A Computer Presentation of the Analytical and Numerical Study of Nonlinear Vibration Response for Porous Functionally Graded Cylindrical Panel

Abstract

The current study uses a new analytical model and numerical method to present a study of free vibration carried out on a cylindrical shell panel that is simply supported and functionally graded. It is anticipated that the FG thickness attributes will be dependent on the porosity level and will change along the thickness axis in accordance with a distribution that follows a power-law. This work makes a contribution by analyzing the performance of porous FGMs, which are employed in a particularly wide variety of biomedical applications. For the purpose of determining the free vibration characteristics as well as the nonlinear vibration response, the governing equations are constructed on a first-order shear deformation theory by utilizing the Galerkin technique with the fourth-order Runge Kutta a close encounter with an incomplete FGM cylindrical shell panel and include different parameters. Parameters included are the power-law index, graded distributions of porosity, and FG thickness. With the help of both the ANSYS 2021-R1 software, a numerical investigation was carried out making use of the finite element approach, and a modal investigation was carried out. This was done in order to verify the analytical strategy.

1 Introduction

In mechanical and construction engineering, investigating and developing new materials and structures is essential. Because of their lightweight nature and excellent mechanical properties, structures composed of advanced materials are often employed in numerous industries, including civil, mechanical, and aerospace engineering. The application in practice is normally in the shape as nanocomposite structures, Functionally Graded Materials (FGM) structures, laminated, and sandwich structures [1]. In recent years, analyzing a new structure has become a popular study area for numerous material and engineering scientists. Many articles have been published about composite materials. Ranging from micro-scale (nanocomposite and FGM) to macro-scale (laminated and sandwich structures), the studies have been conducted using a variety of techniques, scales, and forms [2, 3]. In the dynamics case, numerous investigations have been done on the vibration analysis of structures [4,5,6]. Bagheri et al. [7] studied the geometrically nonlinear dynamic response of joined conical shells constructed of functionally graded material (FGM) subjected to thermal shock. Zhu et al. [8] employed Reddy’s theory of Shear Deformation at Higher Orders with the geometric nonlinearity hypothesis (von Kármán) to investigate the relationship between nonlinear forced vibration properties and nonlinear free vibration properties of viscoelastic plates. Li and Liu [9] examined the thermal free vibration as well as buckling attitude of viscoelastic sandwich of FGM shallow shell having a core constructed of tunable auxetic honeycomb. Singh et al. [10] coupled piezoelectric sensors with time-dependent tri analytical solutions in order to investigate the free vibration of viscoelastic of orthotropic rectangular plates in-plane FG. Sahu et al. [11] presented the free vibration and damping investigation of the sandwich doubly-curved shallow shell with a viscoelastic-FGM layer as suggested by the theory of shear deformation of the first order. Moreover, there has been a significant increase in the number of investigations for shell structures constructed of FGM materials [26]. These published studies investigated the buckling properties as well as the linear and nonlinear vibration behavior in classical shell theory [12, 13]. Throughout the development of the material industry, FGM porous cores employed for lightweight structures are becoming a significant element in civil, mechanical, and aerospace engineering due to their electrical, mechanical, and thermal properties. In particular, the FG porous material has a high strength as well as an excellent energy absorption capability. Ghobadi et al. [14] investigated the influence of the various distributions of porosity on the static and dynamic behavior of sandwich FGM nanostructures under thermo-electro-elastic coupling. Esayas and Kattimani [15] studied the influence of porosity on the dynamic damping of geometrically nonlinear vibrations of an FG magneto-electro-elastic plate. Javid et al. [16] researched the free vibrational characteristics of porous FG micro-cylindrical shells with viscoelastic medium and two skins made of nanocomposite based on Biot’s assumptions. According to third-order shear deformation theory, Keleshteri and Jelovica [17] sandwich panels with FG metal cores of foam were studied to determine their buckling and free vibrational behavior. Kumar H S et al. [18] approaching together transient responses and FG nonlinear free vibration of skew plate under the influence of porosity distribution. Srikarun et al. [19] researched the linear and nonlinear stability of sandwich beams with porous FG cores subjected to various types of distributed loads. Chan et al. [20] Using theory of first-order shear deformation shell, researchers were investigated free vibration with a nonlinear response of the dynamic properties of a porous functionally graded trimmed shell have aconical shap equipped with actuators of piezoelectric in warm settings. Dastjerdi and Behdinan [21] studied the free vibration characteristics of smart sandwich plates with carbon nanotube-reinforced and piezoelectric layers by employing Reddy’s theory of third-order shear deformation. Yadav et al. [22] investigated statics of nonlinear of sandwich circular-cylindrical shells consisting of two carbon nanotube-reinforced face sheets and a porous FG core by utilizing higher-order shear deformation and thickness theory. Binh et al. [23] examined the nonlinear dynamic characteristics of a porous FG toroidal shell of variable thickness subjected to thermal loads and enclosed by a medium that has elasticity. Furthermore, numerous investigations displayed the advent of the porous core in strengthening structures [24, 25]. Based on 3D elasticity, the dynamic analyses and natural frequencies of porous FG cylindrical panels and annular sector plates were determined by Babaei et al. [26]. Li et al. [27] used the differential quadrature method to study the natural vibration properties of metal porous foam for conical shells consisting of two intriguing elastically restrained boundaries. Hung et al. [12] investigated the effect of the porous FG variable thickness for the toroidal shell on the nonlinear stability (buckling and post-buckling) under compressive loads and surrounded by the elastic foundation. Vinh and Huy [28] presented an inclusive analysis of the buckling, static bending and free vibration of the FG plates for sandwiches, including porosity distribution according to the finite element method with new hyperbolic shear deformation theory. Amir and Talha [29] employing finite element and high-order shear deformation technique, this research looked at the vibration of nonlinear thermo-elastic characteristics of FG porous double curve shallow shells. Keleshteri and Jelovica [30] employed high-order bidirectional porosity distributions to examine the free and forced vibration characteristics of FG porous beams. Nevertheless, there have only been a few studies on the FGM free vibration systems with porous metal formation. This investigation’s objective is to carry out a study on the nonlinear analysis of free vibration of a simply-supported two-phase FGM cylindrical shell panel with porous as part of its research. In the present work, suppose that the FGM is constructed from ceramic and metal, mechanical characteristics are varied with reason disparate porosity distributions based on power-law distributions, with changes in the thickness direction. A novel model for the first-order shear deformation theory is formed to discover the nonlinear free vibration characteristics according to different FGM parameters. Utilizing the FEA strategy that is exemplified by ANSYS software, the results of mode and natural frequency forms of the FGM cylindrical shell with porous are provided here. The numerical findings for FG porous materials that are offered here are not found anywhere else in the literature, and as a consequence, should be of relevance to industrial applications. This study is organized into four parts. In the first section, theory of first-order shear deformation criteria is introduced, in addition to constitutive equations, features of FG porous structures, and an analytical vibration analysis of the porous cylindrical panel. The second section introduces numerical analysis and finite element simulation. Results and discussion are included in the third section. The last part includes study summaries and findings.

2 Models of Porous FGM Cylindrical Shell Panel

Consider a thick FGM cylindrical shell panel made of metal and ceramic, in which the lower surface is ceramic-rich and the upper surface is metal-rich, respectively. The FGM cylindrical panel is considered to carry porosities that distribute evenly and unevenly through the shell thickness direction (Fig. 1). The shell’s thickness, Radius, and edges are represented by h, R, a, and b, respectively. To describe the shell’s motion, a cartesian coordinate system (x, y, z) on the center surface of the shell is employed, where z identifies the out-of-plane coordinate, and x and y determine the shell’s in-plane coordinates.

Fig. 1.

The geometry of the cylindrical panel found on the FGM.

In addition, the power law, the sigmoid law, or the exponential law may be used to adequately represent the volume fraction of the FG cylindrical shell layers. Equation 1 makes the assumption that the distribution of the ceramic volume fraction Vc is governed by a power law [31]:

$$ V_{m} + V_{c} = 1V_{c} = V_{c} \left( z \right) = \left( {\frac{2z + h}{{2h}}} \right)^{g} $$

(1)

where, Vc and Vm are volume fractions of ceramic and metal, respectively. g is the power-law index. When the value of g is equal to infinity, it indicates a fully metallic shell, whereas when the value of g is equal to zero, it denotes a fully ceramic shell. The fundamental mechanical properties of the FGM cylindrical shell panels, with a porosity volume ratio of G (G < 1), adopt the modified form of Eq. 2, assuming that porosities disperse equally in the ceramic and metal phases

$$ P\left( z \right) = P_{m} + \left( {P_{c} - P_{m} } \right)\left( {\frac{2z + h}{{2h}}} \right)^{g} - \frac{G}{2}\left( {P_{c} - P_{m} } \right) $$

(2)

Pm and Pc represented the values of the material properties of the metal and ceramic components of the FG shells, respectively. Young’s modulus (E) and mass density (ρ) are taken to change in the thickness direction for our current formulations, while Poisson’s ratio (v) will be assumed to remain constant for simplicity based on earlier research.

2.1 Fundamental Equations

This study takes into account thick porous FGM cylindrical shell panels subjected to external loading with varying boundary conditions. As a result, the system of governing equations is established, and the nonlinear vibration of the cylindrical shell is determined using first-order shear deformation plate theory [32]:

$$ \begin{aligned} & \tilde{u}\left( {x,y,z,t} \right) = u\left( {x,y,t} \right) + z\phi_{x} \left( {x,y,t} \right), \\ & \tilde{v}\left( {x,y,z,t} \right) = v\left( {x,y,t} \right) + z\phi_{y} \left( {x,y,t} \right), \\ & \tilde{w}\left( {x,y,z,t} \right) = w\left( {x,y,t} \right) \\ \end{aligned} $$

(3)

where, $\left({\phi }_{x},{\phi }_{y}\right)$ describes the transverse normal slopes about x- and y-axes at $\left(z = 0\right)$. The following equations serve as the basis for the strain-displacement relationships of the cylindrical shell:

$$ \left\{ {\begin{array}{*{20}c} {\varepsilon_{x} } \\ {\varepsilon_{y} } \\ {\gamma_{xy} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {\varepsilon_{x}^{ \circ } } \\ {\varepsilon_{y}^{ \circ } } \\ {\gamma_{xy}^{ \circ } } \\ \end{array} } \right\} + z\left\{ {\begin{array}{*{20}c} {\lambda_{x} } \\ {\lambda_{y} } \\ {\lambda_{xy} } \\ \end{array} } \right\},\left\{ {\begin{array}{*{20}c} {\gamma_{xz} } \\ {\gamma_{yz} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {\frac{\partial w}{{\partial x}} + \phi_{x} } \\ {\frac{\partial w}{{\partial y}} + \phi_{y} } \\ \end{array} } \right\} $$

(4)

$$ \left\{ {\begin{array}{*{20}c} {\varepsilon_{x}^{ \circ } } \\ {\varepsilon_{y}^{ \circ } } \\ {\gamma_{xy}^{ \circ } } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {\frac{\partial u}{{\partial x}} + \frac{1}{2}\left( {\frac{\partial w}{{\partial x}}} \right)^{2} } \\ {\frac{\partial v}{{\partial y}} + \frac{1}{2}\left( {\frac{\partial w}{{\partial x}}} \right)^{2} } \\ {\frac{\partial u}{{\partial y}} + \frac{\partial v}{{\partial x}} + \frac{\partial w}{{\partial x}}\frac{\partial w}{{\partial y}}} \\ \end{array} } \right\}\;\,{\text{and}}\,\;\left\{ {\begin{array}{*{20}c} {\lambda_{x} } \\ {\lambda_{y} } \\ {\lambda_{xy} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {\frac{{\partial \phi_{x} }}{\partial x}} \\ {\frac{{\partial \phi_{y} }}{\partial y}} \\ {\frac{{\partial \phi_{x} }}{\partial y} + \frac{{\partial \phi_{y} }}{\partial x}} \\ \end{array} } \right\} $$

In the planes $\left({\text{xz}},{\text{yz}}\right)$, the x designates the transverse shear strain components $\left({\upgamma }_{{\text{xz}}}, {\upgamma }_{{\text{yz}}}\right)$. The nonlinear stress-strain constitutive relations at a general point inside the skin of the cylindrical shell can be expressed as:

$$ \left\{ {\begin{array}{*{20}c} {\upsigma _{{\text{x}}}^{{{\text{sh}}}} } \\ {\upsigma _{{\text{y}}}^{{{\text{sh}}}} } \\ {\uptau _{{{\text{xy}}}}^{{{\text{sh}}}} } \\ {\uptau _{{{\text{xz}}}}^{{{\text{sh}}}} } \\ {\uptau _{{{\text{yz}}}}^{{{\text{sh}}}} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {{\text{C}}_{11} } & {{\text{C}}_{12} } & 0 & 0 & 0 \\ {{\text{C}}_{12} } & {{\text{C}}_{22} } & 0 & 0 & 0 \\ 0 & 0 & {{\text{C}}_{44} } & 0 & 0 \\ 0 & 0 & 0 & {{\text{C}}_{55} } & 0 \\ 0 & 0 & 0 & 0 & {{\text{C}}_{66} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\upvarepsilon _{{\text{x}}} } \\ {\upvarepsilon _{{\text{y}}} } \\ {\upgamma _{{{\text{xy}}}} } \\ \end{array} } \\ {\begin{array}{*{20}c} {\upgamma _{{{\text{xz}}}} } \\ {\upgamma _{{{\text{yz}}}} } \\ \end{array} } \\ \end{array} } \right\} $$

(5)

For, ${\text{C}}_{11} = {\text{C}}_{22} = \frac{{{\text{E}}\left( {\text{z}} \right)}}{{1 -\upnu ^{2} }},{\text{ C}}_{12} = \frac{{\upnu {\text{E}}\left( {\text{z}} \right)}}{{1 -\upnu ^{2} }}$, ${\text{C}}_{55} = {\text{C}}_{66} = \frac{{{\text{E}}\left( {\text{z}} \right)}}{{2\left( {1 +\upnu } \right)}}$.

Numerous studies have found the shear correction factor for homogeneous structures. Furthermore, some researchers have given a thickness-shear vibration value of $\frac{{\pi }^{2}}{12}$, but this will result in a different value for the FGM structure due to the material properties continuously changing in the thickness direction. So, the shear correction factor is denoted by $\left({{\text{K}}}_{{\text{s}}}\right)$, and its value is offered $\left({\text{K}}=\frac{5}{6}\right)$ as [33]. The stress and moment resultants of the FGM porous cylindrical shell panel can be represented as,

$$ \begin{gathered} {\text{N}}_{{\text{x}}} = \int_{{ - \frac{{\text{h}}}{2}}}^{{\frac{{\text{h}}}{2}}} {\rm{\sigma }_{{\text{x}}}^{{{\text{sh}}}} {\text{dz}}} {\text{,}}\,{\text{N}}_{{\text{y}}} = \int_{{ - \frac{{\text{h}}}{2}}}^{{\frac{{\text{h}}}{2}}} {\rm{\sigma }_{{\text{y}}}^{{{\text{sh}}}} {\text{dz}}} {\text{, N}}_{{{\text{xy}}}} = \int_{{ - \frac{{\text{h}}}{2}}}^{{\frac{{\text{h}}}{2}}} {\rm{\tau }_{{{\text{xy}}}}^{{{\text{sh}}}} {\text{dz}}} ,\left( {{\text{Q}}_{{\text{x}}} ,{\text{Q}}_{{\text{y}}} } \right) = \int_{{ - \frac{{\text{h}}}{2}}}^{{\frac{{\text{h}}}{2}}} {{\text{K}}_{{\text{s}}} \left( {\rm{\tau }_{{{\text{xz}}}}^{{{\text{sh}}}} ,\rm{\tau }_{{{\text{yz}}}}^{{{\text{sh}}}} } \right){\text{dz}}} ,{\text{M}}_{{\text{x}}} = \int_{{ - \frac{{\text{h}}}{2}}}^{{\frac{{\text{h}}}{2}}} {\rm{\sigma }_{{\text{x}}}^{{{\text{sh}}}} {\text{zdz}}} ,\, \hfill \\ {\text{M}}_{{\text{y}}} = \int_{{ - \frac{{\text{h}}}{2}}}^{{\frac{{\text{h}}}{2}}} {\rm{\sigma }_{{\text{x}}}^{{{\text{sh}}}} {\text{zdz}}} ,\,{\text{~M}}_{{{\text{xy}}}} = \int_{{ - \frac{{\text{h}}}{2}}}^{{\frac{{\text{h}}}{2}}} {\rm{\tau }_{{\text{x}}}^{{{\text{sh}}}} {\text{zdz}}} \hfill \\ \end{gathered} $$

(6)

where,

$$ \begin{aligned} & {\text{N}}_{{\text{x}}} = {\text{I}}_{10}\upvarepsilon _{{\text{x}}}^{ \circ } + {\text{I}}_{20}\upvarepsilon _{{\text{y}}}^{ \circ } + {\text{I}}_{11} \frac{{\partial\upphi _{{\text{x}}} }}{{\partial {\text{x}}}} + {\text{I}}_{21} \frac{{\partial\upphi _{{\text{y}}} }}{{\partial {\text{y}}}},\,{\text{N}}_{{\text{y}}} = {\text{I}}_{20}\upvarepsilon _{{\text{x}}}^{ \circ } + {\text{I}}_{10}\upvarepsilon _{{\text{y}}}^{ \circ } + {\text{I}}_{21} \frac{{\partial\upphi _{{\text{x}}} }}{{\partial {\text{x}}}} + {\text{I}}_{11} \frac{{\partial\upphi _{{\text{y}}} }}{{\partial {\text{y}}}} \\ & {\text{N}}_{{{\text{xy}}}} = {\text{I}}_{30}\upgamma _{{{\text{xy}}}}^{ \circ } + 2{\text{I}}_{31} \left( {\frac{{\partial\upphi _{{\text{x}}} }}{{\partial {\text{y}}}} + \frac{{\partial\upphi _{{\text{y}}} }}{{\partial {\text{x}}}}} \right),\,{\text{M}}_{{\text{x}}} = {\text{I}}_{11}\upvarepsilon _{{\text{x}}}^{ \circ } + {\text{I}}_{21}\upvarepsilon _{{\text{y}}}^{ \circ } + {\text{I}}_{12} \frac{{\partial\upphi _{{\text{x}}} }}{{\partial {\text{x}}}} + {\text{I}}_{22} \frac{{\partial\upphi _{{\text{y}}} }}{{\partial {\text{y}}}} \\ & {\text{M}}_{{\text{y}}} = {\text{I}}_{21}\upvarepsilon _{{\text{x}}}^{ \circ } + {\text{I}}_{11}\upvarepsilon _{{\text{y}}}^{ \circ } + {\text{I}}_{22} \frac{{\partial\upphi _{{\text{x}}} }}{{\partial {\text{x}}}} + {\text{I}}_{12} \frac{{\partial\upphi _{{\text{y}}} }}{{\partial {\text{y}}}},\,{\text{M}}_{{{\text{xy}}}} = {\text{I}}_{31}\upgamma _{{{\text{xy}}}}^{ \circ } + {\text{I}}_{32} \left( {\frac{{\partial\upphi _{{\text{x}}} }}{{\partial {\text{y}}}} + \frac{{\partial\upphi _{{\text{y}}} }}{{\partial {\text{x}}}}} \right) \\ \end{aligned} $$

(7)

${\text{Q}}_{{\text{x}}} = {\text{K}}_{{\text{s}}} {\text{I}}_{30} {\upgamma }_{{{\text{xz}}}}$,${\text{Q}}_{{\text{y}}} = {\text{K}}_{{\text{s}}} {\text{I}}_{30} {\upgamma }_{{{\text{yz}}}}$.

The nonlinear governing motion equations of a porous FGM cylindrical shell panel are given below,

$$ \begin{aligned} &\updelta {\text{u}}:\frac{{\partial {\text{N}}_{{\text{x}}} }}{{\partial {\text{x}}}} + \frac{{\partial {\text{N}}_{{{\text{xy}}}} }}{{\partial {\text{y}}}} = {\text{ I}}_{0} \frac{{\partial^{2} {\text{u}}}}{{\partial {\text{t}}^{2} }} + {\text{ I}}_{1} \frac{{\partial^{2}\upphi _{{\text{x}}} }}{{\partial {\text{t}}^{2} }} \\ &\updelta {\text{v}}:\frac{{\partial {\text{N}}_{{{\text{xy}}}} }}{{\partial {\text{x}}}} + \frac{{\partial {\text{N}}_{{\text{y}}} }}{{\partial {\text{y}}}} = {\text{ I}}_{0} \frac{{\partial^{2} {\text{v}}}}{{\partial {\text{t}}^{2} }} + {\text{ I}}_{1} \frac{{\partial^{2}\upphi _{{\text{y}}} }}{{\partial {\text{t}}^{2} }} \\ &\updelta {\text{w}}:\frac{{\partial {\text{Q}}_{{\text{x}}} }}{{\partial {\text{x}}}} + \frac{{\partial {\text{Q}}_{{\text{y}}} }}{{\partial {\text{y}}}} + {\text{N}}_{{\text{x}}} \frac{{\partial^{2} {\text{w}}}}{{\partial {\text{x}}^{2} }} + 2{\text{N}}_{{{\text{xy}}}} \frac{{\partial^{2} {\text{w}}}}{{\partial {\text{x}}\partial {\text{y}}}} \\ & + {\text{N}}_{{\text{y}}} \frac{{\partial^{2} {\text{w}}}}{{\partial {\text{y}}^{2} }} + {\text{q}} + \frac{{{\text{N}}_{{\text{y}}} }}{{\text{R}}} = {\text{ I}}_{0} \frac{{\partial^{2} {\text{w}}}}{{\partial {\text{t}}^{2} }} \\ & \updelta \upphi _{{\text{x}}} :\frac{{\partial {\text{M}}_{{\text{x}}} }}{{\partial {\text{x}}}} + \frac{{\partial {\text{M}}_{{{\text{xy}}}} }}{{\partial {\text{y}}}} - {\text{Q}}_{{\text{x}}} = {\text{ I}}_{2} \frac{{\partial^{2}\upphi _{{\text{x}}} }}{{\partial {\text{t}}^{2} }} + {\text{ I}}_{1} \frac{{\partial^{2} {\text{u}}}}{{\partial {\text{t}}^{2} }} \\ &\updelta \,\upphi _{{\text{y}}} :\frac{{\partial {\text{M}}_{{{\text{xy}}}} }}{{\partial {\text{x}}}} + \frac{{\partial {\text{M}}_{{\text{y}}} }}{{\partial {\text{y}}}} - {\text{Q}}_{{\text{y}}} = {\text{ I}}_{2} \frac{{\partial^{2}\upphi _{{\text{y}}} }}{{\partial {\text{t}}^{2} }} + {\text{ I}}_{1} \frac{{\partial^{2} {\text{v}}}}{{\partial {\text{t}}^{2} }} \\ \end{aligned} $$

(8)

By submitting stress function f (x, y) as follow:

$$ {\text{N}}_{{\text{x}}} = \frac{{\partial^{2} {\text{f}}}}{{\partial {\text{y}}^{2} }},{ }\,{\text{N}}_{{\text{y}}} = \frac{{\partial^{2} {\text{f}}}}{{\partial {\text{x}}^{2} }},\,{\text{ N}}_{{{\text{xy}}}} = - \frac{{\partial^{2} {\text{f}}}}{{\partial {\text{x}}\partial {\text{y}}}} $$

(9)

Equation (9) is substituted into the first two equations of Eq. (8) to yield;

$$ \frac{{\partial^{2} {\text{u}}}}{{\partial {\text{t}}^{2} }} = - \frac{{{\text{ I}}_{1} }}{{{\text{ I}}_{0} }}\frac{{\partial^{2}\upphi _{{\text{x}}} }}{{\partial {\text{t}}^{2} }},\;\;\,\frac{{\partial^{2} {\text{v}}}}{{\partial {\text{t}}^{2} }} = - \frac{{{\text{ I}}_{1} }}{{{\text{ I}}_{0} }}\frac{{\partial^{2}\upphi _{{\text{y}}} }}{{\partial {\text{t}}^{2} }} $$

(10)

Replacing (10) into the remaining three equations of Eq. (8), get,

$$ {\delta w}:\frac{{\partial {\text{Q}}_{{\text{x}}} }}{{\partial {\text{x}}}} + \frac{{\partial {\text{Q}}_{{\text{y}}} }}{{\partial {\text{y}}}} + {\text{N}}_{{\text{x}}} \frac{{\partial^{2} {\text{w}}}}{{\partial {\text{x}}^{2} }} + 2{\text{N}}_{{{\text{xy}}}} \frac{{\partial^{2} {\text{w}}}}{{\partial {\text{x}}\partial {\text{y}}}} + {\text{N}}_{{\text{y}}} \frac{{\partial^{2} {\text{w}}}}{{\partial {\text{y}}^{2} }} + {\text{q}} + \frac{{{\text{N}}_{{\text{y}}} }}{{\text{R}}} = {\text{ I}}_{0} \frac{{\partial^{2} {\text{w}}}}{{\partial {\text{t}}^{2} }} $$

$$ \begin{aligned} &\updelta \,\upphi _{{\text{x}}} :\frac{{\partial {\text{M}}_{{\text{x}}} }}{{\partial {\text{x}}}} + \frac{{\partial {\text{M}}_{{{\text{xy}}}} }}{{\partial {\text{y}}}} - {\text{Q}}_{{\text{x}}} = \left( {{\text{ I}}_{2} - \frac{{{\text{ I}}_{1}^{2} }}{{{\text{ I}}_{0} }}} \right)\frac{{\partial^{2}\upphi _{{\text{x}}} }}{{\partial {\text{t}}^{2} }} \\ &\updelta \,\upphi _{{\text{y}}} :\frac{{\partial {\text{M}}_{{{\text{xy}}}} }}{{\partial {\text{x}}}} + \frac{{\partial {\text{M}}_{{\text{y}}} }}{{\partial {\text{y}}}} - {\text{Q}}_{{\text{y}}} = \left( {{\text{ I}}_{2} - \frac{{{\text{ I}}_{1}^{2} }}{{{\text{ I}}_{0} }}} \right)\frac{{\partial^{2}\upphi _{{\text{y}}} }}{{\partial {\text{t}}^{2} }} \\ \end{aligned} $$

(11)

Through Eqs. (7) yield,

$$ \begin{aligned} &\upvarepsilon _{{\text{x}}}^{{\text{o}}} = {\text{A}}_{22} {\text{N}}_{{\text{x}}} - {\text{A}}_{12} {\text{N}}_{{\text{y}}} - {\text{B}}_{11} \frac{{\partial\upphi _{{\text{x}}} }}{{\partial {\text{x}}}} - {\text{B}}_{12} \frac{{\partial\upphi _{{\text{y}}} }}{{\partial {\text{y}}}} \\ &\upvarepsilon _{{\text{y}}}^{{\text{o}}} = {\text{A}}_{11} {\text{N}}_{{\text{y}}} - {\text{A}}_{12} {\text{N}}_{{\text{x}}} - {\text{B}}_{21} \frac{{\partial\upphi _{{\text{x}}} }}{{\partial {\text{x}}}} - {\text{B}}_{22} \frac{{\partial\upphi _{{\text{y}}} }}{{\partial {\text{y}}}} \\ &\upgamma _{{{\text{xy}}}}^{{\text{o}}} = {\text{A}}_{66} {\text{N}}_{{{\text{xy}}}} - {\text{B}}_{66} \left( {\frac{{\partial\upphi _{{\text{x}}} }}{{\partial {\text{y}}}} + \frac{{\partial\upphi _{{\text{y}}} }}{{\partial {\text{x}}}}} \right) \\ \end{aligned} $$

(12)

The compatibility equation must be used to add an equation,

$$\frac{{\partial }^{2}{{\upvarepsilon }_{{\text{x}}}}^{\circ }}{\partial {{\text{y}}}^{2}}+\frac{{\partial }^{2}{{\upvarepsilon }_{{\text{y}}}}^{\circ }}{\partial {{\text{x}}}^{2}}-\frac{{\partial }^{2}{\upgamma }_{{\text{xy}}}^{\circ }}{\partial {\text{x}}\partial {\text{y}}}=\frac{{\partial }^{2}{{\text{w}}}^{2}}{\partial {\text{x}}\partial {\text{y}}}-\frac{{\partial }^{2}{\text{w}}}{\partial {{\text{x}}}^{2}}\frac{{\partial }^{2}{\text{w}}}{\partial {{\text{y}}}^{2}}-\frac{1}{{\text{R}}}\frac{{\partial }^{2}{\text{w}}}{\partial {{\text{x}}}^{2}}$$

(13)

Equation (12) is inserted into Eq. (7) and then into Eq. (11), which results in;

$$ \begin{aligned} & {\text{T}}_{11} \left( {\text{w}} \right) + {\text{T}}_{12} \left( {\upphi _{{\text{x}}} } \right) + {\text{T}}_{13} \left( {\upphi _{{\text{y}}} } \right) + {\text{S}}_{1} \left( {{\text{w}},{\text{f}}} \right) + {\text{q}} = {\text{ I}}_{0} \frac{{\partial^{2} {\text{w}}}}{{\partial {\text{t}}^{2} }}, \\ & {\text{T}}_{21} \left( {\text{w}} \right) + {\text{T}}_{22} \left( {\upphi _{{\text{x}}} } \right) + {\text{T}}_{23} \left( {\upphi _{{\text{y}}} } \right) + {\text{R}}_{2} \left( {\text{f}} \right) = \left( {{\text{ I}}_{2} - \frac{{{\text{ I}}_{1}^{2} }}{{{\text{ I}}_{0} }}} \right)\frac{{\partial^{2}\upphi _{{\text{x}}} }}{{\partial {\text{t}}^{2} }}, \\ & {\text{T}}_{31} \left( {\text{w}} \right) + {\text{T}}_{32} \left( {\upphi _{{\text{x}}} } \right) + {\text{T}}_{33} \left( {\upphi _{{\text{y}}} } \right) + {\text{R}}_{3} \left( {\text{f}} \right) = \left( {{\text{ I}}_{2} - \frac{{{\text{ I}}_{1}^{2} }}{{{\text{ I}}_{0} }}} \right)\frac{{\partial^{2}\upphi _{{\text{y}}} }}{{\partial {\text{t}}^{2} }} \\ \end{aligned} $$

(14)

When Eq. (12) is replaced with Eq. (13) and the stress functions, it provides the following formalization for the compatibility of FGM porous cylindrical shell panels

$$ \left( {\begin{array}{*{20}c} {{\text{A}}_{11} \frac{{\partial^{4} {\text{f}}}}{{\partial {\text{x}}^{4} }} + {\text{A}}_{22} \frac{{\partial^{4} {\text{f}}}}{{\partial {\text{y}}^{4} }} + \left( {{\text{A}}_{66} - 2{\text{A}}_{12} } \right)\frac{{\partial^{4} {\text{f}}}}{{\partial {\text{x}}^{2} \partial {\text{y}}^{2} }} - {\text{B}}_{21} \frac{{\partial^{3}\upphi _{{\text{x}}} }}{{\partial {\text{x}}^{3} }} - {\text{B}}_{12} \frac{{\partial^{3}\upphi _{{\text{y}}} }}{{\partial {\text{y}}^{3} }} + } \\ {\left( {{\text{B}}_{66} - {\text{B}}_{11} } \right)\frac{{\partial^{3}\upphi _{{\text{x}}} }}{{\partial {\text{x}}\partial {\text{y}}^{2} }} + \left( {{\text{B}}_{66} - {\text{B}}_{22} } \right)\frac{{\partial^{3}\upphi _{{\text{y}}} }}{{\partial {\text{x}}^{2} \partial {\text{y}}}} - \left( {\frac{{\partial^{2} {\text{w}}^{2} }}{{\partial {\text{x}}\partial {\text{y}}}} - \frac{{\partial^{2} {\text{w}}}}{{\partial {\text{x}}^{2} }}\frac{{\partial^{2} {\text{w}}}}{{\partial {\text{y}}^{2} }} - \frac{1}{{\text{R}}}\frac{{\partial^{2} {\text{w}}}}{{\partial {\text{x}}^{2} }}} \right)} \\ \end{array} } \right) = 0 $$

(15)

2.2 Nonlinear Vibration Analysis

In this paper, suppose that the porous FGM shell is subjected to impact of the uniformly distributed transverse load ${\text{q}}=\mathrm{Q \,sin\,\Omega t}$ with four edges simply supported boundary conditions:

$$ \begin{aligned} & {\text{w}} = {\text{N}}_{{{\text{xy}}}} =\upphi _{{\text{y}}} = 0,\;{\text{at }}\;{\text{x}} = 0,\,{\text{a }} \\ & {\text{w}} = {\text{N}}_{{{\text{xy}}}} =\upphi _{{\text{x}}} = 0,\;{\text{at}}\;{\text{ y}} = 0,\,{\text{b}} \\ \end{aligned} $$

(16)

The next equations are desired to apply to the displacements in the present cases that satisfy the assumed boundary conditions [34]:

$$ \begin{aligned} & {\text{w}}\left( {{\text{x}},{\text{y}},{\text{t}}} \right) = {\text{W}}\left( {\text{t}} \right)\sin\uplambda _{{\text{m}}} {\text{x}}\,\sin\updelta _{{\text{n}}} {\text{y}} \\ &\upphi _{{\text{x}}} \left( {{\text{x}},{\text{y}},{\text{t}}} \right) =\Phi _{{\text{x}}} \left( {\text{t}} \right)\cos\uplambda _{{\text{m}}} {\text{x}}\,\sin\updelta _{{\text{n}}} {\text{y}} \\ &\upphi _{{\text{y}}} \left( {{\text{x}},{\text{y}},{\text{t}}} \right) =\Phi _{{\text{y}}} \left( {\text{t}} \right)\sin\uplambda _{{\text{m}}} {\text{x}}\,\cos\updelta _{{\text{n}}} {\text{y}} \\ \end{aligned} $$

(17)

Replacing (17) with (15), gain;

$$ {\text{f}}\left( {{\text{x}},{\text{y}},{\text{t}}} \right) = {\tilde{\text{A}}}_{1} \left( {\text{t}} \right)\cos 2\uplambda _{{\text{m}}} {\text{x}} + {\tilde{\text{A}}}_{2} \left( {\text{t}} \right)\cos 2\updelta _{{\text{n}}} {\text{y}} + {\tilde{\text{A}}}_{3} \left( {\text{t}} \right)\sin\uplambda _{{\text{m}}} {\text{x}}\,\sin \,\updelta _{{\text{n}}} {\text{y}} $$

$$ \begin{aligned} & {\tilde{\text{A}}}_{1} \left( {\text{t}} \right) = \frac{{\updelta _{{\text{n}}}^{2} }}{{32{\text{A}}_{11}\uplambda _{{\text{m}}}^{2} }}{\text{W}}^{2} ;\,{\tilde{\text{A}}}_{2} \left( {\text{t}} \right) = \frac{{\uplambda _{{\text{m}}}^{2} }}{{32{\text{A}}_{22} {\updelta }_{{\text{n}}}^{2} }}{\text{W}}^{2} \\ & {\tilde{\text{A}}}_{3} \left( {\text{t}} \right) = \frac{{\left( {\frac{{\text{W}}}{{\text{R}}}} \right) + \left( {{\text{B}}_{21}\uplambda _{{\text{m}}}^{3} + \left( {{\text{B}}_{11} - {\text{B}}_{66} } \right)\uplambda _{{\text{m}}}\updelta _{{\text{n}}}^{2} } \right)\Phi _{{\text{x}}} \left( {\text{t}} \right) + \left( {{\updelta }_{{\text{n}}}^{3} {\text{B}}_{12} + \left( {{\text{B}}_{22} - {\text{B}}_{66} } \right)\uplambda _{{\text{m}}}^{2}\updelta _{{\text{n}}} } \right)\Phi _{{\text{y}}} \left( {\text{t}} \right)}}{{\left( {{\text{A}}_{11}\uplambda _{{\text{m}}}^{4} + {\text{A}}_{22}\updelta _{{\text{n}}}^{4} + \left( {{\text{A}}_{66} - 2{\text{A}}_{12} } \right)\uplambda _{{\text{m}}}^{2}\updelta _{{\text{n}}}^{2} } \right)}} \\ \end{aligned} $$

(18)

After minor adjustments, the system of nonlinear motion equations in terms of displacements is derived by introducing the formula (17 and 18) into (14) and using the Galerkin method;

$$ {\text{t}}_{11} {\text{W}} + {\text{t}}_{12}\Phi _{{\text{x}}} + {\text{t}}_{13}\Phi _{{\text{y}}} + {\text{t}}_{14} {\text{W}}\Phi _{{\text{x}}} + {\text{t}}_{15} {\text{W}}\Phi _{{\text{y}}} + {\text{t}}_{16} {\text{W}} + {\text{t}}_{17} {\text{W}}^{2} + {\text{t}}_{18} {\text{W}}^{3} + {\text{L}}_{32} {\text{q}} = {\text{ I}}_{0} \frac{{{\text{d}}^{2} {\text{W}}}}{{{\text{dt}}^{2} }} $$

$$ \begin{aligned} & {\text{t}}_{21} {\text{W}} + {\text{t}}_{22}\Phi _{{\text{x}}} + {\text{t}}_{23}\Phi _{{\text{y}}} + {\text{n}}_{7} {\text{W}} + {\text{n}}_{2} {\text{W}}^{2} = {\tilde{\rho }}_{1} {\ddot{\Phi }}_{{\text{x}}} , \\ & {\text{t}}_{31} {\text{W}} + {\text{t}}_{32}\Phi _{{\text{x}}} + {\text{t}}_{33}\Phi _{{\text{y}}} + {\text{n}}_{9} {\text{W}} + {\text{n}}_{4} {\text{W}}^{2} = {\tilde{\rho }}_{1} {\ddot{\Phi }}_{{\text{y}}} , \\ \end{aligned} $$

(19)

The natural frequencies of the FGM porous cylindrical shell are obtained by solving Eq. (20), setting q = 0, and taking the linear components of Eq. (19):

$$\left|\begin{array}{ccc}{{\text{t}}}_{11}+{{\text{t}}}_{16}+{\mathrm{ I}}_{0}{\upomega }^{2}& {{\text{t}}}_{12}& {{\text{t}}}_{13}\\ {{\text{t}}}_{21}+{{\text{n}}}_{1}& {{\text{t}}}_{22}+{\stackrel{\sim }{\uprho }}_{1}{\upomega }^{2}& {{\text{t}}}_{23}\\ {{\text{t}}}_{31}+{{\text{n}}}_{3}& {{\text{t}}}_{32}& {{\text{t}}}_{33}+{\stackrel{\sim }{\uprho }}_{1}{\upomega }^{2}\end{array}\right|=0$$

(20)

Three answers to Eq. (20) correspond to the axial, circumferential, and radial angular frequencies of the porosity FGM cylindrical shells. One with the lowest frequency is taken into account. The porosity FGM cylindrical shells panel subjected to orderly distributed load $q=Q sin\Omega t$ is considered, Eq. (19) becomes:

$$ {\text{ I}}_{0} \frac{{{\text{d}}^{2} {\text{W}}}}{{{\text{dt}}^{2} }} - {\text{t}}_{11} {\text{W}} - {\text{t}}_{12}\Phi _{{\text{x}}} - {\text{t}}_{13}\Phi _{{\text{y}}} - {\text{t}}_{14} {\text{W}}\Phi _{{\text{x}}} - {\text{t}}_{15} {\text{W}}\Phi _{{\text{y}}} - {\text{t}}_{16} {\text{W}} - {\text{t}}_{17} {\text{W}}^{2} - {\text{t}}_{18} {\text{W}}^{3} = {\text{L}}_{32} {\text{Q}}\,\sin \,\Omega {\text{t}} $$

$$ \begin{aligned} & {\text{t}}_{21} {\text{W}} + {\text{t}}_{22}\Phi _{{\text{x}}} + {\text{t}}_{23}\Phi _{{\text{y}}} + {\text{n}}_{1} {\text{W}} + {\text{n}}_{2} {\text{W}}^{2} = {\tilde{\rho }}_{1} {\ddot{\Phi }}_{{\text{x}}} \\ & {\text{t}}_{31} {\text{W}} + {\text{t}}_{32}\Phi _{{\text{x}}} + {\text{t}}_{33}\Phi _{{\text{y}}} + {\text{n}}_{3} {\text{W}} + {\text{n}}_{4} {\text{W}}^{2} = {\tilde{\rho }}_{1} {\ddot{\Phi }}_{{\text{y}}} \\ \end{aligned} $$

(21)

The nonlinear dynamic responses of a porous FGM cylindrical shell panel can be calculated using the fourth-order Runge-Kutta technique by solving Eq. (21) with the following initial conditions: ${\text{W}}\left( 0 \right) = 0$. When the second and third equations relating to $(\Phi _{{\text{x}}} {,}\Phi _{{\text{y}}} )$, are solved from Eq. (21), the results are then substituted into the first equation to yield:

$$ {\text{ I}}_{0} \frac{{{\text{d}}^{2} {\text{W}}}}{{{\text{dt}}^{2} }} - \left( {{\text{a}}_{1} + {\text{a}}_{2} } \right){\text{W}} - \left( {{\text{a}}_{3} + {\text{a}}_{4} + {\text{a}}_{6} + {\text{r}}_{17} } \right){\text{W}}^{2} - \left( {{\text{a}}_{5} + {\text{r}}_{18} } \right){\text{W}}^{3} = {\text{L}}_{32} {\text{Q}}\,\sin \,\Omega {\text{t}} $$

(22)

The fundamental natural frequencies of the cylindrical shell may be calculated as follows:

$$\upomega _{{{\text{mn}}}} = \sqrt {\frac{{ - \left( {{\text{a}}_{1} + {\text{a}}_{2} } \right)}}{{{\text{ I}}_{0} }}} $$

(23)

3 Numerical Investigation

Using numerical methodologies is one way in which one may validate the precision of the analytical method that has been suggested. Problems may be solved using a wide variety of numerical methodologies, although the method of finite elements (FEM) [35] is considered to be the most accurate. The finite element method (FEM) that is outlined by the ANSYS software (Version 2020 R1) was used in this investigation. The construction of a three-dimensional prototype of the Functionally graded cylindrical shell panel, as shown in Fig. 2, is followed by the application of the matching shell’s sides boundary conditions, which are subjected to modal study. In addition, as can be seen in Fig. 3, the prototype has been meshed with an 8-nodes SOLID186 slices type, which has resulted in a total of 19360 slices and 137922 nodes. This element type is a significant basic element that is used in the representation of structural models. As shown in Fig. 4 [36], the component consists of a higher-order solid element having 20 nodes that has quadratic spatial characteristics and three freedom degrees for translations along the normal axes. The equation that is used to determine the mechanical characteristics of the FGMs layers as follows: (2). The modal analysis for the selected models is carried out to determine the free vibration characteristics both natural frequencies and mode shapes, as shown in Fig. 5. This is done on the basis of the various factors that were described before.

Fig. 2.

FGM cylindrical shell panel

Fig. 3.

Meshed Model

Fig. 4.

Structural geometry of element type SOLID 186

Fig. 5.

View of the modular analysis of the FGM cylindrical shell

4 Results and Discussion

In this study, a novel mathematical form was developed to analyze the natural frequencies and modes of vibration of FGM cylindrical shell panels supported by just a simply support with uniform porosity distribution based on power-law distribution. Influences of different properties on frequency parameters are analyzed. The material properties of numerous individual materials that were used in this investigation are presented in Table 1. The analytical solution was also verified using the (ANSYS 2020 R1) software that is available for purchase, and the results were tabulated and presented using a variety of curves. Assume that the dimensions of the cylindrical shell are R = 3 m, a = b = 0.5 m., porosity factor (0.1, 0.2, and 0.3), the power-law distribution g = 2, and FG thickness (h = 10, 14, and 16 mm).

Table 1. The material properties of many individual materials

Table 2 demonstrates both analytical and numerical findings for the cylindrical panel’s natural frequencies for a variety of porosity factors. According to the information shown in Table 2, the thickness of the FG has a considerable impact on the frequency characteristics. Good agreements are reached between analytical analysis tests and numerical tests when the percentage of difference between the two is less than 0.9%. When the porosity factor goes up, the natural frequencies go down because the material stiffness of the goes down. On the other hand, when the panel FG thickness goes up, the natural frequency goes up because the panels get better as the Functionally graded core thickness goes up. This can be seen as a consequence of the findings in Table 2, which show that the natural frequencies go down when the porosity factor goes up. Accordingly, the first six deflections of three-dimensional mode shapes are shown for FGM in Fig. 6 when the material is simply supported in the form of porous cylindrical shell panels with the following parameters: a gradient index of (g = 0.5), a porosity ratio of (g = 30%), a = b = 0.5 m, R = 3, m = n = 1, and a thickness of 10 mm for the FGM. In a like way, it’s also possible to depict other 3D mode shapes that are supported by a variety of edge conditions.

Figures 7 and 8 illustrate, respectively, the impacts of the material gradient factor (g) on the dynamic response and natural frequency of cylindrical panels with three different porosity values. These panels were designed to test the effectiveness of the material gradient parameter. Due to a reduction in the material’s rigidity, shown that the natural frequency decline whenever the gradient indicator (g) increases, and the dynamic of nonlinear response expands in three porosity magnitudes of (10, 20, and 30%). Additionally, according to Eq. 2, the amount of metal increases while a volume of ceramic decreases, which results in a decrease in shell stiffness. The influence that the porosity factor, denoted by the symbol G, has on the dynamic of nonlinear response is seen in Fig. 9. There are three different levels of porosity that are taken into consideration: G = 10%, G = 20%, and G = 30%. As can be seen, increasing the pores results in a greater amplitude deflection of FGM cylindrical shells. This is something that can be noticed.

Analytical findings of the dynamic nonlinear response of the cylindrical shells with varying FG core thicknesses are shown in Fig. 10. (10, 15, and 20 mm). Figure 9 illustrates that the intensity of the dynamic of nonlinear response diminishes as the Functionally graded thickness of the shells grows. This is due to the fact that the stiffness of the FG panel increases as the Functionally graded thickness does.

Table 2. The FGM cylindrical panels’ natural frequency with a power law index of g = 2.

Fig. 6.

The initial six mode shapes of Porous FGM simply supported cylindrical panels at G = 0.3, g = 2.

Fig. 7.

The result of applying the power law index on porous FGM cylindrical panel was the following natural frequency results

Fig. 8.

A look at how the gradient index of a material affects the dynamic response of porous cylindrical shells made from fibrous graphene (FGM).

Fig. 9.

The influence of porosity on the dynamic response of cylindrical panels

Fig. 10.

Cylindrical shells made with porous FGM vibrate with varying FGM thicknesses.

5 Conclusion

In the current work, the nonlinear free vibration characteristics of an FG two-phase porous cylindrical shell with a simply supported boundary condition based on the FSDT are developed. A simply supported cylindrical shell’s analytical formulation is presented in order to calculate the nonlinear free dynamic behavior. In order to validate the findings of an analytical solution, a numerical analysis was carried out with the assistance of ANSYS 2021 R1. Also, on deflection-time curve and natural frequency, the research findings for material gradient, porous parameter, and FG thickness are displayed. Based on the findings, it can be deduced that the porosity parameter does, in fact, have some kind of influence on the essential natural frequency of the FG cylindrical panels. According to the findings, the natural frequencies go up when the porosity parameter goes down, but they go down when the stiffness of the material goes up. This is because the natural frequencies are more sensitive to changes in the rigidity of the material. In addition, a downward shift in the amplitude-time curve was seen whenever the porosity factor was raised.

Appendix

$$ \begin{aligned} & I_{10} = \frac{{E_{1} }}{{1 - \upsilon^{2} }},I_{20} = \frac{{\upsilon E_{1} }}{{1 - \upsilon^{2} }},I_{30} = \frac{{E_{1} }}{{2\left( {1 + \upsilon } \right)}},I_{11} = \frac{{E_{2} }}{{1 - \upsilon^{2} }},I_{21} = \frac{{\upsilon E_{2} }}{{1 - \upsilon^{2} }},I_{31} = \frac{{E_{2} }}{{2\left( {1 + \upsilon } \right)}}, \\ & I_{12} = \frac{{E_{3} }}{{1 - \upsilon^{2} }},I_{22} = \frac{{\upsilon E_{3} }}{{1 - \upsilon^{2} }},I_{32} = \frac{{E_{3} }}{{2\left( {1 + \upsilon } \right)}}, \\ & A_{11} = \frac{1}{\Delta }I_{10} ,A_{22} = \frac{1}{\Delta }I_{10} ,A_{12} = \frac{{I_{20} }}{\Delta },A_{66} = \frac{1}{{I_{30} }},\Delta = I_{10}^{2} - I_{20}^{2} ,B_{11} = A_{22} I_{11} - A_{12} I_{21} ,B_{22} = A_{11} I_{11} - A_{12} I_{21} , \\ & B_{12} = A_{22} I_{21} - A_{12} I_{11} ,B_{21} = A_{11} I_{21} - A_{12} I_{11} ,B_{66} = \frac{{I_{31} }}{{I_{30} }},D_{11} = I_{12} - B_{11} B_{12} - I_{21} B_{21} ,D_{22} = I_{22} - B_{22} I_{11} - I_{21} B_{12} , \\ & D_{12} = I_{22} - B_{12} I_{11} - I_{21} B_{22} ,D_{21} = I_{22} - B_{21} I_{11} - I_{21} B_{11} ,D_{66} = I_{32} - I_{31} B_{66} , \\ & {\text{ T}}_{11} \left( w \right) = K_{s} I_{30} \frac{{\partial^{2} w}}{{\partial x^{2} }} + K_{s} I_{30} \frac{{\partial^{2} w}}{{\partial y^{2} }},{ T}_{12} \left( {\phi_{x} } \right) = K_{s} I_{30} \frac{{\partial \phi_{x} }}{\partial x}, \\ & {\text{ T}}_{13} \left( {\phi_{y} } \right) = K_{s} I_{30} \frac{{\partial \phi_{y} }}{\partial y},R_{1} \left( {w,f} \right) = \frac{{\partial^{2} f}}{{\partial x^{2} }}\frac{{\partial^{2} w}}{{\partial x^{2} }} - 2\frac{{\partial^{2} f}}{\partial x\partial y}\frac{{\partial^{2} w}}{\partial x\partial y} + \frac{{\partial^{2} f}}{{\partial x^{2} }}\frac{{\partial^{2} w}}{{\partial y^{2} }} + \frac{1}{R}\frac{{\partial^{2} f}}{{\partial x^{2} }}, \\ & {\text{ T}}_{21} \left( w \right) = - K_{s} I_{30} \frac{\partial w}{{\partial x}},{ T}_{22} \left( {\phi_{x} } \right) = D_{11} \frac{{\partial^{2} \phi_{x} }}{{\partial x^{2} }} + D_{66} \frac{{\partial^{2} \phi_{y} }}{{\partial y^{2} }} - K_{s} I_{30} \phi_{x} ,{\text{ T}}_{23} \left( {\phi_{y} } \right) = \left( {D_{12} + D_{66} } \right)\frac{{\partial^{2} \phi_{y} }}{\partial x\partial y}, \\ & R_{2} \left( f \right) = B_{21} \frac{{\partial^{3} f}}{{\partial x^{3} }} + \left( {B_{11} - B_{66} } \right)\frac{{\partial^{3} f}}{{\partial x\partial y^{2} }},{\text{ T}}_{31} \left( w \right) = - K_{s} I_{30} \frac{\partial w}{{\partial y}},{\text{ T}}_{32} \left( {\phi_{x} } \right) = \left( {D_{21} + D_{66} } \right)\frac{{\partial^{2} \phi_{x} }}{\partial x\partial y}, \\ & {\text{ T}}_{33} \left( {\phi_{y} } \right) = D_{22} \frac{{\partial^{2} \phi_{y} }}{{\partial y^{2} }} + D_{66} \frac{{\partial^{2} \phi_{y} }}{{\partial x^{2} }} - K_{s} I_{30} \phi_{y} ,R_{3} \left( f \right) = B_{12} \frac{{\partial^{3} f}}{{\partial y^{3} }} + \left( {B_{22} - B_{66} } \right)\frac{{\partial^{3} f}}{{\partial x^{2} \partial y}}, \\ \end{aligned} $$