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Viscoelastic models revisited: characteristics and interconversion formulas for generalized Kelvin–Voigt and Maxwell models

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Abstract

Generalized Kelvin–Voigt and Maxwell models using Prony series are some of the most well-known models to characterize the behavior of polymers. The simulation software for viscoelastic materials generally implement only some material models. Therefore, for the practice of the engineer, it is very useful to have formulas that establish the equivalence between different models. Although the existence of these relationships is a well-established fact, moving from one model to another involves a relatively long process. This article presents a development of the relationships between generalized Kelvin–Voigt and Maxwell models using the aforementioned series and their respective relaxation and creep coefficients for one and two summations. The relationship between the singular points (maximums, minimums and inflexion points) is also included.

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References

  1. Tschoegl, N.W.: Time dependence in material properties: an overview. Mech. Time Depend. Mater. 1, 3–31 (1997)

    Article  Google Scholar 

  2. Casula, G., Carcione, J.: Generalized mechanical model analogies of linear viscoelastic behaviour. Bolletino di Geofis. Teor. ed Appl. 34, 235–256 (1992)

    Google Scholar 

  3. Menard, K.P., Peter, K.: Dynamic Mechanical Analysis: a Practical Introduction. CRC Press, Washington DC (1999)

    Book  Google Scholar 

  4. Gutierrez-Lemini, D.: Engineering Viscoelasticity. Springer, New York (2014)

    Book  Google Scholar 

  5. Drozdov, A.D.: Finite Elasticity and Viscoelasticity. World Scientific Publishing, Hong Kong (1996)

    Book  Google Scholar 

  6. Chawla, A., Mukherjee, S., Karthikeyan, B.: Characterization of human passive muscles for impact loads using genetic algorithm and inverse finite element methods. Biomech. Model. Mechanobiol. 8, 67–76 (2009)

    Article  Google Scholar 

  7. Fatemifar, F., Salehi, M., Adibipoor, R., et al.: Three-phase modeling of viscoelastic nanofiber-reinforced matrix. J. Mech. Sci. Technol. 28, 1039–1044 (2014)

    Article  Google Scholar 

  8. Matter, Y.S., Darabseh, T.T., Mourad, A.H.I.: Flutter analysis of a viscoelastic tapered wing under bending–torsion loading. Meccanica 53, 3673–3691 (2018)

    Article  MathSciNet  Google Scholar 

  9. Forte, A.E., Gentleman, S.M., Dini, D.: On the characterization of the heterogeneous mechanical response of human brain tissue. Biomech. Model. Mechanobiol. 16, 907–920 (2017)

    Article  Google Scholar 

  10. Ding, H.: Steady-state responses of a belt-drive dynamical system under dual excitations. Acta Mech. Sin. 32, 156–169 (2016)

    Article  MathSciNet  Google Scholar 

  11. Manda, K., Xie, S., Wallace, R.J., et al.: Linear viscoelasticity—bone volume fraction relationships of bovine trabecular bone. Biomech. Model. Mechanobiol. 15, 1631–1640 (2016)

    Article  Google Scholar 

  12. Nantasetphong, W., Jia, Z., Amirkhizi, A., et al.: Dynamic properties of polyurea-milled glass composites. Part I: experimental characterization. Mech. Mater. 98, 142–153 (2016)

    Article  Google Scholar 

  13. Liu, H., Yang, J., Liu, H.: Effect of a viscoelastic target on the impact response of a flat-nosed projectile. Acta Mech. Sin. 34, 162–174 (2018)

    Article  MathSciNet  Google Scholar 

  14. Li, Y., Hong, Y., Xu, G.K., et al.: Non-contact tensile viscoelastic characterization of microscale biological materials. Acta Mech. Sin. 34, 589–599 (2018)

    Article  Google Scholar 

  15. Bai, T., Tsvankin, I.: Time-domain finite-difference modeling for attenuative anisotropic media. Geophysics 81, C69–C77 (2016)

    Article  Google Scholar 

  16. Zhang, Y., Lian, Z., Zhou, M., et al.: Viscoelastic behavior of a casing material and its utilization in premium connections in high-temperature gas wells. Adv. Mech. Eng. 10, 168781401881745 (2018)

    Article  Google Scholar 

  17. Baumgaertel, M., Winter, H.H.: Determination of discrete relaxation and retardation time spectra from dynamic mechanical data. Rheol. Acta 28, 511–519 (1989)

    Article  Google Scholar 

  18. Nikonov, A., Davies, A.R., Emri, I.: The determination of creep and relaxation functions from a single experiment. J. Rheol. 49, 1193–1211 (2005)

    Article  Google Scholar 

  19. Sorvari, J., Malinen, M.: On the direct estimation of creep and relaxation functions. Mech. Time Depend. Mater. 11, 143–157 (2007)

    Article  Google Scholar 

  20. Renaud, F., Dion, J.: A new identification method of viscoelastic behavior: application to the generalized Maxwell model. Mech. Syst. Signal Process. 25, 991–1010 (2011)

    Article  Google Scholar 

  21. Bang, K., Jeong, H.Y.: Combining stress relaxation and rheometer test results in modeling a polyurethane stopper. J. Mech. Sci. Technol. 26, 1849–1855 (2012)

    Article  Google Scholar 

  22. Soo Cho, K.: Power series approximations of dynamic moduli and relaxation spectrum. J. Rheol. 57, 679–697 (2013)

    Article  Google Scholar 

  23. Chen, D.L., Chiu, T.C., Chen, T.C., et al.: Using DMA to simultaneously acquire Young’s relaxation modulus and time-dependent Poisson’s ratio of a viscoelastic material. Procedia Eng. 79, 153–159 (2014)

    Article  Google Scholar 

  24. Pacheco, J.E.L., Bavastri, C.A., Pereira, J.T.: Viscoelastic relaxation modulus characterization using Prony series. Lat. Am. J. Solids Struct. 12, 420–445 (2015)

    Article  Google Scholar 

  25. Kim, M., Bae, J.E., Kang, N., et al.: Extraction of viscoelastic functions from creep data with ringing. J. Rheol. 59, 237–252 (2015)

    Article  Google Scholar 

  26. Jung, J.W., Hong, J.W., Lee, H.K., et al.: Estimation of viscoelastic parameters in Prony series from shear wave propagation. J. Appl. Phys. 119, 234701 (2016)

    Article  Google Scholar 

  27. Bonfitto, A., Tonoli, A., Amati, N.: Viscoelastic dampers for rotors: modeling and validation at component and system level. Appl. Sci. 7, 1181 (2017)

    Article  Google Scholar 

  28. Rubio-Hernández, F.J.: Rheological behavior of fresh cement pastes. Fluids 3, 106 (2018)

    Article  Google Scholar 

  29. Poul, M.K., Zerva, A.: Time-domain PML formulation for modeling viscoelastic waves with Rayleigh-type damping in an unbounded domain: theory and application in ABAQUS. Finite Elem. Anal. Des. 152, 1–16 (2018)

    Article  MathSciNet  Google Scholar 

  30. Gross, B.: On creep and relaxation. J. Appl. Phys. 18, 212–221 (1947)

    Article  MathSciNet  Google Scholar 

  31. Gross, B.: Mathematical Structure of the Theories of Viscoelasticity. Hermann & Co., Paris (1953)

    MATH  Google Scholar 

  32. Loy, R.J., Anderssen, R.S.: Interconversion relationships for completely monotone functions. SIAM J. Math. Anal. 46, 2008–2032 (2014)

    Article  MathSciNet  Google Scholar 

  33. Park, S.W., Schapery, R.A.: Methods of interconversion between linear viscoelastic material functions. Part I: a numerical method based on Prony series. Int. J. Solids Struct. 36, 1653–1675 (1999)

    Article  Google Scholar 

  34. Schapery, R.A., Park, S.W.: Methods of interconversion between linear viscoelastic material functions. Part II: an approximate analytical method. Int. J. Solids Struct. 36, 1677–1699 (1999)

    Article  Google Scholar 

  35. Sorvari, J., Malinen, M.: Numerical interconversion between linear viscoelastic material functions with regularization. Int. J. Solids Struct. 44, 1291–1303 (2007)

    Article  Google Scholar 

  36. Luk-Cyr, J., Crochon, T., Li, C., et al.: Interconversion of linearly viscoelastic material functions expressed as Prony series: a closure. Mech. Time Depend. Mater. 17, 53–82 (2013)

    Article  Google Scholar 

  37. Loy, R.J., de Hoog, F.R., Anderssen, R.S.: Interconversion of Prony series for relaxation and creep. J. Rheol. 59, 1261–1270 (2015)

    Article  Google Scholar 

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Correspondence to J. Menacho.

Appendices

Appendix A

1.1 A.1 Constants of differential equations

1.1.1 A.1.1 Constants of Eq. (27) in the text

$$A = \mathop \sum \limits_{i = 0}^{n} \mathop {\mathop \prod \limits_{j = 0}^{n} }\limits_{j \ne i} E_{{i_{K} }} = \varDelta \cdot \mathop \sum \limits_{i = 0}^{n} \frac{1}{{E_{{i_{K} }} }},$$

with \(\varDelta = \mathop \prod \limits_{i = 0}^{n} E_{{i_{K} }} ,\)

$$B = \frac{\varDelta }{{E_{{0_{K} }} }} \cdot \mathop \sum \limits_{j = 1}^{n} \frac{{\eta_{{j_{K} }} }}{{E_{{j_{K} }} }} + \varDelta \cdot \mathop \sum \limits_{i = 1}^{n} \mathop {\mathop \sum \limits_{j = 1}^{n} }\limits_{j \ne i} \frac{{\eta_{{j_{K} }} }}{{E_{{i_{K} }} E_{{j_{K} }} }},$$
$$C = \frac{\varDelta }{{E_{{0_{K} }} }} \cdot \mathop \sum \limits_{i = 1}^{n - 1} \mathop {\mathop \sum \limits_{j = 2}^{n} }\limits_{j > i} \frac{{\eta_{{i_{K} }} \eta_{{j_{K} }} }}{{E_{{i_{K} }} E_{{j_{K} }} }} + \varDelta \cdot \mathop \sum \limits_{i = 1}^{n} \mathop {\mathop \sum \limits_{j = 1}^{n - 1} }\limits_{j \ne i} \mathop {\mathop {\mathop \sum \limits_{k = 2}^{n} }\limits_{k > j} }\limits_{k \ne i} \frac{{\eta_{{j_{K} }} \eta_{{k_{K} }} }}{{E_{{i_{K} }} E_{{j_{K} }} E_{{k_{K} }} }},$$
$$D = \frac{\varDelta }{{E_{{0_{K} }} }} \cdot \mathop \sum \limits_{i = 1}^{n - 2} \mathop {\mathop \sum \limits_{j = 2}^{n - 1} }\limits_{j > i} \mathop {\mathop \sum \limits_{k = 3}^{n} }\limits_{k > j} \frac{{\eta_{{i_{K} }} \eta_{{j_{K} }} \eta_{{k_{K} }} }}{{E_{{i_{K} }} E_{{j_{K} }} E_{{k_{K} }} }} + \varDelta \cdot \mathop \sum \limits_{i = 1}^{n} \mathop {\mathop \sum \limits_{j = 1}^{n - 2} }\limits_{j \ne i} \mathop {\mathop {\mathop \sum \limits_{k = 2}^{n - 1} }\limits_{k > j} }\limits_{k \ne i} \mathop {\mathop {\mathop \sum \limits_{l = 3}^{n} }\limits_{l > k} }\limits_{l \ne i} \frac{{\eta_{{j_{K} }} \eta_{{k_{K} }} \eta_{{l_{K} }} }}{{E_{{i_{K} }} E_{{j_{K} }} E_{{k_{K} }} E_{{k_{K} }} }},$$
$$\vdots$$
$$Y = \mathop \sum \limits_{i = 1}^{n} \left[ {\left( {E_{{0_{K} }} + E_{{i_{K} }} } \right)\mathop {\mathop \prod \limits_{j = 1}^{n} }\limits_{j \ne i} \eta_{{j_{K} }} } \right],$$
$$Z = \mathop \prod \limits_{i = 1}^{n} \eta_{{i_{K} }} ,$$
$$a = \varDelta ,$$
$$b = \varDelta \cdot \mathop \sum \limits_{i = 1}^{n} \frac{{\eta_{{i_{K} }} }}{{E_{{i_{K} }} }},$$
$$c = \varDelta \cdot \mathop \sum \limits_{i = 1}^{n - 1} \mathop {\mathop \sum \limits_{j = 2}^{n} }\limits_{j > i} \frac{{\eta_{{i_{K} }} \eta_{{j_{K} }} }}{{E_{{i_{K} }} E_{{j_{K} }} }},$$
$$d = \varDelta \cdot \mathop \sum \limits_{i = 1}^{n - 2} \mathop {\mathop \sum \limits_{j = 2}^{n - 1} }\limits_{j > i} \mathop {\mathop \sum \limits_{k = 3}^{n} }\limits_{k > j} \frac{{\eta_{{i_{K} }} \eta_{{j_{K} }} \eta_{{k_{K} }} }}{{E_{{i_{K} }} E_{{j_{K} }} E_{{k_{K} }} }},$$
$$\vdots$$
$$y = E_{{0_{K} }} \cdot \mathop \sum \limits_{i = 1}^{n} E_{{i_{K} }} \mathop {\mathop \prod \limits_{j = 1}^{n} }\limits_{j \ne i} \eta_{{j_{K} }} ,$$
$$z = E_{{0_{K} }} \cdot\mathop \prod \limits_{i = 1}^{n} \eta_{{i_{K} }} .$$

1.1.2 A.1.2 Constants of Eq. (56) in the text

$$A = \mathop \prod \limits_{i = 1}^{n} E_{{i_{M} }} \equiv \varDelta ,$$
$$B = \varDelta \cdot \mathop \sum \limits_{i = 1}^{n} \frac{{\eta_{{i_{M} }} }}{{E_{{i_{M} }} }},$$
$$C = \varDelta \cdot \mathop \sum \limits_{i = 1}^{n - 1} \mathop {\mathop \sum \limits_{j = 2}^{n} }\limits_{j > i} \frac{{\eta_{{i_{M} }} \eta_{{j_{M} }} }}{{E_{{i_{M} }} E_{{j_{M} }} }},$$
$$D = \varDelta \cdot \mathop \sum \limits_{i = 1}^{n - 2} \mathop {\mathop \sum \limits_{j = 2}^{n - 1} }\limits_{j > i} \mathop {\mathop \sum \limits_{k = 3}^{n} }\limits_{k > j} \frac{{\eta_{{i_{M} }} \eta_{{j_{M} }} \eta_{{k_{M} }} }}{{E_{{i_{M} }} E_{{j_{M} }} E_{{k_{M} }} }},$$
$$\vdots$$
$$Y = \mathop \sum \limits_{i = 1}^{n} E_{{i_{M} }} \mathop {\mathop \prod \limits_{j = 1}^{n} }\limits_{j \ne i} \eta_{{j_{M} }} ,$$
$$Z = \mathop \prod \limits_{i = 1}^{n} \eta_{{i_{M} }} ,$$
$$a = E_{{\infty_{M} }} \cdot \varDelta ,$$
$$b = \varDelta \cdot \left( {E_{{\infty_{M} }} \cdot\mathop \sum \limits_{i = 1}^{n} \frac{{\eta_{{i_{M} }} }}{{E_{{i_{M} }} }} + \mathop \sum \limits_{i = 1}^{n} \eta_{{i_{M} }} } \right),$$
$$c = {\varDelta} \cdot \left( {E_{{\infty _{M} }} \sum\limits_{{i = 1}}^{{n - 1}} {\sum\limits_{\begin{subarray}{l} j = 2 \\ j > 1 \end{subarray} }^{n} {\frac{{\eta _{{I_{M} }} \eta _{{j_{M} }} }}{{E_{{i_{M} }} E_{{j_{M} }} }}} + \sum\limits_{{i = 1}}^{n} {\sum\limits_{\begin{subarray}{l} j = 1 \\ j \ne i \end{subarray} }^{n} {\frac{{\eta _{{i_{M} }} \eta _{{j_{M} }} }}{{E_{{j_{M} }} }}} } } } \right),$$
$$d = \varDelta \cdot \left( {E_{\infty _M} \mathop \sum \limits_{i = 1}^{n - 2} \mathop {\mathop \sum \limits_{j = 2}^{n - 1} }\limits_{j > i} \mathop {\mathop \sum \limits_{k = 3}^{n} }\limits_{k > j} \frac{{\eta_{i_M} \eta_{j_M} \eta_{k_M} }}{{E_{{i_{M} }} E_{{j_{M} }} E_{{k_{M} }} }} + \mathop \sum \limits_{i = 1}^{n} \mathop {\mathop \sum \limits_{j = 1}^{n} }\limits_{j \ne i} \mathop {\mathop {\mathop \sum \limits_{k = 1}^{n} }\limits_{k \ne i} }\limits_{k > j} \frac{{\eta_{i_M} \eta_{j_M} \eta_{k_M} }}{{E_{{j_{M} }} E_{{k_{M} }} }}} \right),$$
$$\vdots$$
$$y = E_{\infty _M} \mathop \sum \limits_{i = 1}^{n} E_{i_M} \mathop {\mathop \prod \limits_{j = 1}^{n} }\limits_{j \ne i} \eta_{j_M} + \mathop \sum \limits_{i = 1}^{n} E_{i_M} \eta_{i_M} \mathop {\mathop \sum \limits_{j = 1}^{n} }\limits_{j \ne i} E_{j_M} \mathop {\mathop {\mathop \prod \limits_{k = 1}^{n} }\limits_{k \ne i} }\limits_{k \ne j} \eta_{k_M} ,$$
$$z = \left(E_{\infty _M} + \mathop \sum \limits_{i = 1}^{n} E_{i_M} \right) \mathop \prod \limits_{j = 1}^{n} \eta_{j_M}.$$

1.2 A.2 Derivatives of Prony series

In this Appendix, the mathematical expressions of the first and second order derivatives of Prony series are shown.

1.2.1 A.2.1 Derivatives as a function of time


A.2.1.1 First order derivatives Mathematical expressions to calculate maximums or minimums in relaxation modulus and creep compliance as a function of time, using Prony series, are shown in Eqs. (A.1) and (A.2), respectively

$$\frac{{{\text{d}}Y}}{{{\text{d}}t}} = - Y_{0} \mathop \sum \limits_{i = 1}^{n} \frac{{p_{i} }}{{\tau_{i} }}{\text{e}}^{{ - \frac{t}{{\tau_{i} }}}} ,$$
(A.1)
$$\frac{{{\text{d}}J}}{{{\text{d}}t}} = J_{0} \mathop \sum \limits_{i = 1}^{n} \frac{{q_{i} }}{{\lambda_{i} }}{\text{e}}^{{ - \frac{t}{{\lambda_{i} }}}} .$$
(A.2)

A.2.1.2 Second order derivatives Mathematical expressions to calculate inflexion points in relaxation modulus and creep compliance as a function of time, using Prony series, are shown in equation Eqs. (A.3) and (A.4), respectively

$$\frac{{{\text{d}}^{2} Y}}{{{\text{d}}t^{2} }} = Y_{0} \mathop \sum \limits_{i = 1}^{n} \frac{{p_{i} }}{{\tau_{i}^{2} }}{\text{e}}^{{ - \frac{t}{{\tau_{i} }}}} ,$$
(A.3)
$$\frac{{{\text{d}}^{2} J}}{{{\text{d}}t^{2} }} = - J_{0} \mathop \sum \limits_{i = 1}^{n} \frac{{q_{i} }}{{\lambda_{i}^{2} }}{\text{e}}^{{ - \frac{t}{{\lambda_{i} }}}} .$$
(A.4)

1.2.2 A.2.2 Derivatives as a function of frequency

A.2.2.1 First order derivatives Mathematical expressions to calculate maximums or minimums in storage modulus, loss modulus, storage compliance, loss compliances and tangents of the phase angle as a function of frequency, using Prony series, are shown in equations Eqs. (A.5)–(A.9), respectively

$$\frac{{{\text{d}}G^{\prime}}}{{{\text{d}}\omega }} = 2G_{0} \mathop \sum \limits_{i = 1}^{n} \frac{{p_{i} \tau_{i}^{2} \omega }}{{\left( {1 + \tau_{i}^{2} \omega^{2} } \right)^{2} }} ,$$
(A.5)
$$\frac{{{\text{d}}G^{\prime\prime}}}{{{\text{d}}\omega }} = G_{0} \mathop \sum \limits_{i = 1}^{n} \frac{{p_{i} \tau_{i} \left( {1 - \tau_{i}^{2} \omega^{2} } \right)}}{{\left( {1 + \tau_{i}^{2} \omega^{2} } \right)^{2} }},$$
(A.6)
$$\frac{{{\text{d}}J^{\prime}}}{{{\text{d}}\omega }} = - \,2J_{0} \mathop \sum \limits_{i = 1}^{n} \frac{{q_{i} \lambda_{i}^{2} \omega }}{{\left( {1 + \lambda_{i}^{2} \omega^{2} } \right)^{2} }} ,$$
(A.7)
$$\frac{{{\text{d}}J^{\prime\prime}}}{{{\text{d}}\omega }} = J_{0} \mathop \sum \limits_{i = 1}^{n} \frac{{q_{i} \lambda_{i} \left( {1 - \lambda_{i}^{2} \omega^{2} } \right)}}{{\left( {1 + \lambda_{i}^{2} \omega^{2} } \right)^{2} }},$$
(A.8)
$$\frac{{{\text{d}}\tan \delta }}{{{\text{d}}\omega }} = \frac{{\frac{{{\text{d}}G^{\prime\prime}}}{{{\text{d}}\omega }} \cdot G^{\prime} - G^{\prime\prime} \cdot \frac{{{\text{d}}G^{\prime}}}{{{\text{d}}\omega }}}}{{ {G^{\prime}}^{2} }}.$$
(A.9)

A.2.2.2 Second order derivatives Mathematical expressions to calculate inflexion points in storage modulus, loss modulus, storage compliance, loss compliances and tangents of the phase angle as a function of frequency, using Prony series, are shown in equations Eqs. (A.10)–(A.14), respectively

$$\frac{{{\text{d}}^{2} G '}}{{{\text{d}}\omega^{2} }} = 2G_{0} \mathop \sum \limits_{i = 1}^{n} p_{i} \tau_{i}^{2} \frac{{1 - 3\tau_{i}^{2} \omega^{2} }}{{\left( {1 + \tau_{i}^{2} \omega^{2} } \right)^{3} }} ,$$
(A.10)
$$\frac{{{{\text{d}}^{2}} {G ^{\prime\prime}}}}{{{{\text{d}}\omega^{2}} }} = 2G_{0} \mathop \sum \limits_{i = 1}^{n} {p_{i}} {\tau_{i}^{3}} \frac{{\omega \left( {{\tau_{i}^{2}} {\omega^{2}} - 3} \right)}}{{\left( {1 + {\tau_{i}^{2}} {\omega^{2}} } \right)^{3} }} ,$$
(A.11)
$$\frac{{{\text{d}}^{2} J '}}{{{\text{d}}\omega^{2} }} = - 2J_{0} \mathop \sum \limits_{i = 1}^{n} q_{i} \lambda_{i}^{2} \frac{{1 - 3\lambda_{i}^{2} \omega^{2} }}{{\left( {1 + \lambda_{i}^{2} \omega^{2} } \right)^{3} }} ,$$
(A.12)
$$\frac{{{\text{d}}^{2} J ^{\prime\prime}}}{{{\text{d}}\omega^{2} }} = 2J_{0} \mathop \sum \limits_{i = 1}^{n} q_{i} \lambda_{i}^{3} \frac{{\omega \left( {\lambda_{i}^{2} \omega^{2} - 3} \right)}}{{\left( {1 + \lambda_{i}^{2} \omega^{2} } \right)^{3} }} ,$$
(A.13)
$$\frac{{{\text{d}}^{2} \tan \delta }}{{{\text{d}}\omega^{2} }} = \frac{{G^{\prime}\left( {\frac{{{{\rm d}^{2}} {{{G}}^{\prime\prime}}}}{{{\rm d}\omega^{2} }} \cdot G^{\prime} - G^{\prime\prime} \cdot \frac{{{{\rm d}^{2}} {G^{\prime}}}}{{{\rm d}\omega^{2} }}} \right) - 2\frac{{{\rm d}G^{\prime}}}{{\rm d}\omega } \left( {\frac{{{\rm d}G^{\prime\prime}}}{{\rm d}\omega } \cdot G^{\prime} - G^{\prime\prime} \cdot \frac{{{\rm d}G^{\prime}}}{{\rm d}\omega }} \right)}}{{ {G^{\prime}}^{3} }}.$$
(A.14)

1.3 A.3 Roots of polynomial equations

In this Appendix, the resolution of cubic and quartic polynomial equations is presented.

1.3.1 A.3.1 Roots of the cubic equation \(x^{3} + ax^{2} + bx + c = 0\)

$$x_{1} = \frac{1}{12}\left[ {\frac{{ - 2^{4/3} \left( {1 + \sqrt 3 {\text{i}}} \right)\left( {a^{2} - 3b} \right) + 2^{2/3} \left( { - 1 + \sqrt 3 i} \right)H^{2} - 4Ha}}{H}} \right],$$
(A.15)
$$x_{2} = \frac{1}{12}\left[ {\frac{{2^{4/3} \left( { - 1 + \sqrt 3 {\text{i}}} \right)\left( {a^{2} - 3b} \right) - 2^{2/3} \left( {1 + \sqrt 3 i} \right)H^{2} - 4Ha}}{H}} \right],$$
(A.16)
$$x_{3} = \frac{1}{6}\left[ {\frac{{2^{4/3} \left( {a^{2} - 3b} \right) + 2^{2/3} H^{2} - 2Ha}}{H}} \right],$$
(A.17)

with

$$H = \sqrt[3]{{ - 2a^{3} + 9ab - 27c + 3\sqrt 3 \sqrt { - a^{2} b^{2} + 4b^{3} + 4a^{3} c - 18abc + 27c^{2} } }}.$$
(A.18)

1.3.2 A.3.2 Roots of the quartic equation \(x^{4} + ax^{3} + bx^{2} + cx + d = 0\)

$$x_{1} = - \frac{{3a + \sqrt 3 H_{4} }}{12} - \frac{\sqrt 6 }{12}\cdot\sqrt {3a^{2} - 8b - \frac{{2H_{3} }}{{\sqrt[3]{{H_{2} }}}} - \sqrt[3]{{4H_{2} }} + \frac{{3\sqrt 3 \left( {a^{3} - 4ab + 8c} \right)}}{{H_{4} }},}$$
(A.19)
$$x_{2} = - \frac{{3a + \sqrt 3 H_{4} }}{12} + \frac{\sqrt 6 }{12}\cdot\sqrt {3a^{2} - 8b - \frac{{2H_{3} }}{{\sqrt[3]{{H_{2} }}}} - \sqrt[3]{{4H_{2} }} + \frac{{3\sqrt 3 \left( {a^{3} - 4ab + 8c} \right)}}{{H_{4} }}} ,$$
(A.20)
$$x_{3} = - \frac{{3a - \sqrt 3 H_{4} }}{12} - \frac{\sqrt 6 }{12}\cdot\sqrt {3a^{2} - 8b - \frac{{2H_{3} }}{{\sqrt[3]{{H_{2} }}}} - \sqrt[3]{{4H_{2} }} + \frac{{3\sqrt 3 \left( {a^{3} - 4ab + 8c} \right)}}{{H_{4} }}} ,$$
(A.21)
$$x_{4} = - \frac{{3a - \sqrt 3 H_{4} }}{12} + \frac{\sqrt 6 }{12}\cdot\sqrt {3a^{2} - 8b - \frac{{2H_{3} }}{{\sqrt[3]{{H_{2} }}}} - \sqrt[3]{{4H_{2} }} + \frac{{3\sqrt 3 \left( {a^{3} - 4ab + 8c} \right)}}{{H_{4} }}} ,$$
(A.22)

with

$$H_{1} = - 4\left( {b^{2} - 3ac + 12d} \right)^{3} + \left[ {2b^{3} - 9b\left( {ac + 8d} \right) + 27\left( {c^{2} + a^{2} d} \right)} \right]^{2} ,$$
(A.23)
$$H_{2} = 2b^{3} - 9abc + 27c^{2} + 27a^{2} d - 72bd + \sqrt {H_{1} } ,$$
(A.24)
$$H_{3} = \sqrt[3]{2} \left( {b^{2} - 3ac + 12d} \right),$$
(A.25)
$$H_{4} = \sqrt {3a^{2} - 8b + \frac{{4H_{3} }}{{\sqrt[3]{{H_{2} }}}} + \sqrt[3]{{32H_{2} }}} .$$
(A.26)

Appendix B

2.1 B.1 Relationships with n = 1

See Table 1.

Table 1 Relationships for models with n = 1

2.2 B.2 Relationships with n = 2

See Table 2.

Table 2 Relationships for models with n = 2

With the auxiliary constants: (a) Case “GM parameters known” (1st row in the table) (Eq. (54) in the text)

$$C_{1_M} = \frac{1}{{E_{\infty_M} }},$$
$$C_{2_M} = - \frac{{E_{1_M} + E_{2_M} }}{{E_{{\infty_{M} }} \left( {E_{{\infty_{M} }} + E_{1_M} + E_{2_M} } \right)}},$$
$$C_{3_M} = \frac{\alpha }{{2\eta_{1_M} \eta_{2_M} \left( {E_{{\infty_{M} }} + E_{1_M} + E_{2_M} } \right)}},$$
$$C_{4_M} = \frac{{\sqrt {\alpha^{2} - 4E_{{\infty_{M} }} E_{1_M} E_{2_M} \eta_{1_M} \eta_{2_M} \left( {E_{{\infty_{M} }} + E_{1_M} + E_{2_M} } \right)} }}{{2\eta_{1_M} \eta_{2_M} \left( {E_{{\infty_{M} }} + E_{1_M} + E_{2_M} } \right)}},$$
$$C_{5_M} = \frac{{E_{1_M} E_{2_M} \left( {\eta_{1_M} + \eta_{2_M} } \right)}}{{\eta_{1_M} \eta_{2_M} \left( {E_{1_M} + E_{2_M} } \right)}},$$

with

$$\alpha = E_{{\infty_{M} }} \left( {E_{1_M} \eta_{2_M} + E_{2_M} \eta_{1_M} } \right) + E_{1_M} E_{2_M} \left( {\eta_{1_M} + \eta_{2_M} } \right).$$

(b) Case “relaxation coefficients known” (2nd row in the table)

$$C_{1_R} = \frac{1}{{Y_{0} \left( {1 - p_{1} - p_{2} } \right)}},$$
$$C_{2_R} = - \frac{{p_{1} + p_{2} }}{{Y_{0} \left( {1 - p_{1} - p_{2} } \right)}},$$
$$C_{3_R} = \frac{\beta }{{2\tau_{1} \tau_{2} }},$$
$$C_{4_R} = \frac{{\sqrt {\beta^{2} - 4\tau_{1} \tau_{2} \left( {1 - p_{1} - p_{2} } \right)} }}{{2\tau_{1} \tau_{2} }},$$
$$C_{5_R} = \frac{{p_{1} \tau_{1} + p_{2} \tau_{2} }}{{\tau_{1} \tau_{2} \left( {p_{1} + p_{2} } \right)}},$$

with

$$\beta = \tau_{1} \left( {1 - p_{2} } \right) + \tau_{2} \left( {1 - p_{1} } \right).$$

(c) Case “creep coefficients known” (3rd row in the table)

$$C_{1_C} = \frac{1}{{J_{0} \left( {1 + q_{1} + q_{2} } \right)}},$$
$$C_{2_C} = \frac{{q_{1} + q_{2} }}{{J_{0} \left( {1 + q_{1} + q_{2} } \right)}},$$
$$C_{3_C} = \frac{\delta }{{2\lambda_{1} \lambda_{2} }},$$
$$C_{4_C} = \frac{{\sqrt {\delta^{2} - 4\lambda_{1} \lambda_{2} \left( {1 + q_{1} + q_{2} } \right)} }}{{2\lambda_{1} \lambda_{2} }},$$
$$C_{5_C} = \frac{{q_{1} \lambda_{1} + q_{2} \lambda_{2} }}{{\lambda_{1} \lambda_{2} \left( {q_{1} + q_{2} } \right) }},$$

with

$$\delta = \lambda_{1} \left( {1 + q_{2} } \right) + \lambda_{2} \left( {1 + q_{1} } \right).$$

(d) Case “GKV parameters known” (4th row in the table) (Eq. (17) in the text)

$$C_{1_K} = \frac{{E_{0_K} E_{1_K} E_{2_K} }}{{E_{0_K} E_{1_K} + E_{0_K} E_{2_K} + E_{1_K} E_{2_K} }},$$
$$C_{2_K} = \frac{{E_{0_K}^{2} \left( {E_{1_K} + E_{2_K} } \right)}}{{E_{0_K} E_{1_K} + E_{0_K} E_{2_K} + E_{1_K} E_{2_K} }},$$
$$C_{3_K} = \frac{\gamma }{{2\eta_{1_K} \eta_{2_K} }},$$
$$C_{4_K} = \frac{{\sqrt {\gamma^{2} - 4\eta_{1_K} \eta_{2_K} \left( {E_{0_K} E_{1_K} + E_{0_K} E_{2_K} + E_{1_K} E_{2_K} } \right)} }}{{2\eta_{1_K} \eta_{2_K} }},$$
$$C_{5_K} = \frac{{E_{1_K}^{2} \eta_{2_K} + E_{2_K}^{2} \eta_{1_K} }}{{\eta_{1_K} \eta_{2_K} \left( {E_{1_K} + E_{2_K} } \right)}},$$

with

$$\gamma = E_{0_K} \left( {\eta_{1_K} + \eta_{2_K} } \right) + E_{1_K} \eta_{2_K} + E_{2_K} \eta_{1_K} .$$

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Serra-Aguila, A., Puigoriol-Forcada, J.M., Reyes, G. et al. Viscoelastic models revisited: characteristics and interconversion formulas for generalized Kelvin–Voigt and Maxwell models. Acta Mech. Sin. 35, 1191–1209 (2019). https://doi.org/10.1007/s10409-019-00895-6

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