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Research Papers: Heat and Mass Transfer

Time-Fractional Hygrothermoelastic Problem for a Sphere Subjected to Heat and Moisture Flux

[+] Author and Article Information
Xue-Yang Zhang, Yi Peng

School of Civil Engineering,
Central South University,
Changsha 410075, China

Xian-Fang Li

School of Civil Engineering,
Central South University,
Changsha 410075, China
e-mail: xfli@csu.edu.cn

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received September 2, 2017; final manuscript received August 22, 2018; published online October 8, 2018. Assoc. Editor: Evelyn Wang.

J. Heat Transfer 140(12), 122002 (Oct 08, 2018) (9 pages) Paper No: HT-17-1522; doi: 10.1115/1.4041419 History: Received September 02, 2017; Revised August 22, 2018

In this paper, a non-Fourier model of heat conduction and moisture diffusion coupling is proposed. We study a hygrothermal elastic problem within the framework of time-fractional calculus theory for a centrally symmetric sphere subjected to physical heat and moisture flux at its surface. Analytic expressions for transient response of temperature change, moisture distribution, displacement, and stress components in the sphere are obtained for heat/moisture flux pulse and constant heat/moisture flux at the sphere's surface, respectively, by using the integral transform method. Numerical results are calculated and the effects of fractional order on temperature field, moisture distribution, and hygrothermal stress components are illustrated graphically. Subdiffusive and super-diffusive transport coupling behavior as well as wave-like behavior are shown. When fractional-order derivative reduces to first-order derivative, the usual heat and moisture coupling is recovered, which obeys Fourier heat conduction and Fick's moisture diffusion.

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References

Sih, G. , Shih, M. , and Chou, S. , 1980, “ Transient Hygrothermal Stresses in Composites: Coupling of Moisture and Heat With Temperature Varying Diffusivity,” Int. J. Eng. Sci., 18(1), pp. 19–42. [CrossRef]
Sih, G. C. , Michopoulos, J. , and Chou, S.-C. , 1986, Hygrothermoelasticity, Martinus Niijhoof Publishing, Dordrecht, The Netherlands.
Chang, W.-J. , Chen, T.-C. , and Weng, C.-I. , 1991, “ Transient Hygrothermal Stresses in an Infinitely Long Annular Cylinder: Coupling of Heat and Moisture,” J. Therm. Stresses, 14(4), pp. 439–454. [CrossRef]
Chang, W.-J. , 1994, “ Transient Hygrothermal Responses in a Solid Cylinder by Linear Theory of Coupled Heat and Moisture,” Appl. Math. Modell., 18(8), pp. 467–473. [CrossRef]
Benkhedda, A. , Tounsi, A. , and Addabedia, E. A. , 2008, 2008, “ Effect of Temperature and Humidity on Transient Hygrothermal Stresses During Moisture Desorption in Laminated Composite Plates,” Compos. Struct., 82(4), pp. 629–635. [CrossRef]
Zenkour, A. , 2010, “ Hygro-Thermo-Mechanical Effects on FGM Plates Resting on Elastic Foundations,” Compos. Struct., 93(1), pp. 234–238. [CrossRef]
Zenkour, A. , 2014, “ Hygrothermoelastic Responses of Inhomogeneous Piezoelectric and Exponentially Graded Cylinders,” Int. J. Pressure Vessels Piping, 119, pp. 8–18. [CrossRef]
Chiba, R. , and Sugano, Y. , 2011, “ Transient Hygrothermoelastic Analysis of Layered Plates With One-Dimensional Temperature and Moisture Variations Through the Thickness,” Compos. Struct., 93(9), pp. 2260–2268. [CrossRef]
Ishihara, M. , Ootao, Y. , and Kameo, Y. , 2014, “ Hygrothermal Field Considering Nonlinear Coupling Between Heat and Binary Moisture Diffusion in Porous Media,” J. Therm. Stresses, 37(10), pp. 1173–1200. [CrossRef]
Mitra, K. , Kumar, S. , Vedevarz, A. , and Moallemi, M. K. , 1995, “ Experimental Evidence of Hyperbolic Heat Conduction in Processed Meat,” ASME J. Heat Transfer, 117(3), pp. 568–573. [CrossRef]
Tzou, D. Y. , 1995, “ Experimental Support for the Lagging Behavior in Heat Propagation,” J. Thermophys. Heat Transfer, 9(4), pp. 686–693. [CrossRef]
Hetnarski, R. B. , and Ignaczak, J. , 1999, “ Generalized Thermoelasticity,” J. Therm. Stresses, 22(4–5), pp. 451–476.
Lord, H. W. , and Shulman, Y. , 1967, “ A Generalized Dynamical Theory of Thermoelasticity,” J. Mech. Phys. Solids, 15(5), pp. 299–309. [CrossRef]
Green, A. , and Lindsay, K. , 1972, “ Thermoelasticity,” J. Elast., 2(1), pp. 1–7. [CrossRef]
Hetnarski, R. B. , and Ignaczak, J. , 1996, “ Soliton-Like Waves in a Low Temperature Nonlinear Thermoelastic Solid,” Int. J. Eng. Sci., 34(15), pp. 1767–1787. [CrossRef]
Green, A. , and Naghdi, P. , 1993, “ Thermoelasticity Without Energy Dissipation,” J. Elast., 31(3), pp. 189–208. [CrossRef]
Tzou, D. Y. , 1995, “ A Unified Field Approach for Heat Conduction From Macro- to Micro-Scales,” ASME J. Heat Transfer, 117(1), pp. 8–16. [CrossRef]
Chandrasekharaiah, D. , 1998, “ Hyperbolic Thermoelasticity: A Review of Recent Literature,” Appl. Mech. Rev., 51(12), pp. 705–730. [CrossRef]
Metzler, R. , and Klafter, J. , 2000, “ The Random Walk's Guide to Anomalous Diffusion: A Fractional Dynamics Approach,” Phys. Rep., 339(1), pp. 1–77. [CrossRef]
Kimmich, R. , 2002, “ Strange Kinetics, Porous Media, and NMR,” Chem. Phys., 284(1–2), pp. 253–285. [CrossRef]
Fujita, Y. , 1990, “ Integrodifferential Equation Which Interpolates the Heat Equation and the Wave Equation,” Osaka J. Math., 27(2), pp. 309–321.
Luchko, Y. , Mainardi, F. , and Povstenko, Y. , 2013, “ Propagation Speed of the Maximum of the Fundamental Solution to the Fractional Diffusion-Wave Equation,” Comput. Math. Appl., 66(5), pp. 774–784. [CrossRef]
Chaves, A. , 1998, “ A Fractional Diffusion Equation to Describe Lévy Flights,” Phys. Lett. A, 239(1–2), pp. 13–16. [CrossRef]
Zanette, D. H. , 1998, “ Macroscopic Current in Fractional Anomalous Diffusion,” Phys. A, 252(1-2), pp. 159–164. [CrossRef]
Gorenflo, R. , Mainardi, F. , Moretti, D. , and Paradisi, P. , 2002, “ Time Fractional Diffusion: A Discrete Random Walk Approach,” Nonlinear Dyn., 29(1), pp. 129–143. [CrossRef]
Povstenko, Y. , 2008, “ Time-Fractional Radial Diffusion in a Sphere,” Nonlinear Dyn., 53(1–2), pp. 55–65. [CrossRef]
Povstenko, Y. , 2012, “ Central Symmetric Solution to the Neumann Problem for a Time-Fractional Diffusion-Wave Equation in a Sphere,” Nonlinear Anal. Real World Appl., 13(3), pp. 1229–1238. [CrossRef]
Povstenko, Y. Z. , 2004, “ Fractional Heat Conduction Equation and Associated Thermal Stress,” J. Therm. Stresses, 28(1), pp. 83–102. [CrossRef]
Sherief, H. H. , El-Sayed, A. , and El-Latief, A. A. , 2010, “ Fractional Order Theory of Thermoelasticity,” Int. J. Solids Struct., 47(2), pp. 269–275. [CrossRef]
Youssef, H. M. , 2010, “ Theory of Fractional Order Generalized Thermoelasticity,” ASME J. Heat Transfer, 132(6), p. 061301. [CrossRef]
Ezzat, M. A. , 2011, “ Magneto-Thermoelasticity With Thermoelectric Properties and Fractional Derivative Heat Transfer,” Phys. B, 406(1), pp. 30–35. [CrossRef]
Ezzat, M. A. , and El Karamany, A. S. , 2011, “ Theory of Fractional Order in Electro-Thermoelasticity,” Eur. J. Mech. A. Solids, 30(4), pp. 491–500. [CrossRef]
Jumarie, G. , 2010, “ Derivation and Solutions of Some Fractional Black-Scholes Equations in Coarse-Grained Space and Time. Application to Merton's Optimal Portfolio,” Comput. Math. Appl., 59(3), pp. 1142–1164. [CrossRef]
Povstenko, Y. , 2011, “ Dirichlet Problem for Time-Fractional Radial Heat Conduction in a Sphere and Associated Thermal Stresses,” J. Therm. Stresses, 34(1), pp. 51–67. [CrossRef]
Povstenko, Y. , 2015, “ Time-Fractional Thermoelasticity Problem for a Sphere Subjected to the Heat Flux,” Appl. Math. Comput., 257, pp. 327–334.
Zhang, X.-Y. , and Li, X.-F. , 2017, “ Thermal Shock Fracture of a Cracked Thermoelastic Plate Based on Time–Fractional Heat Conduction,” Eng. Fract. Mech., 171, pp. 22–34. [CrossRef]
Zhang, X.-Y. , and Li, X.-F. , 2017, “ Transient Thermal Stress Intensity Factors for a Circumferential Crack in a Hollow Cylinder Based on Generalized Fractional Heat Conduction,” Int. J. Therm. Sci., 121, pp. 336–347. [CrossRef]
Andarwa, S. , and Tabrizi, H. B. , 2010, “ Non-Fourier Effect in the Presence of Coupled Heat and Moisture Transfer,” Int. J. Heat Mass Transfer, 53(15–16), pp. 3080–3087. [CrossRef]
Silva, F. R. , Gonçalves, G. , Lenzi, M. K. , and Lenzi, E. K. , 2013, “ An Extension of the Linear Luikov System Equations of Heat and Mass Transfer,” Int. J. Heat Mass Transfer, 63, pp. 233–238. [CrossRef]
Zhang, X.-Y. , and Li, X.-F. , 2017, “ Transient Response of a Hygrothermoelastic Cylinder Based on Fractional Diffusion Wave Theory,” J. Therm. Stresses, 40(12), pp. 1575–1594. [CrossRef]
Peng, Y. , Zhang, X.-Y. , Xie, Y.-J. , and Li, X.-F. , 2018, “ Transient Hygrothermoelastic Response in a Cylinder Considering non-Fourier Hyperbolic Heat-Moisture Coupling,” Int. J. Heat Mass Transfer, 126, pp. 1094–1103. [CrossRef]
Kilbas, A. A. A. , Srivastava, H. M. , and Trujillo, J. J. , 2006, Theory and Applications of Fractional Differential Equations, Vol. 204, Elsevier Science Limited, Amsterdam.
Chang, W.-J. , and Weng, C.-I. , 2000, “ An Analytical Solution to Coupled Heat and Moisture Diffusion Transfer in Porous Materials,” Int. J. Heat Mass Transfer, 43(19), pp. 3621–3632. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Schematic of a solid sphere in heat and moisture environment

Grahic Jump Location
Fig. 2

The distribution of dimensionless temperature θ at κ = 0.25 for fundamental solution

Grahic Jump Location
Fig. 3

The distribution of dimensionless moisture ψ at κ = 0.25 for fundamental solution

Grahic Jump Location
Fig. 4

The distribution of dimensionless radial stress component σrr at κ = 0.25 for fundamental solution

Grahic Jump Location
Fig. 5

The distribution of dimensionless tangential stress component σφφ at κ = 0.25 for fundamental solution

Grahic Jump Location
Fig. 6

The distribution of dimensionless temperature θt¯α−1 at κ = 0.5 for constant heat flux

Grahic Jump Location
Fig. 7

The distribution of dimensionless moisture ψt¯α−1 at κ = 0.5 for constant heat flux

Grahic Jump Location
Fig. 8

The distribution of dimensionless radial stress component σrrt¯α−1 at κ = 0.5 for constant heat flux

Grahic Jump Location
Fig. 9

The distribution of dimensionless tangential stress component σφφt¯α−1 at κ = 0.25 for constant heat flux

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