Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-14T07:23:15.192Z Has data issue: false hasContentIssue false
  • English
  • Français

Analysis of Vibrations in Fractional Order Magneto-Thermo-Viscoelasticity with Diffusion

Published online by Cambridge University Press:  05 June 2014

K. K. Kalkal  and
S. Deswal
K. K. Kalkal
Affiliation:
Department of Mathematics, G. J. University of Science and Technology, Hisar-125001, Haryana, India
S. Deswal*
Affiliation:
Department of Mathematics, G. J. University of Science and Technology, Hisar-125001, Haryana, India
*
*[email protected]
Get access

Abstract

Present paper is aimed at studying the effects of fractional order parameter, magnetic field, viscosity and diffusion on the thermoelastic interactions in an infinite body whose surface suffers a mechanical load. Body is assumed to be unstrained and unstressed initially and has uniform temperature. Formulation is applied to the fractional generalization of Lord-Shulman theory and the equations are tackled by employing Laplace and Fourier transforms. Expressions for different fields such as displacement, temperature, stress, concentration and chemical potential in physical domain are obtained using a numerical inversion technique. Finally, numerical solution is carried out for copper material and corresponding graphs are plotted to illustrate and compare theoretical results. Some particular cases of interest have also been deduced from the present study. Numerical results predict finite speeds of propagation for thermoelastic and diffusive waves.

Keywords

Fractional order theoryMagneto-thermo-viscoelasticityDiffusionLaplace-Fourier transformsRamp parameter
Type
Research Article
Information
Journal of Mechanics , Volume 30 , Issue 4 , August 2014 , pp. 383 - 394
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Nowacki, W., “Dynamical Problems of Thermodiffusion in Solids I,” Bulletin De l' Academie Polonaise Des Sciences, Serie Des Sciences Techniques, 22, pp. 55–64 (1974).Google Scholar
2.Nowacki, W., “Dynamical Problems of Thermodiffusion in Solids II,” Bulletin De l' Academie Polonaise Des Sciences, Serie Des Sciences Techniques, 22, pp. 129–135 (1974).Google Scholar
3.Nowacki, W., “Dynamical Problems of Thermodiffusion in Solids III,” Bulletin De l' Academie Polonaise Des Sciences, Serie Des Sciences Techniques, 22, pp. 257–266 (1974).Google Scholar
4.Sherief, H., Hamza, F. and Saleh, H., “The Theory of Generalized Thermoelastic Diffusion,” International Journal of Engineering Science, 42, pp. 591–608 (2004).Google Scholar
5.Singh, B., “Reflection of P and SV Waves from Free Surface of an Elastic Solid with Generalized Thermodiffusion,” Journal of Earth System Science, 114, pp. 159–168 (2005).Google Scholar
6.Aouadi, M., “A Problem for an Infinite Elastic Body with a Spherical Cavity in the Theory of Generalized Thermoelastic Diffusion,” International Journal of Solids and Structures, 44, pp. 5711–5722 (2007).CrossRefGoogle Scholar
7.Ram, P., Sharma, N. and Kumar, R., “Thermomechanical Response of Generalized Thermoelastic Diffusion with One Relaxation Time Due to Time Harmonic Sources,” International Journal of Thermal Sciences, 47, pp. 315–323 (2008).Google Scholar
8.Deswal, S. and Choudhary, S., “Two-Dimensional Interactions Due to Moving Load in Generalized Thermoelastic Solid with Diffusion,” Applied Mathematics and Mechanics, 29, pp. 207–221 (2008).CrossRefGoogle Scholar
9.Deswal, S. and Choudhary, S., “Impulsive Effect on an Elastic Solid with Generalized Thermodiffusion,” Journal of Engineering Mathematics, 63, pp. 79–94 (2009).CrossRefGoogle Scholar
10.Deswal, S. and Kalkal, K., “A Two-Dimensional Generalized Electro-Magneto-Thermo-Viscoelastic Problem for a Half-Space with Diffusion,” International Journal of Thermal Sciences, 50, pp. 749–759 (2011).CrossRefGoogle Scholar
11.Oldham, K. B. and Spanier, J., The Fractional Calculus, Academic Press, New York (1974).Google Scholar
12.Miller, K. S. and Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York (1993).Google Scholar
13.Podlubny, I., Fractional Differential Equations, Academic Press, New York (1999).Google Scholar
14.Povstenko, Y. Z., “Fractional Heat Conduction Equation and Associated Thermal Stress,” Journal of Thermal Stresses, 28, pp. 83–102 (2005).CrossRefGoogle Scholar
15.Povstenko, Y. Z., “Fractional Radial Heat Conduction in an Infinite Medium with a Cylindrical Cavity and Associated Thermal Stresses,” Mechanics Research Communications, 37, pp. 436–440 (2010).Google Scholar
16.Youssef, H. M., “Theory of Fractional Order Generalized Thermoelasticity,” Journal of Heat Transfer, ASME, 132, pp. 1–7 (2010).Google Scholar
17.Sherief, H., El-Sayed, A. M. A. and El-Latief, A. M. A., “Fractional Order Theory of Thermoelasticity,” International Journal of Solids and Structures, 47, pp. 269–275 (2010).Google Scholar
18.Jumarie, G., “Derivation and Solutions of Some Fractional Black-Scholes Equations in Coarse-Grained Space and Time. Application to Merton's Optimal Portfolio,” Computers & Mathematics with Applications, 59, pp. 1142–1164 (2010).CrossRefGoogle Scholar
19.Ezzat, M. A., “Thermoelectric MHD Non-Newtonian Fluid with Fractional Derivative Heat Transfer,” Physica B, 405, pp. 4188–4194 (2010).Google Scholar
20.Ezzat, M. A., “Magneto-Thermoelasticity with Thermoelectric Properties and Fractional Derivative Heat Transfer,” Physica B, 406, pp. 30–35 (2011).Google Scholar
21.Abouelregal, A. E., “Fractional Order Generalized Thermo-Piezoelectric Semi-Infinite Medium with Temperature-Dependent Properties Subjected to a Ramp-Type Heating,” Journal of Thermal Stresses, 34, pp. 1139–1155 (2011).Google Scholar
22.Ezzat, M. A. and El-Karamany, A. S., “Fractional Order Theory of a Perfect Conducting Thermoelastic Medium,” Canadian Journal of Physics, 89, pp. 311–318 (2011).Google Scholar
23.Ezzat, M. A. and Fayik, M. A., “Fractional Order Theory of Thermoelastic Diffusion,” Journal of Thermal Stresses, 34, pp. 851–872 (2011).Google Scholar
24.Sharma, N., Kumar, R. and Ram, P., “Plane Strain Deformation in Generalized Thermoelastic Diffusion,” International Journal of Thermophysics, 29, pp. 1503–1522 (2008).Google Scholar
25.Sharma, J. N. and Kumar, V., “Plane Strain Problems of Transversely Isotropic Thermoelastic Media,” Journal of Thermal Stresses, 20, pp. 463–476 (1997).CrossRefGoogle Scholar