Fractional Viscoelastic Models: Equation Verification Report

Date: 2026-03-30 Sources: Schiessel et al. (1995), Jaishankar & McKinley (2013), Bonfanti et al. (2020), Stankiewicz (2018), Eldred et al. (2015/PMC4658031)

1. Springpot (Scott Blair Element)

User’s equations

  • Constitutive: sigma(t) = V * (d^alpha gamma / dt^alpha)

  • Complex modulus: G*(omega) = V * (i*omega)^alpha

  • V is quasi-property with units [Pa*s^alpha], alpha in [0,1]

  • alpha=0 -> spring (sigma = Ggamma), alpha=1 -> dashpot (sigma = etagamma_dot)

Literature verification

CONFIRMED. Multiple sources agree exactly.

  • Bonfanti et al. (2020, Soft Matter 16, 6002-6020), Eq. (15): The springpot constitutive equation is sigma(t) = c_alpha * D^alpha [epsilon(t)], where D^alpha is the Caputo fractional derivative. The parameter c_alpha (= V in your notation) has units [Pa*s^alpha].

  • Stankiewicz (2018, BIO Web Conf. 10, 02032), Eq. (2): sigma(t) = Etau^alpha * d^alpha epsilon / dt^alpha. Here Etau^alpha plays the role of V (quasi-property). The Scott Blair element interpolates between Hooke spring (alpha=0) and Newton dashpot (alpha=1).

  • Jaishankar & McKinley (2013, Proc. R. Soc. A 469, 20120284): Introduced the term “quasi-property” for V with dimensions [Pas^alpha]. Complex modulus G(omega) = V*(i*omega)^alpha confirmed.

Verdict: CORRECT as stated.

2. Fractional Maxwell Model (FMM) – Two Springpots in Series

User’s equation

G*(omega) = [V1*(iomega)^alpha * V2(iomega)^beta] / [V1(iomega)^alpha + V2(i*omega)^beta]

Literature verification

CONFIRMED. This is the standard series combination rule applied to two springpots.

  • Stankiewicz (2018), Eq. (6): The FMM constitutive equation with two Scott Blair elements (E1, tau1, alpha) and (E2, tau2, beta) in series yields the fractional differential equation: tau^(alpha-beta) * D^(alpha-beta)[sigma] + sigma = E*tau^alpha * D^alpha[epsilon]

  • The complex modulus follows from series combination: for two elements with individual G1*(omega) = V1*(iomega)^alpha and G2(omega) = V2*(iomega)^beta, the series rule gives: 1/G = 1/G1* + 1/G2*, hence G* = (G1* * G2*)/(G1* + G2*).

  • Bonfanti et al. (2020): Confirms that the FMM connects two springpots in series with 0 <= beta < alpha <= 1. The more elastic springpot (lower exponent beta) governs high-frequency behavior; the more viscous one (higher alpha) governs low-frequency behavior.

  • Equivalent parametrization (Stankiewicz 2018): G*(omega) = E*(iomegatau)^alpha / [1 + (iomegatau)^(alpha-beta)] which is algebraically equivalent when V1 = Etau^alpha, V2 = Etau^beta, and appropriate substitutions.

Verdict: CORRECT as stated.

3. Fractional Kelvin-Voigt (FKV) – Two Springpots in Parallel

User’s equation

G*(omega) = V1*(iomega)^alpha + V2(i*omega)^beta

Literature verification

CONFIRMED.

  • Bonfanti et al. (2020), Eq. (28): G*(omega) = A*(iomega)^alpha + B(i*omega)^beta, described as “exactly equivalent to the complex modulus of the fractional Kelvin-Voigt model consisting of two springpots in parallel.”

  • RheoJAX docs (FKV model): G*(omega) = G_e + c_alpha*(i*omega)^alpha, which is the special case where one element is a pure spring (beta=0, V2=G_e).

  • Eldred et al. (PMC4658031): FKVM2 (fractional Kelvin-Voigt with two springpots) uses this parallel combination.

Verdict: CORRECT as stated.

4. Fractional Zener (Standard Linear Solid)

User’s description

Various configurations with springpots and springs.

Literature verification

CONFIRMED – multiple variants exist.

  • Bonfanti et al. (2020), Eq. (32): The fractional standard linear solid (Zener) has relaxation modulus involving the Mittag-Leffler function. Configuration: spring in parallel with a fractional Maxwell arm (spring + springpot in series).

  • RheoJAX implements three Zener variants:

    • FZ-SS (Solid-Solid): Spring || (Spring – SpringPot). G(t) = G_e + G_m * E_alpha(-(t/tau)^alpha)

    • FZ-SL (Solid-Liquid): Spring || (SpringPot – Dashpot). Terminal flow behavior.

    • FZ-LL (Liquid-Liquid): Dashpot || (SpringPot – Dashpot). Double flow.

  • Schiessel, Metzler, Blumen & Nonnenmacher (1995, J. Phys. A 28, 6567-6584): Systematic construction of fractional Zener and Poynting-Thomson models from springpot elements.

Verdict: CORRECT framework. Multiple valid configurations.

5. Relaxation Modulus for FMM

User’s equation

G(t) = G * E_alpha(-(t/tau)^alpha), where E_alpha is the Mittag-Leffler function.

Literature verification

PARTIALLY CORRECT – applies to the special case FMM with one springpot (FMM1), not the general two-springpot FMM.

  • For the general FMM (two springpots, orders alpha and beta), from Stankiewicz (2018), Eqs. (7)-(9): G(t) = E * (t/tau)^{-beta} * E_{alpha-beta, 1-beta}(-(t/tau)^{alpha-beta}) where E_{a,b}(z) is the TWO-parameter Mittag-Leffler function.

  • For the special case where the lower-order element is a spring (beta=0), this reduces to: G(t) = G * E_alpha(-(t/tau)^alpha) which is exactly the user’s formula. This is the Fractional Maxwell Liquid (FML) model.

  • Bonfanti et al. (2020): Confirms that the FMM relaxation modulus involves E_{alpha-beta, 1-beta}, exhibiting stretched-exponential (KWW) at short times and power-law at long times.

Verdict: CORRECT for FML (beta=0 case). For the general two-springpot FMM, the relaxation modulus is G(t) = E(t/tau)^{-beta} * E_{alpha-beta, 1-beta}(-(t/tau)^{alpha-beta}).*

6. Mittag-Leffler Function

User’s equation

E_{alpha,beta}(z) = sum_{k=0}^{inf} z^k / Gamma(alpha*k + beta) Special case: E_{1,1}(z) = e^z

Literature verification

CONFIRMED.

  • Stankiewicz (2018), Eq. (8): E_{phi,mu}(z) = sum_{k=0}^{inf} z^k / Gamma(phi*k + mu) Identical to user’s definition.

  • RheoJAX docs (FML model): E_{alpha,beta}(z) = sum_{k=0}^{inf} z^k / Gamma(alpha*k + beta). “This generalization of the exponential function is essential for fractional viscoelasticity.” E_{1,1}(z) = exp(z) confirmed.

  • One-parameter form: E_alpha(z) = E_{alpha,1}(z) = sum_{k=0}^{inf} z^k / Gamma(alpha*k + 1).

Verdict: CORRECT as stated.

7. Creep Compliance for FMM

User’s equation

J(t) = (1/G) * t^alpha / Gamma(1+alpha), or involves Mittag-Leffler.

Literature verification

PARTIALLY CORRECT – applies to the springpot element, not the full FMM.

  • For a single springpot: J(t) = (1/V) * t^alpha / Gamma(1+alpha). This is the creep compliance of the Scott Blair element alone. CONFIRMED.

  • For the FML (spring + springpot in series, beta=0): J(t) = (1/G) * t^alpha * E_{alpha, 1+alpha}((t/tau)^alpha) from RheoJAX docs. This involves the two-parameter Mittag-Leffler function.

  • For the FKV (spring || springpot): J(t) = (1/G_e) * [1 - E_alpha(-(t/tau_eps)^alpha)] where tau_eps = (c_alpha/G_e)^{1/alpha}. From RheoJAX docs.

  • For the general FMM (two springpots): J(t) = (1/E) * (t/tau)^alpha * E_{alpha-beta, 1+alpha}((t/tau)^{alpha-beta})

*Verdict: The formula J(t) = (1/G)t^alpha/Gamma(1+alpha) is the springpot-only creep compliance. The full FMM creep compliance involves Mittag-Leffler as noted.

Summary Table

Model/Equation

User’s Form

Status

Notes

Springpot constitutive

sigma = V*D^alpha[gamma]

CORRECT

Caputo derivative, V = quasi-property

Springpot G*(omega)

V*(i*omega)^alpha

CORRECT

Jaishankar & McKinley (2013)

FMM G*(omega)

V1V2(iw)^{a+b} / (V1*(iw)^a + V2*(iw)^b)

CORRECT

Series combination rule

FKV G*(omega)

V1*(iw)^a + V2*(iw)^b

CORRECT

Parallel combination

Fractional Zener

Various springpot+spring configs

CORRECT

Multiple variants (SS, SL, LL)

FMM G(t)

G*E_alpha(-(t/tau)^alpha)

PARTIALLY

Only for FML (beta=0). General: involves E_{a-b,1-b}

Mittag-Leffler

sum z^k/Gamma(ak+b)

CORRECT

E_{1,1}=exp confirmed

FMM J(t)

(1/G)*t^a/Gamma(1+a)

PARTIALLY

Springpot-only. Full FMM uses E_{a-b,1+a}

Key References

  1. Schiessel H, Metzler R, Blumen A, Nonnenmacher TF. “Generalized viscoelastic models: their fractional equations with solutions.” J. Phys. A: Math. Gen. 28, 6567-6584 (1995). DOI: 10.1088/0305-4470/28/23/012

  2. Jaishankar A, McKinley GH. “Power-law rheology in the bulk and at the interface: quasi-properties and fractional constitutive equations.” Proc. R. Soc. A 469, 20120284 (2013). DOI: 10.1098/rspa.2012.0284

  3. Bonfanti A, Kaplan JL, Charras G, Kabla A. “Fractional viscoelastic models for power-law materials.” Soft Matter 16, 6002-6020 (2020). DOI: 10.1039/D0SM00354A

  4. Stankiewicz A. “Fractional Maxwell model of viscoelastic biological materials.” BIO Web Conf. 10, 02032 (2018). DOI: 10.1051/bioconf/20181002032

  5. Eldred et al. “Fractional Generalizations of Maxwell and Kelvin-Voigt Models for Biopolymer Characterization.” PLoS ONE 10(11), e0143090 (2015). DOI: 10.1371/journal.pone.0143090

  6. Bagley RL, Torvik PJ. “A theoretical basis for the application of fractional calculus to viscoelasticity.” J. Rheol. 27, 201-210 (1983).

  7. Scott Blair GW. “The role of psychophysics in rheology.” J. Colloid Sci. 2, 21-32 (1947).