Abstract
The purpose of this paper is to compute the relaxation and creep functions from the data of shear complex modulus, G ∗(iν). The experimental data are available in the frequency window ν∈[νmin ,νmax ] in terms of the storage G′(ν) and loss G″(ν) moduli. The loss factor \(\eta( \nu) = \frac{G''( \nu )}{G'(\nu )}\) is asymmetrical function. Therefore, a five-parameter fractional derivative model is used to predict the complex shear modulus, G ∗(iν). The corresponding relaxation spectrum is evaluated numerically because the analytical solution does not exist. Thereby, the fractional model is approximated by a generalized Maxwell model and its rheological parameters (G k ,τ k ,N) are determined leading to the discrete relaxation spectrum G(t) valid in time interval corresponding to the frequency window of the input experimental data. Based on the deterministic approach, the creep compliance J(t) is computed on inversing the relaxation function G(t).
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References
Bagley, R.L., Torvik, P.J.: Fractional calculus—A different approach to the analysis of viscoelasticity damped structures. AIAA J. 21(5), 741–748 (1983)
Bagley, R.L., Torvik, P.J.: On the fractional calculus model of viscoelastic behaviour. J. Rheol. 30(1), 133–155 (1986)
Baumgaertel, M., Winter, H.H.: Determination of discrete relaxation and retardation time line spectra from dynamic mechanical data. Rheol. Acta 28, 511–519 (1989)
Baumgaertel, M., Winter, H.H.: Interrelation between continuous and discrete relaxation time spectra. J. Non-Newtonian Fluid Mech. 44, 15–36 (1992)
Beda, T., Chevalier, Y.: New methods for identifying rheological parameters for fractional derivative modelling of viscoelastic behaviour. Mech. Time Depend. Mater. 8(2), 105–118 (2004)
Caputo, M., Mainardi, F.: Linear models of dissipation in anelastic solids. Riv. Nuovo Cimento 1, 161–198 (1971)
Caracciolo, R., Gasparetto, A., Giovagnoni, M.: Application of causality check and the reduced variables method for experimental determination of Young’s modulus of viscoelastic material. Mech. Mater. 33, 693–703 (2001)
Christensen, R.M.: A nonlinear theory of viscoelasticity for application to elastomers. J. Appl. Mech. 47, 762–768 (1980)
Elester, C., Honerkamp, J., Weese, J.: Using regularization methods for the determination of relaxation and retardation spectra of polymer liquids. Rheol. Acta 30, 161–174 (1991)
Emri, I., Tschoegl, N.W.: Generating line spectra from experimental responses. Part I. Relaxation modulus and creep compliance. Rheol. Acta 31(2), 311–321 (1993)
Emri, I., Von Bernstorff, K., Cvelbar, B.S., Nikonov, A.: Re-examination of the approximate methods of interconversion between frequency-and time-dependent material functions. J. Non-Newtonian Fluids Mech. 129, 75–84 (2005)
Ferry, J.D.: Viscoelastic Properties of Polymers. Wiley, New York (1980)
Fowler, B.L.: Interactive characterization and data base storage of complex data. In: Proceedings of Damping’89, West Palm Beach, FL, vol. 2 (1989), FAA 1-12
Gross, B.: Mathematical Structure of the Theories of Polymers. Hermann, Paris (1953)
Haupt, P., Lion, A., Backauss, E.: On the dynamic behaviour of polymers under finite strains: constitutive modelling and identification of parameters. International. J. Solids Struct. 37, 3633–3646 (2000)
Honerkamp, J.: Ill-posed problems in rheology. Rheol. Acta 28(5), 363–371 (1989)
Honerkamp, J., Wesse, J.: Tikhonov regularization method for ill-posed problems: a comparison of different methods for the determination of the regularization parameter. Contin. Mech. Thermodyn. 2, 17–30 (1990)
Honerkamp, J., Weese, J.: A nonlinear regularization method for the calculation of relaxation spectra. Rheol. Acta 28, 65–73 (1993)
Jackson, J., DeRosa, M., Winter, H.H.: Molecular weight dependence of relaxation time spectra for the entanglement and flow behaviour of monodisperse linear flexible polymers. Macromolecules 27(9), 2426–2471 (1994)
Koeller, R.C.: Applications of fractional calculus to the theory of viscoelasticity. J. Appl. Mech. 51, 299–307 (1984)
Laun, H.M.: Predictions of elastic strains of polymers melts in shear and elongation. J. Rheol. 30, 459–465 (1989)
Lee, S., Knauss, W.G.: A note on the determination of relaxation and creep data ramp tests. Mech. Time-Depend. Mater. 4, 1–7 (2000)
Lubliner, J., Panoskaltis, V.P.: The modified Kuhn model of linear viscoelasticity. Int. J. Solids Struct. 29(24), 3099–3112 (1992)
Malkin, A.Ya.: Some inverse problems in rheology leading to integral equations. Rheol. Acta 29(6), 512–518 (1990)
Malkin, A.Ya.: The use of a continuous relaxation spectrum for describing the viscoelastic properties of polymers. Polymer Sci. A 42(1), 39–45 (2006)
Malkin, A.Ya., Kuznetsov, V.V.: Linearization as a method for determining parameters of relaxation spectra. Rheol. Acta 39(3), 261–271 (2001)
Malkin, A.Ya., Masalova, I.: From dynamic modulus via different relaxation spectra to relaxation and creep function. Rheol. Acta 40, 261–271 (2001)
Olard, F., Di Benedetto, H., Eckmann, B.: Rhéologie des bitumes. Prédiction des résultats des tests de fluage BBR à partir des résultats de module complexe. Bull. Lab. Ponts Chauss. 252–253, 3–15 (2004)
Parrot, J.-M., Duperray, B.: Exact computation of creep compliance and relaxation modulus from complex modulus measurements data. Mech. Mater. 40, 575–565 (2008)
Pritz, T.: Five-parameter fractional derivative model for polymeric damping materials. J. Sound Vib. 265, 935–952 (2003)
Pritz, T.: Unbounded complex modulus of viscoelastic materials and the Kramers-Kronig relations. J. Sound Vib. 279, 687–697 (2005)
Rossikhin, Y.A., Shitikova, M.V.: Analysis of rheological equations involving more than one fractional parameters by use of the simplest mechanical based on theses equations. Mech. Time-Depend. Mater. 5, 131–175 (2001)
Roths, J., Maier, D., Friedrich, C., Marth, M., Honerkamp, J.: Determination of the relaxation time spectrum from dynamic moduli using an edge preserving regularization method. Rheol. Acta 39, 163–173 (2000)
Schapery, S.A., Park, S.W.: Methods of interconversion between linear viscoelastic material functions. Part II—an approximate analytical method. Int. J. Solids Struct. 36, 1653–1675 (1999a)
Schapery, S.A., Park, S.W.: Methods of interconversion between linear viscoelastic material functions. Part I—a numerical method based on Prony series. Int. J. Solids Struct. 36, 1677–1699 (1999b)
Schiessel, H., Blumen, A.: Hierarchical analogues to fractional relaxation equations. J. Phys., A, Math. Gen. 26, 5057–5069 (1993)
Sorvari, J., Malinen, M.: Determination of the relaxation modulus of linearly viscoelastic material. Mech. Time-Depend. Mater. 10(2), 125 (2006)
Sorvari, J., Malinen, M.: Numerical interconversion between linear viscoelastic material functions with regularization. Int. J. Solids Struct. 44, 1291–1303 (2007)
Soula, M., Vinh, T., Chevalier, Y.: Transient response of polymers and elastomers deduced from harmonic responses. J. Sound Vib. 205(2), 185–203 (1997)
Tobolsky, A.V., Catsiff, E.: Elastoviscous properties of polyisobutylene (and other amorphous polymers) from stress-relaxation studies. IX. A summary of results. J. Polym. Sci. 19, 111–121 (1956)
Tschoegl, N.W.: The Phenomenological Theory of Linear Viscoelastic Behaviour. Springer, Heidelberg (1989)
Tschoegl, N.W., Emri, I.: Generating line spectra from experimental responses. Part. II. Storage and loss functions. Rheol. Acta 32, 322–327 (1993)
Winter, H.H.: Analysis of dynamic mechanical data: inversion into a relaxation time spectrum and consistency check. J. Non-Newtonian Fluid Mech. 68(2/3), 225–239 (1997)
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Bechir, H., Idjeri, M. Computation of the relaxation and creep functions of elastomers from harmonic shear modulus. Mech Time-Depend Mater 15, 119–138 (2011). https://doi.org/10.1007/s11043-010-9126-5
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DOI: https://doi.org/10.1007/s11043-010-9126-5