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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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Figures

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Fig. 1

The Generalized Maxwell Model

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Fig. 2

The initial and current configurations of a continuum body

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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

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Fig. 4

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

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