}\, \prescript{C}{0}{}\mathcal{D}_t^{\beta_2-\beta_1} \sigma(t) = \mathbb{E}_2\,\prescript{C}{0}{}\mathcal{D}_t^{\beta_2}\varepsilon(t), \qquad 0<\beta_1<\beta_2<1,$2	$\sigma(t) + \frac{\mathbb{E}_2}{\mathbb{E}_1}\, \prescript{C}{0}{}\mathcal{D}_t^{\beta_2-\beta_1} \sigma(t) = \mathbb{E}_2\,\prescript{C}{0}{}\mathcal{D}_t^{\beta_2}\varepsilon(t), \qquad 0<\beta_1<\beta_2<1,$3

The coexistence of these formulations shows that “four-parameter FMM” is not a single fixed equation but a family of closely related Maxwell-type fractional models distinguished by how the four parameters are assigned to elastic scales, viscous scales, and fractional orders.

2. Fractional elements, series composition, and operator structure

The elementary building block of many FMMs is the Scott–Blair or spring-pot element. In the McKinley-type notation used for dual cross-linked hydrogels, a single element satisfies

$\sigma(t) + \frac{\mathbb{E}_2}{\mathbb{E}_1}\, \prescript{C}{0}{}\mathcal{D}_t^{\beta_2-\beta_1} \sigma(t) = \mathbb{E}_2\,\prescript{C}{0}{}\mathcal{D}_t^{\beta_2}\varepsilon(t), \qquad 0<\beta_1<\beta_2<1,$4

with $\sigma(t) + \frac{\mathbb{E}_2}{\mathbb{E}_1}\, \prescript{C}{0}{}\mathcal{D}_t^{\beta_2-\beta_1} \sigma(t) = \mathbb{E}_2\,\prescript{C}{0}{}\mathcal{D}_t^{\beta_2}\varepsilon(t), \qquad 0<\beta_1<\beta_2<1,$5; $\sigma(t) + \frac{\mathbb{E}_2}{\mathbb{E}_1}\, \prescript{C}{0}{}\mathcal{D}_t^{\beta_2-\beta_1} \sigma(t) = \mathbb{E}_2\,\prescript{C}{0}{}\mathcal{D}_t^{\beta_2}\varepsilon(t), \qquad 0<\beta_1<\beta_2<1,$6 gives a Hookean spring and $\sigma(t) + \frac{\mathbb{E}_2}{\mathbb{E}_1}\, \prescript{C}{0}{}\mathcal{D}_t^{\beta_2-\beta_1} \sigma(t) = \mathbb{E}_2\,\prescript{C}{0}{}\mathcal{D}_t^{\beta_2}\varepsilon(t), \qquad 0<\beta_1<\beta_2<1,$7 gives a dashpot with viscosity $\sigma(t) + \frac{\mathbb{E}_2}{\mathbb{E}_1}\, \prescript{C}{0}{}\mathcal{D}_t^{\beta_2-\beta_1} \sigma(t) = \mathbb{E}_2\,\prescript{C}{0}{}\mathcal{D}_t^{\beta_2}\varepsilon(t), \qquad 0<\beta_1<\beta_2<1,$8 (Dutta et al., 2 Sep 2025). The hydrogels paper builds the FMM by connecting two such elements in series, imposing common stress and additive strains,

$\sigma(t) + \frac{\mathbb{E}_2}{\mathbb{E}_1}\, \prescript{C}{0}{}\mathcal{D}_t^{\beta_2-\beta_1} \sigma(t) = \mathbb{E}_2\,\prescript{C}{0}{}\mathcal{D}_t^{\beta_2}\varepsilon(t), \qquad 0<\beta_1<\beta_2<1,$9

with exponents ordered as $\mathbb{E}_1,\mathbb{E}_2,\beta_1,\beta_2$0 (Dutta et al., 2 Sep 2025).

The same series rule underlies the four-parameter FM model in the return-mapping framework. Each element obeys $\mathbb{E}_1,\mathbb{E}_2,\beta_1,\beta_2$1, and elimination of the internal strains produces a single fractional differential equation for the total strain and common stress (Suzuki et al., 2022). This construction is the direct fractional analogue of the classical Maxwell series connection, except that the discrete spring/dashpot dichotomy is replaced by a continuum of intermediate rheological behaviors indexed by the orders $\mathbb{E}_1,\mathbb{E}_2,\beta_1,\beta_2$2.

In the Hadamard-type modification, the operator itself carries time dependence. The integer-order precursor is

$\mathbb{E}_1,\mathbb{E}_2,\beta_1,\beta_2$3

with linearly varying viscosity $\mathbb{E}_1,\mathbb{E}_2,\beta_1,\beta_2$4 (Garra et al., 2022). Its fractional generalization replaces $\mathbb{E}_1,\mathbb{E}_2,\beta_1,\beta_2$5 by a Hadamard-type fractional power $\mathbb{E}_1,\mathbb{E}_2,\beta_1,\beta_2$6, which is a Caputo-type derivative with respect to the function

$\mathbb{E}_1,\mathbb{E}_2,\beta_1,\beta_2$7

and therefore implements memory in logarithmic time (Garra et al., 2022). This changes the model from a fixed-kernel Caputo FMM to one in which memory and time-dependent viscosity are intertwined.

3. Relaxation, creep, and complex modulus

For the two-springpot FMM, the characteristic material functions are explicit. In normalized form, the relaxation modulus is

$\mathbb{E}_1,\mathbb{E}_2,\beta_1,\beta_2$8

and the creep compliance is

$\mathbb{E}_1,\mathbb{E}_2,\beta_1,\beta_2$9

In the frequency domain, the normalized complex compliance is

$$\sigma(t)+\frac{\mathbb{V}}{\mathbb{G}}\frac{d^{\alpha-\beta}\sigma(t)}{dt^{\alpha-\beta}} = \mathbb{V}\,\frac{d^{\alpha}\gamma(t)}{dt^{\alpha}}, \qquad 0<\beta\le \alpha<1,$$0

so the model exhibits two power-law regimes in both time and frequency (Ferrás et al., 2022).

The FM model of the return-mapping paper gives the same structural result in a non-normalized notation: $$\sigma(t)+\frac{\mathbb{V}}{\mathbb{G}}\frac{d^{\alpha-\beta}\sigma(t)}{dt^{\alpha-\beta}} = \mathbb{V}\,\frac{d^{\alpha}\gamma(t)}{dt^{\alpha}}, \qquad 0<\beta\le \alpha<1,$$1 The paper notes that this yields stretched-exponential behavior at short times and power-law behavior at long times (Suzuki et al., 2022).

In the hydrogel parametrization with shared modulus and time scale, the creep compliance is

$$\sigma(t)+\frac{\mathbb{V}}{\mathbb{G}}\frac{d^{\alpha-\beta}\sigma(t)}{dt^{\alpha-\beta}} = \mathbb{V}\,\frac{d^{\alpha}\gamma(t)}{dt^{\alpha}}, \qquad 0<\beta\le \alpha<1,$$2

while the relaxation modulus is

$$\sigma(t)+\frac{\mathbb{V}}{\mathbb{G}}\frac{d^{\alpha-\beta}\sigma(t)}{dt^{\alpha-\beta}} = \mathbb{V}\,\frac{d^{\alpha}\gamma(t)}{dt^{\alpha}}, \qquad 0<\beta\le \alpha<1,$$3

This representation is explicitly used to fit creep, stress relaxation, and oscillatory shear with the four parameters $$\sigma(t)+\frac{\mathbb{V}}{\mathbb{G}}\frac{d^{\alpha-\beta}\sigma(t)}{dt^{\alpha-\beta}} = \mathbb{V}\,\frac{d^{\alpha}\gamma(t)}{dt^{\alpha}}, \qquad 0<\beta\le \alpha<1,$$4 (Dutta et al., 2 Sep 2025).

A one-order Maxwell form, often used as a target or limiting case, is

$$\sigma(t)+\frac{\mathbb{V}}{\mathbb{G}}\frac{d^{\alpha-\beta}\sigma(t)}{dt^{\alpha-\beta}} = \mathbb{V}\,\frac{d^{\alpha}\gamma(t)}{dt^{\alpha}}, \qquad 0<\beta\le \alpha<1,$$5

which under constant strain gives

$$\sigma(t)+\frac{\mathbb{V}}{\mathbb{G}}\frac{d^{\alpha-\beta}\sigma(t)}{dt^{\alpha-\beta}} = \mathbb{V}\,\frac{d^{\alpha}\gamma(t)}{dt^{\alpha}}, \qquad 0<\beta\le \alpha<1,$$6

Its asymptotics are

$$\sigma(t)+\frac{\mathbb{V}}{\mathbb{G}}\frac{d^{\alpha-\beta}\sigma(t)}{dt^{\alpha-\beta}} = \mathbb{V}\,\frac{d^{\alpha}\gamma(t)}{dt^{\alpha}}, \qquad 0<\beta\le \alpha<1,$$7

and

$$\sigma(t)+\frac{\mathbb{V}}{\mathbb{G}}\frac{d^{\alpha-\beta}\sigma(t)}{dt^{\alpha-\beta}} = \mathbb{V}\,\frac{d^{\alpha}\gamma(t)}{dt^{\alpha}}, \qquad 0<\beta\le \alpha<1,$$8

so the decay is algebraic and the model has no finite equilibrium modulus in this pure Maxwell form (Giusti et al., 2024).

The modified Hadamard-type model changes the asymptotics qualitatively. Under constant strain, in normalized units $$\sigma(t)+\frac{\mathbb{V}}{\mathbb{G}}\frac{d^{\alpha-\beta}\sigma(t)}{dt^{\alpha-\beta}} = \mathbb{V}\,\frac{d^{\alpha}\gamma(t)}{dt^{\alpha}}, \qquad 0<\beta\le \alpha<1,$$9,

$(\mathbb{V},\mathbb{G},\alpha,\beta)$0

and

$(\mathbb{V},\mathbb{G},\alpha,\beta)$1

For $(\mathbb{V},\mathbb{G},\alpha,\beta)$2, the relaxation becomes

$(\mathbb{V},\mathbb{G},\alpha,\beta)$3

The paper explicitly analyzes relaxation and states that creep compliance is not derived (Garra et al., 2022).

4. Generalizations and unifying formulations

The four-parameter FMM sits inside several larger fractional-calculus frameworks. A particularly broad one is the Maxwell–Prabhakar model,

$(\mathbb{V},\mathbb{G},\alpha,\beta)$4

with kernel parameters $(\mathbb{V},\mathbb{G},\alpha,\beta)$5 and mechanical parameters $(\mathbb{V},\mathbb{G},\alpha,\beta)$6 (Giusti et al., 2017). In this formulation, the classical fractional Maxwell model is recovered either by setting $(\mathbb{V},\mathbb{G},\alpha,\beta)$7 or by setting $(\mathbb{V},\mathbb{G},\alpha,\beta)$8, since both reductions collapse the Prabhakar kernel to the Caputo one (Giusti et al., 2017). The same paper also shows formal equivalence, in Laplace-domain creep compliance, to fractional Voigt and fractional Zener models for specific parameter sets, making the Maxwell–Prabhakar equation a unifying super-structure rather than a single named FMM (Giusti et al., 2017).

Another extension is the generalized distributed-order Maxwell model. Its constitutive equation in Laplace form is

$(\mathbb{V},\mathbb{G},\alpha,\beta)$9

If the weighting functions are delta distributions,

$\alpha=1,\beta=0$0

then $\alpha=1,\beta=0$1, $\alpha=1,\beta=0$2, and the generalized model reduces exactly to the two-springpot FMM (Ferrás et al., 2022). The distributed-order formulation therefore replaces the two discrete fractional orders of the FMM by two continuous order spectra.

The Hadamard-type model provides a different generalization strategy. Instead of broadening the order distribution, it changes the time variable itself through

$\alpha=1,\beta=0$3

so that in the transformed variable the model behaves like a standard Caputo-type fractional Maxwell equation, while in physical time it produces ultra-slow, logarithmic relaxation (Garra et al., 2022). This suggests that some nonstandard four-parameter FMMs are best understood not as additional spring–dashpot branches but as Maxwell equations written in a non-Euclidean time scale.

5. Admissibility, network interpretation, and wave propagation

Physical admissibility is a central issue for four-parameter fractional Maxwell-type models. For the fractional Zener form,

$\alpha=1,\beta=0$4

the admissible regime is

$\alpha=1,\beta=0$5

which ensures thermodynamic constraints and monotonic relaxation (Nasholm et al., 2012). In the frequency domain its complex modulus is

$\alpha=1,\beta=0$6

with

$\alpha=1,\beta=0$7

This is precisely the modulus of a fractional Maxwell branch in parallel with a spring (Nasholm et al., 2012).

The same paper also connects the model to a Maxwell–Wiechert continuum. For the admissible case $\alpha=1,\beta=0$8, the generalized compressibility is equivalent to a continuous distribution of Maxwell elements with spectrum

$\alpha=1,\beta=0$9

which has three asymptotic power-law regions (Nasholm et al., 2012). In wave propagation this yields three attenuation regimes,

$$\sigma(t) + \tau_{\epsilon}^{\alpha}\frac{\partial^{\alpha}\sigma(t)}{\partial t^{\alpha}} = E_0\left[\epsilon(t)+\tau_{\sigma}^{\alpha}\frac{\partial^{\alpha}\epsilon(t)}{\partial t^{\alpha}}\right],$$0

at low, intermediate, and high frequencies, respectively (Nasholm et al., 2012). This is one reason the four-parameter Maxwell/Zener family is used in acoustics, elastography, and medical applications (Nasholm et al., 2012).

A different admissibility result appears in the nonlinear Rational Extended Thermodynamics embedding of the fractional Maxwell law. There, exact coincidence with the fractional Maxwell relaxation

$$\sigma(t) + \tau_{\epsilon}^{\alpha}\frac{\partial^{\alpha}\sigma(t)}{\partial t^{\alpha}} = E_0\left[\epsilon(t)+\tau_{\sigma}^{\alpha}\frac{\partial^{\alpha}\epsilon(t)}{\partial t^{\alpha}}\right],$$1

is obtained for a special viscous energy $$\sigma(t) + \tau_{\epsilon}^{\alpha}\frac{\partial^{\alpha}\sigma(t)}{\partial t^{\alpha}} = E_0\left[\epsilon(t)+\tau_{\sigma}^{\alpha}\frac{\partial^{\alpha}\epsilon(t)}{\partial t^{\alpha}}\right],$$2, constant strain $$\sigma(t) + \tau_{\epsilon}^{\alpha}\frac{\partial^{\alpha}\sigma(t)}{\partial t^{\alpha}} = E_0\left[\epsilon(t)+\tau_{\sigma}^{\alpha}\frac{\partial^{\alpha}\epsilon(t)}{\partial t^{\alpha}}\right],$$3, and initial condition $$\sigma(t) + \tau_{\epsilon}^{\alpha}\frac{\partial^{\alpha}\sigma(t)}{\partial t^{\alpha}} = E_0\left[\epsilon(t)+\tau_{\sigma}^{\alpha}\frac{\partial^{\alpha}\epsilon(t)}{\partial t^{\alpha}}\right],$$4 (Giusti et al., 2024). The same work shows that bounded viscous energy requires $$\sigma(t) + \tau_{\epsilon}^{\alpha}\frac{\partial^{\alpha}\sigma(t)}{\partial t^{\alpha}} = E_0\left[\epsilon(t)+\tau_{\sigma}^{\alpha}\frac{\partial^{\alpha}\epsilon(t)}{\partial t^{\alpha}}\right],$$5, while the broader compatibility range is $$\sigma(t) + \tau_{\epsilon}^{\alpha}\frac{\partial^{\alpha}\sigma(t)}{\partial t^{\alpha}} = E_0\left[\epsilon(t)+\tau_{\sigma}^{\alpha}\frac{\partial^{\alpha}\epsilon(t)}{\partial t^{\alpha}}\right],$$6, with $$\sigma(t) + \tau_{\epsilon}^{\alpha}\frac{\partial^{\alpha}\sigma(t)}{\partial t^{\alpha}} = E_0\left[\epsilon(t)+\tau_{\sigma}^{\alpha}\frac{\partial^{\alpha}\epsilon(t)}{\partial t^{\alpha}}\right],$$7 giving the classical Maxwell model (Giusti et al., 2024). This locates the FMM inside a nonlinear, thermodynamically structured theory rather than treating it solely as an empirical kernel.

6. Computation, fitting, applications, and unresolved points

For numerical implementation, the return-mapping framework for fractional visco-elasto-plasticity discretizes the Caputo operators with the implicit L1 scheme and derives a discrete constitutive update for the FM model,

$$\sigma(t) + \tau_{\epsilon}^{\alpha}\frac{\partial^{\alpha}\sigma(t)}{\partial t^{\alpha}} = E_0\left[\epsilon(t)+\tau_{\sigma}^{\alpha}\frac{\partial^{\alpha}\epsilon(t)}{\partial t^{\alpha}}\right],$$8

with model-dependent projection term

$$\sigma(t) + \tau_{\epsilon}^{\alpha}\frac{\partial^{\alpha}\sigma(t)}{\partial t^{\alpha}} = E_0\left[\epsilon(t)+\tau_{\sigma}^{\alpha}\frac{\partial^{\alpha}\epsilon(t)}{\partial t^{\alpha}}\right],$$9

in the plastic correction phase (Suzuki et al., 2022). The framework is fully implicit for linear viscoelastic models, semi-implicit for the quasi-linear case, and the reported numerical behavior is “at least first-order accurate for general loading conditions,” with a “reduction of $E_0,\tau_\sigma,\tau_\epsilon,\alpha$0 in CPU time” relative to existing approaches in the visco-plastic range (Suzuki et al., 2022).

The hydrogel study provides a direct materials application of a four-parameter FMM. The model is fitted simultaneously to stress relaxation, creep, and oscillatory shear using the parameters $E_0,\tau_\sigma,\tau_\epsilon,\alpha$1, with creep and stress relaxation fitted over $E_0,\tau_\sigma,\tau_\epsilon,\alpha$2 to $E_0,\tau_\sigma,\tau_\epsilon,\alpha$3 s and oscillatory shear over $E_0,\tau_\sigma,\tau_\epsilon,\alpha$4 to $E_0,\tau_\sigma,\tau_\epsilon,\alpha$5 rad/s (Dutta et al., 2 Sep 2025). The cost function is a weighted sum of logarithmic least-squares residuals with

$E_0,\tau_\sigma,\tau_\epsilon,\alpha$6

and optimization is performed in Excel using the Generalized Reduced Gradient method (Dutta et al., 2 Sep 2025). For the PMA–Fe$E_0,\tau_\sigma,\tau_\epsilon,\alpha$7 dual-crosslinked hydrogels, $E_0,\tau_\sigma,\tau_\epsilon,\alpha$8 increases from $E_0,\tau_\sigma,\tau_\epsilon,\alpha$9 to $\alpha=\beta$0 as salt concentration rises from $\alpha=\beta$1 to $\alpha=\beta$2 M, $\alpha=\beta$3 decreases from about $\alpha=\beta$4 to about $\alpha=\beta$5, $\alpha=\beta$6 is approximately $\alpha=\beta$7–$\alpha=\beta$8 at low/intermediate salt and about $\alpha=\beta$9 at $(\mathbb{V},\mathbb{G},\alpha,\beta)$00 M, and $(\mathbb{V},\mathbb{G},\alpha,\beta)$01 remains in the range $(\mathbb{V},\mathbb{G},\alpha,\beta)$02–$(\mathbb{V},\mathbb{G},\alpha,\beta)$03 (Dutta et al., 2 Sep 2025).

Several unresolved points recur across the literature. In the hydrogel application, “The physical meaning of the two Scott Blair elements is, at present, somewhat obscure,” and the suggested mapping to covalent and ionic subnetworks is explicitly described as tentative (Dutta et al., 2 Sep 2025). In the Maxwell–Prabhakar model, a complete characterization of physically acceptable parameter sets is “left for future work” (Giusti et al., 2017). In the Hadamard-type model, only relaxation is worked out explicitly, and no creep formula is written (Garra et al., 2022). In the wave-propagation literature, the microscopic physical origins of fractional behavior are described as still debated, even though the Maxwell–Wiechert equivalence gives a concrete interpretation in terms of many relaxation processes with a power-law distribution of relaxation times (Nasholm et al., 2012).

Taken together, these results place the four-parameter FMM at the center of modern fractional rheology: it is simultaneously a compact constitutive model, a special case of broader operator families, a reduced representation of distributed Maxwell networks, and a practical fitting model for materials with broad relaxation spectra. Its precise mathematical form varies across subfields, but the persistent features are a Maxwell backbone, four independent constitutive parameters, and non-exponential memory encoded by fractional calculus.