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Steady state thermal and mechanical stresses of a poro-piezo-FGM hollow sphere

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Abstract

In this study, an analytical method is developed to obtain mechanical and thermal stress and electrical potential functions, electrical and mechanical displacement in two dimensional steady (r,θ) stat a functionally graded piezo electric porous material hollow sphere (FGPPM). It is assumed that properties of poro, piezoelectric and FGM material is changed through thickness according to power law functions, Heat conduction equation is obtained for obtaining temperature distribution and Navier equations analytically using Legendre polynomials and Euler differential equations system for investigating displacements changes and stress and potential functions for different power indices law and is drawn on a graph. These results are confirmed with the obtained in formations in the paper.

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Author information

Authors and Affiliations

  1. South Tehran Branch, Islamic Azad University, Tehran, Iran

    M. Jabbari

  2. Department of Mechanical Engineering, Omidiye Branch, Islamic Azad University, Omidiye, Iran

    S. Karampour

  3. Mechanical Department, Distinguished Thermoelastic Center, Amirkabir University of Technology, Tehran, Iran

    M. R. Eslami

Authors
  1. M. Jabbari
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  2. S. Karampour
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  3. M. R. Eslami
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Corresponding author

Correspondence to M. R. Eslami.

Appendices

Appendix A

Appendix B

Consider the Legendre differential equation [18] as

$$\bigl(1 - x^{2}\bigr)y'' (x) - 2xy' (x) + n(n + 1)y(x) = 0, $$

where x and y are independent and dependent variables, respectively. Solution of foregoing differential equation is

$$y(x) = P_{n} (x), $$

where P n (x) is the Legendre polynomial and may be written as

$$P_{n} (x) = \frac{1}{2^{n}n!} \frac{d''}{dx^{n}} \bigl(x^{2} - 1\bigr)^{n} $$

Utilizing Eqs. (46)–(48), following relations may be derived:

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Jabbari, M., Karampour, S. & Eslami, M.R. Steady state thermal and mechanical stresses of a poro-piezo-FGM hollow sphere. Meccanica 48, 699–719 (2013). https://doi.org/10.1007/s11012-012-9625-3

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  • DOI: https://doi.org/10.1007/s11012-012-9625-3

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