# 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 = G*gamma), alpha=1 -> dashpot (sigma = eta*gamma_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) = E*tau^alpha * d^alpha epsilon / dt^alpha. Here E*tau^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 [Pa*s^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*(i*omega)^alpha * V2*(i*omega)^beta] / [V1*(i*omega)^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*(i*omega)^alpha and G2*(omega) = V2*(i*omega)^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*(i*omega*tau)^alpha / [1 + (i*omega*tau)^(alpha-beta)] which is algebraically equivalent when V1 = E*tau^alpha, V2 = E*tau^beta, and appropriate substitutions. **Verdict: CORRECT as stated.** --- ## 3. Fractional Kelvin-Voigt (FKV) -- Two Springpots in Parallel ### User's equation G*(omega) = V1*(i*omega)^alpha + V2*(i*omega)^beta ### Literature verification **CONFIRMED.** - Bonfanti et al. (2020), Eq. (28): G*(omega) = A*(i*omega)^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) | V1*V2*(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).