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Two-dimensional solution of functionally graded piezoelectric-layered beams

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Abstract

This study presents an exact elasticity solution for the functionally graded piezoelectric beams subjected to sinusoidally distributed transverse electromechanical loads. The elasticity solution can reflect the symmetrical exponential variation of materials about the interface of perfectly bonded two layers. The effects of the material gradation, beam span length to thickness ratio, intensity of external mechanical/electrical loads, and material properties on the axial normal stresses, transverse normal stress, transverse shear stress, axial and vertical displacements, electric displacements, and electrical potential are comprehensively investigated. The proposed elasticity solution can be used for the verification/comparison purposes of numerical procedures as a reference procedure.

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Authors and Affiliations

  1. Civil Engineering Department, Karadeniz Technical University, Trabzon, Turkey

    İsa Çömez

  2. Civil Engineering Department, Istanbul Medipol University, Istanbul, Turkey

    Umit N. Aribas & Mehmet H. Omurtag

  3. Civil Engineering Department, Istanbul Technical University, Istanbul, Turkey

    Akif Kutlu

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  1. İsa Çömez
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  2. Umit N. Aribas
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  3. Akif Kutlu
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Correspondence to Akif Kutlu.

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Appendix A

Appendix A

Expressions of \(L_{1ji}\) \((j = 1,2,...6)\) appearing in Eq. (10) are given as follows:

$$ L_{11i} = 3\gamma_{i} /\beta $$
(A1)
$$ \begin{aligned} L_{12i} & = (e_{330i} ((2c_{440i} e_{310i} + 2c_{130i} (e_{150i} + e_{310i} ) - c_{110i} e_{330i} )\beta^{2} + 3c_{440i} e_{330i} \gamma_{i}^{2} ) \\ & - c_{330i} \beta^{2} ((e_{150i} + e_{310i} )^{2} + c_{440i} \in_{110i} ) + (c_{130i}^{2} + 2c_{130i} c_{440i} )\beta^{2} \in_{330i} \\ & + c_{330i} \beta^{2} ( - c_{110i} \beta^{2} + 3c_{440i} \gamma_{i}^{2} ) \in_{330i} )/(c_{440i} \beta^{2} (e_{330i}^{2} + c_{330i} \in_{330i} )) \\ \end{aligned} $$
(A2)
$$ \begin{aligned} L_{13i} & = (e_{330i} (2(2c_{440i} e_{310i} + 2c_{130i} (e_{150i} + e_{310i} ) - c_{110i} e_{330i} )\beta^{2} + 3c_{440i} e_{330i} \gamma_{i}^{2} ) \\ & - 2c_{330i} \beta^{2} ((e_{150i} + e_{310i} )^{2} + c_{440i} \in_{110i} ) + 2(c_{130i}^{2} + 2c_{130i} c_{440i} )\beta^{2} \in_{330i} \\ & + c_{330i} \beta^{2} ( - 2c_{110i} \beta^{2} + c_{440i} \gamma_{i}^{2} ) \in_{330i} )/(c_{440i} \beta^{2} (e_{330i}^{2} + c_{330i} \in_{330i} )) \\ \end{aligned} $$
(A3)
$$ \begin{aligned} L_{14i} & = - (\beta^{2} (c_{130i}^{2} {\kern 1pt} \in_{110i} {\kern 1pt} - c_{110i} (2e_{150i} e_{330i} + c_{330i} \in_{110i} ) \\ & + 2c_{130i} (e_{150i} (e_{150i} + e_{310i} ) + c_{440i} \in_{110i} ) - c_{440i} (c_{310i}^{2} {\kern 1pt} + c_{110i} \in_{330i} )) \\ & + \gamma_{i}^{2} ( - e_{330i} (3c_{130i} e_{150i} + 2c_{130i} e_{310i} + 3c_{440i} e_{310i} - c_{110i} e_{330i} ) \\ & - (c_{130i}^{2} + 3c_{130i} c_{440i} ) \in_{330i} + c_{330i} (e_{150i}^{2} + 3e_{150i} e_{310i} \\ & + e_{310i}^{2} + c_{440i} \in_{110i} + c_{110i} \in_{330i} ))/(c_{440} \xi^{2} (e_{330}^{2} + c_{330} \in_{330} )) \\ \end{aligned} $$
(A4)
$$ \begin{aligned} L_{15i} & = - \gamma_{i} (\beta^{2} (c_{130i}^{2} {\kern 1pt} \in_{110i} {\kern 1pt} - c_{110i} (2e_{150i} e_{330i} + c_{330i} \in_{110i} ) \\ & + 2c_{130i} (e_{150i} (e_{150i} + e_{310i} ) + c_{440i} \in_{110i} ) + \\ & - c_{440i} (e_{310i}^{2} + c_{110i} \in_{330i} ))(c_{110} + c_{440} ){\kern 1pt} \in_{330} ) \\ & + \gamma_{i}^{2} (c_{330i} e_{310i} e_{150i} - c_{440i} e_{310i} e_{330i} \\ & - c_{130i} (e_{150i} e_{330i} + c_{440i} \in_{330i} )))/(c_{440i} \beta^{2} (e_{330i}^{2} + c_{330i} \in_{330i} )) \\ \end{aligned} $$
(A5)
$$ \begin{aligned} L_{16i} & = ( - c_{110i} \beta^{2} (e_{150i}^{2} {\kern 1pt} + c_{440i} \in_{110i} ) \\ & - \gamma_{i}^{2} (c_{130i} (e_{150i}^{2} {\kern 1pt} + c_{440i} \in_{110i} ))) \\ & /(c_{440i} \beta^{2} (e_{330i}^{2} + c_{330i} \in_{330i} )). \\ \end{aligned} $$
(A6)

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Çömez, İ., Aribas, U.N., Kutlu, A. et al. Two-dimensional solution of functionally graded piezoelectric-layered beams. J Braz. Soc. Mech. Sci. Eng. 44, 101 (2022). https://doi.org/10.1007/s40430-022-03414-0

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