THERMAL SCIENCE

International Scientific Journal

Feng Gao

,

Xiao-Jun Yang

,

Syed Tauseef Mohyud-Din

ON LINEAR VISCOELASTICITY WITHIN GENERAL FRACTIONAL DERIVATIVES WITHOUT SINGULAR KERNEL

ABSTRACT
The Riemann-Liouville and Caputo-Liouville fractional derivatives without singular kernel are proposed as mathematical tools to describe the mathematical models in line viscoelasticity in the present article. The fractional mechanical models containing the Maxwell and Kelvin-Voigt elements are graphically discussed with the Laplace transform. The results are accurate and efficient to reveal the complex behaviors of the real materials.
KEYWORDS
fractional derivatives without singular kernel, laplace transform, Maxwell element, Kelvin-Voigt element, linear viscoelasticity
PAPER SUBMITTED: 2017-03-08
PAPER REVISED: 2017-04-25
PAPER ACCEPTED: 2017-05-18
PUBLISHED ONLINE: 2017-09-09
DOI REFERENCE: https://doi.org/10.2298/TSCI170308197G
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2017, VOLUME 21, ISSUE Supplement 1, PAGES [S335 - S342]
REFERENCES
  1. Yang, X. J., et al., Local Fractional Integral Transforms and Their Applications, Academic Press, New York, USA, 2015
  2. Li, C., Deng, W., Remarks on Fractional Derivatives, Applied Mathematics and Computation, 187 (2007), 2, pp. 777-784
  3. Tarasov, V. E., Zaslavsky, G. M., Nonholonomic Constraints with Fractional Derivatives, Journal of Physics A: Mathematical and General,39 (2006), 31, pp.97
  4. Ortigueira, M. D., et al., On the Computation of the Multidimensional Mittag-Leffler function, Communications in Nonlinear Science and Numerical Simulation, 53 (2017), pp. 278-287
  5. Liu, F., et al., Numerical Solution of the Space Fractional Fokker-Planck Equation, Journal of Computational and Applied Mathematics, 166 (2004), 1, pp. 209-219
  6. Baleanu, D., Trujillo, J. I., A New Method of Finding the Fractional Euler-Lagrange and Hamilton Equations within Caputo Fractional Derivatives, Communications in Nonlinear Science and Numerical Simulation, 15 (2010), 5, pp. 1111-1115.
  7. Mainardi, F., An Historical Perspective on Fractional Calculus in Linear Viscoelasticity, Fractional Calculus and Applied Analysis, 15 (2012), 4, pp. 712-717
  8. Bagley, R. L., Torvik, P. J. On the Fractional Calculus Model of Viscoelastic Behavior, Journal of Rheology, 30 (1986). 1, pp. 133-155
  9. Jaishankar, A., Mckinley, G. H., Power-law Rheology in the Bulk and at the Interface: Quasi-properties and Fractional Constitutive Equations, Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences, 469 (2012), pp. 2149
  10. Fukunaga, M., Shimizu, N., Role of Prehistories in the Initial Value Problems of Fractional Viscoelastic Equations, Nonlinear Dynamics, 38 (2004), 1-4, pp. 207-220
  11. Zhou, H. W., et al., A Creep Constitutive Model for Salt Rock Based on Fractional Derivatives, International Journal of Rock Mechanics and Mining Sciences, 48 (2011), 1, pp. 116-121
  12. Hristov, J., Transient Heat Diffusion with a Non-singular Fading Memory: from the Cattaneo Constitutive Equation with Jeffrey's Kernel to the Caputo-Fabrizio Time-fractional Derivative, Thermal Science, 20 (2016), 2, pp. 765-770
  13. Atangana, A., Baleanu, D., Caputo-Fabrizio Derivative Applied to Groundwater Flow within Confined Aquifer, Journal of Engineering Mechanics, 2016, D4016005
  14. Gómez-Aguilar, J. F., et al., Analytical Solutions of the Electrical RLC Circuit Via Liouville-Caputo Operators with Local and Non-local Kernels, Entropy, 18 (2016), 8, pp. 402
  15. Yang, X. J., et al., A New Fractional Derivative without Singular Kernel: Application to the Modelling of the Steady Heat Flow, Thermal Science, 20 (2016), pp. 753-756
  16. Yang, A. M., et al., On Steady Heat Flow Problem Involving Yang-Srivastava-Machado Fractional Derivative without Singular Kernel, Thermal Science, 20 (2016), 3, pp. S717-S721
  17. Yang, X. J., et al., Some New Applications for Heat and Fluid Flows Via Fractional Derivatives without Singular Kernel, Thermal Science, 20 (2016), S3, pp. 833-839
  18. Yang, X. J., et al., Anomalous Diffusion Models with General Fractional Derivatives within the Kernels of the Extended Mittag-Leffler Type Functions, Romanian Reports in Physics, 2017, in press
  19. Yang, X. J., Fractional Derivatives of Constant and Variable Orders Applied to Anomalous Relaxation Models in Heat-transfer Problems, Thermal Science, 21 (2017), 3, 2017, pp.1161-1171
  20. Yang, X. J., New Rheological Problems Involving General Fractional Derivatives within Nonsingular Power-law Kernel, Proceedings of the Romanian Academy - Series A, 2017, in press
  21. Yang, X. J., New General Fractional-order Rheological Models within Kernels of Mittag-Leffler Functions, Romanian Reports in Physics, 2017, in press
  22. Yang, et al., New Rheological Models within Local Fractional Derivative, Romanian Reports in Physics, 69 (2017), 3, pp.113
  23. Gao, F., et al., Fractional Maxwell Fluid with Fractional Derivative without Singular Kernel, Thermal Science, 20 (2016), pp. 871-877
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On linear viscoelasticity within general fractional derivatives without singular kernel

© 2024 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence