Hygrothermal vibration of a cross-ply composite plate with magnetostrictive layers, viscoelastic faces, and a homogeneous core

Abstract

In this article, a quasi-3D trigonometric shear deformation plate theory is utilized to study the vibration response of an advanced cross-ply multilayered composite plate that contains a homogenous core and viscoelastic faces subjected to a hygrothermal loading and embedded in a viscoelastic foundation. Two actuating layers of magnetostrictive material are employed for controlling and enhancing the vibration damping via a constant velocity feedback gain distributed control. The layers of the viscoelastic material are modelled using the Kelvin–Voigt model. The dynamic system is obtained employing Hamilton’s principle and solved analytically based on Navier’s approach. The influences of important factors on eigenfrequency values and deflection of the proposed multilayered plate are investigated. Of these, the effect of the feedback control gain magnitude, aspect ratio, magnetostrictive layer location, thickness ratio, viscoelastic layer thickness-to-core thickness ratio, half-wave numbers, magnetostrictive layer thickness-to-core thickness ratio, orientations of the viscoelastic layer’s fiber, and viscoelastic foundation and material. Numerical results proved that the stiffness of viscoelastic foundations, location of the smart layers, feedback control gain value, and magnetostrictive thickness-to-core thickness ratio have strong roles in the improvement of vibration damping characteristics.

Appendices

Appendix 1

The coefficients $\overline{Q}_{ij}^{\left( r \right)}$, $\overline{q}_{ij}$, $\tilde{\alpha }_{l}$ and $\tilde{\beta }_{l}$, $l = xx,yy,zz,$ appeared in Eqs. (11)–(14) are expanded as

$$\overline{Q}_{11}^{\left( r \right)} = Q_{11}^{\left( r \right)} \cos^{4} \theta^{\left( r \right)} + 2\left( {Q_{12}^{\left( r \right)} + 2Q_{66}^{\left( r \right)} } \right)\cos^{2} \theta^{\left( r \right)} \sin^{2} \theta^{\left( r \right)} + Q_{22}^{\left( r \right)} \sin^{4} \theta^{\left( r \right)} ,$$

$$\overline{Q}_{12}^{\left( r \right)} = \left( {Q_{11}^{\left( r \right)} + Q_{22}^{\left( r \right)} - 4Q_{66}^{\left( r \right)} } \right)\cos^{2} \theta^{\left( r \right)} \sin^{2} \theta^{\left( r \right)} + Q_{12}^{\left( r \right)} \left( {\sin^{4} \theta^{\left( r \right)} + \cos^{4} \theta^{\left( r \right)} } \right),$$

$$\overline{Q}_{22}^{\left( r \right)} = Q_{11}^{\left( r \right)} \sin^{4} \theta^{\left( r \right)} + 2\left( {Q_{12}^{\left( r \right)} + 2Q_{66}^{\left( r \right)} } \right)\cos^{2} \theta^{\left( r \right)} \sin^{2} \theta^{\left( r \right)} + Q_{22}^{\left( r \right)} \cos^{4} \theta^{\left( r \right)} ,$$

$$\overline{Q}_{23}^{\left( r \right)} = Q_{23}^{\left( r \right)} \cos^{2} \theta^{\left( r \right)} + Q_{13}^{\left( r \right)} \sin^{2} \theta^{\left( r \right)} ,\quad { }\overline{Q}_{33}^{\left( r \right)} = Q_{33}^{\left( r \right)} ,$$

$$\overline{Q}_{44}^{\left( r \right)} = Q_{44}^{\left( r \right)} \cos^{2} \theta^{\left( r \right)} + Q_{55}^{\left( r \right)} \sin^{2} \theta^{\left( r \right)} ,$$

$$\overline{Q}_{55}^{\left( r \right)} = Q_{55}^{\left( r \right)} \cos^{2} \theta^{\left( r \right)} + Q_{44}^{\left( r \right)} \sin^{2} \theta^{\left( r \right)} ,$$

$$\overline{Q}_{66}^{\left( r \right)} = \left( {Q_{11}^{\left( r \right)} + Q_{22}^{\left( r \right)} - 2Q_{12}^{\left( r \right)} - 2Q_{66}^{\left( r \right)} } \right)\sin^{2} \theta^{\left( r \right)} \cos^{2} \theta^{\left( r \right)} + Q_{66}^{\left( r \right)} \left( {\sin^{4} \theta^{\left( r \right)} + \cos^{4} \theta^{\left( r \right)} } \right),$$

$$Q_{11}^{\left( r \right)} = \frac{{E_{1} \left( {1 - {\upnu }_{23}^{\left( r \right)} {\upnu }_{32}^{\left( r \right)} } \right)}}{\Delta },\quad { }Q_{12}^{\left( r \right)} = \frac{{E_{1} \left( {{\upnu }_{21}^{\left( r \right)} + {\upnu }_{31}^{\left( r \right)} {\upnu }_{23}^{\left( r \right)} } \right)}}{\Delta },\quad { }Q_{13}^{\left( r \right)} = \frac{{E_{1} \left( {{\upnu }_{31}^{\left( r \right)} + {\upnu }_{21}^{\left( r \right)} {\upnu }_{32}^{\left( r \right)} } \right)}}{\Delta },$$

$$Q_{22}^{\left( r \right)} = \frac{{E_{2} \left( {1 - {\upnu }_{13}^{\left( r \right)} {\upnu }_{31}^{\left( r \right)} } \right)}}{\Delta },\quad { }Q_{23}^{\left( r \right)} = \frac{{E_{2} \left( {{\upnu }_{32}^{\left( r \right)} + {\upnu }_{12}^{\left( r \right)} {\upnu }_{31}^{\left( r \right)} } \right)}}{\Delta },\quad { }Q_{33}^{\left( r \right)} = \frac{{E_{3} \left( {1 - {\upnu }_{12}^{\left( r \right)} {\upnu }_{21}^{\left( r \right)} } \right)}}{\Delta }{ },$$

$$Q_{44}^{\left( r \right)} = G_{23}^{\left( r \right)} ,\quad { }Q_{55}^{\left( r \right)} = G_{13}^{\left( r \right)} ,\quad { }Q_{66}^{\left( r \right)} = G_{12}^{\left( r \right)} ,$$

$$\Delta = 1 - \nu_{21}^{\left( r \right)} \nu_{12}^{\left( r \right)} - \nu_{23}^{\left( r \right)} \nu_{32}^{\left( r \right)} - \nu_{13}^{\left( r \right)} \nu_{31}^{\left( r \right)} - 2\nu_{21}^{\left( r \right)} \nu_{13}^{\left( r \right)} \nu_{32}^{\left( r \right)} ,$$

$$\nu_{21}^{\left( r \right)} = \frac{{\nu_{12}^{\left( r \right)} E_{22}^{\left( r \right)} }}{{E_{1}^{\left( r \right)} }},\quad \nu_{31}^{\left( r \right)} = \frac{{\nu_{13}^{\left( r \right)} E_{3}^{\left( r \right)} }}{{E_{1}^{\left( r \right)} }},\quad \nu_{32}^{\left( r \right)} = \frac{{\nu_{23}^{\left( r \right)} E_{3}^{\left( r \right)} }}{{E_{2}^{\left( r \right)} }},$$

$$\tilde{\alpha }_{xx} = \alpha_{xx} \cos^{2} \theta + \alpha_{yy} \sin^{2} \theta ,\quad { }\tilde{\alpha }_{yy} = \alpha_{yy} \cos^{2} \theta + \alpha_{xx} \sin^{2} \theta ,\quad { }\tilde{\alpha }_{zz} = \alpha_{zz} ,$$

$$\tilde{\alpha }_{xy} = \left( {\alpha_{xx} - \alpha_{yy} } \right)\sin \theta \cos \theta ,$$

$$\tilde{\beta }_{xx} = \beta_{xx} \cos^{2} \theta + \beta_{yy} \sin^{2} \theta ,\quad { }\tilde{\beta }_{yy} = \beta_{yy} \cos^{2} \theta + \beta_{xx} \sin^{2} \theta ,\quad { }\tilde{\beta }_{zz} = \beta_{zz} ,$$

$$\tilde{\beta }_{xy} = \left( {\beta_{xx} - \beta_{yy} } \right)\sin \theta \cos \theta ,$$

$$\overline{q}_{31} = q_{31} \cos^{2} \theta + q_{32} \sin^{2} \theta ,\quad { }\overline{q}_{32} = q_{32} \cos^{2} \theta + q_{31} \sin^{2} \theta ,\quad { }\overline{q}_{33} = q_{33} ,$$

$$\overline{q}_{14} = \left( {q_{15} - q_{24} } \right)\sin \theta \cos \theta ,\quad { }\overline{q}_{24} = q_{24} \cos^{2} \theta + q_{15} \sin^{2} \theta ,$$

$$\overline{q}_{15} = q_{15} \cos^{2} \theta + q_{24} \sin^{2} \theta ,\quad { }\overline{q}_{25} = \left( {q_{15} - q_{24} } \right)\sin \theta \cos \theta ,$$

$$\overline{q}_{36} = \left( {q_{31} - q_{32} } \right)\sin \theta \cos \theta ,$$

where $v_{ij}$, ${ }E_{i}$ and $G_{ij}$ represent, respectively, Poisson’s ratios, Young’s moduli, and shear moduli. Further, the coefficients $\alpha_{ij}$, $\beta_{ij}$ and $q_{ij}$ are the thermal coefficients, hygroscopic expansion coefficients, and the magnetostrictive modules.

Appendix 2

The constants $\hat{S}_{ij}$, $\hat{M}_{ij}$ and $\hat{C}_{ij}$ ($i = 1, 2, 3, 4$) which appear in Eqs. (42) and (44) are expressed as:

$$\begin{aligned} \hat{S}_{11} & = \left( {1 + \left. {\text{g}\frac{\partial }{\partial t}} \right|_{{r = {\text{face}}}} } \right)\left[ {D_{11} \left( {\frac{n\pi }{a}} \right)^{4} + D_{22} \left( {\frac{m\pi }{b}} \right)^{4} + 2\left( {D_{12} + 2D_{66} } \right)\left( {\frac{n\pi }{a}} \right)^{2} \left( {\frac{m\pi }{b}} \right)^{2} } \right] \\ & \quad + \left( {F_{x} + K_{{\text{P}}} } \right)\left( {\frac{n\pi }{a}} \right)^{2} + \left( {F_{y} + K_{{\text{P}}} } \right)\left( {\frac{m\pi }{b}} \right)^{2} + K_{{\text{W}}} , \\ \end{aligned}$$

$$\hat{S}_{12} = c_{2} \left( {1 + \left. {\text{g}\frac{\partial }{\partial t}} \right|_{{r = {\text{face}}}} } \right)\left[ {T_{13}^{1} \left( {\frac{n\pi }{a}} \right)^{2} + T_{23}^{1} \left( {\frac{m\pi }{b}} \right)^{2} } \right],$$

$$\hat{S}_{13} = - \left( {1 + \left. {\text{g}\frac{\partial }{\partial t}} \right|_{{r = {\text{face}}}} } \right)\left[ {E_{11}^{1} \left( {\frac{n\pi }{a}} \right)^{3} + \left( {E_{21}^{1} + 2E_{66}^{1} } \right)\frac{n\pi }{a}\left( {\frac{m\pi }{b}} \right)^{2} } \right],$$

$$\hat{S}_{14} = - \left( {1 + \left. {\text{g}\frac{\partial }{\partial t}} \right|_{{r = {\text{face}}}} } \right)\left[ {\left( {E_{12}^{1} + 2E_{66}^{1} } \right)\left( {\frac{n\pi }{a}} \right)^{2} \frac{m\pi }{b} + E_{22}^{1} \left( {\frac{m\pi }{b}} \right)^{3} } \right],$$

$$\hat{S}_{22} = c_{2}\left( {1 + \left. {\text{g}\frac{\partial }{\partial t}} \right|_{{r = {\text{face}}}} } \right) \left[ {T_{33} + E_{44}^{3} \left( {\frac{m\pi }{b}} \right)^{2} + E_{55}^{3} \left( {\frac{n\pi }{a}} \right)^{2} } \right],$$

$$\hat{S}_{23} = -c_{2} \left( {1 + \left. {\text{g}\frac{\partial }{\partial t}} \right|_{{r = {\text{face}}}} } \right) \left( {E_{55}^{3} - T_{31}^{3} } \right)\frac{n\pi }{a},$$

$$\hat{S}_{24} = - c_{2}\left( {1 + \left. {\text{g}\frac{\partial }{\partial t}} \right|_{{r = {\text{face}}}} } \right) \left( {E_{44}^{3} - T_{32}^{3} } \right)\frac{m\pi }{b},$$

$$\hat{S}_{33} = \left( {1 + \left. {\text{g}\frac{\partial }{\partial t}} \right|_{{r = {\text{face}}}} } \right)\left[ {E_{11}^{3} \left( {\frac{n\pi }{a}} \right)^{2} + E_{66}^{3} \left( {\frac{m\pi }{b}} \right)^{2} + E_{55}^{3} } \right],$$

$$\hat{S}_{34} = \left( {1 + \left. {\text{g}\frac{\partial }{\partial t}} \right|_{{r = {\text{face}}}} } \right)\left( {E_{12}^{3} + E_{66}^{3} } \right)\frac{n\pi }{a}\frac{m\pi }{b},$$

$$\hat{S}_{44} = \left( {1 + \left. {\text{g}\frac{\partial }{\partial t}} \right|_{{r = {\text{face}}}} } \right)\left[ {E_{66}^{3} \left( {\frac{n\pi }{a}} \right)^{2} + E_{22}^{3} \left( {\frac{m\pi }{b}} \right)^{2} + E_{44}^{3} } \right],$$

$$\hat{M}_{11} = - \beta_{31} \left( {\frac{n\pi }{a}} \right)^{2} - \beta_{32} \left( {\frac{m\pi }{b}} \right)^{2} + c_{{\text{d}}} ,\quad \hat{M}_{12} = - c_{2} \left( {\tilde{\beta }_{31} + \tilde{\beta }_{32} } \right)\left( {\frac{n\pi }{a}} \right)^{2} ,\quad \hat{M}_{21} = - { }c_{2} \mu ,$$

$$\hat{M}_{22} = - { }c_{2} \tilde{\mu },\quad \hat{M}_{31} = \gamma_{31} \frac{n\pi }{a},\quad \hat{M}_{32} = c_{2} \tilde{\gamma }_{31} \frac{n\pi }{a},$$

$$\hat{M}_{41} = \gamma_{32} \frac{m\pi }{b},\quad \hat{M}_{42} = \tilde{\gamma }_{32} \frac{m\pi }{b},$$

$$\hat{M}_{13} = { }\hat{M}_{23} = { }\hat{M}_{33} = \hat{M}_{43} = \hat{M}_{14} = \hat{M}_{24} = { }\hat{M}_{34} = \hat{M}_{44} = 0,$$

$$\hat{C}_{11} = I_{2} \left[ {\left( {\frac{n\pi }{a}} \right)^{2} + \left( {\frac{m\pi }{b}} \right)^{2} } \right] + I_{0} ,\quad \hat{C}_{22} = c_{2}^{2} I_{gg} ,\quad \hat{C}_{33} = I_{e}^{2} ,\quad \hat{C}_{44} = I_{e}^{2} ,{ }$$

$$\hat{C}_{12} = c_{2} I_{g} ,\quad \hat{C}_{13} = - I_{e} \frac{n\pi }{a},\quad \hat{C}_{14} = - I_{e} \frac{m\pi }{b},\quad \hat{C}_{23} = \hat{C}_{24} = \hat{C}_{34} = 0.$$