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.

References

1.
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
.
2.
Sih
,
G. C.
,
Michopoulos
,
J.
, and
Chou
,
S.-C.
,
1986
,
Hygrothermoelasticity
, Martinus Niijhoof Publishing, Dordrecht, The Netherlands.
3.
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
.
4.
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
.
5.
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
.
6.
Zenkour
,
A.
,
2010
, “
Hygro-Thermo-Mechanical Effects on FGM Plates Resting on Elastic Foundations
,”
Compos. Struct.
,
93
(
1
), pp.
234
238
.
7.
Zenkour
,
A.
,
2014
, “
Hygrothermoelastic Responses of Inhomogeneous Piezoelectric and Exponentially Graded Cylinders
,”
Int. J. Pressure Vessels Piping
,
119
, pp.
8
18
.
8.
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
.
9.
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
.
10.
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
.
11.
Tzou
,
D. Y.
,
1995
, “
Experimental Support for the Lagging Behavior in Heat Propagation
,”
J. Thermophys. Heat Transfer
,
9
(
4
), pp.
686
693
.
12.
Hetnarski
,
R. B.
, and
Ignaczak
,
J.
,
1999
, “
Generalized Thermoelasticity
,”
J. Therm. Stresses
,
22
(
4–5
), pp.
451
476
.
13.
Lord
,
H. W.
, and
Shulman
,
Y.
,
1967
, “
A Generalized Dynamical Theory of Thermoelasticity
,”
J. Mech. Phys. Solids
,
15
(
5
), pp.
299
309
.
14.
Green
,
A.
, and
Lindsay
,
K.
,
1972
, “
Thermoelasticity
,”
J. Elast.
,
2
(
1
), pp.
1
7
.
15.
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
.
16.
Green
,
A.
, and
Naghdi
,
P.
,
1993
, “
Thermoelasticity Without Energy Dissipation
,”
J. Elast.
,
31
(
3
), pp.
189
208
.
17.
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
.
18.
Chandrasekharaiah
,
D.
,
1998
, “
Hyperbolic Thermoelasticity: A Review of Recent Literature
,”
Appl. Mech. Rev.
,
51
(
12
), pp.
705
730
.
19.
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
.
20.
Kimmich
,
R.
,
2002
, “
Strange Kinetics, Porous Media, and NMR
,”
Chem. Phys.
,
284
(
1–2
), pp.
253
285
.
21.
Fujita
,
Y.
,
1990
, “
Integrodifferential Equation Which Interpolates the Heat Equation and the Wave Equation
,”
Osaka J. Math.
,
27
(
2
), pp.
309
321
.
22.
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
.
23.
Chaves
,
A.
,
1998
, “
A Fractional Diffusion Equation to Describe Lévy Flights
,”
Phys. Lett. A
,
239
(
1–2
), pp.
13
16
.
24.
Zanette
,
D. H.
,
1998
, “
Macroscopic Current in Fractional Anomalous Diffusion
,”
Phys. A
,
252
(
1-2
), pp.
159
164
.
25.
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
.
26.
Povstenko
,
Y.
,
2008
, “
Time-Fractional Radial Diffusion in a Sphere
,”
Nonlinear Dyn.
,
53
(
1–2
), pp.
55
65
.
27.
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
.
28.
Povstenko
,
Y. Z.
,
2004
, “
Fractional Heat Conduction Equation and Associated Thermal Stress
,”
J. Therm. Stresses
,
28
(
1
), pp.
83
102
.
29.
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
.
30.
Youssef
,
H. M.
,
2010
, “
Theory of Fractional Order Generalized Thermoelasticity
,”
ASME J. Heat Transfer
,
132
(
6
), p.
061301
.
31.
Ezzat
,
M. A.
,
2011
, “
Magneto-Thermoelasticity With Thermoelectric Properties and Fractional Derivative Heat Transfer
,”
Phys. B
,
406
(
1
), pp.
30
35
.
32.
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
.
33.
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
.
34.
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
.
35.
Povstenko
,
Y.
,
2015
, “
Time-Fractional Thermoelasticity Problem for a Sphere Subjected to the Heat Flux
,”
Appl. Math. Comput.
,
257
, pp.
327
334
.
36.
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
.
37.
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
.
38.
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
.
39.
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
.
40.
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
.
41.
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
.
42.
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.
43.
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
.
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