Hygrothermal forced vibration of a viscoelastic laminated plate with magnetostrictive actuators resting on viscoelastic foundations

Abstract

The article presents a vibration analysis of a laminated composite sandwich plate containing viscoelastic layers in the core and faces of the plate. The sandwich plate includes magnetostrictive actuating layers, rests on a three-parameter viscoelastic medium, and undergoes to hygrothermal environmental conditions and in-plane forces in $x$ and $y$ directions. The viscoelastic layers are considered as Kelvin–Voigt viscoelastic model. The partial differential equations system is deduced according to the dynamic version of the principle of virtual displacements and is based on a theory that accounts for the exponential distribution of transverse shear deformation. The analytical Navier’s solution type is obtained to investigate the parametric effect of the location of smart layers, lamination schemes, modes, the feedback gain control value, viscoelastic structural damping, thickness ratio, the aspect ratio, viscoelastic layer thickness-to-magnetostrictive layer thickness ratio, foundation, in-plane forces, temperature and moisture on vibration characteristics of the sandwich plate. The outcomes show that the deformation and vibration in the structure can be controlled by changing the smart layer position and value of the feedback gain control parameter as well as the studied system stability can be affected by the value of viscoelastic structural damping and foundation constants. Besides, the absorption of moisture in matrix material increases the deflection of structure and increases the damping time. The results of this study can contribute to the development of the design of smart structural applications, especially those that require foundations to support them and deal with their dynamic response as well as provide a reference framework for studies of the impacts of humid environments on such structures in the presence of in-plane forces, especially, in the applications subjected to various forces and environmental conditions. Many of these systems require foundations to support the structures and deal with their dynamic response such as civil and mechanical structural applications and the aerospace and aeronautical vehicles.

Appendices

Appendix 1

The coefficients $\overline{Q}_{ij}^{\left( k \right)}$, $\overline{q}_{ij}$, $\tilde{\alpha }_{l}$ and $\tilde{\beta }_{l} ,l = xx,yy, xy$ appeared in Eqs. (9), (10), (11) and Eq. (15) are expanded as

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

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

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

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

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

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

$$Q_{11}^{\left( k \right)} = \frac{{E_{1} \left( {1 - \nu_{23}^{\left( k \right)} \nu_{32}^{\left( k \right)} } \right)}}{\Delta }, Q_{12}^{\left( k \right)} = \frac{{E_{1} \left( {\nu_{21}^{\left( k \right)} + \nu_{31}^{\left( k \right)} \nu_{23}^{\left( k \right)} } \right)}}{\Delta },$$

$$Q_{22}^{\left( k \right)} = \frac{{E_{2} \left( {1 - \nu_{13}^{\left( k \right)} \nu_{31}^{\left( k \right)} } \right)}}{\Delta },$$

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

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

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

$$\begin{aligned} &\tilde{\alpha }_{xx} = \alpha_{xx} \cos^{2} \theta + \alpha_{yy} \sin^{2} \theta , & \tilde{\alpha }_{yy} = \alpha_{yy} \cos^{2} \theta + \alpha_{xx} \sin^{2} \theta , \end{aligned}$$

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

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

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

$$\begin{aligned} \overline{q}_{31} = q_{31} \cos^{2} \theta + q_{32} \sin^{2} \theta , & \overline{q}_{32} = q_{32} \cos^{2} \theta + q_{31} \sin^{2} \theta , \end{aligned}$$

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

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

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

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

Appendix 2

The coefficients $\hat{S}_{ij}$, $\hat{M}_{ij}$ and $\hat{C}_{ij}$ ($i = 1, 2, 3$) appeared in Eq. (37) are expanded as the following

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

$$\hat{S}_{12} = - \left( {1 + \left. {g\frac{\partial }{\partial t}} \right|_{{k = {\text{viscoelastic}}}} } \right)\left[ {E_{11}^{1} \left( {\frac{n\pi }{a}} \right)^{3} + \left( {E_{21}^{1} + 2E_{66}^{1} } \right)\frac{n\pi }{a}\left( {\frac{m\pi }{b}} \right)^{2} } \right],$$

$$\hat{S}_{13} = - \left( {1 + \left. {g\frac{\partial }{\partial t}} \right|_{{k = {\text{viscoelastic}}}} } \right)\left[ {E_{22}^{1} \left( {\frac{m\pi }{b}} \right)^{3} + \left( {E_{12}^{1} + 2E_{66}^{1} } \right)\left( {\frac{n\pi }{a}} \right)^{2} \frac{m\pi }{b}} \right],$$

$$\hat{S}_{22} = \left( {1 + \left. {g\frac{\partial }{\partial t}} \right|_{{k = {\text{viscoelastic}}}} } \right)\left[ {E_{11}^{3} \left( {\frac{n\pi }{a}} \right)^{2} + E_{66}^{3} \left( {\frac{m\pi }{b}} \right)^{2} + E_{55}^{3} } \right],$$

$$\hat{S}_{23} = \hat{S}_{23} = \left( {1 + \left. {g\frac{\partial }{\partial t}} \right|_{{k = {\text{viscoelastic}}}} } \right)\left[ {\left( {E_{12}^{3} + E_{66}^{3} } \right)\frac{n\pi }{a}\frac{m\pi }{b}} \right],$$

$$\hat{S}_{33} = \left( {1 + \left. {g\frac{\partial }{\partial t}} \right|_{{k = {\text{viscoelastic}}}} } \right)\left[ {E_{66}^{2} \left( {\frac{n\pi }{a}} \right)^{2} + E_{22}^{3} \left( {\frac{m\pi }{b}} \right)^{2} + E_{44}^{3} } \right],$$

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

$$\hat{M}_{12} = \hat{M}_{13} = \hat{M}_{22} = \hat{M}_{23} = \hat{M}_{32} = \hat{M}_{33} = 0,$$

$$\hat{C}_{11} = I_{2} \left[ {\left( {\frac{n\pi }{a}} \right)^{2} + \left( {\frac{m\pi }{b}} \right)^{2} } \right] + I_{0} , \quad \hat{C}_{12} = - I_{e} \frac{n\pi }{a},\quad \hat{C}_{13} = - I_{e} \frac{m\pi }{b},$$

$$\hat{C}_{22} = I_{e}^{2} ,\quad \hat{C}_{23} = 0,\quad \hat{C}_{33} = I_{e}^{2} .$$