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Thermal vibration and buckling analysis of magneto-electro-elastic functionally graded porous higher-order nanobeams using nonlocal strain gradient theory

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Abstract

In this paper, free vibration analysis and temperature-dependent buckling behavior of porous functionally graded magneto-electro-thermo-elastic material consisting of cobalt ferrite and barium titanate were modeled and analyzed. A high-order sinusoidal shear deformation theory was used to accurately model the anisotropic material behavior. The study examined the porosity role variation across thickness in the buckling and free vibration behavior of nanobeams, as well as the effects of magneto-electro-elastic coupling, thermal stresses, nonlocal properties, externally applied electric and magnetic field potential, and porosity volume fraction.

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Authors and Affiliations

  1. Engineering Faculty, Mechanical Engineering Department, Sakarya University, 54187, Sakarya, Turkey

    Mustafa Eroğlu

  2. Engineering Faculty, Mechanical Engineering Department, Karabük University, 78050, Karabük, Turkey

    İsmail Esen

  3. Technology Faculty, Mechatronics Engineering Department, Sakarya Applied Sciences University, 54187, Sakarya, Turkey

    Mehmet Akif Koç

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  1. Mustafa Eroğlu
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Appendix A

Appendix A

The eigenvalue equations for trigonometric solution considering general boundary conditions presented in Table 2.

$$ \begin{gathered} \left[ {A_{11} \left( {r_{3} + l_{m}^{2} r_{10} } \right) + I_{0} \omega_{n}^{2} \left( {r_{1} + \left( {e_{0} a} \right)^{2} \left( {r_{3} } \right)} \right)} \right]U_{mn} - \left[ {B_{11} \left( {r_{3} + l_{m}^{2} r_{10} } \right) + I_{1} \omega_{n}^{2} r_{1} + \left( {e_{0} a} \right)^{2} I_{1} \omega_{n}^{2} (r_{3} )} \right]W_{bmn} \hfill \\ + [ - B_{11}^{s} \left( {r_{3} + l_{m}^{2} r_{10} } \right) + J_{1} \omega_{n}^{2} \left( {r_{1} + \left( {e_{0} a} \right)^{2} \left( {r_{3} } \right)} \right)W_{smn} + A_{31}^{e} \left( {r_{1} + l_{m}^{2} r_{3} } \right) \phi_{mn} + A_{31}^{m} \left( {r_{1} + l_{m}^{2} r_{3} } \right) Y_{mn} = 0 \hfill \\ \end{gathered} $$
(A.1)
$$ \begin{aligned} &\left[ {B_{11} \left( {r_{9} + l_{m}^{2} r_{11} } \right) + I_{1} \omega_{n}^{2} \left( {r_{7} + \left( {e_{0} a} \right)^{2} r_{9} } \right)} \right]U_{mn} \\&+ \left[ - D_{11} \left( {r_{9} + l_{m}^{2} r_{11} } \right) - I_{2} \omega_{n}^{2} \left( {r_{7} + \left( {e_{0} a} \right)^{2} r_{9} } \right)\right.\\ &\left.\quad + I_{0} \omega_{n}^{2} \left( {r_{5} + \left( {e_{0} a} \right)^{2} r_{7} } \right) + \left( {N^{E} + N^{H} + N^{T} } \right) \left( {r_{7}+ \left( {e_{0} a} \right)^{2} r_{9} } \right) \right]W_{bmn} \hfill \\ &\quad + \left[ - D_{11}^{s} \left( {r_{9} + l_{m}^{2} r_{11} } \right) \right.\\ &\left.\quad - I_{0} \omega_{n}^{2} \left( {r_{5} + \left( {e_{0} a} \right)^{2} r_{7} } \right)+ J_{2} \omega_{n}^{2} \left( {r_{7} + \left( {e_{0} a} \right)^{2} r_{9} } \right) + \left( {N^{E} + N^{H} + N^{T} } \right)\left( {r_{7} + \left( {e_{0} a} \right)^{2} r_{9} } \right) \right] W_{smn} \hfill \\ &\quad+ \left[ {(E_{31}^{e} - A_{15}^{e} )\left( {r_{7} + l_{m}^{2} r_{9} } \right)} \right] \phi_{mn} + \left[ {(E_{31}^{m} - A_{15}^{m} ) \left( {r_{7} + l_{m}^{2} r_{9} } \right)} \right]Y_{mn} = 0 \end{aligned} $$
(A.2)
$$ \begin{gathered} \left[ {B_{11}^{s} \left( {r_{9} + l_{m}^{2} r_{11} } \right) + J_{1} \omega_{n}^{2} \left( {r_{7} + \left( {e_{0} a} \right)^{2} r_{9} } \right)} \right]U_{mn} + \left[ { - D_{11}^{s} \left( {r_{9} + l_{m}^{2} r_{11} } \right) - J_{2} \omega_{n}^{2} \left( {r_{7} + \left( {e_{0} a} \right)^{2} r_{9} } \right) - I_{0} \omega_{n}^{2} \left( {r_{5} + \left( {e_{0} a} \right)^{2} r_{7} } \right) + )} \right. \hfill \\ \left. { - D_{11}^{s} \left( {r_{9} + l_{m}^{2} r_{11} } \right) - I_{0} \omega_{n}^{2} \left( {r_{5} + \left( {e_{0} a} \right)^{2} r_{7} } \right) + (N^{E} + N^{H} + N^{T} )\left( {r_{7} + \left( {e_{0} a} \right)^{2} r_{9} } \right)} \right]W_{bmn} \hfill \\ + \left[ {A_{44}^{s} \left( {r_{7} + l_{m}^{2} r_{9} } \right) - H_{11}^{s} \left( {r_{9} + l_{m}^{2} r_{11} } \right) + (N^{E} + N^{H} + N^{T} } \right)\left( {r_{7} + \left( {e_{0} a} \right)^{2} r_{9} } \right)) \hfill \\ \left. { - I_{0} \omega_{n}^{2} \left( {r_{5} + \left( {e_{0} a} \right)^{2} r_{7} } \right) - K_{2} \omega_{n}^{2} \left( {\left( {r_{7} + \left( {e_{0} a} \right)^{2} r_{9} } \right)} \right)} \right]W_{smn} \hfill \\ + \left[ {(F_{31}^{e} - A_{15}^{e} ) \left( {r_{7} + l_{m}^{2} r_{9} } \right)} \right]\phi_{mn} + \left( {F_{31}^{m} - A_{15}^{m} } \right) \left( {r_{7} + l_{m}^{2} r_{9} } \right)Y_{mn} = 0 \hfill \\ \end{gathered} $$
(A.3)
$$ \begin{gathered} A_{31}^{e} \left( {r_{1} + l_{m}^{2} r_{3} } \right) {\text{U}}_{mn} + \left[ { - E_{31}^{e} \left( {r_{1} + l_{m}^{2} r_{3} } \right)} \right]{\text{W}}_{bmn} + \left[ {\left( {F_{31}^{e} - A_{15}^{e} } \right) \left( {r_{7} + l_{m}^{2} r_{9} } \right)} \right]{\text{W}}_{smn} \hfill \\ + \left[ {F_{11}^{e} \left( {r_{7} + l_{m}^{2} r_{9} } \right) - F_{33}^{e} \left( {r_{5} + l_{m}^{2} r_{7} } \right)} \right]\phi_{mn} + \left[ {F_{11}^{m} \left( {r_{7} + l_{m}^{2} r_{9} } \right) - F_{33}^{m} \left( {r_{5} + l_{m}^{2} r_{7} } \right)} \right]{\text{Y}}_{mn} = 0 \hfill \\ \end{gathered} $$
(A.4)
$$ \begin{aligned} &A_{31}^{m} \left( {r_{1} + l_{m}^{2} r_{3} } \right){\text{ U}}_{mn} - E_{31}^{m} \left( {r_{1} + l_{m}^{2} r_{3} } \right){\text{W}}_{bmn} + \left( {E_{15}^{m} - F_{31}^{m} } \right)\left( {r_{7} + l_{m}^{2} r_{9} } \right){\text{W}}_{smn}\\ &\qquad + \left[ {F_{11}^{m} \left( {r_{7} + l_{m}^{2} r_{9} } \right) - F_{33}^{m} \left( {r_{5} + l_{m}^{2} r_{7} } \right)} \right]\psi_{mn} + \left[ {X_{11}^{m} \left( {r_{7} + l_{m}^{2} r_{9} } \right) - X_{33}^{m} \left( {r_{5} + l_{m}^{2} r_{7} } \right)} \right]\gamma_{mn} = 0 \hfill \\ \end{aligned} $$
(A.5)

with

$$\begin{aligned}&{r}_{1}={\int }_{0}^{L}{X}_{m}{\prime}{X}_{m}{\prime}dx, {r}_{3}={\int }_{0}^{L}{X}_{m}^{{\prime}{\prime}{\prime}}{X}_{m}{\prime}dx, {r}_{5}={\int }_{0}^{L}{X}_{m}{X}_{m}dx, {r}_{7}={\int }_{0}^{L}{X}_{m}^{{\prime}{\prime}}{X}_{m}dx, \\ &{r}_{9}={\int }_{0}^{L}{X}_{m}^{{\prime}{\prime}{\prime}{\prime}}{X}_{m}dx, {r}_{10}={\int }_{0}^{L}{X}_{m}^{{\prime}{\prime}{\prime}{\prime}{\prime}}{X}_{m}{\prime}dx, {r}_{11}={\int }_{0}^{L}{X}_{m}^{{\prime}{\prime}{\prime}{\prime}{\prime}{\prime}}{X}_{m}dx\end{aligned}$$
(A.6)

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Eroğlu, M., Esen, İ. & Koç, M.A. Thermal vibration and buckling analysis of magneto-electro-elastic functionally graded porous higher-order nanobeams using nonlocal strain gradient theory. Acta Mech 235, 1175–1211 (2024). https://doi.org/10.1007/s00707-023-03793-y

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