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Hygrothermal Vibration and Damping Behavior of Magnetostrictive Sandwich Plate Resting On Pasternak’s Foundations

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Abstract

In the present study, the vibration behavior of laminated sandwich plate resting on a two-parameter elastic foundation, containing embedded magnetostrictive actuating layers and homogeneous core, is investigated. The plate is subjected to three different types of hygrothermal environments: uniform, linear and nonlinear distributions. The system associated with the vibration problem for the simply supported rectangular plate under the hygrothermal effect is derived based on Hamilton’s principle. The system solutions are obtained depending on Navier’s technique. Parametric effects due to the elastic foundation, smart layer location, lamination schemes, mode numbers, feedback gain control value, thickness ratio, aspect ratio, core thickness-to-fiber reinforced layer thickness ratio, magnetostrictive layer thickness to fiber-reinforced layer thickness ratio, temperature and moisture factors on vibration characteristics of the structure, are analyzed and discussed in detail. Numerical results can be useful as a benchmark for future studies of hygro-thermo-dynamic influences on structural applications in various fields as well as the suggested model can be contributed to the development of vibration control of advanced structural applications.

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Acknowledgements

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Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

    Ashraf M. Zenkour & Hela D. El-Shahrany

  2. Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafrelsheikh, 33516, Egypt

    Ashraf M. Zenkour

  3. Department of Mathematics, Faculty of Science, University of Bisha, Bisha, 61922, Saudi Arabia

    Hela D. El-Shahrany

Authors
  1. Ashraf M. Zenkour
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Contributions

Conception or design of the work – Dr. H. D. El-Shahrany, Prof. A. M. Zenkour.

Data collection – Dr. H. D. El-Shahrany.

Data analysis and interpretation – Prof. A. M. Zenkour.

Drafting the article – Dr. H. D. El-Shahrany.

Critical revision of the article – Prof. A. M. Zenkour.

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Correspondence to Ashraf M. Zenkour.

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Appendices

Appendix A

The coefficients \({\overline{Q} }_{ij}^{\left(r\right)}\), \({\overline{q} }_{ij}\), \({\stackrel{\sim }{\alpha }}_{l}\) and \({\stackrel{\sim }{\beta }}_{l}\), \(l=xx,yy, xy\) appeared in Eq. (7) and Eq. (9) are described as.

$${\overline{Q} }_{11}^{\left(r\right)}={Q}_{11}^{\left(r\right)}{{\mathrm{cos}}^{4}\theta }^{\left(r\right)}+2\left({Q}_{12}^{\left(r\right)}{+2Q}_{66}^{\left(r\right)}\right){{\mathrm{cos}}^{2}\theta }^{\left(r\right)}{{\mathrm{sin}}^{2}\theta }^{\left(r\right)}+{Q}_{22}^{\left(r\right)}{{\mathrm{sin}}^{4}\theta }^{\left(r\right)},$$
$${\overline{Q} }_{12}^{\left(r\right)}=\left({Q}_{11}^{\left(r\right)}+{Q}_{22}^{\left(r\right)}-{4Q}_{66}^{\left(r\right)}\right){{\mathrm{cos}}^{2}\theta }^{\left(r\right)}{{\mathrm{sin}}^{2}\theta }^{\left(r\right)}+{Q}_{12}^{\left(r\right)}\left({{\mathrm{sin}}^{4}\theta }^{\left(r\right)}+{{\mathrm{cos}}^{4}\theta }^{\left(r\right)}\right),$$
$${\overline{Q} }_{22}^{\left(r\right)}={Q}_{11}^{\left(r\right)}{{\mathrm{sin}}^{4}\theta }^{\left(r\right)}+2\left({Q}_{12}^{\left(r\right)}{+2Q}_{66}^{\left(r\right)}\right){{\mathrm{cos}}^{2}\theta }^{\left(r\right)}{{\mathrm{sin}}^{2}\theta }^{\left(r\right)}+{Q}_{22}^{\left(r\right)}{{\mathrm{cos}}^{4}\theta }^{\left(r\right)},$$
$${\overline{Q} }_{44}^{\left(r\right)}={Q}_{44}^{\left(r\right)}{{\mathrm{cos}}^{2}\theta }^{\left(r\right)}+{Q}_{55}^{\left(r\right)}{{\mathrm{sin}}^{2}\theta }^{\left(r\right)},$$
$${\overline{Q} }_{55}^{\left(r\right)}={Q}_{55}^{\left(r\right)}{{\mathrm{cos}}^{2}\theta }^{\left(r\right)}+{Q}_{44}^{\left(r\right)}{{\mathrm{sin}}^{2}\theta }^{\left(r\right)},$$
$${\overline{Q} }_{66}^{\left(r\right)}={\left({Q}_{11}^{\left(r\right)}+{Q}_{22}^{\left(r\right)}-{2Q}_{12}^{\left(r\right)}-{2Q}_{66}^{\left(r\right)}\right){{\mathrm{sin}}^{2}\theta }^{\left(r\right)}{{\mathrm{cos}}^{2}\theta }^{\left(r\right)}+Q}_{66}^{\left(r\right)}\left({{\mathrm{sin}}^{4}\theta }^{\left(r\right)}+{{\mathrm{cos}}^{4}\theta }^{\left(r\right)}\right),$$
$${Q}_{11}^{\left(r\right)}=\frac{{E}_{1}\left(1-{\upnu }_{23}^{\left(r\right)}{\upnu }_{32}^{\left(r\right)}\right)}{\Delta },$$
$${Q}_{12}^{\left(r\right)}=\frac{{E}_{1}\left({\upnu }_{21}^{\left(r\right)}+{\upnu }_{31}^{\left(r\right)}{\upnu }_{23}^{\left(r\right)}\right)}{\Delta },$$
$${Q}_{22}^{\left(r\right)}=\frac{{E}_{2}\left(1-{\upnu }_{13}^{\left(r\right)}{\upnu }_{31}^{\left(r\right)}\right)}{\Delta },$$
$${Q}_{44}^{\left(r\right)}={G}_{23}^{\left(r\right)},\;\; {Q}_{55}^{\left(r\right)}={G}_{13}^{\left(r\right)},\;\; {Q}_{66}^{\left(r\right)}={G}_{12}^{\left(r\right)},$$
$$\Delta =1- {\upnu }_{21}^{\left(r\right)}{\upnu }_{12}^{\left(r\right)}-{\upnu }_{23}^{\left(r\right)}{\upnu }_{32}^{\left(r\right)}-{\upnu }_{13}^{\left(r\right)}{\upnu }_{31}^{\left(r\right)} -2{\upnu }_{21}^{\left(r\right)}{\upnu }_{13}^{\left(r\right)}{\upnu }_{32}^{\left(r\right)},$$
$${\upnu }_{21}^{\left(r\right)}=\frac{{\upnu }_{12}^{\left(r\right)}{E}_{2-}^{\left(r\right)}}{{E}_{1}^{\left(r\right)}},\;\; {\upnu }_{31}^{\left(r\right)}=\frac{{\upnu }_{13}^{\left(r\right)}{E}_{3}^{\left(r\right)}}{{E}_{1}^{\left(r\right)}},\;\; {\upnu }_{32}^{\left(r\right)}=\frac{{\upnu }_{23}^{\left(r\right)}{E}_{3}^{\left(r\right)}}{{E}_{2}^{\left(r\right)}},$$
$${\stackrel{\sim }{\alpha }}_{xx}={\alpha }_{xx}{\mathrm{cos}}^{2}\theta +{\alpha }_{yy}{\mathrm{sin}}^{2}\theta ,$$
$${\stackrel{\sim }{\alpha }}_{yy}={\alpha }_{yy}{\mathrm{cos}}^{2}\theta +{\alpha }_{xx}{\mathrm{sin}}^{2}\theta ,$$
$${\stackrel{\sim }{\alpha }}_{xy}=\left({\alpha }_{xx}-{\alpha }_{yy}\right)\mathrm{sin}\theta \; \mathrm{cos}\theta ,$$
$${\stackrel{\sim }{\beta }}_{xx}={\beta }_{xx}{\mathrm{cos}}^{2}\theta +{\beta }_{yy}{\mathrm{sin}}^{2}\theta ,$$
$${\stackrel{\sim }{\beta }}_{yy}={\beta }_{yy}{\mathrm{cos}}^{2}\theta +{\beta }_{xx}{\mathrm{sin}}^{2}\theta ,$$
$${\stackrel{\sim }{\beta }}_{xy}=\left({\beta }_{xx}-{\beta }_{yy}\right)\mathrm{sin}\theta \; \mathrm{cos}\theta ,$$
$${\overline{q} }_{31}={q}_{31}{\mathrm{cos}}^{2}\theta +{q}_{32}{\mathrm{sin}}^{2}\theta ,$$
$${\overline{q} }_{32}={q}_{32}{\mathrm{cos}}^{2}\theta +{q}_{31}{\mathrm{sin}}^{2}\theta ,$$
$${\overline{q} }_{14}=\left({q}_{15}-{q}_{24}\right)\mathrm{sin}\theta \; \mathrm{cos}\theta ,$$
$${\overline{q} }_{24}={q}_{24}{\mathrm{cos}}^{2}\theta +{q}_{15}{\mathrm{sin}}^{2}\theta ,$$
$${\overline{q} }_{15}={q}_{15}{\mathrm{cos}}^{2}\theta +{q}_{24}{\mathrm{sin}}^{2}\theta ,$$
$${\overline{q} }_{25}=\left({q}_{15}-{q}_{24}\right)\mathrm{sin}\theta \; \mathrm{cos}\theta ,$$
$${\overline{q} }_{36}=\left({q}_{31}-{q}_{32}\right)\mathrm{sin}\theta \; \mathrm{cos}\theta ,$$

in which \({E}_{i}\), \({v}_{ij}\), \({G}_{ij}\), \({\alpha }_{ij}\) and \({\beta }_{ij}\) are Young’s moduli, Poisson’s ratios, shear moduli, the thermal and hygroscopic expansion coefficients, respectively. Besides, the coefficients \({q}_{ij}\) are the magnetostrictive modules.

Appendix B

The coefficients \({\widehat{S}}_{ij},{\widehat{M}}_{ij}\) and \({\widehat{C}}_{ij}\) (\(i=1, 2, 3\)) appeared in Eq. (35) are obtained as the following.

$${\widehat{S}}_{11}={D}_{11}{\left(\frac{n\pi }{a}\right)}^{4}+{D}_{22}{\left(\frac{m\pi }{b}\right)}^{4}+\left(2{D}_{12}+4{D}_{66}\right){\left(\frac{n\pi }{a}\right)}^{2}{\left(\frac{m\pi }{b}\right)}^{2}+\left({K}_{P}+{F}_{x}\right){\left(\frac{n\pi }{a}\right)}^{2}$$
$$+\left({K}_{P}+{F}_{y}\right){\left(\frac{m\pi }{b}\right)}^{2}+{K}_{W},$$
$${\widehat{S}}_{12}=-\frac{n\pi }{a}\left[{E}_{11}^{1}{\left(\frac{n\pi }{a}\right)}^{2}+\left({E}_{21}^{1}+2{E}_{66}^{1}\right){\left(\frac{m\pi }{b}\right)}^{2}\right],$$
$${\widehat{S}}_{13}=-\frac{m\pi }{b}\left[{E}_{22}^{1}{\left(\frac{m\pi }{b}\right)}^{2}+\left({E}_{12}^{1}+2{E}_{66}^{1}\right){\left(\frac{n\pi }{a}\right)}^{2}\right],$$
$${\widehat{S}}_{22}={E}_{11}^{3}{\left(\frac{n\pi }{a}\right)}^{2}+{E}_{66}^{3}{\left(\frac{m\pi }{b}\right)}^{2}+{E}_{55}^{3},$$
$${\widehat{S}}_{23}={\widehat{S}}_{23}=\left({E}_{12}^{3}+{E}_{66}^{3}\right)\frac{n\pi }{a}\frac{m\pi }{b},$$
$${\widehat{S}}_{33}={E}_{66}^{2}{\left(\frac{n\pi }{a}\right)}^{2}+{E}_{22}^{3}{\left(\frac{m\pi }{b}\right)}^{2}+{E}_{44}^{3},$$
$${\widehat{M}}_{11}=-{\beta }_{31}{\left(\frac{n\pi }{a}\right)}^{2}-{\beta }_{32}{\left(\frac{m\pi }{b}\right)}^{2}, \;\;\; {\widehat{M}}_{21}={\gamma }_{31}\frac{n\pi }{a}, \;\;\; {\widehat{M}}_{31}={\gamma }_{32}\frac{m\pi }{b},$$
$${\widehat{M}}_{12}={\widehat{M}}_{13}={\widehat{M}}_{22}={\widehat{M}}_{23}={\widehat{M}}_{32}={\widehat{M}}_{33}=0,$$
$${\widehat{C}}_{11}={I}_{2}\left[{\left(\frac{n\pi }{a}\right)}^{2}+{\left(\frac{m\pi }{b}\right)}^{2}\right]+{I}_{0}, \;\;\; {\widehat{C}}_{12}=-{I}_{e}\frac{n\pi }{a}, \;\;\; {\widehat{C}}_{13}=-{I}_{e}\frac{m\pi }{b},$$
$${\widehat{C}}_{22}={I}_{e}^{2}, \;\;\; {\widehat{C}}_{23}=0, \;\;\; {\widehat{C}}_{33}={I}_{e}^{2}.$$

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Zenkour, A.M., El-Shahrany, H.D. Hygrothermal Vibration and Damping Behavior of Magnetostrictive Sandwich Plate Resting On Pasternak’s Foundations. Appl Compos Mater 29, 803–828 (2022). https://doi.org/10.1007/s10443-021-09970-3

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