Abstract
The present article reports on a study on nonlinear coupled thermoelasticity based on Green-Naghdi type III model in one-dimensional form. Unlike other research in which temperature change is low and the strain relations are linear, in the present study the nonlinear finite strain assumption is governed. Two coupled nonlinear partial differential equations, namely; energy and linear momentum are established. The basic equations are formulated in material coordinates and then transformed into dimensionless form. The homotopy analysis method is used to analyze the one-dimensional structure. Two different cases are studied. The thermo-mechanical wave propagation with reflection from the boundary, and the influence of the damping parameter–presented in Green-Naghdi type III model–on temperature. The linear and nonlinear equations are compared. It should be noted that the nonlinear finite strain theory governs better when the loading temperature is high.
Similar content being viewed by others
References
D. S. Chandrasekharaiah, Thermoelasticity with second sound: A review, Applied Mechanics Reviews, 39 (3) (1986) 355–376.
M. Aouadi, B. Lazzari and R. Nibbi, A theory of thermoelasticity with diffusion under GreenNaghdi models, ZAMM Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 94 (10) (2014) 837–852.
D. S. Chandrasekharaiah, A note on the uniqueness of solution in the linear theory of thermoelasticity without energy dissipation, Journal of Elasticity, 43 (3) (1996) 279–283.
D. Iesan, On the theory of thermoelasticity without energy dissipation, Journal of Thermal Stresses, 21 (3–4) (1998) 295–307.
R. Quintanilla and B. Straughan, A note on discontinuity waves in type III thermoelasticity, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 460 (2044) April (2004) 1169–1175, The Royal Society.
R. Quintanilla, Spatial stability for the quasi–static problem of thermoelasticity, Journal of Elasticity, 76 (2) (2004) 93–105.
R. Quintanilla and R. Racke, Stability for thermoelasticity of type III, Discrete and Continuous Dynamical Systems, B (3) (2003) 383–400.
M. N. Allam, K. A. Elsibai and A. E. Abouelregal, Thermal stresses in a harmonic field for an infinite body with a circular cylindrical hole without energy dissipation, Journal of Thermal Stresses, 25 (1) (2002) 57–67.
R. B. Hetnarski and J. Ignaczak, Soliton–like waves in a low temperature nonlinear thermoelastic solid, International Journal of Engineering Science, 34 (15) (1996) 1767–1787.
S. Chakraborty, S. C. Mandal, A. K. Das, N. Sarkar and A. Lahiri, Plane wave propagation in a 3D anisotropic halfspace under Green–Naghdi theory II, Mathematical Models in Engineering, 2 (2) (2016) 114–134.
A. Bagri and M. R. Eslami, A unified generalized thermoelasticity; solution for cylinders and spheres, International Journal of Mechanical Sciences, 49 (12) (2007) 1325–1335.
S. Bargmann and P. Steinmann, Finite element approaches to non–classical heat conduction in solids, Comput. Model. Eng. Sci., 9 (2) (2005) 133–150.
H. Taheri, S. J. Fariborz and M. R. Eslami, Thermoelastic analysis of an annulus using the Green–Naghdi model, Journal of Thermal Stresses, 28 (9) (2005) 911–927.
M. I. Othman, S. Y. Atwa and R. M. Farouk, The effect of diffusion on two–dimensional problem of generalized thermoelasticity with Green–Naghdi theory, International Communications in Heat and Mass Transfer, 36 (8) (2009) 857–864.
M. Shariyat, Nonlinear transient stress and wave propagation analyses of the FGM thick cylinders, employing a unified generalized thermoelasticity theory, International Journal of Mechanical Sciences, 65 (1) (2012) 24–37.
Y. Kiani and M. R. Eslami, The GDQ approach to thermally nonlinear generalized thermoelasticity of a hollow sphere, International Journal of Mechanical Sciences, 118 (2016) 195–204.
Y. Kiani and M. R. Eslami, Nonlinear generalized thermoelasticity of an isotropic layer based on Lord–Shulman theory, European Journal of Mechanics–A/Solids, 61 (2017) 245–253.
A. Jafarian, P. Ghaderi, A. K. Golmanichaneh and D. Baleanu, On a one–dimensional nonlinear coupled system of equations in the theory of thermoelasticity, Rom. J. Phys, 58 (2013) 694–702.
M. Mirazadeh, M. Eslami and A. Biswas, Acomputational method for the solution of one–dimensional nonlinear thermoelasticity, Pramana, 85 (4) (2015) 701–712.
M. E. Gurtin, E. Fried and L. Anand, The Mechanics and Thermodynamics of Continua, Cambridge University Press (2010).
R. B. Hetnarski and M. R. Eslami, Basic Laws of Thermoelasticity, Springer Netherlands (2009) 1–41.
A. E. Green and P. M. Naghdi, A re–examination of the basic postulates of thermomechanics, Proc. R. Soc. Lond. A, 432 (1885) (1991) 171–194.
A. E. Green and P. M. Naghdi, On undamped heat waves in an elastic solid, Journal of Thermal Stresses, 15 (2) (1992) 253–264.
S. Liao, Homotopy Analysis Method in Nonlinear Differential Equations, Beijing: Higher Education Press (2012) 153–165.
D. S. Chandrasekharaiah and K. S. Srinath, Onedimensional waves in a thermoelastic half–space without energy dissipation, International Journal of Engineering Science, 34 (13) (1996) 1447–1455.
Author information
Authors and Affiliations
Corresponding author
Additional information
Recommended by Associate Editor Jun-Sik Kim
Mohammad Hosein Nejabat received his M.S. degree in mechanical engineering from Yazd University, Iran, in 2018. His current research interests are in the nonlinear elasticity and thermoelasticity, and vibration.
Ali Reza Fotuhi received his B.S. from Sharif University of Technology in 1998 and M.S. and Ph.D. in Mechanical Engineering from Amirkabir University of Technology in 2000 and 2007, respectively. He is an Associate Professor of Mechanical Engineering, Yazd University, Iran.
Rights and permissions
About this article
Cite this article
Nejabat Meimandi, M.H., Fotuhi, A.R. & Fazel, M.R. One-dimensional nonlinear vibration analysis and coupled thermoelasticity based on Green-Naghdi model. J Mech Sci Technol 33, 721–728 (2019). https://doi.org/10.1007/s12206-019-0126-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12206-019-0126-3