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A Fractional Calculus Approach to the Prediction of Free Recovery Behaviors of Amorphous Shape Memory Polymers

Published online by Cambridge University Press:  27 October 2015

C.-Q. Fang ,
H.-Y. Sun  and
J.-P. Gu
C.-Q. Fang
Affiliation:
State Key Laboratory of Mechanics and Control of Mechanical Structures Nanjing University of Aeronautics and Astronautics Nanjing, China
H.-Y. Sun*
Affiliation:
State Key Laboratory of Mechanics and Control of Mechanical Structures Nanjing University of Aeronautics and Astronautics Nanjing, China
J.-P. Gu
Affiliation:
State Key Laboratory of Mechanics and Control of Mechanical Structures Nanjing University of Aeronautics and Astronautics Nanjing, China Department of Materials Engineering Nanjing Institute of Technology Nanjing, China
*
* Corresponding author (hysun@nuaa.edu.cn)
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Abstract

A fractional model generalized from the Zener model is proposed for the prediction of temperature-dependent free recovery behaviors of amorphous shape memory polymers (SMPs). This model differs from the Zener model in that it involves nonlinear differential equations of fractional, not integer, order. The theoretical solution based on this fractional model is utilized to simulate the isothermal and nonisothermal free recovery of an amorphous SMP compared with the one based on the Zener model. The results show a reasonable improvement in the prediction of the strain recovery response of SMP by the fractional calculus method.

Keywords

Shape memory polymersFractional calculusConstitutive modelFree recovery
Type
Research Article
Information
Journal of Mechanics , Volume 32 , Issue 1 , February 2016 , pp. 11 - 17
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2016 

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References

1.Behl, M. and Lendlein, A., “Shape-Memory Polymers,” Materials Today, 10, pp. 20–28 (2007).Google Scholar
2.Lendlein, A. and Kelch, S., “Shape-Memory Polymers,” Angewandte Chemie International Edition, 41, pp. 2034–2057 (2002).Google Scholar
3.Ratna, D. and Karger-Kocsis, J., “Recent Advances in Shape Memory Polymers and Composites: A Review,” Journal of Materials Science, 43, pp. 254–269 (2008).CrossRefGoogle Scholar
4.Leng, J. S., Lan, X., Liu, Y. J. and Du, S. Y. ,“Shape-Memory Polymers and their Composites: Stimulus Methods and Applications,” Progress in Materials Science, 56, pp. 1077–1135 (2011).CrossRefGoogle Scholar
5.Liu, Y. J.Du, H. Y., Liu, L. W. and Leng, J. S., “Shape Memory Polymer Composites and their Applications in Aerospace: A Review,” Smart Materials and Structures, 23, p. 023001 (2014).Google Scholar
6.Chen, J. G., Liu, L. W., Liu, Y. J. and Leng, J. S., “Thermoviscoelastic Shape Memory Behavior for Epoxy-Shape Memory Polymer,” Smart Materials and Structures, 23, p. 055025 (2014).Google Scholar
7.Liu, Y. P., Gall, K., Dunn, M. L., Greenberg, A. R. and Diani, J., “Thermomechanics of Shape Memory Polymers: Uniaxial Experiments and Constitutive Modeling,” International Journal of Plasticity, 22, pp. 279–313 (2006).Google Scholar
8.Qi, H. J., Nguyen, T. D., Castro, F., Yakacki, C. M. and Shandas, R., “Finite Deformation Thermo-Mechanical Behavior of Thermally Induced Shape Memory Polymers,” Journal of Mechanics and Physics of Solids, 56, pp. 1730–1751 (2008).Google Scholar
9.Westbrook, K. K., Kao, P. H., Castro, F., Ding, Y. F. and Qi, H. J., “A 3D Finite Deformation Constitutive Model for Amorphous Shape Memory Polymers: A Multi-branch Modeling Approach for Nonequilibrium Relaxation Processes,” Mechanics of Materials, 14, pp. 853–869 (2011).CrossRefGoogle Scholar
10.Srivastava, V., Chester, S. A. and Anand, L., “Thermally Actuated Shape Memory Polymers: Experiments, Theory, and Numerical Simulations,” Journal of Mechanics and Physics of Solids, 58, pp.1100–1124 (2010).Google Scholar
11.Nguyen, T. D., Qi, H. J., Castro, F. and Long, K. N., “A Thermoviscoelastic Model for Amorphous Shape Memory Polymers: Incorporating Structural and Stress Relaxation,” Journal of Mechanics and Physics of Solids, 56, pp. 2792–2814 (2008).CrossRefGoogle Scholar
12.Chen, Y. C. and Lagoudas, D. C., “A Constitutive Theory for Shape Memory Polymers. Part I-Large Deformations,” Journal of Mechanics and Physics of Solids, 56, pp. 1752–1765 (2008).Google Scholar
13.Chen, Y. C. and Lagoudas, D. C., “A Constitutive Theory for Shape Memory Polymers. Part II-A Linearized Model for Small Deformations,” Journal of Mechanics and Physics of Solids, 56, pp. 1766–1778 (2008).Google Scholar
14.Ge, Q., Yu, K., Ding, Y. F. and Qi, H. J., “Prediction of Temperature-Dependent Free Recovery Behaviors of Amorphous Shape Memory Polymers,” Soft Matter, 8, p. 11098 (2012).Google Scholar
15.Bagley, R. L. and Torvik, P. J., “A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity,” Journal of Rheology, 11, pp. 201–210 (1983).Google Scholar
16.Nutting, P. G., “A New Generalized Law of Deformation,” Journal of the Franklin Institute, 191, pp. 679–685 (1921).Google Scholar
17.Gemant, A., “A Method of Analyzing Experimental Results Obtained from Elasto-Viscous Bodies,” Journal of Applied Physics, 7, pp. 311–317 (1936).Google Scholar
18.Bagley, R. L. and Torvik, J., “Fractional Calculus-a Different Approach to the Analysis of Viscoelastically Damped Structures,” AIAA Journal, 21, pp. 741–748 (1983).Google Scholar
19.Rogers, L., “Operators and Fractional Derivatives for Viscoelastic Constitutive Equations,” Journal of Rheology, 27, pp. 351–372 (1983).Google Scholar
20.Bagley, R. L. and Torvik, P. J., “On the Fractional Calculus Model of Viscoelastic Behavior,” Journal of Rheology, 30, pp. 133–155 (1986).Google Scholar
21.Pritz, T., “Analysis of Four-Parameter Fractional Derivative Model of Real Solid Materials,” Journal of Sound and Vibration, 195, pp. 103–115 (1996).Google Scholar
22.Shimizu, N. and Zhang, W., “Fractional Calculus Approach to Dynamic Problems of Viscoelastic Materials,” JSME International Journal. Series C, Mechanical Systems, Machine Elements and Manufacturing, 42, pp. 825–837 (1999).Google Scholar
23.Koeller, R. C., “Applications of Fractional Calculus to the Theory of Viscoelasticity,” Journal of Applied Mechanics, 51, pp. 299–307 (1984).Google Scholar
24.Welch, S. W. J., Rorrer, R. A. L. and Duren, J. R. G., “Application of Time-Based Fractional Calculus Methods to Viscoelastic Creep and Stress Relaxation of Materials,” Mechanics of Time-Dependent Materials, 3, pp. 279–303 (1999).Google Scholar
25.Pritz, T., “Five-Parameter Fractional Derivative Model for Polymeric Damping Materials,” Journal of Sound and Vibration, 265, pp. 935–952 (2003).Google Scholar
26.Podlubny, I., Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, California (1998).Google Scholar
27.Lendlein, A. and Langer, R., “Biodegradable, Elastic Shape-Memory Polymers for Potential Biomedical Applications,” Science, 296, pp. 1673–1676 (2002).Google Scholar
28.Yu, K., Ge, Q. and Qi, H. J., “Reduced Time as a Unified Parameter Determining Fixity and Free Recovery of Shape Memory Polymers,” Nature Communications, 5, p. 3066 (2014).Google Scholar
29.Williams, M. L., Landel, R. F. and Ferry, J. D., “The Temperature Dependence of Relaxation Mechanism in Apmorphous Polymers and Other Glass-Forming Liquids,” Journal of the American Chemical Society, 77, pp. 3701–3707 (1955).Google Scholar
30.Marzio, E. A. D. and Yang, A. J. M., “Configurational Entropy Approach to the Kinetics of Glasses,” Journal of Research of the National Institute of Standards and Technology, 102, pp. 135–157 (1997).Google Scholar
31.Raja, M. A. Z., Khan, J. A. and Qureshi, I. M., “Solution of Fractional Order System of Bagley-Torvik equation using Evolutionary Computational Intelligence,” Mathematical Problems in Engineering, 2011, p. 675075(2011).Google Scholar
32.Kanetakis, J., Fytas, G., Kremer, F. and Pakula, T., “Segmental Dynamics in Homogeneous 1, 4-Polyisoprene-1, 2-Polybutadiene Diblock Copolymers,” Macromolecules, 25, pp. 3484–3491 (1992).Google Scholar
33.Apitz, D. and Johansen, P. M., “Limitations of the Stretched Exponential Function for Describing Dynamics in Disordered Solid Materials,” Journal of Applied Physics, 97, p. 063507 (2005).Google Scholar
34.Nordström, J., Nilsson, E., Jarvol, P., Nayeri, M., Palmqvist, A., Bergenholtz, J. and Matic, A., “Concentration and pH-Dependence of Highly Alkaline Sodium Silicate Solutions,” Journal of Colloid and Interface Science, 356, pp. 37–45 (2011).Google Scholar