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Meta-material beams with magnetostrictive coatings: vibrational characteristics

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Abstract

The main objective of this work is to examine the vibrational characteristics of a nanobeam exhibiting auxetic activity, achieved by the incorporation of a magnetostrictive material called Terfenol-D. The nanobeam is postulated to include three distinct layers, whereby the central layer is comprised of a magnetostrictive substance, while the outside layers are formed of auxetic material. On the other hand, the use of higher-order parabolic shear deformation beam theory is utilized to obtain the kinematic relations. Moreover, Eringen’s nonlocal theory is used to include the impact of small-scale phenomena. The governing equations are derived by the application of Hamilton’s principle and then solved using analytical techniques. This work presents a thorough analysis and elucidation of the influence of several parameters, such as auxetic inclination angle, auxetic rib length, and feedback gain, on the investigated system. Based on current research, evidence suggests that adding auxetic facesheets to magnetostrictive beam results in decreasing dimensionless natural frequency. In order to establish the accuracy and dependability of the present study, a comparison examination has been undertaken to juxtapose our results with the existing body of scholarly literature. The results obtained from the present study have the potential to provide a valuable contribution to the advancement and improved understanding of nano-systems, namely nano-sensors and nano-actuators. Furthermore, the conclusions acquired from this study might potentially serve as a fundamental framework for future research.

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Acknowledgements

The authors would like to thank the reviewers for their comments and suggestions to improve this article’s clarity.

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  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

$$ K_{11} = - A_{11} \left( {o^{2} } \right) $$
$$ K_{12} = - B_{11} \left( {o^{2} } \right) + \alpha S_{11} \left( {o^{2} } \right) $$
$$ K_{13} = \alpha S_{11} \left( {o^{3} } \right) $$
$$ K_{21} = - B_{11} \left( {o^{2} } \right) + \alpha S_{11} \left( {o^{2} } \right) $$
$$ K_{22} = - D_{11} \left( {o^{2} } \right) + 2\alpha M_{11} \left( {o^{2} } \right) - \alpha^{2} L_{11} \left( {o^{2} } \right) - A_{55} + 2\beta D_{55} - \beta^{2} M_{55} $$
$$ K_{23} = \alpha M_{11} \left( {o^{3} } \right) - \alpha^{2} L_{11} \left( {o^{3} } \right) + 2\beta D_{55} \left( o \right) - \beta^{2} M_{55} \left( o \right) - A_{55} \left( o \right) $$
$$ K_{31} = \alpha S_{11} \left( {o^{3} } \right) $$
$$ K_{32} = \alpha M_{11} \left( {o^{3} } \right) - \alpha^{2} L_{11} \left( {o^{3} } \right) + 2\beta D_{55} \left( o \right) - \beta^{2} M_{55} \left( o \right) - A_{55} \left( o \right) $$
$$ K_{33} = - \alpha^{2} L_{11} \left( {o^{4} } \right) - \beta^{2} M_{55} \left( {o^{2} } \right) + 2\beta D_{55} \left( {o^{2} } \right) - A_{55} \left( {o^{2} } \right) - K_{w} - K_{g} \left( {o^{2} } \right) - K_{w} \left( {\mu^{2} o^{2} } \right) - K_{g} \left( {\mu^{2} o^{4} } \right) $$
$$ M_{11} = I_{0} \left( {1 + \mu^{2} o^{2} } \right) $$
$$ M_{12} = - \left( {I_{1} - I_{3} \alpha^{ } + \mu^{2} I_{1} o^{2} + I_{3} \alpha^{ } \mu^{2} o^{2} } \right) $$
$$ M_{13} = - \left( { - I_{3} \alpha \left( o \right) - \mu^{2} I_{3} \alpha \left( {o^{3} } \right)} \right) $$
$$ M_{21} = - \left( {I_{1} - I_{3} \alpha^{ } + \mu^{2} I_{1} o^{2} + I_{3} \alpha^{ } \mu^{2} o^{2} } \right) $$
$$ M_{22} = - \left( {I_{2} - 2I_{4} \alpha + I_{6} \alpha^{2} + I_{2} \mu^{2} o^{2} - 2I_{4} \alpha o^{2} + I_{6} \mu^{2} \alpha^{2} o^{2} } \right) $$
$$ M_{23} = - \left( {I_{6} \alpha^{2} o - I_{4} \alpha \left( o \right) + I_{6} \alpha^{2} \mu^{2} o^{3} - I_{4} \alpha \mu^{2} o^{3} } \right) $$
$$ M_{31} = - \left( { - I_{3} \alpha \left( o \right) - \mu^{2} I_{3} \alpha \left( {o^{3} } \right)} \right) $$
$$ M_{32} = - \left( {I_{6} \alpha^{2} o - I_{4} \alpha \left( o \right) + I_{6} \alpha^{2} \mu^{2} o^{3} - I_{4} \alpha \mu^{2} o^{3} } \right) $$
$$ M_{33} = - \left( {I_{0} + I_{6} o^{2} \alpha^{2} + I_{0} \mu^{2} o^{2} + I_{6} \mu^{2} o^{4} \alpha^{2} } \right) $$
$$ A_{11} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} C_{11}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} C_{11}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} C_{11}^{f} {\text{d}}z $$
$$ A_{12} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} C_{12}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} C_{12}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} C_{12}^{f} {\text{d}}z $$
$$ A_{22} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} C_{22}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} C_{22}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} C_{22}^{f} {\text{d}}z $$
$$ A_{44} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} C_{44}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} C_{44}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} C_{44}^{f} {\text{d}}z $$
$$ A_{55} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} C_{55}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} C_{55}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} C_{55}^{f} {\text{d}}z $$
$$ A_{66} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} C_{66}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} C_{66}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} C_{66}^{f} {\text{d}}z $$
$$ B_{11} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} zC_{11}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} zC_{11}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} zC_{11}^{f} {\text{d}}z $$
$$ B_{12} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} zC_{12}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} zC_{12}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} zC_{12}^{f} {\text{d}}z $$
$$ B_{22} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} zC_{22}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} zC_{22}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} zC_{22}^{f} {\text{d}}z $$
$$ B_{44} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} zC_{44}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} zC_{44}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} zC_{44}^{f} {\text{d}}z $$
$$ B_{55} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} zC_{55}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} zC_{22}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} zC_{55}^{f} {\text{d}}z $$
$$ B_{66} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} zC_{66}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} zC_{66}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} zC_{66}^{f} {\text{d}}z $$
$$ D_{11} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} z^{2} C_{11}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} z^{2} C_{11}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} z^{2} C_{11}^{f} {\text{d}}z $$
$$ D_{12} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} z^{2} C_{12}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} z^{2} C_{12}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} z^{2} C_{12}^{f} {\text{d}}z $$
$$ D_{22} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} z^{2} C_{22}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} z^{2} C_{22}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} z^{2} C_{22}^{f} {\text{d}}z $$
$$ D_{44} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} z^{2} C_{44}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} z^{2} C_{44}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} z^{2} C_{44}^{f} {\text{d}}z $$
$$ D_{55} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} z^{2} C_{55}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} z^{2} C_{55}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} z^{2} C_{55}^{f} {\text{d}}z $$
$$ D_{66} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} z^{2} C_{66}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} z^{2} C_{66}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} z^{2} C_{66}^{f} {\text{d}}z $$
$$ S_{11} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} z^{3} C_{11}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} z^{3} C_{11}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} z^{3} C_{11}^{f} {\text{d}}z $$
$$ S_{12} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} z^{3} C_{12}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} z^{3} C_{12}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} z^{3} C_{12}^{f} {\text{d}}z $$
$$ S_{22} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} z^{3} C_{22}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} z^{3} C_{22}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} z^{3} C_{22}^{f} {\text{d}}z $$
$$ S_{44} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} z^{3} C_{44}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} z^{3} C_{44}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} z^{3} C_{44}^{f} {\text{d}}z $$
$$ S_{55} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} z^{3} C_{55}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} z^{3} C_{55}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} z^{3} C_{55}^{f} {\text{d}}z $$
$$ S_{66} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} z^{3} C_{66}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} z^{3} C_{66}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} z^{3} C_{66}^{f} {\text{d}}z $$
$$ M_{11} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} z^{4} C_{11}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} z^{4} C_{11}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} z^{4} C_{11}^{f} {\text{d}}z $$
$$ M_{12} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} z^{4} C_{12}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} z^{4} C_{12}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} z^{4} C_{12}^{f} {\text{d}}z $$
$$ M_{22} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} z^{4} C_{22}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} z^{4} C_{22}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} z^{4} C_{22}^{f} {\text{d}}z $$
$$ M_{44} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} z^{4} C_{44}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} z^{4} C_{44}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} z^{4} C_{44}^{f} {\text{d}}z $$
$$ M_{55} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} z^{4} C_{55}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} z^{4} C_{22}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} z^{4} C_{55}^{f} {\text{d}}z $$
$$ M_{66} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} z^{4} C_{66}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} z^{4} C_{66}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} z^{4} C_{66}^{f} {\text{d}}z $$
$$ L_{11} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} z^{6} C_{11}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} z^{6} C_{11}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} z^{6} C_{11}^{f} {\text{d}}z $$
$$ L_{12} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} z^{6} C_{12}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} z^{6} C_{12}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} z^{6} C_{12}^{f} {\text{d}}z $$
$$ L_{22} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} z^{6} C_{22}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} z^{6} C_{22}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} z^{6} C_{22}^{f} {\text{d}}z $$
$$ L_{44} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} z^{6} C_{44}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} z^{6} C_{44}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} z^{6} C_{44}^{f} {\text{d}}z $$
$$ L_{55} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} z^{6} C_{55}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} z^{6} C_{55}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} z^{6} C_{55}^{f} {\text{d}}z $$
$$ L_{66} = \mathop \int \limits_{{\frac{{ - h_{{\text{c}}} }}{2}}}^{{\frac{{h_{{\text{c}}} }}{2}}} z^{6} C_{66}^{c} {\text{d}}z + \mathop \int \limits_{{ - \frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}}^{{\frac{{ - h_{{\text{c}}} }}{2}}} z^{6} C_{66}^{f} {\text{d}}z + \mathop \int \limits_{{\frac{{h_{{\text{c}}} }}{2}}}^{{\frac{{\left( {h_{{\text{c}}} + h_{f} } \right)}}{2}}} z^{6} C_{66}^{f} {\text{d}}z $$

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Ebrahimi, F., Ahari, M.F. Meta-material beams with magnetostrictive coatings: vibrational characteristics. Acta Mech 235, 2495–2512 (2024). https://doi.org/10.1007/s00707-023-03845-3

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