Thermal vibration of functionally graded porous beams with classical and non-classical boundary conditions using a modified Fourier method

Abstract

In this study, the modified Fourier method is developed to analyze thermal vibration of functionally graded porous beams based on the third-order shear deformation theory. It can provide accurate results due to satisfying both force and geometrical boundary conditions. Three different forms including uniform, linear and nonlinear temperature distribution are considered. Some important effects including patterns of porous distribution, porous coefficient, temperature rises, spring constants, and classical and non-classical boundary conditions are investigated. Based on this extensive investigation, it is worth revealing that the frequencies of the beams carrying pores with uniform distribution decrease to the same point at the critical buckling temperature, while the beams with symmetrical and un-symmetrical forms illustrate different manners in which the details are explained in this study.

Appendix 1

Supplementary functions and their derivatives with respect to x from the first to fourth derivatives defined by the symbol of prime are provided in Table 7.

The constants of the supplementary functions and their derivatives are expressed as follows:

Table 7 The first and second function and their derivatives

For the first function:

$$\begin{aligned} & d_{11}^{0} = \frac{{L^{3} }}{{\pi^{3} }} + \frac{9L}{{4\pi }},\;\;d_{12}^{0} = - \frac{{L^{3} }}{{3\pi^{3} }} - \frac{L}{12\pi },\;\;d_{11}^{1} = \frac{{L^{2} }}{{2\pi^{2} }} + \frac{9}{8}, \\ & d_{12}^{1} = - \frac{{L^{2} }}{{2\pi^{2} }} - \frac{1}{8},\;\;d_{11}^{2} = - \frac{L}{4\pi } - \frac{9\pi }{{16L}}, \\ & d_{12}^{2} = \frac{3L}{{4\pi }} + \frac{3\pi }{{16L}},\;\;d_{11}^{3} = - \frac{1}{8} - \frac{{9\pi^{2} }}{{32L^{2} }},\;\;d_{12}^{3} = \frac{9}{8} + \frac{{9\pi^{2} }}{{32L^{2} }}, \\ & d_{11}^{4} = \frac{\pi }{16L} + \frac{{9\pi^{3} }}{{64L^{3} }},\;\;d_{12}^{4} = - \frac{27\pi }{{16L}} - \frac{{27\pi^{3} }}{{64L^{3} }}, \\ \end{aligned}$$

For the second function:

$$\begin{aligned} & d_{21}^{0} = - \frac{{L^{3} }}{{\pi^{3} }} - \frac{9L}{{4\pi }},\;\;d_{22}^{0} = - \frac{{L^{3} }}{{3\pi^{3} }} - \frac{L}{12\pi }, \\ & d_{21}^{1} = \frac{{L^{2} }}{{2\pi^{2} }} + \frac{9}{8},\;\;d_{22}^{1} = \frac{{L^{2} }}{{2\pi^{2} }} + \frac{1}{8},\;\;d_{21}^{2} = \frac{L}{4\pi } + \frac{9\pi }{{16L}}, \\ & d_{22}^{2} = \frac{3L}{{4\pi }} + \frac{3\pi }{{16L}},\;\;d_{21}^{3} = - \frac{1}{8} - \frac{{9\pi^{2} }}{{32L^{2} }},\;\;d_{22}^{3} = - \frac{9}{8} - \frac{{9\pi^{2} }}{{32L^{2} }}, \\ & d_{21}^{4} = - \frac{\pi }{16L} - \frac{{9\pi^{3} }}{{64L^{3} }},\;\;d_{22}^{4} = - \frac{27\pi }{{16L}} - \frac{{27\pi^{3} }}{{64L^{3} }}. \\ \end{aligned}$$

Fourier cosine series expansion coefficients related to the first and second functions written in the above Table of this appendix can be expressed as:

$$\sin \left( {\frac{\pi }{2L}x} \right) = \sum\limits_{m = 0}^{M} {\alpha_{1}^{m} \cos \lambda_{m} x\quad \Rightarrow } \quad \alpha_{1}^{m} = \left\{ \begin{gathered} \;\frac{2}{\pi }\quad \quad \quad \quad \;\,m = 0 \hfill \\ \;\frac{4}{{(1 - 4m^{2} )\pi }}\quad m \ne 0 \hfill \\ \end{gathered}, \right.$$

$$\sin \left( {\frac{3\pi }{{2L}}x} \right) = \sum\limits_{m = 0}^{M} {\alpha_{2}^{m} \cos \lambda_{m} x\quad \Rightarrow } \quad \alpha_{2}^{m} = \left\{ \begin{gathered} \;\frac{2}{3\pi }\quad \quad \quad \quad \;\,m = 0 \hfill \\ \;\frac{12}{{(9 - 4m^{2} )\pi }}\quad m \ne 0 \hfill \\ \end{gathered}, \right.$$

$$\cos \left( {\frac{\pi }{2L}x} \right) = \sum\limits_{m = 0}^{M} {\alpha_{3}^{m} \cos \lambda_{m} x\quad \Rightarrow } \quad \alpha_{3}^{m} = \left\{ \begin{gathered} \;\frac{2}{\pi }\quad \quad \quad \quad \;\,m = 0 \hfill \\ \;\frac{{4( - 1)^{m} }}{{(1 - 4m^{2} )\pi }}\quad m \ne 0 \hfill \\ \end{gathered}, \right.$$

$$\cos \left( {\frac{3\pi }{{2L}}x} \right) = \sum\limits_{m = 0}^{M} {\alpha_{4}^{m} \cos \lambda_{m} x\quad \Rightarrow } \quad \alpha_{4}^{m} = \left\{ \begin{gathered} \; - \frac{2}{3\pi }\quad \quad \quad \;\,m = 0 \hfill \\ \;\frac{{12( - 1)^{m + 1} }}{{(9 - 4m^{2} )\pi }}\quad m \ne 0 \hfill \\ \end{gathered}, \right.$$

$$\sin \lambda_{m} x = \sum\limits_{i = 0}^{M} {\beta_{i}^{m} \cos \lambda_{i} x\quad \Rightarrow } \quad \beta_{i}^{m} = \left\{ \begin{gathered} \;0\quad \quad \quad \quad \;\,\quad \;\,m = 0 \hfill \\ \;0\quad \quad \quad \quad \quad \quad m = i \hfill \\ \;\frac{{1 - ( - 1)^{m} }}{m\pi }\quad \quad \;\;\,m \ne 0,\;i = 0 \hfill \\ \;\frac{{2m\left[ {( - 1)^{m + i} - 1} \right]}}{{(i^{2} - m^{2} )\pi }}\;\;m \ne 0,\;i \ne 0 \hfill. \\ \end{gathered} \right.$$

By using the above formulations, one can obtain the coefficients of Fourier cosine series associated with the supplementary functions and their derivatives, which are utilized in Eqs. (30.1–3), as in Table 8.

Table 8 Coefficients of the first and second function