Skip to main content
SpringerLink
Log in

Transient response analysis of sandwich cylindrical panel with FGM core subjected to thermal shock

  • Published:
International Journal of Mechanics and Materials in Design Aims and scope Submit manuscript

Abstract

In the present work, transient response of sandwich cylindrical panel with functionally graded material (FGM) core under thermal shock was studied employing generalized coupled thermoelasticity in the framework of the Lord–Shulman formulation. Using Fourier series solution along the axial and circumferential coordinates along with state space formulation for space domain and applying Laplace transform for time domain results in state space first order differential equations that can be solved analytically. Solutions are then changed to time domain via employing inverse Laplace transform. In numerical illustration, influence of relaxation time constant, amount of thermal shock, stacking sequence and mid radius to thickness ratio on transient behavior of sandwich cylindrical panel under thermal shock are examined.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  • Abo-Daha, S.M., Abbas, L.S.: I.A: model on thermal shock problem of generalized magneto-thermoelasticity for an infinitely long annular cylinder with variable thermal conductivity. Applied Mathematical Modelling 35, 3759–3768 (2011)

    Article  MathSciNet  Google Scholar 

  • Alibeigloo, A.: Three-dimensional thermo-elasticity solution of sandwich cylindrical panel with functionally graded core. Compos. Struct. 107, 458–468 (2014)

    Article  Google Scholar 

  • Alibeigloo, A.: Three dimensional coupled thermoelasticity solution of sandwich plate with FGM core under thermal shock. Compos. Struct. 177, 96–103 (2017)

    Article  Google Scholar 

  • Alibeigloo, A.: Coupled thermoelasticity analysis of carbon nano tube reinforced composite rectangular plate subjected to thermal shock. Compos. B 153, 445–455 (2018)

    Article  Google Scholar 

  • Birman, V., Kardomateas, G.A., Simitses, G.J., Li, R.: Response of a sandwich panel subject to fire or elevated temperature on one of the surfaces. Composites: Part A 37, 981–988 (2006)

    Article  Google Scholar 

  • Chakrabarti, A., Mukhopadhyay, B., Bera, R.K.: Nonlinear stability of a shallow unsymmetrical heated orthotropic sandwich shell of double curvature with orthotropic core. Int J of Solids and Structures 44, 5412–5424 (2007)

    Article  Google Scholar 

  • Chang, D.M., Liu, X.F., Wang, B.L., Liu, L., Wang, T.G., Wang, Q., Han, J.X.: Exact solutions to magneto-electro-thermo-elastic fields for a cracked cylinder composite during thermal shock. Int. J. Mech. Mater. Des. 16, 3–18 (2020)

    Article  Google Scholar 

  • Chitikireddy, R., Datta, S.K., Shah, A.H., Bai, H.: Transient thermoelastic waves in an anisotropic hollow cylinder due to localized heating. Int J of Solids and Structures 48, 3063–3074 (2011)

    Article  Google Scholar 

  • Cho, H., Kardomateas, G.A.: Thermal shock stresses due to heat convection at a bounding surfaces in a thick orthotropic cylindrical shell. Int. J. of Solids and Structures 38, 2769–2788 (2001a)

    Article  Google Scholar 

  • Cho, H., Kardomateas, G.A.: Thermal shock stresses due to heat convection at a boundary surface in a thick shell. Int. J. Solids Struct. 38, 2769–2788 (2001b)

    Article  Google Scholar 

  • Dai, H.L., Wang, X.: Stress wave propagation in laminated piezoelectric spherical shells under thermal shock and electric excitation. European Journal of Mechanics A/solids 24, 263–276 (2005)

    Article  Google Scholar 

  • Darabseh, T., Yilmaz, N., Bataineh, M.: Transient thermoelasticity analysis of functionally graded thick hollow cylinder based on Green-Lindsay model. Int. J. Mech. Mater. Des. 8, 247–255 (2012)

    Article  Google Scholar 

  • Desai, P., Kant, T.: A mixed semi analytical solution for functionally graded (FG) finite length cylinders of orthotropic materials subjected to thermal load. Int. J. Mech. Mater. Des. 8, 89–100 (2012)

    Article  Google Scholar 

  • Frostig, Y., Thomsen, O.T.: Non-linear thermo-mechanical behavior of delaminated curved sandwich panels with a compliant core. Int J of Solids and Structures 48, 2218–2237 (2011)

    Article  Google Scholar 

  • Hohe, J., Librescu, L.: A nonlinear theory for doubly curved anisotropic sandwich shells with transversely compressible core. Int. J. of Solids and Structures 40, 1059–1088 (2003)

    Article  Google Scholar 

  • Jordan, P.M., Puri, P.: Thermal stresses in a spherical shell under three thermoelastic models. J. of Thermal Stresses 24(1), 47–70 (2006)

    Google Scholar 

  • Kant, T., Swaminathan, K.: Analytical solutions for the static analysis of laminated composite and sandwich plates based on a higher order refined theory. Compos. Struct. 56, 329–344 (2002)

    Article  Google Scholar 

  • Kar, A., Kanoria, M.: Generalized thermoelastic functionally graded orthotropic hollow sphere under thermal shock with three-phase-lag effect. European Journal of Mechanics A/solids 28, 757–767 (2009)

    Article  Google Scholar 

  • Kashtalyan, M., Menshykova, M.: Three-dimensional elasticity solution for sandwich panels with a functionally graded core. Compos. Struct. 87, 36–43 (2009)

    Article  Google Scholar 

  • Keibolahi, A., Kiani, Y., Eslami, M.R.: Dynamic snap-through of shallow arches under thermal shock. Aerosp. Sci. Technol. 77, 545–554 (2018)

    Article  Google Scholar 

  • Ootao, Y., Tanigawa, Y.: Three-dimensional transient piezo-thermo-elasticity in functionally graded rectangular plate bonded to a piezoelectric layer. Int. J. of Solids and Structures 37, 4377–4401 (2000)

    Article  Google Scholar 

  • Othman, M.I.A., Abbas, I.A.: Generalized thermoelasticity of thermal shock problem in a non-homogenous isotropic hollow cylinder with energy dissipation. Int. J. of Thermophysics 33(5), 913–923 (2012)

    Article  Google Scholar 

  • Ranjbar, J., Alibeigloo, A.: Response of functionally graded spherical shell to thermo-mechanical shock. Aerosp. Sci. Technol. 51, 61–69 (2016)

    Article  Google Scholar 

  • Safari-Kahnaki, A., Hosseini, S.M., Tahani, M.: Thermal shock analysis and thermo-elastic stress waves in functionally graded thick hollow cylinders using analytical method. Int. J. Mech. Mater. Des. 7, 167–181 (2011)

    Article  Google Scholar 

  • Santos, H., Mota Soares, C.M., Mota Soares, C.A., Reddy, J.N.: A semi-analytical finite element model for the analysis of cylindrical shells made of functionally graded materials under thermal shock. Compos. Struct. 86, 10–21 (2008)

    Article  Google Scholar 

  • Shariyat, M.: Non-linear dynamic thermo-mechanical buckling analysis of the imperfect laminated and sandwich cylindrical shells based on a global-local theory inherently suitable for nonlinear analyses. Int J of Non-Linear Mechanics 46, 253–271 (2011)

    Article  Google Scholar 

  • Singh, A.V.: Free vibration analysis of deep doubly curved panels. Comput. Struct. 73, 385–394 (1992)

    Article  Google Scholar 

  • Singh, W., Kumar, V., Saini, A., Khosla, P.K., Mishra, S.: Response analysis of MEMS based high-g acceleration threshold switch under mechanical shock. Int. J. Mech. Mater. Des. 17, 137–151 (2021)

    Article  Google Scholar 

  • Sobhani Aragh, B., Yas, M.H.: Effect of continuously grading fiber orientation face sheets on vibration of sandwich panels with FGM core. Int J of Mechanical Sciences 53, 628–638 (2011)

    Article  Google Scholar 

  • Theodosios, K.P., Panos, A.G., Francesco, D.C.: Finite element simulation of a gradient elastic half-space subjected to thermal shock on the boundary. Appl. Math. Model. 40, 10181–10198 (2016)

    Article  MathSciNet  Google Scholar 

  • Vel, S.S., Batra, R.C.: Three-dimensional analysis of transient thermal stresses in functionally graded plates. Int J. of Solids and Structures 40, 7181–7196 (2003)

    Article  Google Scholar 

  • Ying, J., Wang, H.M.: Axisymmetric thermoelastic analysis in a finite hollow cylinder due to nonuniform thermal shock. Int. J. Press. Vessels Pip. 87, 714–720 (2010)

    Article  Google Scholar 

  • Zenkour, A.M.: Bending analysis of piezoelectric exponentially graded fiber-reinforced composite cylinders in hygrothermal environments. Int. J. Mech. Mater. Des. 13, 515–529 (2017)

    Article  Google Scholar 

  • Zhang, L.W., Song, Z.G., Liew, K.M.: State-space Levy method for vibration analysis of FG-CNT composite plates subjected to in-plane loads based on higher-order shear deformation theory. Compos. Struct. 134, 989–1003 (2015)

    Article  Google Scholar 

  • Zhou, F.X., Li, S.R., Lai, Y.M.: Three-dimensional analysis for transient coupled thermoelastic response of a functionally graded rectangular plate. J. Sound Vib. 330, 3990–4001 (2011)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Faculty of Engineering, Tarbiat Modares University, 14115-143, Tehran, Iran

    A. Alibeigloo

Authors
  1. A. Alibeigloo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Alibeigloo.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

$$G_{f} = \left[ {\begin{array}{*{20}c} {g_{11} } & {g_{12} } & {g_{13} } & {g_{14} } & {\frac{R}{L}\overline{p}_{n} } & {\frac{{\overline{p}_{m} }}{{\overline{r}\theta_{m} }}} & {g_{17} } & 0 \\ 0 & 0 & 0 & { - \frac{R}{L}\overline{p}_{n} } & {g_{25} } & 0 & 0 & 0 \\ 0 & 0 & {\frac{1}{{\overline{r}}}} & { - \frac{{\overline{p}_{m} }}{{\overline{r}\theta_{m} }}} & 0 & {g_{25} } & 0 & 0 \\ {g_{41} } & {g_{42} } & {g_{43} } & {g_{44} } & 0 & 0 & {g_{47} } & 0 \\ { - g_{42} } & {g_{52} } & {g_{53} } & {g_{54} } & {\frac{{ - \left( {m_{1} + 1} \right)}}{{\overline{r}}}} & 0 & {g_{57} } & 0 \\ {g_{61} } & {g_{53} } & {g_{63} } & {g_{13} } & 0 & {\frac{{ - \left( {m_{1} + 2} \right)}}{{\overline{r}}}} & {g_{67} } & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & {\frac{1}{{\overline{k}_{i} }}} \\ {g_{81} } & {g_{82} } & {g_{83} } & {g_{84} } & 0 & 0 & {g_{87} } & 0 \\ \end{array} } \right]$$

where

$$\begin{gathered} g_{11} = - \frac{1}{{\overline{r}}}\left( {m_{1} + \frac{1 - 2\nu }{{1 - \nu }}} \right),\;g_{12} = - \frac{h}{L}\frac{{\overline{E}_{i} \nu \overline{p}_{n} }}{{\overline{r}\left( {1 - \nu^{2} } \right)}},\;g_{13} = - \frac{h}{R}\frac{{\overline{p}_{m} }}{{\theta_{m} }}\frac{{\overline{E}_{i} }}{{\overline{r}^{2} \left( {1 - \nu^{2} } \right)}},\; \hfill \\ g_{14} = \frac{h}{R}\left( {\frac{{\overline{E}_{i} }}{{\overline{r}^{2} \left( {1 - \nu^{2} } \right)}} + \frac{{k_{o}^{2} \overline{\rho }_{i} }}{{R^{2} Y_{o} \rho_{o} c_{o}^{2} }} \times \left( {\frac{{\overline{r}}}{{\overline{R} + \overline{h}_{m} }}} \right)^{{\left( {m_{4} - m_{1} } \right)}} \times \frac{{\partial^{2} }}{{\partial \overline{t}^{2} }}} \right),\;g_{17} = - \frac{{\overline{E}_{i} \overline{\alpha }_{i} }}{{\overline{r}\left( {1 - \nu } \right)}}\left( {\frac{{\overline{r}}}{{\overline{R} + \overline{h}_{m} }}} \right)^{{m_{2} }} \hfill \\ \end{gathered}$$

\(\begin{gathered} g_{25} = \frac{{2\left( {1 + \nu } \right)}}{{\overline{E}_{i} }}\frac{R}{h},\;g_{41} = \frac{R}{h}\frac{{\left( {1 + \nu } \right)\left( {1 - 2\nu } \right)}}{{\overline{E}_{i} \left( {1 - \nu } \right)}}, \hfill \\ g_{42} = \frac{R}{L}\frac{{\nu \overline{p}_{n} }}{1 - \nu }\;,g_{43} = \frac{{\overline{p}_{m} }}{{\theta_{m} }}\frac{\nu }{{\overline{r}\left( {1 - \nu } \right)}},\;g_{44} = \frac{ - \nu }{{\overline{r}\left( {1 - \nu } \right)}}\; \hfill \\ \end{gathered}\)

$$\begin{gathered} g_{52} = \frac{h}{R}\left( {\frac{{\overline{E}_{i} }}{1 + \nu }\left( {\left( \frac{R}{L} \right)^{2} \times \frac{{\overline{p}_{n}^{2} }}{1 - \nu } + \frac{1}{2}\left( {\frac{{\overline{p}_{m} }}{{\overline{r}\theta_{m} }}} \right)^{2} } \right) + \frac{{k_{o}^{2} \overline{\rho }_{i} }}{{R^{2} Y_{o} \rho_{o} c_{o}^{2} }} \times \left( {\frac{{\overline{r}}}{{\overline{R} + \overline{h}_{m} }}} \right)^{{\left( {m_{4} - m_{1} } \right)}} \times \frac{{\partial^{2} }}{{\partial \overline{t}^{2} }}} \right)\; \hfill \\ g_{47} = \frac{R}{h}\frac{{\overline{\alpha }_{i} \left( {1 + \nu } \right)}}{{\left( {1 - \nu } \right)}}\left( {\frac{{\overline{r}}}{{\overline{R} + \overline{h}_{m} }}} \right)^{{m_{2} }} ,\;g_{53} = \frac{h}{L}\frac{{\overline{p}_{m} }}{{\theta_{m} }}\frac{{\overline{p}_{n} \overline{E}_{i} }}{{2\overline{r}\left( {1 - \nu } \right)}},\;g_{54} = - \frac{h}{L}\frac{{\overline{p}_{n} \overline{E}_{i} \nu }}{{\overline{r}\left( {1 - \nu^{2} } \right)}},\; \hfill \\ g_{57} = \frac{R}{L}\frac{{\overline{E}_{i} \overline{\alpha }_{i} \overline{p}_{n} }}{1 - \nu }\left( {\frac{{\overline{r}}}{{\overline{R} + \overline{h}_{m} }}} \right)^{{m_{2} }} ,\;g_{61} = - \frac{{\overline{p}_{m} }}{{\theta_{m} }}\frac{\nu }{{\overline{r}\left( {1 - \nu } \right)}},\;g_{67} = \frac{{\overline{p}_{m} }}{{\theta_{m} }}\frac{{\overline{E}_{i} \overline{\alpha }_{i} }}{{\overline{r}\left( {1 - \nu } \right)}}\left( {\frac{{\overline{r}}}{{\overline{R} + \overline{h}_{m} }}} \right)^{{m_{2} }} ,\; \hfill \\ g_{63} = \frac{h}{R}\left( {\frac{{\overline{E}_{i} }}{1 + \nu }\left( {\left( \frac{R}{L} \right)^{2} \times \frac{{\overline{p}_{n}^{2} }}{2} + \frac{1}{1 - \nu }\left( {\frac{{\overline{p}_{m} }}{{\overline{r}\theta_{m} }}} \right)^{2} } \right) + \frac{{k_{o}^{2} \overline{\rho }_{i} }}{{R^{2} Y_{o} \rho_{o} c_{o}^{2} }} \times \left( {\frac{{\overline{r}}}{{\overline{R} + \overline{h}_{m} }}} \right)^{{\left( {m_{4} - m_{1} } \right)}} \times \frac{{\partial^{2} }}{{\partial \overline{t}^{2} }}} \right) \hfill \\ \end{gathered}$$

\(g_{57} = \frac{R}{L}\frac{{\overline{E}_{i} \overline{\alpha }_{i} \overline{p}_{n} }}{1 - \nu }\left( {\frac{{\overline{r}}}{{\overline{R} + \overline{h}_{m} }}} \right)^{{m_{2} }}\), \(g_{61} = - \frac{{\overline{p}_{m} }}{{\theta_{m} }}\frac{\nu }{{\overline{r}\left( {1 - \nu } \right)}}\), \(g_{67} = \frac{{\overline{p}_{m} }}{{\theta_{m} }}\frac{{\overline{E}_{i} \overline{\alpha }_{i} }}{{\overline{r}\left( {1 - \nu } \right)}}\left( {\frac{{\overline{r}}}{{\overline{R} + \overline{h}_{m} }}} \right)^{{m_{2} }}\)

$$g_{63} = \frac{h}{R}\left( {\frac{{\overline{E}_{i} }}{1 + \nu }\left( {\left( \frac{R}{L} \right)^{2} \times \frac{{\overline{p}_{n}^{2} }}{2} + \frac{1}{1 - \nu }\left( {\frac{{\overline{p}_{m} }}{{\overline{r}\theta_{m} }}} \right)^{2} } \right) + \frac{{k_{o}^{2} \overline{\rho }_{i} }}{{R^{2} Y_{o} \rho_{o} c_{o}^{2} }} \times \left( {\frac{{\overline{r}}}{{\overline{R} + \overline{h}_{m} }}} \right)^{{\left( {m_{4} - m_{1} } \right)}} \times \frac{{\partial^{2} }}{{\partial \overline{t}^{2} }}} \right)$$
$$g_{81} = D\frac{1 + \nu }{{\overline{E}_{i} }},\;g_{82} = - D\frac{h}{L}\overline{p}_{n} ,\;g_{83} = - D\frac{h}{R}\frac{{\overline{p}_{m} }}{{\overline{r}\theta_{m} }},\;g_{84} = D\frac{h}{R}\frac{1}{{\overline{r}}}$$

Where

$$D = \frac{1}{1 - \nu }\frac{{\alpha_{o}^{2} Y_{o} T_{o} \overline{E}_{i} \overline{\alpha }_{i} }}{{\rho_{o} c_{o} }}\left( {\frac{{\overline{r}}}{{\overline{R} + \overline{h}_{m} }}} \right)^{{m_{1} + m_{2} }} \left( {\frac{\partial }{{\partial \overline{t}}} + \overline{\tau }_{0} \frac{{\partial^{2} }}{{\partial \overline{t}^{2} }}} \right)$$
$$\begin{gathered} g_{87} = \overline{k}_{i} \left( {\frac{{\overline{r}}}{{\overline{R} + \overline{h}_{m} }}} \right)^{{m_{3} }} \left[ {\left( {\frac{R}{L}\overline{p}_{n} } \right)^{2} + \left( {\frac{{\overline{p}_{m} }}{{\overline{r}\theta_{m} }}} \right)^{2} } \right] + \left[ {\overline{c}_{i} \overline{\rho }_{i} \left( {\frac{{\overline{r}}}{{\overline{R} + \overline{h}_{m} }}} \right)^{{m_{4} }} } \right. + \frac{1 + \nu }{{\left( {1 + \nu } \right)\left( {1 - 2\nu } \right)}}\frac{{\alpha_{o}^{2} Y_{o} T_{o} \overline{E}_{i} \overline{\alpha }_{i} }}{{\rho_{o} c_{o} }} \hfill \\ \, \left. { \times \left( {\frac{{\overline{r}}}{{\overline{R} + \overline{h}_{m} }}} \right)^{{\left( {m_{1} + m_{2} } \right)}} } \right]\left( {\frac{\partial }{{\partial \overline{t}}} + \overline{\tau }_{0} \frac{{\partial^{2} }}{{\partial \overline{t}^{2} }}} \right) \hfill \\ \end{gathered}$$
$$F_{f} = \left[ {\begin{array}{*{20}c} {\frac{\nu }{1 - \nu }} & { - \frac{h}{L}\frac{{\overline{E}_{0} \overline{p}_{n} }}{{1 - \nu^{2} }}} & { - \frac{h}{{R_{m} }}\frac{{\overline{p}_{m} }}{{\theta_{m} }}\frac{{\overline{E}_{0} \nu }}{{\overline{r}\left( {1 - \nu^{2} } \right)}}} & {\frac{h}{{R_{m} }}\frac{{\overline{E}_{0} \nu }}{{\overline{r}\left( {1 - \nu^{2} } \right)}}} \\ {\frac{\nu }{1 - \nu }} & { - \frac{h}{L}\frac{{\overline{E}_{0} \overline{p}_{n} \nu }}{{1 - \nu^{2} }}} & { - \frac{h}{{R_{m} }}\frac{{\overline{p}_{m} }}{{\theta_{m} }}\frac{{\overline{E}_{0} }}{{\overline{r}\left( {1 - \nu^{2} } \right)}}} & {\frac{h}{{R_{m} }}\frac{{\overline{E}_{0} }}{{\overline{r}\left( {1 - \nu^{2} } \right)}}} \\ 0 & {\frac{h}{{2R_{m} }}\frac{{\overline{p}_{m} }}{{\theta_{m} }}\frac{{\overline{E}_{0} }}{{\overline{r}\left( {1 + \nu } \right)}}} & {\frac{h}{2L}\frac{{\overline{E}_{0} \overline{p}_{n} }}{{\overline{r}\left( {1 + \nu } \right)}}} & 0 \\ \end{array} } \right]$$
$$F_{i} = \left[ {\begin{array}{*{20}c} {\frac{\nu }{1 - \nu }} & { - \frac{h}{L}\frac{{\overline{E}_{i} \overline{p}_{n} }}{{1 - \nu^{2} }}} & { - \frac{h}{{R_{m} }}\frac{{\overline{p}_{m} }}{{\theta_{m} }}\frac{{\overline{E}_{i} \nu }}{{\overline{r}\left( {1 - \nu^{2} } \right)}}} & {\frac{h}{{R_{m} }}\frac{{\overline{E}_{i} \nu }}{{\overline{r}\left( {1 - \nu^{2} } \right)}}} \\ {\frac{\nu }{1 - \nu }} & { - \frac{h}{L}\frac{{\overline{E}_{i} \overline{p}_{n} \nu }}{{1 - \nu^{2} }}} & { - \frac{h}{{R_{m} }}\frac{{\overline{p}_{m} }}{{\theta_{m} }}\frac{{\overline{E}_{i} }}{{\overline{r}\left( {1 - \nu^{2} } \right)}}} & {\frac{h}{{R_{m} }}\frac{{\overline{E}_{i} }}{{\overline{r}\left( {1 - \nu^{2} } \right)}}} \\ 0 & {\frac{h}{{2R_{m} }}\frac{{\overline{p}_{m} }}{{\theta_{m} }}\frac{{\overline{E}_{i} }}{{\overline{r}\left( {1 + \nu } \right)}}} & {\frac{h}{2L}\frac{{\overline{E}_{i} \overline{p}_{n} }}{{\overline{r}\left( {1 + \nu } \right)}}} & 0 \\ \end{array} } \right]\;i = m \, ,{\text{ c}}$$
$$I = \left[ {\begin{array}{*{20}c} {\left( {\frac{{\overline{r}_{o} }}{{\overline{r}_{i} + \overline{h}_{m} }}} \right)^{{m_{1} }} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\left( {\frac{{\overline{r}_{o} }}{{\overline{r}_{i} + \overline{h}_{m} }}} \right)^{{m_{1} }} } & 0 \\ 0 & 0 & 0 & 0 & 0 & {\left( {\frac{{\overline{r}_{o} }}{{\overline{r}_{i} + \overline{h}_{m} }}} \right)^{{m_{1} }} } \\ \end{array} } \right]$$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alibeigloo, A. Transient response analysis of sandwich cylindrical panel with FGM core subjected to thermal shock. Int J Mech Mater Des 17, 707–719 (2021). https://doi.org/10.1007/s10999-021-09554-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10999-021-09554-w

Keywords

Search

Navigation