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Research Papers

Nonlinear Fractional Derivative Models of Viscoelastic Impact Dynamics Based on Entropy Elasticity and Generalized Maxwell Law

[+] Author and Article Information
Masataka Fukunaga1

College of Engineering, Nihon University, Koriyama, Fukushima 963-8642, Japanfukunaga@apple.ifnet.or.jp

Nobuyuki Shimizu

Department of Mechanical Systems and Design Engineering, Iwaki Meisei University, Iwaki 970-8551, Japannshim@iwakimu.ac.jp

1

Corresponding author.

J. Comput. Nonlinear Dynam 6(2), 021005 (Oct 21, 2010) (6 pages) doi:10.1115/1.4002383 History: Received August 20, 2009; Revised March 26, 2010; Published October 21, 2010; Online October 21, 2010

Two types of models are proposed for describing nonlinear fractional derivative dynamical behavior of viscoelastic materials subject to impulse forces. The models are derived based on the thermodynamic elasticity in terms of entropy and on the “scale-free response of the material” under the basic assumption that the viscoelastic materials consist of stable coils of polymers, which we refer to as blobs. The blobs, which may be connected to each other by chemical bonds or physical bonds, are considered here as the elementary constituent of viscoelastic materials from which the nonlinear fractional derivative models are derived. Responses of individual blobs can determine the net collective response of the viscoelastic material to impulse forces. From the above consideration, two types of models are proposed in which the force elements or the stress elements are connected by the generalized Maxwell law.

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Figures

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Figure 1

Deformation of a rectangular volume element. The arrow indicates the applied force.

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Figure 2

The generalized Maxwell model

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Figure 3

A rough sketch of blobs in a gel before and after the compression. The points A, B, C, and D represent the junctions where the blobs connect.

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Figure 4

Change in the x-surface by deformation. The circles indicate the cross sections of the blobs. The thick frame shows the surface of the volume elements where the stress is estimated. (a) The volume element of fixed size. (b) The volume element varying with the deformation of the gel.

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