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Heat conduction and heat wave propagation in functionally graded thick hollow cylinder base on coupled thermoelasticity without energy dissipation

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Abstract

In this paper, heat wave propagation and coupled thermoelasticity without energy dissipation in functionally graded thick hollow cylinder is presented based on Green–Naghdi theory. The material properties are supposed to vary as a power function of radius across the thickness of cylinder. The cylinder is considered in axisymmetry and plane strain conditions and it is divided to many sub-cylinders (layers) across the thickness. Each sub-cylinder is considered to be made of isotropic material and functionally graded property can be created by suitable arrangement of layers. The Galerkin finite element method and Newmark finite difference method are employed to solve the problem. The time history of second sounds and displacement wave propagation are obtained for various values of power function. Computed results agree well with the published data.

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Authors and Affiliations

  1. Mechanical Engineering Department, Khorasan Research Institute of Sciences and Food Technology, P.O. Box 91735-139, Mashhad, Iran

    S. M. Hosseini

  2. Mechanical Engineering Department, Amirkabir University of Technology, P.O. Box 15875-4413, Hafez Avenue, Tehran, Iran

    M. Akhlaghi & M. Shakeri

Authors
  1. S. M. Hosseini
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  2. M. Akhlaghi
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  3. M. Shakeri
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Corresponding author

Correspondence to M. Akhlaghi.

Appendix

Appendix

The components of mass matrices for each element

$$ {\begin{array}{*{20}l} {M_{11} =-\alpha_5} & {M_{12} =0} & {M_{13} =-\alpha_6} & {M_{14} =0} \\ \end{array}} $$
$$ {\begin{array}{*{20}l} {M_{21} =\frac{\varepsilon^\ast \alpha_{13}}{R}-\frac{\varepsilon^\ast R}{3}} & {M_{22} =-\frac{c}{c_1}\alpha_5} & {M_{23} =-\frac{\varepsilon^\ast \alpha_{13}}{R}-\varepsilon^\ast \alpha_8} & {M_{24} =-\frac{c}{c_1}\alpha_6} \\ \end{array}} $$
$$ {\begin{array}{*{20}l} {M_{31} =-\alpha_6} & {M_{32} =0} & {M_{33} =-\alpha_{11} } & {M_{34} =0} \\ \end{array}} $$
$$ {\begin{array}{*{20}l} {M_{41} =\frac{\varepsilon^\ast \alpha_{14}}{R}-\varepsilon^\ast \alpha _8} & {M_{42} =-\frac{c}{c_1}\alpha_6} & {M_{43} =-\frac{\varepsilon^\ast \alpha_{14}}{R}-\frac{\varepsilon^\ast R}{3}} & {M_{44} =-\frac{c}{c_1}\alpha_{11}} \\ \end{array}} $$

where R = r i+1 −r i . For simplicity, we assumed that \(\bar{r}=r\) in the “Appendix”.

The components of stiffness matrices for each element

$$\begin{array}{ll} {K_{11} =-\frac{C_p^2r_i}{R}-\frac{C_p^2}{2}-\left({C_p^2-C_s^2} \right)\alpha_1} & {K_{12} =\frac{C_p^2\alpha _{13}}{R}} \\ {K_{13} =\frac{C_p^2r_i}{R}+\frac{C_p^2}{2}-\left({C_p^2-C_s^2} \right)\alpha_2} & {K_{14} =-\frac{C_p^2\alpha_{13}}{R}} \end{array}$$
$$\begin{array}{ll} {K_{21} =0} &{K_{22} =-\frac{C_T^2r_i }{R}-\frac{C_T^2}{2}} \\ {K_{23} =0} & {K_{24} = \frac{C_T^2r_i}{R}+\frac{C_T^2}{2}} \end{array}$$
$$\begin{array}{ll} {K_{31} =\frac{C_p^2r_{i+1}}{R}-\frac{C_p^2}{2}-\left({C_p^2-C_s^2} \right)\alpha_2} & {K_{32} =\frac{C_p^2\alpha _{14}}{R}} \\ {K_{33} =-\frac{C_p^2r_{i+1}}{R}+\frac{C_p^2}{2}-\left({C_p^2-C_s^2} \right)\alpha_9} & {K_{34} =-\frac{C_p^2\alpha_{14}}{R}} \end{array}$$
$$\begin{array}{ll} {K_{41} =0} & {K_{42} =\frac{C_T^2r_{i+1} }{R}-\frac{C_T^2}{2}} \\ {K_{43} =0} & {K_{44} =-\frac{C_T^2r_{i+1}}{R}+\frac{C_T^2}{2}} \end{array}$$

where

$$ \begin{aligned} \alpha_1&\,=\,\int\limits_{r_i}^{r_{i+1}} {\frac{N_1^2}{r}dr\,=\,} \frac{1}{R^2}\left\{ {r_{i+1}^2\ln \frac{r_{i+1}}{r_i}-2r_{i+1} R+\frac{R}{2}\left({r_{i+1} +r_i} \right)} \right\}\\ \alpha_2 &\,=\,\int\limits_{r_i}^{r_{i+1}} {\frac{N_1 N_2}{r}dr\,=\,} \frac{1}{R^2}\left\{{-r_{i+1} r_i \ln \frac{r_{i+1}}{r_i }+\frac{R}{2}\left({r_{i+1} +r_i} \right)} \right\}\\ \alpha_3&\,=\,\int\limits_{r_i}^{r_{i+1}} {\frac{N_1^2}{r^2}dr\,=\,} \frac{1}{R^2}\left\{{-2r_{i+1} \ln \frac{r_{i+1}}{r_i}+R+R\frac{r_{i+1} }{r_i}} \right\}\\ \alpha_4 &\,=\,\int\limits_{r_i}^{r_{i+1}} {\frac{N_1 N_2}{r^2}dr\,=\,} \frac{1}{R^2}\left\{{\left({r_{i+1} +r_i} \right)\ln \frac{r_{i+1}}{r_i }-2R} \right\}\\ \alpha_5 &\,=\,\int\limits_{r_i}^{r_{i+1}} {N_1^2rdr\,=\,} \frac{1}{R^2}\left\{ {\frac{r_{i+1}^2R}{2}\left({r_{i+1} +r_i} \right)-\frac{2r_{i+1} }{3}\left({r_{i+1}^3-r_i^3} \right)+\frac{1}{4}\left({r_{i+1}^4-r_i^4} \right)} \right\}\\ \alpha_6 &\,=\,\int\limits_{r_i}^{r_{i+1}} {N_1 N_2 rdr\,=\,} \frac{1}{R^2}\left\{ {-\frac{r_{i+1} r_i R}{2}\left({r_{i+1} +r_i} \right)+\frac{\left( {r_{i+1} +r_i} \right)}{3}\left({r_{i+1}^3-r_i^3} \right)-\frac{1}{4}\left({r_{i+1}^4-r_i^4} \right)} \right\}\\ \alpha_7 &\,=\,\int\limits_{r_i}^{r_{i+1}} {N_1^2dr\,=\,} \frac{R}{3}\\ \alpha_8 &\,=\,\int\limits_{r_i}^{r_{i+1}} {N_1 N_2 dr\,=\,} \frac{1}{R^2}\left\{ {\frac{R}{2}\left({r_{i+1} +r_i} \right)^2-\frac{1}{3}\left({r_{i+1} ^3-r_i^3} \right)-r_i r_{i+1} R} \right\}\\ \alpha_9 &\,=\,\int\limits_{r_i}^{r_{i+1}} {\frac{N_2^2}{r}dr\,=\,} \frac{1}{R^2}\left\{ {r_i^2\ln \frac{r_{i+1}}{r_i}-2r_i R+\frac{R}{2}\left({r_{i+1} +r_i} \right)} \right\}\\ \alpha_{10} &\,=\,\int\limits_{r_i}^{r_{i+1}} {\frac{N_2^2}{r^2}dr\,=\,} \frac{1}{R^2}\left\{{R-2r_i \ln \frac{r_{i+1}}{r_i}+R\frac{r_i}{r_{i+1} }} \right\}\\ \alpha_{11} &\,=\,\int\limits_{r_i}^{r_{i+1}} {N_2^2rdr\,=\,\frac{1}{R^2}} \left\{{\frac{r_i^2R}{2}\left({r_{i+1} +r_i} \right)-\frac{2r_i }{3}\left({r_{i+1}^3-r_i^3} \right)+\frac{1}{4}\left({r_{i+1}^4-r_i^4} \right)} \right\}\\ \alpha_{12} &\,=\,\int\limits_{r_i}^{r_{i+1}} {N_2^2dr\,=\,} \frac{R}{3}\\ \alpha_{13} &\,=\,\int\limits_{r_i}^{r_{i+1}} {N_1 rdr\,=\,} \frac{1}{R}\left\{ {\frac{R}{2}r_{i+1} \left({r_i +r_{i+1}} \right)-\frac{1}{3}\left( {r_{i+1}^3-r_i^3} \right)} \right\}\\ \alpha_{14} &\,=\,\int_{r_i}^{r_{i+1}} {N_1 rdr\,=\,} \frac{1}{R}\left\{ {-\frac{R}{2}r_i \left({r_i +r_{i+1}} \right)+\frac{1}{3}\left({r_{i+1} ^3-r_i^3} \right)} \right\}\\ \end{aligned} $$

Force matrices for each element

$$ {\begin{array}{*{20}l} {f_{11} =\left. {-C_p^2\frac{\partial \bar{u}}{\partial \bar{r}}} \right|_{r=r_i} .r_i} & & & {f_{21} =\left. {-C_T ^2\frac{\partial \bar{u}}{\partial \bar{r}}} \right|_{r=r_i} .r_i} \\ \end{array}} $$
$$ {\begin{array}{*{20}l} {f_{31} =\left. {C_p^2\frac{\partial \bar{u}}{\partial \bar{r}}} \right|_{r=r_{i+1}} .r_{i+1}} & & & {f_{41} =\left. {C_T^2\frac{\partial \bar{u}}{\partial \bar{r}}} \right|_{r=r_{i+1}} .r_{i+1}} \\ \end{array}} $$

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Hosseini, S.M., Akhlaghi, M. & Shakeri, M. Heat conduction and heat wave propagation in functionally graded thick hollow cylinder base on coupled thermoelasticity without energy dissipation. Heat Mass Transfer 44, 1477–1484 (2008). https://doi.org/10.1007/s00231-008-0381-9

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