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A generalization of the Becker model in linear viscoelasticity: creep, relaxation and internal friction

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Abstract

We present a new rheological model depending on a real parameter \(\nu \in [0,1]\), which reduces to the Maxwell body for \(\nu =0\) and to the Becker body for \(\nu =1\). The corresponding creep law is expressed in an integral form in which the exponential function of the Becker model is replaced and generalized by a Mittag–Leffler function of order \(\nu \). Then the corresponding non-dimensional creep function and its rate are studied as functions of time for different values of \(\nu \) in order to visualize the transition from the classical Maxwell body to the Becker body. Based on the hereditary theory of linear viscoelasticity, we also approximate the relaxation function by solving numerically a Volterra integral equation of the second kind. In turn, the relaxation function is shown versus time for different values of \(\nu \) to visualize again the transition from the classical Maxwell body to the Becker body. Furthermore, we provide a full characterization of the new model by computing, in addition to the creep and relaxation functions, the so-called specific dissipation \(Q^{-1}\) as a function of frequency, which is of particular relevance for geophysical applications.

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Notes

  1. Richard Becker (1887–1955) was a German theoretical physicist who made significant contributions in thermodynamics, statistical mechanics, electromagnetism, superconductivity, and quantum electrodynamics. He was professor formerly in Berlin and then in Göttingen. For more details see https://en.wikipedia.org/wiki/Richard_Becker_(physicist).

  2. This series is a particular realization of the so-called Dirichlet \(\eta \) function (Olver et al. 2010). The latter is part of a broad class of function series, known as Dirichlet series, rather known in rheology as Prony series, which have recently found new physical applications in the so-called Bessel models; see e.g. Giusti and Mainardi (2016), Colombaro et al. (2017) and Giusti (2017).

References

  • Becker, R.: Elastische Nachwirkung und Plastizität. Z. Phys. 33, 185–213 (1925)

    Article  MATH  Google Scholar 

  • Becker, R., Doring, W.: Ferromagnetismus. Springer, Berlin (1939)

    Book  MATH  Google Scholar 

  • Caputo, M., Mainardi, F.: Linear models of dissipation in anelastic solids. Riv. Nuovo Cimento 1, 161–198 (1971)

    Article  Google Scholar 

  • Christensen, R.M.: Theory of Viscoelasticity, 2nd edn. Academic Press, New York (1982) [1st edn. (1972)]

    Google Scholar 

  • Colombaro, I., Giusti, A., Mainardi, F.: A class of linear viscoelastic models based on Bessel functions. Meccanica 52(4–5), 825–832 (2017). arXiv:1602.04664. https://doi.org/10.1007/s11012-016-0456-5

    Article  MathSciNet  MATH  Google Scholar 

  • Garra, R., Mainardi, F., Spada, G.: A generalization of the Lomnitz logarithmic creep law via fractional calculus. Chaos Solitons Fractals (2017). arXiv:1701.03068. https://doi.org/10.1016/j.chaos.2017.03.032

    MathSciNet  Google Scholar 

  • Giusti, A.: On infinite order differential operators in fractional viscoelasticity. Fract. Calc. Appl. Anal. 20(4), 854–867 (2017). arXiv:1701.06350. https://doi.org/10.1515/fca-2017-0045

    Article  MathSciNet  MATH  Google Scholar 

  • Giusti, A., Mainardi, F.: A dynamic viscoelastic analogy for fluid-filled elastic tubes. Meccanica 51(10), 2321–2330 (2016). arXiv:1505.06694. https://doi.org/10.1007/s11012-016-0376-4

    Article  MathSciNet  MATH  Google Scholar 

  • Glöckle, W.G., Nonnenmacher, T.: Fractional integral operators and Fox functions in the theory of viscoelasticity. Macromolecules 24, 6426–6434 (1991)

    Article  Google Scholar 

  • Gorenflo, R., Kilbas, A.A., Mainardi, F., Rogosin, S.V.: Mittag–Leffler Functions, Related Topics and Applications. Springer, Berlin (2014)

    Book  MATH  Google Scholar 

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

    MATH  Google Scholar 

  • Hanyga, A.: Viscous dissipation and completely monotonic relaxation moduli. Rheol. Acta 44, 614–621 (2005)

    Article  Google Scholar 

  • Hanyga, A.: Attenuation and shock waves in linear hereditary viscoelastic media; Strick–Mainardi, Jeffreys–Lomnitz–Strick and Andrade creep compliances. Pure Appl. Geophys. 171(9), 2097–2109 (2014)

    Article  Google Scholar 

  • Holenstein, R., Nieder, P.: A viscoelastic model for use in predicting arterial pulse waves. J. Biomech. Eng. 102, 318–325 (1980)

    Article  Google Scholar 

  • Jellinek, H.H.G., Brill, R.: Viscoelastic properties of ice. J. Appl. Phys. 27(10), 1198–1209 (1956)

    Article  Google Scholar 

  • Knopoff, L.: Q. Rev. Geophys. 2(4), 625–660 (1964)

    Article  Google Scholar 

  • Lubliner, J., Panoskaltsis, V.P.: The modified Kuhn model of linear viscoelasticty. Int. J. Solids Struct. 29(24), 3099–3112 (1992)

    Article  MATH  Google Scholar 

  • Mainardi, F.: Fractional calculus, some basic problems in continuum and statistical mechanics. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 291–348. Springer, Wien (1997). arXiv:1201.0863

    Chapter  Google Scholar 

  • Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press, London (2010)

    Book  MATH  Google Scholar 

  • Mainardi, F., Spada, G.: Creep, relaxation and viscosity properties for basic fractional models in rheology. Eur. Phys. J. Spec. Top. 193, 133–160 (2011). arXiv:1110.3400

    Article  Google Scholar 

  • Mainardi, F., Spada, G.: Becker and Lomnitz rheological models: a comparison. In: D’Amore, A., Grassia, L., Acierno, D. (eds.) Proceedings of the International Conference TOP (Times of Polymers & Composites), Ischia, Italy, 10–14 June 2012. AIP Conf. Proc., vol. 1459, pp. 132–135 (2012a). arXiv:1210.5717

    Google Scholar 

  • Mainardi, F., Spada, G.: On the viscoelastic characterization of the Jeffreys–Lomnitz law of creep. Rheol. Acta 51, 783–791 (2012b). E-print. http://arxiv.org/abs/1112.5543

    Article  Google Scholar 

  • Masina, E., Mainardi, F.: On the generalized exponential integral. Preprint, Department of Physics, University of Bologna (2017)

  • Metzler, R., Schick, W., Kilian, H.G., Nonnenmacher, T.F.: Relaxation in filled polymers: a fractional calculus approach. J. Chem. Phys. 103(16), 7180–7186 (1995)

    Article  Google Scholar 

  • Molinari, A.: Viscoélasticité linéaire et fonctions complètement monotones. J. Méc. 12, 541–553 (1973)

    MATH  Google Scholar 

  • Neubert, H.K.P.: A simple model representing internal damping in solid materials. Aeronaut. Q. 14(2), 187–197 (1963)

    Article  Google Scholar 

  • Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST Digital Library of Mathematical Functions. Cambridge Univ. Press, Cambridge (2010)

    Google Scholar 

  • Orowan, E.: Seismic damping and creep in the mantle. Geophys. J. R. Astron. Soc. 14, 191–218 (1967)

    Article  Google Scholar 

  • Pipkin, A.C.: Lectures on Viscoelastic Theory, 2nd edn. Springer, New York (1986). First Edition, 1972

    Book  Google Scholar 

  • Sasso, M., Palmieri, G.: Application of fractional derivative models in linear viscoelastic problems. Mech. Time-Depend. Mater. 15(4), 367–387 (2011)

    Article  Google Scholar 

  • Schelkunoff, S.A.: Proposed symbols for the modified cosine and integral exponential integral. Q. Appl. Math. 2, 90 (1944)

    Article  MathSciNet  MATH  Google Scholar 

  • Schilling, R.L., Song, R., Vondracek, Z.: Bernstein Functions: Theory and Applications. De Gruyter, Berlin (2012)

    Book  MATH  Google Scholar 

  • Strick, E., Mainardi, F.: On a general class of constant \(Q\) solids. Geophys. J. R. Astron. Soc. 69, 415–429 (1982)

    Article  Google Scholar 

  • Tschoegl, N.W.: The Phenomenological Theory of Linear Viscoelastic Behavior. Springer, Heidelberg (1989)

    Book  MATH  Google Scholar 

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

    Article  Google Scholar 

  • Widder, D.V.: The Laplace Transform. Princeton Univ. Press, Princeton (1946)

    Google Scholar 

Download references

Acknowledgements

The work of F.M. has been carried out in the framework of the activities of the National Group of Mathematical Physics (GNFM, INdAM). The work of G.S. has been carried out in the framework of the activities of the Department of Pure and Applied Sciences (DiSPeA) of the Urbino University “Carlo Bo”. The authors would like to thank Alexander Apelblat, Andrea Giusti, Andrzej Hanyga and Nanna Karlsson for valuable comments, advice and discussions. Furthermore, the authors are grateful to the anonymous referees for careful reading of our paper and for making several important suggestions which improve the presentation.

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Correspondence to Francesco Mainardi.

Appendix A: Essentials of linear viscoelasticity

Appendix A: Essentials of linear viscoelasticity

We recall that in the linear theory of viscoelasticity, based on the hereditary theory by Volterra, a viscoelastic body is characterized by two distinct but interrelated material functions, causal in time (i.e. vanishing for \(t<0\)): the creep compliance \(J(t)\) (the strain response to a unit step of stress) and the relaxation modulus \(G(t)\) (the stress response to a unit step of strain). For more details, see e.g. Christensen (1982), Pipkin (1986), Tschoegl (1989), Tschoegl (1997) and Mainardi (2010).

By taking \(J(0^{+})=J_{0} >0\) so that \(G(0^{+})= G_{0} =1/J_{0}\), the body is assumed to exhibit a non vanishing instantaneous response both in the creep and in the relaxation tests. As a consequence, we find it convenient to introduce two dimensionless quantities \(\psi (t)\) and \(\phi (t)\) as follows:

$$ J(t)= J_{0}\bigl[1+q \psi (t)\bigr], \qquad G(t) = G_{0} \phi (t), $$
(A.1)

where \(\psi (t)\) is a non-negative increasing function with \(\psi (0) =0\) and \(\phi (t)\) is a non-negative decreasing function with \(\phi (0)=1\). We have assumed, without loss of generality \(\tau_{0}=1\), but we have kept the non-dimensional quantity \(q\) for a suitable scaling of the strain, according to convenience in experimental rheology. At this stage, viscoelastic bodies may be distinguished in solid-like and fluid-like whether \(J(+\infty )\) is finite or infinite so that \(G(+\infty )= 1/J(+\infty )\) is non zero or zero, correspondingly.

As pointed out in most treatises on linear viscoelastity, e.g. in Pipkin (1986), Tschoegl (1989), Mainardi (2010), the relaxation modulus \(G(t)\) can be derived from the corresponding creep compliance \(J(t)\) through the Volterra integral equation of the second kind

$$ G(t)= \frac{1}{J_{0}}-\frac{1}{J_{0}} \int_{0}^{t} \frac{dJ}{dt'} G \bigl(t-t'\bigr)\, dt' ; $$
(A.2)

then, as a consequence of Eq. (A.1), the non-dimensional relaxation function \(\phi (t)\) obeys the Volterra integral equation

$$ \phi (t)=1-q \int_{0}^{t}\frac{d\psi }{dt'} \phi \bigl(t-t'\bigr)\, dt'. $$
(A.3)

In linear viscoelasticity, it is quite common to require the existence of positive retardation and relaxation spectra for the material functions \(J(t)\) and \(G(t)\), as pointed out by Gross in his 1953 monograph on the mathematical structure of the theories of viscoelasticity (Gross 1953). This implies, as formerly proved in Molinari (1973) and revisited in Hanyga (2005), see also Mainardi (2010), that \(J(t)\) and \(G(t)\) and consequently the functions \(\psi (t)\) and \(\phi (t)\) turn out to be Bernstein and Completely Monotonic (CM) functions, respectively.

Here we recall that a CM function \(f(t)\) is a non-negative, infinitely derivable function with derivatives alternating in sign for \(t>0\) like \(\exp (-t)\), whereas a Bernstein function is a non-negative function whose derivative is CM, like \(1-\exp (-t)\). Then a necessary and sufficient condition to be a CM function is provided by the Bernstein theorem according to which \(f(t)\) is the Laplace transform of a non-negative real function. For more details on these mathematical properties the interested reader is referred to the excellent monograph by Schilling et al. (2012).

For the rate of creep, we write

$$ \frac{d\psi }{dt} (t)= \int_{0}^{\infty } \mathrm{e}^{-rt} K(r)\, dr = \int_{0}^{\infty } \mathrm{e}^{-t/\tau } H(\tau )\, d\tau , $$
(A.4)

where \(K(r)\) and \(H(\tau )\) are the required spectra in frequency (\(r\)) and in time (\(\tau =1/r\)), respectively.

The frequency spectrum can be determined from the Laplace transform of the rate of creep by the Titchmarsh formula that reads in an obvious notation, if \(\psi (0^{+})=0\),

$$ K(r) = \pm \frac{1}{\pi } \Im \bigl[s\tilde{ \psi }(s)\bigr] \bigm|_{s =r \mathrm{e}^{\mp i\pi }}. $$
(A.5)

This a consequence of the fact that the Laplace transform of the rate of creep is the iterated Laplace transform of the frequency spectrum, that is, the Stieltjes transform of \(K(r)\) and henceforth the Titchmarsh formula provides the inversion of the Stieltjes transform; see e.g. Widder (1946). As a consequence, the time spectrum can be determined using the transformation \(\tau = 1/r\), so that

$$ H(\tau ) = \frac{K(1/\tau )}{\tau^{2}}\,. $$
(A.6)

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Mainardi, F., Masina, E. & Spada, G. A generalization of the Becker model in linear viscoelasticity: creep, relaxation and internal friction. Mech Time-Depend Mater 23, 283–294 (2019). https://doi.org/10.1007/s11043-018-9381-4

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