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Magnetostriction-assisted active control of the multi-layered nanoplates: effect of the porous functionally graded facesheets on the system’s behavior

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Abstract

The current study aims to analyze the vibrational behavior of a three-layered composite sandwich of magnetostrictive nanoplate integrated with functionally graded material facesheets. The core layer is supposed to be consisted of Terfenol-D and the top and the bottom plies are supposed to be from functionally graded material. To consider the small-scale effect, the Eringen’s nonlocal theory is utilized. On the other hand, kinematic relations of the nanoplate, i.e. rested on a Winkler–Pasternak medium, are expressed based on the first-order shear deformation theory. The governing equations are derived by employing the Hamilton’s principle and solved analytically by applying the Navier’s method for the simply supported boundary condition. The effects of various parameters such as nonlocal parameter, Winkler and Pasternak foundation, gradient index, feedback gain, and aspect ratio on the dynamic behaviors of the system are monitored in detail. To exhibit the accuracy and validity of the present study, our results are compared to those available in the literature. The results indicate when the thickness of the magnetostrictive layer increase, natural frequencies also increase. The results of the present study can be utilized in designing mechanical nanosensors, actuators, vibration cancellation, and smart nanovalves in injectors.

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Abbreviations

FGM:

Functionally graded material

FSDT:

First-order shear deformation theory

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Authors and Affiliations

  1. Department of Mechanical Engineering, Faculty of Engineering, Imam Khomeini International University, Qazvin, Iran

    Farzad Ebrahimi & Mehrdad Farajzadeh Ahari

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  1. Farzad Ebrahimi
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Appendix 1

Appendix 1

$$K_{11} = - A_{11} \left( {\alpha^{2} } \right) - A_{66} \left( {\beta^{2} } \right)\,\,K_{12} = - A_{12} \left( {\alpha \beta } \right) - A_{66} \left( {\alpha \beta } \right)$$
$$K_{13} = 0\,\,K_{14} = - B_{11} \left( {\alpha^{2} } \right) - B_{66} \left( {\beta^{2} } \right)$$
$$K_{15} = - B_{12} \left( {\alpha \beta } \right) - B_{66} \left( {\alpha \beta } \right)$$
$$K_{21} = - A_{21} \left( {\alpha \beta } \right) - A_{66} \left( {\alpha \beta } \right)\,\,K_{22} = - A_{22} \left( {\alpha^{2} } \right) - A_{66} \left( {\beta^{2} } \right)$$
$$K_{23} = 0\,\,K_{24} = - B_{21} \left( {\alpha \beta } \right) - B_{66} \left( {\alpha \beta } \right)$$
$$K_{25} = - B_{22} \left( {\alpha^{2} } \right) - B_{66} \left( {\beta^{2} } \right)$$
$$K_{31} = 0\,\,K_{32} = 0$$
$$K_{33} = - A_{55} \left( {\alpha^{2} } \right) - A_{44} \left( {\beta^{2} } \right) - K_{w} - K_{g} \left( {\alpha^{2} } \right) - K_{g} \left( {\beta^{2} } \right) - K_{w} \left( {\mu^{2} \alpha^{2} } \right) - K_{w} \left( {\mu^{2} \beta^{2} } \right) - K_{g} \left( {\mu^{2} \alpha^{4} } \right) - K_{g} \left( {\mu^{2} \beta^{4} } \right) - 2K_{g} \left( {\mu^{2} \alpha^{2} \beta^{2} } \right)$$
$$K_{34} = - A_{55} \left( \alpha \right)\,\,K_{35} = - A_{44} \left( \beta \right)$$
$$K_{41} = - B_{11} \left( {\alpha^{2} } \right) - B_{66} \left( {\beta^{2} } \right)\,\,K_{42} = - B_{21} \left( {\alpha \beta } \right) - B_{66} \left( {\alpha \beta } \right)$$
$$K_{43} = - A_{55} \left( \alpha \right)\,\,K_{44} = - D_{11} \left( {\alpha^{2} } \right) - D_{66} \left( {\beta^{2} } \right) - A_{55}$$
$$K_{45} = - D_{12} \left( {\alpha \beta } \right) - D_{66} \left( {\alpha \beta } \right)$$
$$K_{51} = - B_{12} \left( {\alpha \beta } \right) - B_{66} \left( {\alpha \beta } \right)\,\,K_{52} = - B_{22} \left( {\alpha^{2} } \right) - B_{66} \left( {\beta^{2} } \right)$$
$$K_{53} = - A_{44} \left( \beta \right)\,\,K_{54} = - D_{12} \left( {\alpha \beta } \right) - D_{66} \left( {\alpha \beta } \right)$$
$$K_{55} = - D_{22} \left( {\beta^{2} } \right) - D_{66} \left( {\alpha^{2} } \right) - A_{44}$$
$$M_{11} = I_{0} \left( {1 + \mu^{2} \alpha^{2} + \mu^{2} \beta^{2} } \right)\,\,M_{14} = I_{1} \left( {1 + \mu^{2} \alpha^{2} + \mu^{2} \beta^{2} } \right)$$
$$M_{22} = I_{0} \left( {1 + \mu^{2} \alpha^{2} + \mu^{2} \beta^{2} } \right)\,\,M_{25} = I_{1} \left( {1 + \mu^{2} \alpha^{2} + \mu^{2} \beta^{2} } \right)$$
$$M_{33} = I_{0} \left( {1 + \mu^{2} \alpha^{2} + \mu^{2} \beta^{2} } \right)\,\,M_{41} = I_{1} \left( {1 + \mu^{2} \alpha^{2} + \mu^{2} \beta^{2} } \right)$$
$$M_{44} = I_{2} \left( {1 + \mu^{2} \alpha^{2} + \mu^{2} \beta^{2} } \right)\,\,M_{52} = I_{1} \left( {1 + \mu^{2} \alpha^{2} + \mu^{2} \beta^{2} } \right)$$
$$M_{55} = I_{2} \left( {1 + \mu^{2} \alpha^{2} + \mu^{2} \beta^{2} } \right)$$
$$C_{11} = C_{12} = C_{14} = C_{15} = 0\,\,C_{13} = \mathop \int \limits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} e_{31} \left( \alpha \right)K_{c} C\left( t \right)dz$$
$$C_{21} = C_{22} = C_{24} = C_{25} = 0\,\,C_{23} = \mathop \int \limits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} e_{32} \left( \beta \right)K_{c} C\left( t \right)dz$$
$$C_{31} = \mathop \int \limits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} e_{31} \left( \alpha \right)K_{c} C\left( t \right)dz\,\,C_{32} = \mathop \int \limits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} e_{31} \left( \beta \right)K_{c} C\left( t \right)dz$$
$$C_{33} = 0\,\,C_{34} = \mathop \int \limits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} ze_{31} \left( \alpha \right)K_{c} C\left( t \right)dz$$
$$C_{35} = \mathop \int \limits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} ze_{32} \left( \beta \right)K_{c} C\left( t \right)dz$$
$$C_{41} = C_{42} = C_{44} = C_{45} = 0\,\,C_{43} = \mathop \int \limits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} ze_{31} \left( \alpha \right)K_{c} C\left( t \right)dz$$
$$C_{51} = C_{52} = C_{54} = C_{55} = 0\,\,C_{53} = \mathop \int \limits_{{ - \frac{{h_{c} }}{2}}}^{{\frac{{h_{c} }}{2}}} ze_{32} \left( \beta \right)K_{c} C\left( t \right)dz$$
$$A_{11} = \mathop \int \limits_{{ - h_{c} /2}}^{{h_{c} /2}} C_{11}^{c} dz + \mathop \int \limits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} C_{11}^{f} dz + \mathop \int \limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} C_{11}^{f} dz$$
$$A_{12} = \mathop \int \limits_{{ - h_{c} /2}}^{{h_{c} /2}} C_{12}^{c} dz + \mathop \int \limits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} C_{12}^{f} dz + \mathop \int \limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} C_{12}^{f} dz$$
$$A_{22} = \mathop \int \limits_{{ - h_{c} /2}}^{{h_{c} /2}} C_{22}^{c} dz + \mathop \int \limits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} C_{22}^{f} dz + \mathop \int \limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} C_{22}^{f} dz$$
$$A_{44} = \mathop \int \limits_{{ - h_{c} /2}}^{{h_{c} /2}} C_{44}^{c} dz + \mathop \int \limits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} C_{44}^{f} dz + \mathop \int \limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} C_{44}^{f} dz$$
$$A_{55} = \mathop \int \limits_{{ - h_{c} /2}}^{{h_{c} /2}} C_{55}^{c} dz + \mathop \int \limits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} C_{55}^{f} dz + \mathop \int \limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} C_{55}^{f} dz$$
$$A_{66} = \mathop \int \limits_{{ - h_{c} /2}}^{{h_{c} /2}} C_{66}^{c} dz + \mathop \int \limits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} C_{66}^{f} dz + \mathop \int \limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} C_{66}^{f} dz$$
$$B_{11} = \mathop \int \limits_{{ - h_{c} /2}}^{{h_{c} /2}} zC_{11}^{c} dz + \mathop \int \limits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} zC_{11}^{f} dz + \mathop \int \limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} zC_{11}^{f} dz$$
$$B_{12} = \mathop \int \limits_{{ - h_{c} /2}}^{{h_{c} /2}} zC_{12}^{c} dz + \mathop \int \limits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} zC_{12}^{f} dz + \mathop \int \limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} zC_{12}^{f} dz$$
$$B_{22} = \mathop \int \limits_{{ - h_{c} /2}}^{{h_{c} /2}} zC_{22}^{c} dz + \mathop \int \limits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} zC_{22}^{f} dz + \mathop \int \limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} zC_{22}^{f} dz$$
$$B_{44} = \mathop \int \limits_{{ - h_{c} /2}}^{{h_{c} /2}} zC_{44}^{c} dz + \mathop \int \limits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} zC_{44}^{f} dz + \mathop \int \limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} zC_{44}^{f} dz$$
$$B_{55} = \mathop \int \limits_{{ - h_{c} /2}}^{{h_{c} /2}} zC_{55}^{c} dz + \mathop \int \limits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} zC_{22}^{f} dz + \mathop \int \limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} zC_{55}^{f} dz$$
$$B_{66} = \mathop \int \limits_{{ - h_{c} /2}}^{{h_{c} /2}} zC_{66}^{c} dz + \mathop \int \limits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} zC_{66}^{f} dz + \mathop \int \limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} zC_{66}^{f} dz$$
$$D_{11} = \mathop \int \limits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{2} C_{11}^{c} dz + \mathop \int \limits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} z^{2} C_{11}^{f} dz + \mathop \int \limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{2} C_{11}^{f} dz$$
$$D_{12} = \mathop \int \limits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{2} C_{12}^{c} dz + \mathop \int \limits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} z^{2} C_{12}^{f} dz + \mathop \int \limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{2} C_{12}^{f} dz$$
$$D_{22} = \mathop \int \limits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{2} C_{22}^{c} dz + \mathop \int \limits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} z^{2} C_{22}^{f} dz + \mathop \int \limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{2} C_{22}^{f} dz$$
$$D_{44} = \mathop \int \limits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{2} C_{44}^{c} dz + \mathop \int \limits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} z^{2} C_{44}^{f} dz + \mathop \int \limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{2} C_{44}^{f} dz$$
$$D_{55} = \mathop \int \limits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{2} C_{55}^{c} dz + \mathop \int \limits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} z^{2} C_{55}^{f} dz + \mathop \int \limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{2} C_{55}^{f} dz$$
$$D_{66} = \mathop \int \limits_{{ - h_{c} /2}}^{{h_{c} /2}} z^{2} C_{66}^{c} dz + \mathop \int \limits_{{ - \frac{{\left( {hc + h_{f} } \right)}}{2}}}^{{\frac{ - hc}{2}}} z^{2} C_{66}^{f} dz + \mathop \int \limits_{{\frac{{h_{c} }}{2}}}^{{\frac{{\left( {h_{c} + h_{f} } \right)}}{2}}} z^{2} C_{66}^{f} dz$$

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Ebrahimi, F., Ahari, M.F. Magnetostriction-assisted active control of the multi-layered nanoplates: effect of the porous functionally graded facesheets on the system’s behavior. Engineering with Computers 39, 269–283 (2023). https://doi.org/10.1007/s00366-021-01539-9

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  • DOI: https://doi.org/10.1007/s00366-021-01539-9

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