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Elastic analysis of thick-walled pressurized spherical vessels coated with functionally graded materials

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Abstract

In recent years, functionally graded material (FGM) has been widely explored in coating technology amongst both academic and industry communities. FGM coatings are suitable substitutes for many typical conventional coatings which are susceptible to cracking, debonding and eventual functional failure due to the mismatch of material properties at the coating/substrate interface. In this study, a thick spherical pressure vessel with an inner FGM coating subjected to internal and external hydrostatic pressure is analyzed within the context of three-dimensional elasticity theory. Young’s modulus of the coating is assumed to vary linearly or exponentially through the thickness, while Poisson’s ratio is considered as constant. A comparative numerical study of FGM versus homogeneous coating is conducted for the case of vessel under internal pressure, and the dependence of stress and displacement fields on the type of coating is examined and discussed.

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Acknowledgments

This work was supported by the Italian Ministry of Education, University and Research (MIUR). Project No. 2009XWLFKW: “Multi-scale modeling of materials and structures”.

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Authors and Affiliations

  1. Department of Civil, Chemical and Environmental Engineering, University of Genoa, Genoa, Italy

    S. A. Atashipour & R. Sburlati

  2. Division of Structural and Construction Engineering-Timber Structures, Department of Civil, Environmental and Natural Resources Engineering, Luleå University of Technology, Luleå, Sweden

    S. R. Atashipour

  3. Department of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran

    S. R. Atashipour

Authors
  1. S. A. Atashipour
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  2. R. Sburlati
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  3. S. R. Atashipour
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Corresponding author

Correspondence to R. Sburlati.

Appendix

Appendix

Components of coefficients matrices in Eqs. (24–28):

HMV/HMC (n = 1):

$$ {\mathcal{A}}_{1}^{1} = \frac{{2\left( {1 + \nu } \right)}}{{\left( {1 - 2\nu } \right)}}G_{o} , \quad {\mathcal{A}}_{2}^{1} = - 4\frac{{G_{o} }}{{R_{o}^{3} }} $$
$$ {\mathcal{B}}_{1}^{1} = {\mathcal{A}}_{1}^{1} \left( {\frac{G}{{G_{o} }}} \right) ,\quad {\mathcal{B}}_{2}^{1} = {\mathcal{A}}_{2}^{1} \frac{G}{{G_{o} }}\left( {\frac{{R_{i} }}{{R_{o} }}} \right)^{3} $$
$$ {\mathcal{C}}_{1}^{1} = R_{c} , \quad {\mathcal{C}}_{2}^{1} = \frac{1}{{R_{c}^{2} }},\quad {\mathcal{C}}_{3}^{1} = - R_{c} ,\quad {\mathcal{C}}_{4}^{1} = - \frac{1}{{R_{c}^{2} }} $$
$$ {\mathcal{D}}_{1}^{1} = {\mathcal{A}}_{1}^{1} ,\quad {\mathcal{D}}_{2}^{1} = {\mathcal{A}}_{2}^{1} \left( {\frac{{R_{o} }}{{R_{c} }}} \right)^{3} ,\quad {\mathcal{D}}_{3}^{1} = - {\mathcal{A}}_{1}^{1} \left( {\frac{G}{{G_{o} }}} \right),\quad {\mathcal{D}}_{4}^{1} = - {\mathcal{A}}_{2}^{1} \left( {\frac{G}{{G_{o} }}} \right)\left( {\frac{{R_{o} }}{{R_{c} }}} \right)^{3} $$

HMV/L-FGC (n = 2):

$$ {\mathcal{A}}_{1}^{2} = \frac{{2\left( {1 + \nu } \right)}}{{\left( {1 - 2\nu } \right)}}G_{o} ,\quad {\mathcal{A}}_{2}^{2} = - 4\frac{{G_{o} }}{{R_{o}^{3} }} $$
$$ {\mathcal{B}}_{1}^{2} = - \frac{{2\left( {A + B \cdot R_{i} } \right)\left( {1 - \nu } \right)}}{{\left( {1 - 2\nu } \right)R_{i}^{2 + N} }}\left[ { f_{1} \left( {R_{i} } \right) + \left( {\left( {1 + N} \right) - \frac{2\nu }{{\left( {1 - \nu } \right)}}} \right)f_{3} \left( {R_{i} } \right) } \right] $$
$$ {\mathcal{B}}_{2}^{2} = - \frac{{2\left( {A + B \cdot R_{i} } \right)\left( {1 - \nu } \right)}}{{\left( {1 - 2\nu } \right)R_{i}^{2 - N} }}\left[ { f_{2} \left( {R_{i} } \right) + \left( {\left( {1 - N} \right) - \frac{2\nu }{{\left( {1 - \nu } \right)}}} \right)f_{4} \left( {R_{i} } \right) } \right] $$
$$ {\mathcal{C}}_{1}^{2} = R_{c} ,\quad {\mathcal{C}}_{2}^{2} = \frac{1}{{R_{c}^{2} }},\quad {\mathcal{C}}_{3}^{2} = - \frac{1}{{R_{c}^{1 + N} }}f_{3} \left( {R_{c} } \right),\quad {\mathcal{C}}_{4}^{2} = - \frac{1}{{R_{c}^{1 - N} }}f_{4} \left( {R_{c} } \right) $$
$$ {\mathcal{D}}_{1}^{2} = \frac{{2G_{o} \left( {1 + \nu } \right)}}{{\left( {1 - 2\nu } \right)}},\quad {\mathcal{D}}_{2}^{2} = - \frac{{4G_{o} }}{{R_{c}^{3} }} $$
$$ {\mathcal{D}}_{3}^{2} = \frac{{2\left( {A + B \cdot R_{c} } \right)\left( {1 - \nu } \right)}}{{\left( {1 - 2\nu } \right)R_{c}^{2 + N} }}\left[ { \left( {1 - N} \right)\left( {2 + N} \right)f_{1} \left( {R_{c} } \right) + \left( {\left( {1 + N} \right) - \frac{2\nu }{{\left( {1 - \nu } \right)}}} \right)f_{3} \left( {R_{c} } \right) } \right] $$
$$ {\mathcal{D}}_{4}^{2} = \frac{{2\left( {A + B \cdot R_{c} } \right)\left( {1 - \nu } \right)}}{{\left( {1 - 2\nu } \right)R_{c}^{2 - N} }}\left[ { \left( {1 + N} \right)\left( {2 - N} \right)f_{2} \left( {R_{c} } \right) + \left( {\left( {1 - N} \right) - \frac{2\nu }{{\left( {1 - \nu } \right)}}} \right)f_{4} \left( {R_{c} } \right) } \right] $$

HMV/E-FGC (n = 3):

$$ {\mathcal{A}}_{1}^{3} = \frac{{2\left( {1 + \nu } \right)}}{1 - 2\nu }G_{o} ,\quad {\mathcal{A}}_{2}^{3} = - 4\frac{{G_{o} }}{{R_{o}^{3} }} $$
$$ {\mathcal{B}}_{1}^{3} = \frac{{2\alpha \cdot e^{{\left( {\frac{\beta }{{2R_{i} }}} \right)}} }}{{\left( {1 - 2\nu } \right)R_{i}^{2} }}\left[ {\left( {1 + \nu } \right)g_{1} \left( {R_{i} } \right) - \nu g_{2} \left( {R_{i} } \right)} \right] $$
$$ {\mathcal{B}}_{2}^{3} = \frac{{2\alpha \cdot e^{{\left( {\frac{\beta }{{2R_{i} }}} \right)}} }}{{\left( {1 - 2\nu } \right)R_{i}^{2} }}\left[ { - \left( {1 - \nu } \right)g_{3} \left( {R_{i} } \right) - \nu g_{4} \left( {R_{i} } \right)} \right] $$
$$ {\mathcal{C}}_{1}^{3} = R_{c} ,\quad {\mathcal{C}}_{2}^{3} = \frac{1}{{R_{c}^{2} }},\quad {\mathcal{C}}_{3}^{3} = - \frac{{e^{{ - \frac{\beta }{{2R_{c} }}}} }}{{R_{c} }}g_{2} \left( {R_{c} } \right),\quad {\mathcal{C}}_{4}^{3} = - \frac{{e^{{ - \frac{\beta }{{2R_{c} }}}} }}{{R_{c} }}g_{4} \left( {R_{c} } \right) $$
$$ {\mathcal{D}}_{1}^{3} = \frac{{2G_{o} \left( {1 + \nu } \right)}}{{\left( {1 - 2\nu } \right)}}, \quad {\mathcal{D}}_{2}^{3} = - \frac{{4G_{o} }}{{R_{c}^{3} }} $$
$$ {\mathcal{D}}_{3}^{3} = \frac{{2\alpha \cdot e^{{\left( {\frac{\beta }{{2R_{c} }}} \right)}} }}{{\left( {1 - 2\nu } \right)R_{c}^{2} }}\left[ {\left( {1 + \nu } \right)g_{1} \left( {R_{c} } \right) - \nu g_{2} \left( {R_{c} } \right)} \right] $$
$$ {\mathcal{D}}_{4}^{3} = \frac{{2\alpha \cdot e^{{\left( {\frac{\beta }{{2R_{c} }}} \right)}} }}{{\left( {1 - 2\nu } \right)R_{c}^{2} }}\left[ { - \left( {1 - \nu } \right)g_{1} \left( {R_{c} } \right) - \nu g_{2} \left( {R_{c} } \right)} \right] $$

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Atashipour, S.A., Sburlati, R. & Atashipour, S.R. Elastic analysis of thick-walled pressurized spherical vessels coated with functionally graded materials. Meccanica 49, 2965–2978 (2014). https://doi.org/10.1007/s11012-014-0047-2

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