Analytical solution for coupled non-Fickian diffusion-thermoelasticity and thermoelastic wave propagation analysis

Document Type : Article


1 Mechanical Engineering Department, Faculty of Engineering, Ferdowsi University of Mashhad, PO Box: 91775-1111, Mashhad, Iran

2 Industrial Engineering Department, Faculty of Engineering, Ferdowsi University of Mashhad, PO Box: 91775-1111, Mashhad, Iran

3 Lean Production Engineering Research Center, Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, P.O. Box 91775-1111, Iran


The time history analysis and propagation of molar concentration, temperature and displacement waves are studied in details using an analytical method. The method is applied to coupled non-Fickian diffusion-thermoelasticity analysis of a strip. The governing equations are derived using non-Fickian theory of diffusion and classic theories for coupled thermoelasticity. Molar concentration and thermoelastic wave propagations are considered to be of finite speed. The governing equations are first transferred to the frequency domain using Laplace transform technique. The unknown parameters are then obtained in analytical forms proposed by the presented method. By employing the Talbot technique, the unknown parameters are eventually determined in time domain. It can be concluded that the presented analytical method has a high capability for dynamic and transient analysis of coupled diffusion-thermoelasticity problems. The wave fronts in displacement, temperature and molar concentration fields can be tracked at various time instants employing the presented analytical method.


Main Subjects

1. Levine, I.N., Physical Chemistry, 6th Ed., McGraw
Hill, New York (2009).
2. Kumar, S., Yildirim, A., Khan, Y., and Wei, L.
A fractional model of the di usion equation and its
analytical solution using Laplace transform", Scientia
Iranica B, 19(4), pp. 1117-1123 (2012).
3. Kuang, Z.B. Some variational principles in elastic dielectric
and elastic magnetic material", European Journal
of Mechanics-A/Solids, 27, pp. 504-514 (2008).
4. Kuang, Z.B. Variational principles of generalized dynamical
theory of thermopiezoelectricity", Acta Mech.,
203, pp. 1-11 (2009).
5. Yang, Q.S., Qin, Q.H., Ma, L.H., Lu, X.Z., and Cui,
C.Q. A theoretical model and nite element formulation
for coupled thermo-electro-chemo-mechanical media",
Mechanics of Materials, 42, pp. 148-156 (2010).
6. Kuang, Z.B. Internal energy variational principle
and governing equations in electroelastic analysis",
International Journal of Solids and Structures, 46, pp.
902-911 (2009).
7. Branco, J.R., Ferreira, J.A., and da Silva, P. Non-
Fickian delay reaction-di usion equation: Theoretical
and numerical study", Applied Numerical Mathematics,
60, pp. 531-549 (2010).
8. Peret, T., Clement, A., Freour, S., and Jacquemin,
F. Numerical transient hygro-elastic analyses of reinforced
Fickian and non-Fickian polymers", Composite
Structures, 116, pp. 395-403 (2014).
9. Barbeiro, S., Bardeji, S.G.H., and Ferreira, J.A.
Laplace transform- nite element method for non-
ows", Computer Methods in Applied Mechanics
and Engineering, 261-262, pp. 16-23 (2013).
10. Kuang, Z.B. Variational principles for generalized
thermodi usion theory in pyroelectricity", Acta Mech.,
214, pp. 275-289 (2010).
11. Lord, H.W. and Shulman, Y. A generalized dynamical
theory of thermoelasticity", Journal of the Mechanics
and Physics of Solids, 15, pp. 299-309 (1967).
12. Green, A.E. and Lindsay, K.A. Thermoelasticity",
Journal of Elasticity, 2, pp. 1-7 (1972).
13. Green, A.E. and Naghdi, P.M. Thermoelasticity without
energy dissipation", Journal of Elasticity, 31, pp.
189-208 (1993).
14. Guo, S.H. The comparisons of thermo-elastic waves
for Lord-Shulman mode and Green-Lindsay mode
based on Guo's eigen theory", Acta Mech., 222, pp.
199-208 (2011).
15. Hosseini, S.M. and Abolbashari, M.H. Analytical
solution for thermoelastic waves propagation analysis
in thick hollow cylinder based on Green-Naghdi
model of coupled thermoelasticity", Journal of Thermal
Stresses, 35, pp. 363-376 (2012).
16. Hosseini, S.M. Coupled thermoelasticity and second
sound in nite length functionally graded thick hollow
cylinders (without energy dissipation)", Materials and
Design, 30, pp. 2011-2023 (2009).
17. Hosseini, S.M. Shock-induced thermoelastic wave
propagation analysis in a thick hollow cylinder without
energy dissipation using mesh-free generalized nite
di erence (GFD) method", Acta Mech., 224, pp. 465-
478 (2013).
18. Lee, Z.Y. Generalized coupled transient thermoelastic
problem of multilayered hollow cylinder with hybrid
boundary conditions", International Communications
in Heat and Mass Transfer, 33, pp. 518-528 (2006).
19. Xia, R., Tian, X., and Shen, Y. Dynamic response
of two-dimensional generalized thermoelastic coupling
problem subjected to a moving heat source", Acta
Mechanica Solida Sinica, 27, pp. 300-305 (2014).
20. Shariyat, M. Nonlinear transient stress and wave
propagation analyses of the FGM thick cylinders, employing
a uni ed generalized thermoelasticity theory",
International Journal of Mechanical Sciences, 65, pp.
24-37 (2012).
21. Shariyat, M., Lavasani, S.M.H., and Khaghani, M.
Nonlinear transient thermal stress and elastic wave
propagation analyses of thick temperature-dependent
FGM cylinders using a second-order point-collocation
method", Applied Mathematical Modelling, 34, pp.
898-918 (2010).
22. Shariyat, M., Khaghani, M., and Lavasani, S.M.H.
Nonlinear thermoelasticity, vibration, and stress wave
propagation analyses of thick FGM cylinders with
temperature-dependent material properties", European
Journal of Mechanics - A/Solids, 29, pp. 378-391
23. Bagri, A. and Eslami, M.R. A uni ed generalized
thermoelasticity, solution for cylinders and spheres",
International Journal of Mechanical Sciences, 49, pp.
1325-1335 (2007).
24. Bagri, A. and Eslami, M.R. A uni ed generalized
thermoelasticity formulation; application to thick
functionally graded cylinders", Journal of Thermal
Stresses, 30, pp. 911-930 (2007).
25. Sherief, H.H. and Abd El-Latief, A.M. Application
of fractional order theory of thermoelasticity to a 2D
problem for a half-space", Applied Mathematics and
Computation, 248, pp. 584-592 (2014).
26. Hosseini, S.A., Abolbashari, M.H., and Hosseini, S.M.
Shock induced molar concentration wave propagation
and coupled non-Fick di usion-elasticity analysis using
an analytical method", Acta Mech., 225, pp. 3591-
3599 (2014).
2086 S.A. Hosseini et al./Scientia Iranica, Transactions B: Mechanical Engineering 25 (2018) 2077{2086
27. Kumar, R. and Chawla, V. A study of fundamental
solution in orthotropic thermodi usive elastic media",
International Communications in Heat and Mass
Transfer, 38, pp. 456-462 (2011).
28. Deswal, S. and Choudhary, S. Impulsive e ect on
an elastic solid with generalized Thermodi usion",
Journal of Engineering Mathematics, 63, pp. 79-94
29. Aouadi, M. Uniqueness and reciprocity theorems
in the theory of generalized thermoelastic di usion",
Journal of Thermal Stresses, 30, pp. 665-678 (2007).
30. Aouadi, M. Qualitative aspects in the coupled theory
of thermoelastic di usion", Journal of Thermal
Stresses, 31, pp. 706-727 (2008).
31. Singh, B., Singh, L., and Deswal, S. Re
of plane waves in thermo-di usion elasticity without
dissipation under the e ect of rotation", Mechanics
of Advanced Materials and Structures, 23, pp. 74-79
32. Allam, A.A., Omar, M.A., and Ramadan, K.T. A
thermoelastic di usion interaction in an in nitely long
annular cylinder", Archive of Applied Mechanics, 84,
pp. 953-965 (2014).
33. Suo, Y. and Shen, S. Dynamical theoretical model
and variational principles for coupled temperaturedi
usion-mechanics", Acta Mech, 223, pp. 29-41
34. Suo, Y. and Shen, S. Analytical solution for onedimensional
coupled non-Fick di usion and mechanics",
Archive of Applied Mechanics, 83, pp. 397-411
35. Hosseini, S.M., Sladek, J., and Sladek, V. Two
dimensional analysis of coupled non-Fick di usionelastodynamics
problems in functionally graded materials
using meshless local Petrov-Galerkin (MLPG)
method", Applied Mathematics and Computation,
268, pp. 937-946 (2015).
36. Hosseini, S.M., Sladek, J., and Sladek, V. Two dimensional
transient analysis of coupled non-Fick di usionthermoelasticity
based on Green-Naghdi theory using
the meshless local Petrov-Galerkin (MLPG) method",
International Journal of Mechanical Sciences, 82, pp.
74-80 (2014).
37. Cohen, A.M., Numerical Methods for Laplace Transform
Inversion, Springer US (2007).