A mixed semi analytical solution for functionally graded (FG) finite length cylinders of orthotropic materials subjected to thermal load

Abstract

A simplified and accurate analytical cum numerical model is presented here to investigate the behavior of functionally graded (FG) cylinders of finite length subjected to thermal load. A diaphragm supported FG cylinder under symmetric thermal load which is considered as a two dimensional (2D) plane strain problem of thermoelasticity in (r, z) direction. The boundary conditions are satisfied exactly in axial direction (z) by taking an analytical expression in terms of Fourier series expansion. Fundamental (basic) dependent variables are chosen in the radial coordinate of the cylinder. First order simultaneous ordinary differential equations are obtained as mathematical model which are integrated through an effective numerical integration technique by first transforming the boundary value problem into a set of initial value problems. For FG cylinders, the material properties have power law dependence in the radial coordinate. Effect of non homogeneity parameters and orthotropy of the materials on the stresses and displacements of FG cylinder are studied. The numerical results obtained are also first validated with existing literature for their accuracy. Stresses and displacements in axial and radial directions in cylinders having various l/r _i and r _o/r _i ratios parameter are presented for future reference.

Appendix: 1D formulation for orthotropic cylinder under thermal loading

$$ \frac{{{\text{d}}\sigma_{r} }}{{{\text{d}}r}} + \frac{1}{r}\left( {\sigma_{r} - \sigma_{\theta } } \right) = 0,\quad \varepsilon_{r} = \frac{\partial u}{\partial r} ,\quad \varepsilon_{\theta } = \frac{u}{r}, $$

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$$ \begin{array}{l} {\sigma_{r} = C_{11} \left( {\varepsilon_{r} - \alpha_{r} T} \right) + C_{12} \left( {\varepsilon_{\theta } - \alpha_{\theta } T} \right)} \\ {\sigma_{\theta } = C_{12} \left( {\varepsilon_{r} - \alpha_{r} T} \right) + C_{22} \left( {\varepsilon_{\theta } - \alpha_{\theta } T} \right)} \\ \end{array} ,\begin{array}{l} {\sigma_{r} = C_{11} \frac{{{\text{d}}u}}{{{\text{d}}r}} - C_{11} \alpha_{r} T + C_{12} \frac{u}{r} - C_{12} \alpha_{\theta } T} \\ {\sigma_{\theta } = C_{21} \frac{{{\text{d}}u}}{{{\text{d}}r}} - C_{21} \alpha_{r} T + C_{22} \frac{u}{r} - C_{22} \alpha_{\theta } T} \\ \end{array} , $$

$$ \begin{aligned} & \frac{{{\text{d}}u}}{{{\text{d}}r}} = \frac{{\sigma_{r} }}{{C_{11} }} + \alpha_{r} T - \frac{{C_{12} }}{{C_{11} }}\frac{u}{r} + \frac{{C_{12} }}{{C_{11} }}\alpha_{\theta } T, \\ & \frac{{{\text{d}}\sigma_{r} }}{{{\text{d}}r}} = \frac{{\sigma_{r} }}{r}\left( {\frac{{C_{21} }}{{C_{11} }} - 1} \right) + \frac{u}{{r^{2} }}\left( {C_{22} - \frac{{C_{21} C_{12} }}{{C_{11} }}} \right) + \frac{{\alpha_{\theta } T}}{r}\left( {\frac{{C_{21} C_{12} }}{{C_{11} }} - C_{22} } \right) \hfill \\ \end{aligned} $$

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where,

$$ \begin{aligned} & \nu_{r\theta } = \frac{{\nu_{\theta r} }}{{E_{\theta } }}E_{r} ,\quad C_{11} = \frac{{E_{r} }}{{\left( {1 - \upsilon_{r\theta } \upsilon_{\theta r} } \right)}},\quad C_{12} = \frac{{\upsilon_{r\theta } E_{\theta } }}{{\left( {1 - \upsilon_{r\theta } \upsilon_{\theta r} } \right)}},\quad C_{22} = \frac{{E_{\theta } }}{{\left( {1 - \upsilon_{r\theta } \upsilon_{\theta r} } \right)}},\quad C_{21} = C_{12} , \\ & C_{ij} = C_{ij}^{0} \xi^{n} ,\quad \alpha_{i} = \alpha_{i}^{0} \xi^{n} \\ \end{aligned} $$