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Analytical and numerical analysis of an internally and/or externally pressurized thick-walled sphere made of radially nonuniform material

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Abstract

In this work, a thick-walled spherical vessel made of nonhomogeneous materials subjected to internal and/or external pressure was analyzed within the context of three-dimensional elasticity theory. A closed-form analytical solution was obtained for computing the displacement and stress fields. It has been assumed that the elastic stiffness is varying through thickness of the functionally graded material according to a nonlinear general expression, while Poisson’s ratio is considered as constant. In order to check the relevance of the analytical solution, a finite element model of the pressurized vessel were constructed, taking into account variations in Young's modulus. Very good agreement has been found between the numerical results and the predictions of the analytical solution, which confirms the accuracy of our model. Thus, the inhomogeneity in material properties can be exploited to optimize the distribution of displacement and stress fields.

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Abbreviations

E in :

Young modulus at inner radius

m, s :

FGM grading parameters

M :

Kummer’s function

r :

Radius

P in :

Internal pressure

P out :

External pressure

R in :

Inner radius

R out :

Outer radius

\({\varvec{u}}\) :

Displacement vector

\(u_{r} ,\;u_{\theta } ,\;u_{\phi }\) :

Components of the displacement vector

\(\nu\) :

Poisson’s ratio

\(\gamma\) :

Grading parameter

\({\varvec{\sigma}}\) :

Cauchy stress tensor

\(\sigma_{rr} ,\;\sigma_{\theta \theta } ,\;\sigma_{\phi \phi }\) :

Components of the Cauchy stress tensor

\({\varvec{\varepsilon}}\) :

Stress tensor

\(\varepsilon_{rr} ,\;\varepsilon_{\theta \theta } ,\;\varepsilon_{\phi \phi }\) :

Components of the strain tensor

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

Lamé’s coefficients

\(\delta_{ij}\) :

Symbol of Kronecker

in, out :

Inner (internal), Outer (outside)

r, θ, ϕ :

Radial, circumferential and meridional direction

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Acknowledgements

We thank the editor and anonymous reviewers for their careful reading of our manuscript and their many insightful comments and suggestions, which helped us to improve the manuscript. The authors thank the General Directorate of Scientific Research and Technological Development (DGRSDT/MESRS-Algeria) for their financial support.

Author information

Authors and Affiliations

  1. Laboratoire de Mécanique, Matériaux Et Energétique (L2ME), Faculté de Technologie, Université de Bejaia, 06000, Bejaia, Algérie

    Abdelhakim Benslimane & Mounir Methia

  2. INSA CVL, LaMé, Univ. Tours, Univ. Orléans, 3 Rue de la Chocolaterie, BP 3410, 41034, Blois Cedex, France

    Mounir Methia

  3. Institut Clément Ader (ICA), ISAE-SUPAERO, INSA, IMT MINES ALBI, UT III, CNRS, Université de Toulouse, Toulouse, France

    Roselane Hammoum

  4. Civil Engineering Department, College of Engineering, Prince Sattam Bin Abdulaziz University, Al-Kharj, Saudi Arabia

    Mohamed Amine Khadimallah

Authors
  1. Abdelhakim Benslimane
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  2. Mounir Methia
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Correspondence to Abdelhakim Benslimane.

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Appendix

Appendix

The development of Eq. (17) can be written in the form:

$$ M\left( {a_{i} ,b_{i} ,\gamma \left( {\frac{r}{{R_{in} }}} \right)^{s} } \right) = 1 + \frac{{\alpha_{i} \left( {\gamma \left( {\frac{r}{{R_{in} }}} \right)^{s} } \right)}}{{\beta_{i} }} + \frac{{\alpha_{i} \left( {1 + \alpha_{i} } \right)\left( {\gamma \left( {\frac{r}{{R_{in} }}} \right)^{s} } \right)}}{{\beta_{i} \left( {1 + \beta_{i} } \right)2!}} + .... + \frac{{\left( {\alpha_{i} } \right)_{n} \left( {\gamma \left( {\frac{r}{{R_{in} }}} \right)^{s} } \right)^{n} }}{{\left( {\beta_{i} } \right)_{n} n!}} + ... $$
(1.1)

The derivative form of Kummer’s function (Eq. 17) is given by:

$$ \frac{{dM\left( {\alpha ,\beta ,\gamma \left( {\frac{r}{{R_{in} }}} \right)^{s} } \right)}}{dr} = \frac{\alpha }{\beta }\frac{{\gamma \left( {\frac{r}{{R_{in} }}} \right)^{s} }}{r}M\left( {\alpha + 1,\beta + 1,\gamma \left( {\frac{r}{{R_{in} }}} \right)^{s} } \right) $$
(1.2)

The derivative of the displacement function Eq. (16) is calculated as follows:

$$ \frac{{du_{r} }}{dr} = \frac{1}{r}\sum\limits_{k = 1}^{2} {A_{k,NH} \,r^{{\lambda_{k} }} \,e^{{ - \gamma \left( {\frac{r}{{R_{in} }}} \right)^{s} }} } \left[ {f_{k} \left( r \right)\,g_{k} \left( r \right) + h_{k} \left( r \right)} \right] $$
(1.3)

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Benslimane, A., Methia, M., Hammoum, R. et al. Analytical and numerical analysis of an internally and/or externally pressurized thick-walled sphere made of radially nonuniform material. Int J Interact Des Manuf 17, 637–648 (2023). https://doi.org/10.1007/s12008-022-00976-0

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  • DOI: https://doi.org/10.1007/s12008-022-00976-0

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