Coupled Thermoelasticity of FGM Truncated Conical Shells

Document Type : Research Paper


1 M.Sc. Student, Department of Mechanical Engineering, Amirkabir University of Technology, Tehran, Iran,

2 PhD Student, Department of Mechanical Engineering, Amirkabir University of Technology, Tehran, Iran

3 Corresponding author, Department of Mechanical Engineering, Amirkabir University of Technology, Tehran, Iran


This article presents the coupled thermoelasticity of a truncated functionally
graded conical shell under thermal shock load. The classical
coupled thermoelasticity theory is employed to set the partial differential
equations of motion of the conical shell. The shell governing
equations are based on the first-order shear deformation shell theory
that accounts for the transverse shear strains and rotations. The solution
is obtained by transforming the governing equations into the
Laplace domain and using the Galerkin finite element method in the
Laplace domain to calculate the displacement components. The physical
displacement components in real time domain are obtained by
the numerical inversion of the Laplace transform. Temperature distribution
is assumed to be linear across the shell thickness. Radial
displacement, axial stress, axial force, and temperature versus time
are calculated and the effect of relaxation time and power law index
are examined. Comparison indicates that an increase in the radial vibration
amplitude and a decrease of vibration frequency occur when
changing the material from ceramic to metal. The results are validated
with the known data in the literature.


Main Subjects

[1] Ignaczak, J., and Nowacki, W. ”Transversal Vibration of a Plate, Produced by Heating”,
Archiwum Mechaniki Stosowanej, pp. 650-667, (1961).
[2] Jones, J.P., ”Thermoelastic Vibrations of a Beam”, The Journal of the Acoustical Society
of America, Vol. 39, No. 3, pp. 542-548, (1966).
[3] Manoach, E., and Ribeiro, P., ”Coupled, Thermoelastic, Large Amplitude Vibrations of
Timoshenko Beams”, International Journal of Mechanical Sciences, Vol. 46, No. 11, pp.
1589-1606, (2004).
[4] Babaei, M.H., Abbasi, M., and Eslami, M.R., ”Coupled Thermoelasticity of Functionally
Graded Beams”, Journal of Thermal Stresses, Vol. 31(8), pp. 680-697, (2008).
[5] Inan, E., ”Coupled Theory of Thermoelastic Plates”, Acta Mechanica, Vol. 14, No. 1, pp.
1-29, (1972).
[6] Chang, W.P., and Wan, S.M., ”Thermomechanically Coupled Non-linear Vibration of
Plates”, International Journal of Non-Linear Mechanics, Vol. 21, No. 5, pp. 375-389,
[7] Trajkovski, D., and Cuki, R., ”A Coupled Problem of Thermoelastic Vibrations of a Circular
Plate with Exact Boundary Conditions”, Mechanics Research Communications, Vol.
26, No. 2, pp. 217-224, (1999).
[8] Yeh, Y.L., ”The Effect of Thermo-mechanical Coupling for a Simply Supported Orthotropic
Rectangular Plate on Non-linear Dynamics”, Thin-Walled Structures, Vol. 43,
No. 8, pp. 1277-1295, (2005).
[9] Akbarzadeh, A.H., Abbasi, M., and Eslami, M.R., ”Coupled Thermoelasticity of Functionally
Graded Plates Based on the Third-order Shear Deformation Theory”, Thin-Walled
Structures, Vol. 53, pp. 141-155, (2012).
[10] Jafarinezhad, M.R., and Eslami, M.R., ”Coupled Thermoelasticity of FGM Annular Plate
under Lateral Thermal Shock”, Composite Structures, Vol. 168, pp. 758-771, (2017).
[11] McQuillen, E.J., and Brull, M.A., ”Dynamic Thermoelastic Response of Cylindrical
Shells”, Journal of Applied Mechanics, Vol. 37, No. 3, pp. 661-670, (1970).
[12] Eslami, M.R., Shakeri, M., and Sedaghati, R., ”Coupled Thermoelasticity of an Axially
Symmetric Cylindrical Shell”, Journal of Thermal Stresses, Vol. 17, No. 1, pp. 115-135,
[13] Chang, J.S., and Shyong, J.W., ”Thermally Induced Vibration of Laminated Circular
Cylindrical Shell Panels”, Composites Science and Technology, Vol. 51, No. 3, pp. 419-
427, (1994).
[14] Eslami, M.R., Shakeri, M., Ohadi, A.R., and Shiari, B., ”Coupled Thermoelasticity of
Shells of Revolution: Effect of Normal Stress and Coupling”, AIAA Journal, Vol. 37(4),
pp. 496-504, (1999).
[15] Bahtui, A., and Eslami M.R., ”Coupled Thermoelasticity of Functionally Graded Cylindrical
Shells”, Mechanics Research Communications, Vol. 34, No. 1, pp. 1-18, (2007).
[16] Kraus, H., ”Thermally Induced Vibrations of Thin Nonshallow Spherical Shells”, AIAA
Journal, Vol. 4, No. 3, pp. 500-505, (1966).
[17] Amiri, M., Bateni, M., and Eslami, M.R, ”Dynamic Coupled Thermoelastic Response
of Thin Spherical Shells”, Journal of Structural Engineering & Applied Mechanics,
[18] Soltani, N., Abrinia, K., Ghaderi, P., and Hakimelahi, B., ”A Numerical Solution for the
Coupled Dynamic Thermoelasticity of Axisymmetric Thin Conical Shells”, Mathematical
and Computer Modelling, Vol. 55, pp. 608-621, (2012).
[19] Heydarpour, Y., and Aghdam, M.M., ”Transient Analysis of Rotating Functionally Graded
Truncated Conical Shells Based on the Lord-Shulman Model”, Thin-Walled Structures,
Vol. 104, pp. 168-184, (2016).
[20] Reddy, J.N. Theory and Analysis of Elastic Plates and Shells, CRC Press, 2006.
[21] Hetnarski, R.B. , and Eslami, M.R. Thermal Stresses: Advanced Theory and Applications,
Second Edition, Springer, Switzerland, 2019.
[22] Bateni, M., and Eslami, M.R., ”Thermally Nonlinear Generalized Thermoelasticity of a
Layer”, Journal of Thermal Stresses, Vol. 40, No. 10, pp. 1320-1338, (2017).
[23] Bateni, M., and Eslami, M.R., ”Thermally Nonlinear Generalized Thermoelasticity: A
Note on the Thermal Boundary Conditions”, Acta Mechanica, Vol. 229, No. 2, pp. 807-
826, (2018).
[24] Eslami, M.R. Finite Elements Methods in Mechanics, Springer, Switzerland, 2014.
[25] Brancik ”The Fast Computing Method of Numerical Inversion of Laplace Transforms Using
FFT Algorithm”, Proc. of 5th EDS 98 Int. Conf., Brno, Czech Republic., pp. 97-100,