Research Papers

Fractional Derivative Constitutive Models for Finite Deformation of Viscoelastic Materials

[+] Author and Article Information
Masataka Fukunaga

P.T. Lecturer
College of Engineering,
Nihon University,
Koriyama, Fukushima 963-8642, Japan
e-mail: fukunaga@apple.ifnet.or.jp

Nobuyuki Shimizu

MotionLabo Inc.,
TSK Boulevard 302, 9-1, Kanari-Sunada,
Onahama, Iwaki,
Fukushima 971-8135, Japan
e-mail: nshim@motionlabo.com

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received March 23, 2014; final manuscript received August 25, 2014; published online April 9, 2015. Assoc. Editor: J. A. Tenreiro Machado.

J. Comput. Nonlinear Dynam 10(6), 061002 (Nov 01, 2015) (8 pages) Paper No: CND-14-1080; doi: 10.1115/1.4028438 History: Received March 23, 2014; Revised August 25, 2014; Online April 09, 2015

A methodology to derive fractional derivative constitutive models for finite deformation of viscoelastic materials is proposed in a continuum mechanics treatment. Fractional derivative models are generalizations of the models given by the objective rates. The method of generalization is applied to the case in which the objective rate of the Cauchy stress is given by the Truesdell rate. Then, a fractional derivative model is obtained in terms of the second Piola–Kirchhoff stress tensor and the right Cauchy-Green strain tensor. Under the assumption that the dynamical behavior of the viscoelastic materials comes from a complex combination of elastic and viscous elements, it is shown that the strain energy of the elastic elements plays a fundamental role in determining the fractional derivative constitutive equation. As another example of the methodology, a fractional constitutive model is derived in terms of the Biot stress tensor. The constitutive models derived in this paper are compared and discussed with already existing models. From the above studies, it has been proved that the methodology proposed in this paper is fully applicable and effective.

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Gemant, A., 1936, “A Method of Analyzing Experimental Results Obtained From Elasto-Viscous Bodies,” J. Appl. Phys., 7(8), pp. 311–317.
Blair, S. G. W., 1944, “Analytical and Integrative Aspects of the Stress–Strain–Time Problems,” J. Sci. Instrum., 21(5), pp. 80–84. [CrossRef]
Rabotnov, Y. N., 1980, Elements of Hereditary Solid Mechanics, Mir Publishers, Moscow, Russia.
Bagley, R. L., and Torvik, P. J., 1983, “A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity,” J. Rheol., 27(3), pp. 201–210. [CrossRef]
Bagley, R. L., and Torvik, P. J., 1983, “Fractional Calculus—A Different Approach to the Analysis of Viscoelasticity Damped Structures,” AIAA J., 21(5), pp. 741–748. [CrossRef]
Koeller, R. C., 1984, “Application of Fractional Calculus to the Theory of Viscoelasticity,” ASME J. Appl. Math., 51(2), pp. 299–307. [CrossRef]
Oldham, K. H., and Spanier, J., 1974, The Fractional Calculus, Mineola, NY.
Podlubny, I., 1999, Fractional Differential Equations, Academic Press, NY.
Kilbas, A. A., Srivastava, H. M., and Trujillo, J. J., 2006, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam.
Caputo, M., 1967, “Linear Model of Dissipation Whose q is Almost Frequency Independent—II,” Geophys. J. R. Astron. Soc., 13(5), pp. 529–539. [CrossRef]
Lodge, A. S., 1964, Elastic Liquids, Academic Press, London.
Freed, A. D., and Diethelm, K., 2006, “Fractional Calculus in Biomechanics: A 3D Viscoelastic Model Using Regularlized Fractional Derivative Kernels With Application to the Human Calcaneal Fat Pad,” Biomech. Modell. Mechanobiol., 5(4), pp. 203–215. [CrossRef]
Fukunaga, M., and Shimizu, N., 2009, “Analysis of Impulse Response of a Gel by Nonlinear Fractional Derivative Model,” ASME DETC 2009, San Diego, CA, Aug. 30–Sept. 2, ASME Paper No. DETC2009/86803. [CrossRef]
Bird, R. B., Armstrong, R. C., and Hassager, O., 1987, Dynamics Elastic Liquids, Vol. 1, Wiley, NY.
Bird, R. B., Curtiss, C. F., Armstrong, R. C., and Hassager, O., 1987, Dynamics Elastic Liquids, Vol. 2, Wiley, NY.
Drozdov, A. D., 1997, “Fractional Differential Models in Finite Viscoelasticity,” Acta Mech., 124(1–4), pp. 155–180. [CrossRef]
Haupt, P., and Lion, A., 2002, “On Finite Linear Viscoelasticity of Incompressible Isotropic Materials,” Acta Mech., 159(1–4), pp. 87–124. [CrossRef]
Adolfsson, K., and Enelund, M., 2003, “Fractional Derivative Viscoelasticity at Large Deformations,” Nonlinear Dyn., 33(3), pp. 301–321. [CrossRef]
Adolfsson, K., 2004, “Nonlinear Fractional Order Viscoelasticity at Large Deformations,” Nonlinear Dyn., 38(1–4), pp. 233–246. [CrossRef]
Freed, A. D., and Diethelm, K., 2007, “Caputo Derivatives in Viscoelasticity: A Non-Linear Finite-Deformation Theory for Tissue,” Fractional Calculus Appl. Anal., 10(3), pp. 219–248.
Nasuno, H., 2009, “Nonlinear Viscoelastic Finite Element Analysis by Means of Fractional Calculus,” Ph.D. dissertation, Graduate Course of Science and Engineering, Iwaki Meisei University, Iwaki, Japan.
Xiao, H., Bruhns, O. T., and Meyers, A., 1997, “Hypo-Elasticity Model Based Upon the Logarithmic Stress Rate,” J. Elasticity, 47(1), pp. 51–68. [CrossRef]
Schiessel, H., and Blumen, A., 1993, “Hierarchical Analogues to Fractional Relaxation Equations,” J. Phys. A, 26(19), pp. 5057–5069. [CrossRef]
Heymans, N., and Bauwens, J. C., 1994, “Fractional Rheological Models and Fractional Differential Equations for Viscoelastic Behavior,” Rheol. Acta, 33(3), pp. 210–210. [CrossRef]
Heymans, N., 1996, “Hierarchical Models for Viscoelasticity: Dynamic behavior in the Linear Range,” Rheol. Acta, 35(5), pp. 508–519. [CrossRef]
Rouse, P. E., Jr., 1953, “A Theory of the Linear Viscoelastic Properties of Dilute Solutions of Coiling Polymers,” J. Chem. Phys., 21(7), pp. 1272–1280. [CrossRef]
Zimm, B. H., 1956, “Dynamics of Polymer Molecules in Dilute Solution: Viscoelasticity, Flow Birefringence and Dielectric Loss,” J. Chem. Phys., 24(2), pp. 269–278. [CrossRef]
Coleman, B. D., and Noll, W., 1961, “Foundation of Linear Viscoelasticity,” Rev. Mod. Phys., 33(2), pp. 239–249. [CrossRef]
Caputo, M., and Mainardi, F., 1971, “Linear Models of Dissipation in Anelastic Solids,” Riv. del Nuovo Cimento, 1(2), pp. 161–198. [CrossRef]
Strobl, G. R., 1996, The Physics of Polymers, Springer, Berlin, Germany.
Mainardi, F., 2010, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, UK.
Lu, S. C. H., and Pister, K. S., 1975, “Decomposition of Deformation and Representation of the Free Energy Function for Isotropic Thermoelastic Solids,” Int. J. Solids Struct., 11(7,8), pp. 927–934. [CrossRef]
Lubliner, J., 1985, “A Model of Rubber Viscoelasticity,” Mech. Res. Commun., 12(2), pp. 93–99. [CrossRef]
Simo, J. C., 1987, “On a Fully Three-Dimensional Finite-Strain Viscoelastic Damage Model: Formulation and Computational Aspects,” Comput. Method Appl. Mech. Eng., 60(2), pp. 153–173. [CrossRef]
Govindjee, S., and Simo, C., 1992, “Mullins' Effect and the Strain Amplitude Dependence of the Storage Modus,” Int. J. Solids Struct., 29(14,15), pp. 1737–1751. [CrossRef]
Holzapfel, G. A., and Simo, J. C., 1996, “A New Viscoelastic Constitutive Model for Continuous Media at Finite Thermomechanical Changes,” Int. J. Solids Struct., 33(20–22), pp. 3019–3034. [CrossRef]
Simo, J. C., and Hughes, T. J. R., 1998, Computational Inelasticity. Springer, NY.
Lion, A., 1998, “Thixotropic Behavior of Rubber Under Dynamic Loading Histories: Experimental Results and Theory,” J. Mech. Phys. Solids, 46(5), pp. 895–930. [CrossRef]
Holzapfel, G. A., 2000, Nonlinear Solid Mechanics, Wiley, Chichester, NY.
Haupt, P., and Seldan, K., 2002, “Viscoelasticity of Elastic Materials: Experimental Facts and Constitutive Modelling,” Acta Mech., 159(1–4), pp. 87–124. [CrossRef]
de Gennes, P., 1979, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, NY.
Treloar, L. R. G., 1975, The Physics of Rubber Elasticity, Third Edition, Oxford Classic Texts in the Physical Sciences, Clarendon Press, Oxford, UK.
Fukunaga, M., and Shimizu, N., 2011, “Nonlinear Fractional Derivative Models of Viscoelastic Impact Dynamics Based on Viscoelasticity And Generalized Maxwell Law,” ASME J. Comput. Nonlinear Dyn., 6(2), p. 021005. [CrossRef]
Fukunaga, M., Shimizu, N., and Nasuno, H., 2009, “A Nonlinear Fractional Derivative Models of Impulse Motion for Viscoelastic Materials,” Phys. Scr., T136, p. 014010. [CrossRef]
Lubliner, J., 1969, “On Fading Memory in Materials of Evolutionary Type,” Acta Mech., 8(1,2), pp. 75–81. [CrossRef]
Valanis, K. C., 1971, “A Theory of Viscoelasticity With a Yield Surface Part I—General Theory,” Arch. Mech., 23(4), pp. 517–533.
Truesdell, C., and Noll, W., 2004, The Non-Linear Field Theories of Mechanics, 3rd ed., Springer, Heidelberg, Germany.
Fukunaga, M., and Shimizu, N., 2011, “Three-Dimensional Fractional Derivative Models for Finite Deformation,” ASME 2011 IDETC/CIE 2011, Washington, DC, Aug. 28–31, ASME Paper No. IDETC2011/47552. [CrossRef]
Bonet, J., and Wood, R. D., 1997, Nonlinear Continuum Mechanics for Finite Element Analysis, Cambridge University Press, NY.
Biot, M. A., 1965, Mechanics of Incremental Deformations, Wiley, NY.
Dienes, J. K., 1979, “On the Analysis of Rotation and Stress Rate in Deforming Bodies,” Acta Mech., 32(4), pp. 217–232. [CrossRef]
Kleiber, M., 1986, “On Errors Inherent in Commonly Accepted Rate Forms of Elastic Constitutive Laws,” Arch. Mech., 38(3), pp. 271–277.
Meyers, A., Xiao, H., and Bruhns, O. T., 2006, “Choice of Objective Rate in Single Parameter Hypoelastic Deformation Cycles,” Comput. Struct., 84(17,18), pp. 1134–1140. [CrossRef]
Fukunaga, M., and Shimizu, N., 2014, “Comparison of Fractional Derivative Models for Finite Deformation With Experiments of Impulse Response,” J. Vib. Control, 20(7), pp. 1033–1041. [CrossRef]


Grahic Jump Location
Fig. 1

The Generalized Maxwell Model

Grahic Jump Location
Fig. 3

Infinitesimal vectors directed from the point P to the point Q fixed to the body at the initial time dX and the current time dx

Grahic Jump Location
Fig. 2

The initial and current configurations of a continuum body

Grahic Jump Location
Fig. 4

The response of the fictitious specimen to the uni-axial compression with constant rate




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