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Free and forced vibration analysis of a sandwich beam considering porous core and SMA hybrid composite face layers on Vlasov’s foundation

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Abstract

This paper aims to investigate the free and forced vibration behavior of a sandwich beam with functionally graded porous core and composite face layers embedded with shape memory alloy (SMA). Three different porosity patterns are assumed through the thickness direction of the core, and composite face layers are reinforced with carbon nanotubes (CNT). The sandwich beam is resting on an elastic foundation which is simulated by Vlasov’s model. By using Hamilton’s principle and first-order shear deformation theory, the governing equations of motion are derived. The analytical solution is presented to solve the equations of motion using Navier’s solution. Results are verified with corresponding literatures. Finally, the effects of important parameters such as temperature, volume fraction of SMA, porosity distribution, weight fraction of CNT, and geometric parameters are explored in detail.

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Acknowledgements

The authors would like to thank the referees for their valuable comments. Also, they are thankful to the Iranian Nanotechnology Development Committee for their financial support and the University of Kashan for supporting this work by Grant No. 891238/7.

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran

    Kazem Alambeigi, Mehdi Mohammadimehr & Mostafa Bamdad

  2. Chair of Computational Mechanics, Bauhaus University Weimar, Marienstrasse 15, 99423, Weimar, Germany

    Timon Rabczuk

  3. Institute of Research and Development, Duy Tan University, Da Nang, Vietnam

    Timon Rabczuk

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  1. Kazem Alambeigi
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  2. Mehdi Mohammadimehr
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  3. Mostafa Bamdad
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  4. Timon Rabczuk
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Correspondence to Mehdi Mohammadimehr.

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Appendix

Appendix

1.1 Appendix A

$$\begin{aligned} \psi \left( z \right)= & {} \left\{ \begin{array}{ll} \cos \left( \frac{{\uppi z}}{{h}_{c}} \right) &{}\qquad { \rightarrow \text { Symmetry }} \\ \cos \left( \frac{{\uppi z}}{{2}{h}_{c}}+\frac{{\uppi }}{{4}} \right) &{}\qquad { \rightarrow \text { Asymmetry }} \\ \psi &{}\qquad \rightarrow \text { Uniform } \\ \end{array} \right. \nonumber \\ {\uplambda }\left( {z} \right)= & {} \left\{ {\begin{array}{ll} 1-{e}_{\mathrm {m}}\psi \left( z \right) &{} \qquad \quad \text {Symmetry, Asymmetry } \\ \sqrt{1-{e}_{{0}}\psi }&{}\qquad \quad \mathrm {Uniform } \\ \end{array}} \right. \nonumber \\ \psi= & {} \frac{{1}}{{e}_{{0}}}- \frac{{1}}{{e}_{{0}}}\left( \frac{{2}}{{\uppi }}\sqrt{1-{e}_{{0}}} -\frac{{2}}{{\uppi }}{+1} \right) ^{{2}} , \quad \quad {e}_{{m}}=1-\sqrt{1- e_{0}}. \end{aligned}$$

1.2 Appendix B

$$\begin{aligned} Q_{11}= & {} \frac{E_{m}}{1-\nu _{m}^{2}},\\ Q_{55}= & {} G_{m}=\frac{E_{m}}{2(1+\nu _{m})},\\ C_{11}= & {} \frac{E_{c}(z)}{1-{\nu _{c}(z)}^{2}},\\ C_{55}= & {} G_{c}=\frac{E_{c}}{2(1+\nu _{c})}. \end{aligned}$$

1.3 Appendix C

$$\begin{aligned} D_{11}= & {} \frac{E_{f}(\rho )}{1-{\nu _{f}(\rho )}^{2}},\\ D_{12}= & {} D_{21}=\frac{E_{f}(\rho )\nu _{f}(\rho )}{1-{\nu _{f}(\rho )}^{2}},\\ D_{55}= & {} G_{f}=\frac{E_{f}(\rho )}{2(1+\nu _{f}(\rho ))}. \end{aligned}$$

Subscript “f” denotes the foundation.

1.4 Appendix D

$$\begin{aligned}&{{{\delta }} }{{{u}}}_{\mathbf {0}}:\quad -\left( \frac{\partial ^{2}}{\partial t^{2}}u\left( x,t \right) \right) {uc}_{110}+\left( \frac{\partial ^{2}}{\partial t^{2}}\phi \left( x,t \right) \right) {uc}_{111}+2\left( \frac{\partial }{\partial x}u\left( x,t \right) \right) T_{0}\theta -2\left( \frac{\partial }{\partial x}u\left( x,t \right) \right) T\theta \\&\quad -\left( \frac{\partial ^{2}}{\partial x^{2}}\phi \left( x,t \right) \right) {uft}_{111}-\left( \frac{\partial ^{2}}{\partial x^{2}}u\left( x,t \right) \right) {ufb}_{110}-\left( \frac{\partial ^{2}}{\partial x^{2}}\phi \left( x,t \right) \right) {ufb}_{111}\\&\quad -\left( \frac{\partial ^{2}}{\partial x^{2}}u\left( x,t \right) \right) {uft}_{110}+2\left( \frac{\partial }{\partial x}u\left( x,t \right) \right) \xi _{s}\varepsilon _{L}\xi +\left( \frac{\partial }{\partial x}u\left( x,t \right) \right) T_{sur}\alpha _{f}{ufb}_{110} \\&\quad +\left( \frac{\partial }{\partial x}u\left( x,t \right) \right) T_{sur}\alpha _{f}{uft}_{110}+\left( \frac{\partial }{\partial x}u\left( x,t \right) \right) C\beta _{f}{ufb}_{110}+\left( \frac{\partial }{\partial x}u\left( x,t \right) \right) C\beta _{f}{uft}_{110} \\&\quad +\left( \frac{\partial }{\partial x}u\left( x,t \right) \right) C\beta {uc}_{110}+\left( \frac{\partial }{\partial x}u\left( x,t \right) \right) T_{sur}\alpha {uc}_{110}-\left( \frac{\partial ^{2}}{\partial t^{2}}u\left( x,t \right) \right) {Ic}_{0}-\left( \frac{\partial ^{2}}{\partial t^{2}}\phi \left( x,t \right) \right) {Ic}_{1} \\&\quad -\left( \frac{\partial ^{2}}{\partial t^{2}}u\left( x,t \right) \right) {Ifb}_{0}-\left( \frac{\partial ^{2}}{\partial t^{2}}\phi \left( x,t \right) \right) {Ifb}_{1}-\left( \frac{\partial ^{2}}{\partial t^{2}}u\left( x,t \right) \right) {Ift}_{0}-\left( \frac{\partial ^{2}}{\partial t^{2}}\phi \left( x,t \right) \right) {Ift}_{1}=0, \\&\varvec{\delta }\varvec{w}_{\mathbf {0}}:-\left( \frac{\partial }{\partial x}\phi \left( x,t \right) \right) {uc}_{550}-\left( \frac{\partial ^{2}}{\partial x^{2}}w\left( x,t \right) \right) {uc}_{550}-w\left( x,t \right) K_{1f}-\left( \frac{\partial ^{2}}{\partial x^{2}}w\left( x,t \right) \right) K_{2f} \\&\quad -\left( \frac{\partial }{\partial x}\phi \left( x,t \right) \right) {uft}_{550}-\left( \frac{\partial ^{2}}{\partial t^{2}}w\left( x,t \right) \right) {Ic}_{0}-\left( \frac{\partial ^{2}}{\partial t^{2}}w\left( x,t \right) \right) {Ift}_{0} \\&\quad -\left( \frac{\partial ^{2}}{\partial x^{2}}w\left( x,t \right) \right) {ufb}_{550}\left( \frac{\partial ^{2}}{\partial x^{2}}w\left( x,t \right) \right) {uft}_{550}-\left( \frac{\partial }{\partial x}\phi \left( x,t \right) \right) {ufb}_{550} =0, \end{aligned}$$
$$\begin{aligned}&{{{\delta }} }{{{\phi }} }_{\mathbf {0}}:\quad \left( \frac{\partial }{\partial x}\phi \left( x,t \right) \right) \xi _{s}\varepsilon _{L}\xi {usb}_{1}+\left( \frac{\partial }{\partial x}\phi \left( x,t \right) \right) \xi _{s}\varepsilon _{L}\xi {ust}_{1}+\left( \frac{\partial }{\partial x}w\left( x,t \right) \right) {uc}_{550} \\&\quad +\phi \left( x,t \right) {uc}_{550}+\phi \left( x,t \right) {ufb}_{550}+\phi \left( x,t \right) {uft}_{550}+\left( \frac{\partial }{\partial x}w\left( x,t \right) \right) {uft}_{550} \\&\quad +\left( \frac{\partial }{\partial x}w\left( x,t \right) \right) {ufb}_{550}-\left( \frac{\partial ^{2}}{\partial t^{2}}u\left( x,t \right) \right) {Ic}_{1}-\left( \frac{\partial ^{2}}{\partial t^{2}}\phi \left( x,t \right) \right) {Ic}_{2}-\left( \frac{\partial ^{2}}{\partial x^{2}}u\left( x,t \right) \right) {ufb}_{111} \\&\quad -\left( \frac{\partial ^{2}}{\partial x^{2}}\phi \left( x,t \right) \right) {ufb}_{112}-\left( \frac{\partial ^{2}}{\partial x^{2}}u\left( x,t \right) \right) {uc}_{111}-\left( \frac{\partial ^{2}}{\partial x^{2}}\phi \left( x,t \right) \right) {uc}_{112} \\&\quad -\left( \frac{\partial ^{2}}{\partial x^{2}}\phi \left( x,t \right) \right) {uft}_{112}-\left( \frac{\partial ^{2}}{\partial x^{2}}u\left( x,t \right) \right) {uft}_{111}-\left( \frac{\partial ^{2}}{\partial t^{2}}u\left( x,t \right) \right) {Ifb}_{1}-\left( \frac{\partial ^{2}}{\partial t^{2}}\phi \left( x,t \right) \right) {Ifb}_{2} \\&\quad -\left( \frac{\partial ^{2}}{\partial t^{2}}u\left( x,t \right) \right) {Ift}_{1}-\left( \frac{\partial ^{2}}{\partial t^{2}}\phi \left( x,t \right) \right) {Ift}_{2}+\left( \frac{\partial }{\partial x}\phi \left( x,t \right) \right) \beta _{f}C{ufb}_{111} \\&\quad +\left( \frac{\partial }{\partial x}\phi \left( x,t \right) \right) \beta _{f}C{uft}_{111}+\left( \frac{\partial }{\partial x}\phi \left( x,t \right) \right) T_{sur}\alpha _{f}{uft}_{111}-\left( \frac{\partial }{\partial x}\phi \left( x,t \right) \right) T\theta {usb}_{1}\\&\quad -\left( \frac{\partial }{\partial x}\phi \left( x,t \right) \right) T\theta {ust}_{1}+\left( \frac{\partial }{\partial x}\phi \left( x,t \right) \right) T_{0}\theta {usb}_{1}+\left( \frac{\partial }{\partial x}\phi \left( x,t \right) \right) T_{0}\theta {ust}_{1}\\&\quad +\left( \frac{\partial }{\partial x}\phi \left( x,t \right) \right) T_{sur}\alpha _{f}{ufb}_{111}+\left( \frac{\partial }{\partial x}\phi \left( x,t \right) \right) \beta C{uc}_{111}+\left( \frac{\partial }{\partial x}\phi \left( x,t \right) \right) T_{sur}\alpha {uc}_{111}=0, \end{aligned}$$
$$\begin{aligned}&{uc}_{110}=\int _{-h_{c}/2}^{h_{c}/2} {c_{11}\left( z \right) \mathrm{d}z},&{uc}_{111}=\int _{-h_{c}/2}^{h_{c}/2} {c_{11}\left( z \right) \times z\mathrm{d}z}, \\&{uc}_{112}=\int _{-h_{c}/2}^{h_{c}/2} {c_{11}\left( z \right) \times z^{2}\mathrm{d}z},&{uc}_{113}=\int _{-h_{c}/2}^{h_{c}/2} {c_{11}\left( z \right) \times z^{3}\mathrm{d}z}, \\&{uc}_{114}=\int _{-h_{c}/2}^{h_{c}/2} {c_{11}\left( z \right) \times z^{4}\mathrm{d}z},&{uc}_{116}=\int _{-h_{c}/2}^{h_{c}/2} {c_{11}\left( z \right) \times z^{6}\mathrm{d}z},\\&{uc}_{550}=\int _{-h_{c}/2}^{h_{c}/2} {c_{55}\left( z \right) \mathrm{d}z},&{uc}_{552}=\int _{-h_{c}/2}^{h_{c}/2} {c_{11}\left( z \right) \times z^{5}\mathrm{d}z},\\&{uc}_{554}=\int _{-h_{c}/2}^{h_{c}/2} {c_{55}\left( z \right) \times z^{4}\mathrm{d}z},&{ufb}_{110}=\int _{-\frac{h_{c}}{2}-h_{f}}^{{-h}_{c}/2} {c_{11}^{fb}\left( z \right) \mathrm{d}z}, \\&{ufb}_{111}=\int _{-\frac{h_{c}}{2}-h_{f}}^{{-h}_{c}/2} {c_{11}^{fb}\left( z \right) \times z\mathrm{d}z},&{ufb}_{112}=\int _{-\frac{h_{c}}{2}-h_{f}}^{{-h}_{c}/2} {c_{11}^{fb}\left( z \right) \times z^{2}\mathrm{d}z}, \\&{ufb}_{113}=\int _{-\frac{h_{c}}{2}-h_{f}}^{{-h}_{c}/2} {c_{11}^{fb}\left( z \right) \times z^{3}\mathrm{d}z},&{ufb}_{114}=\int _{-\frac{h_{c}}{2}-h_{f}}^{{-h}_{c}/2} {c_{11}^{fb}\left( z \right) \times z^{4}\mathrm{d}z}, \\&{ufb}_{116}=\int _{-\frac{h_{c}}{2}-h_{f}}^{{-h}_{c}/2} {c_{11}^{fb}\left( z \right) \times z^{6}\mathrm{d}z},&{ufb}_{550}=\int _{-\frac{h_{c}}{2}-h_{f}}^{{-h}_{c}/2} {c_{55}^{fb}\left( z \right) \mathrm{d}z}, \\&{ufb}_{552}=\int _{-\frac{h_{c}}{2}-h_{f}}^{{-h}_{c}/2} {c_{55}^{fb}\left( z \right) \times z^{2}\mathrm{d}z},&{ufb}_{554}=\int _{-\frac{h_{c}}{2}-h_{f}}^{{-h}_{c}/2} {c_{55}^{fb}\left( z \right) \times z^{4}\mathrm{d}z}, \\&{uft}_{110}=\int _\frac{h_{c}}{2}^{\frac{h_{c}}{2}+h_{f}} {c_{11}^{ft}\left( z \right) \mathrm{d}z},&{uft}_{111}=\int _\frac{h_{c}}{2}^{\frac{h_{c}}{2}+h_{f}} {c_{11}^{ft}\left( z \right) \times z\mathrm{d}z}, \\&{utf}_{112}=\int _\frac{h_{c}}{2}^{\frac{h_{c}}{2}+h_{f}} {c_{11}^{ft}\left( z \right) \times z^{2}\mathrm{d}z},&{uft}_{113}=\int _\frac{h_{c}}{2}^{\frac{h_{c}}{2}+h_{f}} {c_{11}^{ft}\left( z \right) \times z^{3}\mathrm{d}z}, \\&{uft}_{114}=\int _\frac{h_{c}}{2}^{\frac{h_{c}}{2}+h_{f}} {c_{11}^{ft}\left( z \right) \times z^{4}\mathrm{d}z},&{uft}_{116}=\int _\frac{h_{c}}{2}^{\frac{h_{c}}{2}+h_{f}} {c_{11}^{ft}\left( z \right) \times z^{6}\mathrm{d}z}, \\&{uft}_{550}=\int _\frac{h_{c}}{2}^{\frac{h_{c}}{2}+h_{f}} {c_{55}^{ft}\left( z \right) \mathrm{d}z},&{uft}_{552}=\int _\frac{h_{c}}{2}^{\frac{h_{c}}{2}+h_{f}} {c_{55}^{ft}\left( z \right) \times z^{2}\mathrm{d}z}, \\&{uft}_{554}=\int _\frac{h_{c}}{2}^{\frac{h_{c}}{2}+h_{f}} {c_{55}^{ft}\left( z \right) \times z^{4}\mathrm{d}z},&{Ic}_{0}=\int _{-\frac{h_{c}}{2}-h_{f}-h_{p}}^{{-h}_{c}/2-h_{f}} {\rho _{c}(z)\mathrm{d}z}, \\&{Ic}_{1}=\int _{-\frac{h_{c}}{2}-h_{f}-h_{p}}^{{-h}_{c}/2-h_{f}} {\rho _{c}(z)\times {z}\mathrm{d}z},&{Ic}_{2}=\int _{-\frac{h_{c}}{2}-h_{f}-h_{p}}^{{-h}_{c}/2-h_{f}} {\rho _{c}(z)\times z^{2}\mathrm{d}z}, \\&{Ic}_{3}=\int _{-\frac{h_{c}}{2}-h_{f}-h_{p}}^{{-h}_{c}/2-h_{f}} {\rho _{c}(z)\times z^{3}\mathrm{d}z},&{Ic}_{4}=\int _{-\frac{h_{c}}{2}-h_{f}-h_{p}}^{{-h}_{c}/2-h_{f}} {\rho _{c}(z)\times z^{4}\mathrm{d}z}, \\&{Ic}_{6}=\int _{-\frac{h_{c}}{2}-h_{f}-h_{p}}^{{-h}_{c}/2-h_{f}} {\rho _{c}(z)\times z^{6}\mathrm{d}z},&{Ifb}_{0}=\int _{-\frac{h_{c}}{2}-h_{f}}^{{-h}_{c}/2} {\rho _{bf}(z)\mathrm{d}z}, \\&{Ifb}_{1}=\int _{-\frac{h_{c}}{2}-h_{f}}^{{-h}_{c}/2} {\rho _{bf}(z)\times z\mathrm{d}z},&{Ifb}_{2}=\int _{-\frac{h_{c}}{2}-h_{f}}^{{-h}_{c}/2} {\rho _{bf}(z)\times z^{2}\mathrm{d}z}, \\&{Ifb}_{3}=\int _{-\frac{h_{c}}{2}-h_{f}}^{{-h}_{c}/2} {\rho _{bf}(z)\times z^{3}\mathrm{d}z},&{Ifb}_{4}=\int _{-\frac{h_{c}}{2}-h_{f}}^{{-h}_{c}/2} {\rho _{bf}(z)\times z^{4}\mathrm{d}z},\\&{Ifb}_{6}=\int _{-\frac{h_{c}}{2}-h_{f}}^{{-h}_{c}/2} {\rho _{bf}(z)\times z^{6}\mathrm{d}z},&{Ift}_{0}=\int _\frac{h_{c}}{2}^{\frac{h_{c}}{2}+h_{f}} {\rho _{tf}(z)\mathrm{d}z}, \\&{Ift}_{1}=\int _\frac{h_{c}}{2}^{\frac{h_{c}}{2}+h_{f}} {\rho _{tf}(z)\times z\mathrm{d}z},&{Ift}_{2}=\int _\frac{h_{c}}{2}^{\frac{h_{c}}{2}+h_{f}} {\rho _{tf}(z)\times z^{2}\mathrm{d}z}, \\&{Ift}_{3}=\int _\frac{h_{c}}{2}^{\frac{h_{c}}{2}+h_{f}} {\rho _{tf}(z)\times z^{3}\mathrm{d}z},&{Ift}_{4}=\int _\frac{h_{c}}{2}^{\frac{h_{c}}{2}+h_{f}} {\rho _{tf}(z)\times z^{4}\mathrm{d}z}, \\&{Ift}_{6}=\int _\frac{h_{c}}{2}^{\frac{h_{c}}{2}+h_{f}} {\rho _{tf}(z)\times z^{6}\mathrm{d}z}. \\ \end{aligned}$$

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Alambeigi, K., Mohammadimehr, M., Bamdad, M. et al. Free and forced vibration analysis of a sandwich beam considering porous core and SMA hybrid composite face layers on Vlasov’s foundation. Acta Mech 231, 3199–3218 (2020). https://doi.org/10.1007/s00707-020-02697-5

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