On the Analysis of Free Vibrations of Nonlocal Elastic Sphere of FGM Type in Generalized Thermoelasticity

Abstract

Purpose

Free vibrations of non-homogenous (functionally graded material (FGM)) axisymmetric nonlocal thermoelastic hollow sphere has been taken into consideration for investigation. The material of nonlocal thermoelastic sphere is supposed to be graded using power law in radial direction.

Methods

The solution of continued power series is employed to resolve the differential equations and to investigate the analytical solutions for field functions i.e. temperature, stress and displacement.

Results and Conclusions

The analytical results have been authenticated using numerical computations with computer based software like MATLAB. The numerical results have been shown graphically for the comparison of frequency shift and thermoelastic damping for local and nonlocal elastic materials. Deduction of results has been validated with the already published literature.

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Abbreviations

\(T(r,\,\,t),\,\,u(r,\,\,t)\) :

Temperature, displacement

\(\alpha_{T}\) :

Coefficient of linear thermal expansion

\(\rho\) :

Mass density

\(C_{e}\) :

Specific heat at constant strain

\(\lambda,\,\,\mu\) :

Lame’s parameters

\(\overline{K}\) :

Thermal conductivity

\(t_{0}\) :

Thermal relaxation time

\(\beta = (3\lambda + 2\mu )\alpha_{T}\) :

Thermoelastic coupling constant

\(\sigma_{ij},\,\,e_{ij} \,;\,(i,j = r,\,\,t)\) :

Stress, strain components

\(\Omega^{*}\) :

Characteristic frequency of vibrations

\(\omega\) :

Circular frequency of vibrations

\(T_{0}\) :

Reference temperature

\(\omega^{*}\) :

Thermoelastic characteristic frequency

\(R_{O} = \hbar \,R_{I}\) :

Outer radius

\(R_{I}\) :

Inner radius

\(\xi_{0} = e_{0} a\) :

Nonlocal elastic parameter, where \(\,e_{0} \,\,\) is a material constant and \(\,a\,\,\) is internal characteristic length.

\(\hbar \,\,\) :

Ratio of outer to inner radius

\(Q^{ - 1}\) :

Quality factor (thermoelastic damping)

\(\varepsilon_{T}\) :

Thermoelastic coupling parameter

\(\Omega_{shift}\) :

Frequency shift

\(c_{2}\) :

Velocity of shear wave

\(c_{1}\) :

Velocity of longitudinal wave

\(D\) :

Dissipation factor

\(f_{n}\) :

Natural frequency

\(m\) :

Mode Number

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Correspondence to Nantu Sarkar.

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Appendix

Appendix

See Eqs. 3640.

$$ \left\{ \begin{gathered} H_{11}^{k} (p_{j} ) = \frac{{\Omega^{2} }}{{\left( {(p_{j} + k + 2)^{2} - n^{2} } \right)}},\,\,\,\,\,\,\,\,\,\,\,H_{22}^{k} (p_{j} ) = \frac{{\Omega^{*} \Omega^{2} \tilde{\tau }_{0} }}{{\left( {(p_{j} + k + 2)^{2} - (a^{*} )^{2} } \right)}} \hfill \\ H_{12}^{k} (p_{j} ) = \frac{{\overline{\varepsilon }(s_{j} + k + 1 + b^{*} )}}{{\left( {(p_{j} + k + 2)^{2} - \eta^{2} } \right)}},\,\,\,\,\,\,\,\,\,\,\,H_{21}^{k} (p_{j} ) = \frac{{ - m_{4} \Omega^{2} \tilde{\tau }_{0} (p_{j} + k + 2 + b^{*} )}}{{\left( {(p_{j} + k + 2)^{2} - (a^{*} )^{2} } \right)}}\,\,\,\, \hfill \\ \end{gathered} \right., $$
(36)
$$ \begin{gathered} d_{11}^{2} (p_{j} ) = \left\{ {H_{12}^{0} (p_{j} )d_{21}^{1} (p_{j} ) - H_{11}^{0} (p_{j} )} \right\}\,\, \hfill \\ d_{22}^{2} (p_{j} ) = \left\{ {H_{21}^{0} (p_{j} )d_{12}^{1} (p_{j} ) - H_{22}^{0} (p_{j} )} \right\}, \hfill \\ \end{gathered} $$
(37)
$$ \begin{gathered} d_{12}^{3} (p_{j} ) = \left\{ {\, - \,\,H_{12}^{1} (p_{j} )d_{22}^{2} (p_{j} ) + H_{11}^{2} (p_{j} )d_{12}^{1} (p_{j} )} \right\} \hfill \\ d_{21}^{3} (p_{j} ) = \left\{ { - \,\,\,H_{21}^{1} (p_{j} )d_{11}^{2} (p_{j} ) + H_{22}^{1} (p_{j} )d_{21}^{1} (p_{j} )} \right\}, \hfill \\ \end{gathered} $$
(38)
$$ \left. \begin{gathered} d_{11}^{2k} (p_{j} ) = \left\{ {H_{12}^{2k - 2} (p_{j} )d_{21}^{2k - 1} (p_{j} ) - H_{11}^{2k - 2} (p_{j} )d_{11}^{2k - 2} (p_{j} )} \right\} \hfill \\ d_{22}^{2k} (p_{j} ) = \left\{ {H_{21}^{2k - 2} (p_{j} )d_{12}^{2k - 1} (p_{j} ) - H_{22}^{2k - 2} (p_{j} )d_{22}^{2k - 2} (p_{j} )} \right\} \hfill \\ d_{12}^{2k + 1} (p_{j} ) = \left\{ {\,\, - \,\,H_{12}^{2k - 1} (p_{j} )d_{22}^{2k} (p_{j} ) + H_{11}^{2k - 1} (p_{j} )d_{12}^{2k - 1} (p_{j} )} \right\} \hfill \\ d_{21}^{2k + 1} (p_{j} ) = \left\{ {\, - \,\,H_{21}^{2k - 1} (p_{j} )d_{11}^{2k} (p_{j} ) + H_{22}^{2k - 1} (p_{j} )d_{21}^{2k - 1} (p_{j} )} \right\}\,\, \hfill \\ \end{gathered} \right\}, $$
(39)
$$ \left( {\begin{array}{*{20}c} {d_{11}^{0} (p_{j} )} & 0 \\ 0 & {d_{22}^{0} (p_{j} )} \\ \end{array} } \right)\,\, = \,\,\,\left( {\begin{array}{*{20}c} 1 & 0 \\ 0 & 1 \\ \end{array} } \right),\,\,\,\,\,\,\,j = 1,\,\,2,\,\,3,\,\,4. $$
(40)

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Sharma, D.K., Bachher, M., Sharma, M.K. et al. On the Analysis of Free Vibrations of Nonlocal Elastic Sphere of FGM Type in Generalized Thermoelasticity. J. Vib. Eng. Technol. (2020). https://doi.org/10.1007/s42417-020-00217-2

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Keywords

  • Functionally graded material
  • Nonlocal elasticity
  • Fröbenius method
  • Frequency equations
  • Frequency shift
  • Thermoelastic damping