Abstract
We use fractional viscoelastic models that result from the application of fractional calculus to the linear viscoelastic theory to characterize thermorheologically simple linear viscoelastic materials. Model parameters are obtained through an optimization procedure that simultaneously determines the time–temperature shift factors. We present analytical interconversion based on the fractional viscoelastic model between the main viscoelastic functions (relaxation modulus, creep compliance, storage modulus, and loss modulus) and the analytical forms of the relaxation and retardation spectra. We show that the fractional viscoelastic model can be approximated by a Prony series to any desired level of accuracy. This property allows the efficient determination of the fractional viscoelastic model response to any loading history using the well-known recursive relationships of Prony series models.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adolfsson K, Enelund M, Larsson S (2004) Adaptive discretization of fractional order viscoelasticity using sparse time history. Comput Meth Appl Mech Eng 193(42–44):4567–4590 doi:10.1016/j.cma.2004.03.006
Adolfsson K, Enelund M, Olsson P (2005) On the fractional order model of viscoelasticity. Mechanics of Time-dependent Materials 9(1):15–34 doi:10.1007/s11043-005-3442-1
Bagley RL, Torvik (1983) A theoretical basis for the application of fractional calculus to viscoelasticity. J Rheol 27(3):201–210 doi:10.1122/1.549724
Baumgaertel M, Winter (1989) Determination of discrete relaxation and retardation time spectra from dynamic mechanical data. Rheol Acta 28(6):511–519 doi:10.1007/bf01332922
Caputo M (1967)4 Linear models of dissipation whose Q is almost frequency independent—II. Geophys J Roy Astron Soc 13(5):529–539 doi:10.1111/j.1365-246X.1967.tb02303.x
Caputo M, Mainardi F (1971) Linear models of dissipation in anelastic solids. Rivista Del Nuovo Cimento 1(2):161–198 doi:10.1007/BF02820620
Enelund M, Lesieutre (1999) Time domain modeling of damping using anelastic displacement fields and fractional calculus. Int J Solids Struct 36(29):4447–4472 doi:10.1016/S0020-7683(98), 00194-2
Ford NJ, Simpson(2001) The numerical solution of fractional differential equations: speed versus accuracy. Numer Algorithm 26(4):333–346 doi:10.1023/A:1016601312158
Friedrich C (1991) Relaxation and retardation functions of the Maxwell model with fractional derivatives. Rheol Acta 30(2):151–158 doi:10.1007/BF01134604
Friedrich C, Braun H, Weese J (1995) Determination of relaxation time spectra by analytical inversion using a linear viscoelastic model with fractional derivatives. Polym Eng Sci 35(21):1661–1669 doi:10.1002/pen.760352102
Glöckle WG, Nonnenmacher TF (1991) Fractional integral operators and Fox functions in the theory of viscoelasticity. Macromolecules 24(24):6426–6434. doi:10.1021/ma00024a009
Haj Ali RM, Muliana (20041) Numerical finite element formulation of the Schapery non linear viscoelastic material model. Int J Numer Methods Eng 59(1):25–45 doi:10.1002/nme.861
Henriksen M (1984) Nonlinear viscoelastic stress analysis—a finite element approach. Comput Struct 18(1):133–139 doi:10.1016/0045-7949(84), 90088-9
Heymans N (1996) Hierarchical models for viscoelasticity: dynamic behaviour in the linear range. Rheol Acta 35(5):508–519 doi:10.1007/BF00369000
Heymans N, Bauwens (1994) Fractal rheological models and fractional differential equations for viscoelastic behavior. Rheol Acta 33(3):210–219 doi:10.1007/BF00437306
Honerkamp J, Weese J (1989) Determination of the relaxation spectrum by a regularization method. Macromolecules 22(11):4372–4377 doi:10.1021/ma00201a036
Honerkamp J, Weese J (1993) A nonlinear regularization method for the calculation of relaxation spectra. Rheol Acta 32(1):65–73 doi:10.1007/bf00396678
Kaschta J, Schwarzl (1994) Calculation of discrete retardation spectra from creep data—I. Method. Rheol Acta 33(6):517–529 doi:10.1007/bf00366336
Katicha SW, Flintsch (2011) Use of Fractional Order Viscoelastic Models to Characterize Asphalt Concrete. In: Proceedings of the 1st T&DI Congress, Chicago, IL, March 2011. ASCE, pp 677–687. doi:10.1061/41167(398)65
Koeller R (1984) Application of fractional calculus to the theory of viscoelasticity. J Appl Mech 51(2):299–307 doi:10.1115/1.3167616
Koeller R (1986) Polynomial operators, Stieltjes convolution, and fractional calculus in hereditary mechanics. Acta Mech 58(3):251–264 doi:10.1007/BF01176603
Lai J, Bakker A (1996) 3-D Schapery representation for non-linear viscoelasticity and finite element implementation. Comput Mech 18(3):182–191 doi:10.1007/BF00369936
Mainardi F (2010a) An historical perspective on fractional calculus in linear viscoelasticity. Arxiv preprint arXiv:10072959
Mainardi F (2010b) Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models. Imperial College, London
Mainardi F, Gorenflo R (2000) On Mittag-Leffler-type functions in fractional evolution processes. J Comput Appl Math 118(1–2):283–299 doi:10.1016/S0377-0427(00), 00294-6
Mainardi F, Spada G (2011) Creep, relaxation and viscosity properties for basic fractional models in rheology. Eur Phys J Spec Top 193(1):133–160
Malkin (2006) The use of a continuous relaxation spectrum for describing the viscoelastic properties of polymers. Polym Sci Series A 48(1):39–45 doi:10.1134/S0965545X06010068
Mead D (1994) Numerical interconversion of linear viscoelastic material functions. J Rheol 38(6):1769–1795 doi:10.1122/1.550526
Oeser M, Freitag S (2009) Modeling of materials with fading memory using neural networks. Int J Numer Methods Eng 78(7):843–862 doi:10.1002/nme.2518
Padovan J (1987) Computational algorithms for FE formulations involving fractional operators. Comput Mech 2(4):271–287 doi:10.1007/BF00296422
Papoulia KD, Panoskaltsis VP, Kurup NV, Korovajchuk I (2010) Rheological representation of fractional order viscoelastic material models. Rheol Acta 49(4):381–400 doi:10.1007/s00397-010-0436-y
Rossikhin YA, Shitikova MV (2010) Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent results. Appl Mech Rev 63(1):1–52 doi:10.1115/1.4000563
Roy S, Reddy J (1988) A finite element analysis of adhesively bonded composite joints with moisture diffusion and delayed failure. Comput Struct 29(6):1011–1031 doi:10.1016/0045-7949(88), 90327-6
Rudin W (1976) Principles of mathematical analysis. McGraw-Hill, New York
Schiessel H, Blumen A (1995) Mesoscopic pictures of the sol-gel transition: ladder models and fractal networks. Macromolecules 28(11):4013–4019 doi:10.1021/ma00115a038
Schiessel H, Metzler R, Blumen A, Nonnenmacher T (1995) Generalized viscoelastic models: their fractional equations with solutions. J Phys A Math Gen 28(23):6567 doi:10.1088/0305-4470/28/23/012
Sharma R, Cherayil (2010) Polymer melt dynamics: microscopic roots of fractional viscoelasticity. Phys Rev E 81(2):021804 doi:10.1103/PhysRevE.81.021804
Stadler F, Bailly C (2009) A new method for the calculation of continuous relaxation spectra from dynamic-mechanical data. Rheol Acta 48(1):33–49 doi:10.1007/s00397-008-0303-2
Taylor RL, Pister KS, Goudreau (1970) Thermomechanical analysis of viscoelastic solids. Int J Numer Methods Eng 2(1):45–59 doi:10.1002/nme.1620020106
Weese J (1993) A regularization method for nonlinear ill-posed problems. Comput Phys Commun 77(3):429–440 doi:10.1016/0010-4655(93)90187-h
Acknowledgements
We would like to acknowledge the anonymous reviewers for their comments and suggestions to improve the paper and for bringing to our attention an error in our initial definition of the Caputo derivative.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Katicha, S.W., Flintsch, G.W. Fractional viscoelastic models: master curve construction, interconversion, and numerical approximation. Rheol Acta 51, 675–689 (2012). https://doi.org/10.1007/s00397-012-0625-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00397-012-0625-y