Advertisement

JMST Advances

, Volume 1, Issue 4, pp 233–248 | Cite as

Analysis of functionally graded plates resting on elastic foundation and subjected to non-linear hygro-thermo-mechanical loading

  • Shantaram M. Ghumare
  • Atteshamuddin S. SayyadEmail author
Letters
  • 155 Downloads

Abstract

In the present study, a new fifth-order shear and normal deformation theory is developed and applied for the bending analysis of functionally graded (FG) plates resting on two-parameter Winkler–Pasternak elastic foundation subjected to non-linear hygro-thermo-mechanical loading. The theory involves the effects of transverse shear and normal deformations, i.e. thickness stretching. Navier’s solution technique is used to obtain analytical solutions for simply supported FG plates. The results are presented in non-dimensional form and are compared with previously published results in the literature. The present study has the following novelties. The present polynomial-type theory is computationally simpler than non-polynomial-type plate theories which are mathematically complicated, tedious and more cumbersome. For the accurate structural analysis of composite plates under hygro-thermal loading, consideration of thickness coordinate up to third-order polynomial is not sufficient. Therefore, in the present theory, thickness coordinate is expanded up to fifth-order polynomial to get the accurate displacements and stresses. Transverse normal stress/strain plays an important role in the modeling of thick plates which is neglected by many theories available in the literature.

Keywords

FG plate Fifth order Shear deformation Normal deformation Hygro-thermo-mechanical loading Elastic foundation 

1 Introduction

Grading hygro-thermo-mechanical properties of advanced composite materials by continuously varying the volume fractions of their constituents is becoming a powerful tool in designing new materials for specific applications. Functionally graded materials (FGMs) are similar kind of advanced composite materials which are now widely utilized in various engineering structures subjected to hygro-thermo-mechanical loading. This motivates the researchers to develop new structural theories for the accurate analysis of FG plates subjected to hygro-thermo-mechanical loading. Analysis of plates can be carried out using classical and refined shear deformation theories. Since classical plate theory (CPT) of Kirchhoff [1] and first-order shear deformation theory (FSDT) of Mindlin [2] are not accurate for predicting bending behavior of thick plates subjected to hygro-thermo-mechanical loads, refined computational models have been developed by researchers to predict more accurate response of the plate under these loads. One can refer recent literature review articles published [3, 4, 5, 6, 7, 8] to know the various classical and refined computational models available in the literature for the analysis composite beams and plates.

A good number of research papers are available in the literature on the bending analysis of simply supported FG plates subjected to mechanical loads [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. Recently, Ghumare and Sayyad [22, 23] and Naik and Sayyad [24, 25, 26] have developed a new fifth-order shear and normal deformation theory for the analysis of functionally graded and laminated composite beams and plates. The stability analysis of FG plates subjected to linear and non-linear thermal loadings using various refined theories also reported by many researchers [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38].

Behavior of FG plates resting on elastic foundation is studied by many researchers. Zidi et al. [39] studied the bending response of FG plates resting on elastic foundation and subjected to hygro-thermo-mechanical loading using four variable plate theories. Daouadji et al. [40] investigated flexural bending analysis of perfect and imperfect FG plates under hygro-thermo-mechanical loadings and resting on elastic foundation. Zenkour et al. [41] discussed the effects of transverse shear and normal strains on bending of FG plates resting on elastic foundations and subjected to hygro-thermo-mechanical loading. Recently, Sayyad and Ghugal [42] developed a simple four unknown exponential shear deformation theories for the bending of FGM rectangular plates resting on two-parameter elastic foundation and subjected to non-linear hygro-thermo-mechanical loading.

Reddy [43] developed a simple higher order theory for square and cross ply laminates subjected to sinusoidal and uniform load. This theory is well known and popularly used by many researchers for investigations on the beams and plates. It satisfies parabolic variations of the transverse shear strains through the thickness. Zenkour [44] presented a static bending response for FG rectangular plate under a transverse uniform load using a generalized shear deformation theory. Winkler [45] provided the series of springs to study the effect of soil on the plate surface, whereas Pasternak [46] has introduced the shear layer over the springs as a two-parametric mechanical model. In combination, these two models are popularly called as Winkler–Pasternak elastic foundation. Arefi and Rahimi [47, 48] presented the non-linear analysis of a functionally graded square and annular plates with piezoelectric layers under normal pressure. Arefi and Rahimi [49] addressed the non-linear response of circular plate which is rested on Winkler–Pasternak foundation. Arefi et al. [50] reported the dynamic analyses of a sandwich microbeam and free vibration using the higher order sinusoidal shear deformation theory.

Arefi and Arani [51] investigated the FG nanobeam subjected to magneto-electro-elastic loads using third-order theory. An investigation is carried out for bending of FG nanobeams considering the influence of material parameters and loadings. Arefi and Zenkour [52] studied a sandwich piezoelectric micro-beam for bending using electro-elastic relations under transverse loads, applied voltage, etc. which is resting on Pasternak’s foundation. Mohammadi et al. [53] analyzed the response of two-dimensional thermo-elastic FG carbon nanotube-reinforced composite cylindrical pressure vessels using the third-order shear deformation theory. Using a rule of mixture, the properties of FG-CNTRC cylindrical pressure vessels are addressed considering the various patterns of reinforcement. The effect with type of pattern, Pasternak parameters and the volume fraction of CNTs are studied under the thermo-elastic response. Arefi and Zenkour [54, 55] presented a transient formulation for a three-layer curved nanobeam in thermo-magneto-elastic environments using the sinusoidal shear deformation theory. The study reveals that increase in nonlocal parameter lowers the stiffness of the curved nanobeam which leads to increase transverse and radial deflections. Arefi [56] presented an analytical solution for small-scale doubly curved piezoelectric shell considering the modified couple stress formulation based on FSDT and resting on Pasternak’s foundation. Arefi and Rabczuk [57] addressed the electro-elastic analysis for piezoelectric a doubly curved nano-shell using higher order shear deformation theory. Arefi and Zenkour [58] used the sinusoidal shear deformation plate theory to address the nonlocal thermo-magneto-electro-mechanical bending analysis for three-layered nano-plates. Linear temperature variation is assumed across the thickness. Arefi et al. [59] analyzed the FG piezoelectric plate using the sinusoidal shear deformation theory. Zenkour and Radwan [60] presented the buckling of FG plate resting on elastic foundations using a quasi-3D model for hygro-thermo-mechanical effect.

In the present theory, hygro-thermo-mechanical properties are assumed to be graded in the thickness direction using power-law distribution. The elastic foundation is modeled as two-parameter Winkler–Pasternak foundation. Navier’s solution technique is used to obtain Closed-form solutions for simply supported FG plates subjected to non-linear hygro-thermo-mechanical loads and non-dimensional results are compared with Kirchhoff [1], Mindlin [2], Zidi et al. [39], Reddy [43] and Zenkour [44]. It is concluded that the present theory is accurate and efficient in predicting the bending responses of FG plates subjected to non-linear hygro-thermo-mechanical loading.

1.1 Novelty of the present theory

In this work, a new polynomial type fifth-order shear and normal deformation theory (FOSNDT) is developed and applied for bending analysis of FG plates subjected to non-linear hygro-thermo-mechanical loading. The novelties of the present study are as follows:
  1. 1.

    From the aforementioned review, it can be observed that the numerous literature are available on mechanical or thermo-mechanical analysis of functionally graded plates, but studies on non-linear hygro-thermo-mechanical response of FG plates are limited which will be the main focus of the present study.

     
  2. 2.

    The present theory falls under polynomial type and computationally simpler than non-polynomial type plate theories which are mathematically complicated, tedious and more cumbersome.

     
  3. 3.

    For the accurate structural analysis of composite plates under hygro-thermal loading, consideration of thickness coordinate up to third-order polynomial is not sufficient. Therefore, in the present theory, thickness coordinate is expanded up to fifth-order polynomial to get the accurate displacements and stresses.

     
  4. 4.

    Transverse normal stress/strain plays an important role in the modeling of thick plates which is neglected by many theories available in the literature. The present theory considers the effects of both transverse shear and normal deformations.

     
  5. 5.

    The displacement field of the theory enforces the realistic variation of the transverse shear stresses (parabolic) across the thickness of the plate.

     

2 Gradation of functionally graded material

The power law used for material gradation follows the simple linear rule of mixture and properties varying across the thickness direction of FG plate stated in Eq. (1). This law is employed by many researchers such as Arefi and Rahimi [47, 48, 49], Arefi et al. [59], Zenkour and Radwan [60], etc. for the gradation of the material properties. The most commonly used constituents to form FG materials are ceramic and metal:
$$ \begin{aligned} E\left( z \right) = E_{m} + \left( {E_{c} - E_{m} } \right)\left( {\frac{1}{2} + \frac{z}{h}} \right)^{p} , \hfill \\ \alpha \left( z \right) = \alpha_{m} + \left( {\alpha_{c} - \alpha_{m} } \right)\left( {\frac{1}{2} + \frac{z}{h}} \right)^{p} , \hfill \\ \beta \left( z \right) = \beta_{m} + \left( {\beta_{c} - \beta_{m} } \right)\left( {\frac{1}{2} + \frac{z}{h}} \right)^{p} \hfill \\ \end{aligned} $$
(1)
where, p is the power law index which governs the volume fraction gradation; \( \mu \) represents the poisson’s ratio; z is the distance from the mid surface of the plate, h is the overall thickness of the plate, \( \alpha \) is the coefficient of thermal expansion, \( \beta \) is the coefficient of moisture concentration. Subscripts c and m represent the ceramic and metallic constituents, respectively. The value of p is equal to zero represents a fully ceramic plate, whereas infinite p indicates a fully metallic plate. The Poisson’s ratio is assumed to be constant since the variation of Poisson’s ratio \( (\mu ) \) on the response of FG plate is very small. Variation of modulus of elasticity through the thickness of plate is shown in Fig. 1. Similar variation can be observed for coefficient of thermal expansion and moisture concentration.
Fig. 1

Variations of elastic modulus through the thickness of the plate for different values of power law index

2.1 Modeling of elastic foundation

Plate structures are often found to be resting on earth in various engineering applications. This motivated many researchers to analyze the behavior of plates on elastic foundations. To describe the interaction between plate and foundation, the simplest model is Winkler model [45] which regards the foundation as a series of separated spring without coupling effects between each other. It is also called as one parameter model. This model is improved by Pasternak [46] by adding a shear layer to simulate the interactions between the separated springs in the Winkler model. The Pasternak, i.e. two-parameter model, is widely used to describe the mechanical behavior of structure-foundation interactions. The applications of two-parameter elastic foundation under the bending response of FG plates is also reported by Zenkour and Radwan [60], Chilton and Wekezer [61], Sheng [62], Najafi et al. [63], Daloglu and Ozgan [64], Zenkour and Sobhy [65], Kiani et al. [66], Sayyad and Ghugal [67, 68], etc. In the present work, FG rectangular plate is assumed to be rested on two-parameter Winkler–Pasternak elastic foundations with the Winkler spring stiffness kw and Pasternak shear layer stiffnesses kp1 and kp2 (Fig. 2). The reactive force (R) due to elastic foundation is given by the following equation as mentioned below:
Fig. 2

FG plate resting on Winkler–Pasternak type elastic foundation

$$ R = k_{w} w(x,y) + k_{p1} \frac{{\partial^{2} w(x,y)}}{{\partial x^{2} }} + k_{p2} \frac{{\partial^{2} w(x,y)}}{{\partial y^{2} }}. $$
(2)

3 Development of the present theory

3.1 Kinematical assumptions in the present formulation

The development of the present theory is based on the following kinematical assumptions:
  1. 1.

    The in-plane displacements in the x- and y-directions consist of extension, bending and shear components. The shear component is expanded up to fifth-order polynomial.

     
  2. 2.

    The transverse displacement (w) in z-direction consists of bending and transverse normal components, i.e. w is assumed to be a function of coordinates x and z.

     
  3. 3.

    Since 3D Hooke’s law is used to obtained stresses associated with the present theory, it accurately describes the state of stress in 3D continuum.

     
  4. 4.

    By implementing the double Fourier series technique, analytical solutions are obtained

     
  5. 5.

    The FG material properties vary across the thickness according to power law distribution.

     
  6. 6.

    The plate is subjected to mechanical and hygro-thermal loads on the top surface. Hygro-thermal loads are varying linearly and non-linearly across the thickness.

     

3.2 Kinematics of the present theory

Based on the aforementioned assumptions, the displacement field of the present theory is shown in Eq. (3) includes the effects of transverse shear deformation and transverse normal deformation. The displacement field shown in Eq. (3) is employed by the authors in their earlier published articles for the analysis of FG beams and plates under mechanical load [22, 23]. One can also refer the articles by Naik and Sayyad [24, 25, 26] has addressed the bending analysis of laminated beams and plates using the same displacement field.
$$ \begin{aligned} u(x,y,z) = u_{0} - z \frac{{\partial w_{0} }}{\partial x} + f_{1} (z) \phi_{x} + f_{2} (z) \psi_{x} , \hfill \\ v(x,y,z) = v_{0} - z \frac{{\partial w_{0} }}{\partial y} + f_{1} (z) \phi_{y} + f_{2} (z) \psi_{y} \hfill \\ w(x,y,z) = w_{0} + f_{1}^{\prime} (z) \phi_{z} + f_{2}^{\prime} (z) \psi_{z} \hfill \\ \end{aligned} $$
(3)
where \( u_{0} \),\( v_{0} \),\( w_{0} \) are the x, y and z directional displacements of the mid-plane; \( \phi_{x} ,\psi_{x} ,\phi_{y} ,\psi_{y} \) are the unknowns associated with the transverse shear deformation and \( \phi_{z} ,\psi_{z} \) are the unknowns associated with the transverse normal deformation; \( f_{1} (z) = z - \left( {4/3} \right)\left( {z^{3} /h^{2} } \right) \) and \( f_{2} (z) = z - \left( {16/5} \right)\left( {z^{5} /h^{4} } \right) \) are the shape functions assigned according to the shearing stress distribution through the thickness of the plate.
The present theory gives rise to three-dimensional state of strain at a point. Therefore, strain components are obtained using 3D strain displacement relationship using linear theory of elasticity:
$$ \begin{aligned}& \varepsilon_{x} = \varepsilon_{x}^{0} + zk_{x}^{b} + f_{1} (z)\varepsilon_{x}^{1} + f_{2} (z)\varepsilon_{x}^{2} ,\\&\varepsilon_{y} = \varepsilon_{y}^{0} + zk_{y}^{b} + f_{1} (z)\varepsilon_{y}^{1} + f_{2} (z)\varepsilon_{y}^{2} , \\& \varepsilon_{z} = f_{1}^{\prime\prime} (z)\phi_{z} + f_{2}^{\prime\prime} (z)\psi_{z} ,\\&\gamma_{xy} = \gamma_{xy}^{0} + z k_{xy}^{b} + f_{1} (z) \gamma_{xy}^{{_{1} }} + f_{2} (z) \gamma_{xy}^{{_{2} }} , \\& \gamma_{xz} = f_{1}^{\prime} (z) \gamma_{xz}^{{s_{1} }} + f_{2}^{\prime} (z) \gamma_{xz}^{{s_{2} }} , \\&\gamma_{yz} = f_{1}^{\prime} (z) \gamma_{yz}^{{s_{1} }} + f_{2}^{\prime} (z) \gamma_{yz}^{{s_{2} }} \hfill \\ \end{aligned} $$
(4)
where
$$ \begin{aligned}& \varepsilon_{x}^{0} = \frac{{\partial u_{0} }}{\partial x}, k_{x}^{b} = - \frac{{\partial^{2} w_{0} }}{{\partial x^{2} }}, \varepsilon_{x}^{1} = \frac{{\partial \phi_{x} }}{\partial x}, \varepsilon_{x}^{2} = \frac{{\partial \psi_{x} }}{\partial x}, \\&\gamma_{xz}^{{s_{1} }} = \left( {\phi_{x} + \frac{{\partial \phi_{z} }}{\partial x}} \right), \gamma_{xz}^{{s_{2} }} = \left( {\psi_{x} + \frac{{\partial \psi_{z} }}{\partial x}} \right), \varepsilon_{y}^{0} = \frac{{\partial v_{0} }}{\partial y}, \\&k_{y}^{b} = - \frac{{\partial^{2} w_{0} }}{{\partial y^{2} }}, \varepsilon_{y}^{1} = \frac{{\partial \phi_{y} }}{\partial y}, \varepsilon_{x}^{2} = \frac{{\partial \psi_{y} }}{\partial y}, \\&\gamma_{xy}^{{s_{1} }} = \left( {\phi_{x} + \frac{{\partial \phi_{y} }}{\partial y}} \right), \gamma_{xy}^{{s_{2} }} = \left( {\psi_{x} + \frac{{\partial \psi_{y} }}{\partial y}} \right), \\& \gamma_{yz}^{{s_{1} }} = \left( {\phi_{y} + \frac{{\partial \phi_{z} }}{\partial z}} \right), \gamma_{xy}^{{s_{2} }} = \left( {\psi_{y} + \frac{{\partial \psi_{z} }}{\partial z}} \right), \\& f_{1}^{\prime} (z) = 1 - 4\left( {z^{2} /h^{2} } \right), f_{1}^{\prime\prime} (z) = - 8\left( {z/h^{2} } \right), \\& f_{2}^{\prime} (z) = 1 - 16\left( {z^{4} /h^{4} } \right), f_{2}^{\prime\prime} (z) = - 64\left( {z^{3} /h^{4} } \right). \hfill \\ \end{aligned} $$
(5)
3D Hooke’s law is used to obtained state of stress at a point including hygro-thermal strains shown in the following equation:
$$ \left\{ \begin{aligned} \sigma_{x} \hfill \\ \sigma_{y} \hfill \\ \sigma_{z} \hfill \\ \tau_{xy} \hfill \\ \tau_{xz} \hfill \\ \tau_{yz} \hfill \\ \end{aligned} \right\} = \left[ \begin{aligned} Q_{11} \;\;\;Q_{12} \;\;\;Q_{13} \;\;\;0\;\;\;\;\;0\;\;\;\;\;\;0 \hfill \\ Q_{12} \;\;\;Q_{22} \;\;\;Q_{23} \;\;\;0\;\;\;\;\;0\;\;\;\;\;\;0\; \hfill \\ Q_{13} \;\;\;Q_{23} \;\;\;Q_{33} \;\;\;0\;\;\;\;\;0\;\;\;\;\;\;0 \hfill \\ 0\;\;\;\;\;\;0\;\;\;\;\;\;0\;\;\;\;\;Q_{66} \;\;\;0\;\;\;\;\;\;0\; \hfill \\ 0\;\;\;\;\;\;0\;\;\;\;\;\;0\;\;\;\;\;\;0\;\;\;\;\;Q_{44} \;\;\;0 \hfill \\ 0\;\;\;\;\;\;0\;\;\;\;\;\;0\;\;\;\;\;\;0\;\;\;\;\;\;0\;\;\;\;Q_{55} \hfill \\ \end{aligned} \right]\;\left\{ \begin{aligned} \varepsilon_{x} - \alpha_{x} \Delta T - \beta_{x} \Delta C \hfill \\ \varepsilon_{y} - \alpha_{y} \Delta T - \beta_{y} \Delta C \hfill \\ \varepsilon_{z} - \alpha_{z} \Delta T - \beta_{z} \Delta C \hfill \\ \gamma_{xy} \hfill \\ \gamma_{xz} \hfill \\ \gamma_{yz} \hfill \\ \end{aligned} \right\} $$
(6)
where, Qij are the stiffness coefficients stated in Eq. (7); \( \left( {\sigma_{x} ,\sigma_{y} ,\sigma_{z} } \right) \) are the normal stress components; \( \left( {\tau_{xy} ,\tau_{yz} ,\tau_{xz} } \right) \) are the shear stress components;\( \alpha_{x} ,\alpha_{y} ,\alpha_{z} {\text{ and }}\beta_{x} ,\beta_{y} ,\beta_{z} \) are the coefficients of linear thermal expansion and moisture concentration in x, y and z directions, respectively.
$$ \begin{aligned} &Q_{11} = Q_{22} = Q_{33} = \frac{E(z) (1 - \mu )}{(1 + \mu ) (1 - 2\mu )}, \hfill \\ &Q_{12} = Q_{13} = Q_{21} = Q_{23} = Q_{31} = Q_{32} = \frac{ E(z) \mu }{(1 + \mu )(1 - 2\mu )}, \hfill \\&Q_{44} = Q_{55} = Q_{66} = \frac{E(z)}{2(1 + \mu )}. \hfill \\ \end{aligned} $$
(7)
In most of the cases, plate structures are subjected to change in temperature and moisture content which leads to the additional stresses in the structures. Change in temperature and moisture profiles are the solutions of heat conduction problems. However, many researchers have assumed the variation profiles of the temperature and moisture content which are either constant, linear or non-linear through the thickness (see Fig. 3). With reference to previous studies published by Zidi et al. [39], Daouadji et al. [40], Sayyad and Ghugal [42], etc. the following field for change in temperature and moisture concentration across the thickness is defined:
Fig. 3

Temperature variations across the thickness of FG plate

$$ \begin{aligned} \Delta T(x,y,z) = T_{0} (x,y) + \frac{z}{h}T_{1} (x,y) + \frac{{f_{1} (z)}}{h}T_{2} (x,y) + \frac{{f_{2} (z)}}{h}T_{3} (x,y), \hfill \\ \Delta C(x,y,z) = C_{0} (x,y) + \frac{z}{h}C_{1} (x,y) + \frac{{f_{1} (z)}}{h}C_{2} (x,y) + \frac{{f_{2} (z)}}{h}C_{3} (x,y), \hfill \\ \end{aligned} $$
(8)
where, \( T_{0,} \,T_{1,} \,T_{2,} \,{\text{and}} T_{3} \) are the terms used for thermal loads and \( C_{0,} C_{1,} C_{2,} {\text{and}} C_{3} \) are the terms used for change in moisture concentration. T0 terms associated for constant temperature, T1 terms associated for linear temperature and \( T_{2} ,T_{3} \) are the non-linear temperature. Similar notations are used for moisture concentration.

3.3 Governing equations

The principle of virtual displacement is employed to derive nine variationally consistent governing equations of the present theory. Equation (9) represents the principle of virtual work which stated that the internal work done is equal to external work done. Internal work done is due to internal forces, i.e. stresses whereas external work done is due to external loads. Many researchers such as Reddy [43], Arefi and Arani [51], Arefi and Zenkour [54, 55], Arefi and Rabczuk [57], Arefi and Zenkour [58], Arefi et al. [59], Zenkour and Radwan [60], etc. have used this principle to derive governing differential equations of the theory:
$$ \begin{aligned}& \int\limits_{0}^{a} {} \int\limits_{0}^{b} {} \int\limits_{ - h/2}^{h/2} {} \left( {\sigma_{x} \delta \varepsilon_{x} + \sigma_{y} \delta \varepsilon_{y} + \sigma_{z} \delta \varepsilon_{z} + \tau_{xy} \delta \gamma_{xy}}\right.\\&\quad\left.{ + \tau_{xz} \delta \gamma_{xz} + \tau_{yz} \delta \gamma_{yz} } \right) \,{\text{d}}z\,{\text{d}}y\,{\text{d}}x \\& \quad = \int\limits_{0}^{a} {} \int\limits_{0}^{b} {} \left( {q\left( {x,y} \right) + k_{w} w - k_{p1} \frac{{\partial^{2} w}}{{\partial x^{2} }} - k_{p2} \frac{{\partial^{2} w}}{{\partial y^{2} }}} \right) \\&\quad\quad\times\,\left( {\delta w_{0} + f_{1}^{\prime} (z) \delta \phi_{z} + f_{2}^{\prime} (z) \delta \psi_{z} } \right){\text{d}}y\,{\text{d}}x, \hfill \\ \end{aligned} $$
(9)
where, \( \delta \) denotes variational parameters. Substitution of strain components from Eq. (4) into the Eq. (9) leads to the following equation:
$$ \begin{aligned} &\int\limits_{0}^{a} {} \int\limits_{0}^{b} {} \left( {N_{x} \frac{{\partial \delta u_{0} }}{\partial x} - M_{x}^{b} \frac{{\partial^{2} \delta w_{0} }}{{\partial x^{2} }} + M_{x}^{{S_{1} }} \frac{{\partial \delta \phi_{x} }}{\partial x} + M_{x}^{{S_{2} }} \frac{{\partial \delta \psi_{x} }}{\partial x} }\right.\\&\quad\left.{+ N_{y} \frac{{\partial \delta v_{0} }}{\partial y} - M_{y}^{b} \frac{{\partial^{2} w_{0} }}{{\partial y^{2} }} + M_{y}^{{S_{1} }} \frac{{\partial \delta \phi_{y} }}{\partial y}} \right. + M_{y}^{{S_{2} }} \frac{{\partial \delta \psi_{y} }}{\partial y} \\&\quad+ Q_{z}^{1} \delta \phi_{z} + Q_{z}^{2} \delta \psi_{z} + N_{xy} \frac{{\partial \delta u_{0} }}{\partial y} + N_{xy} \frac{{\partial \delta v_{0} }}{\partial x} - 2M_{xy}^{b} \frac{{\partial^{2} \delta w_{0} }}{\partial x\partial y} \\&\quad+ M_{xy}^{{S_{1} }} \frac{{\partial \delta \phi_{x} }}{\partial y} + M_{xy}^{{S_{1} }} \frac{{\partial \delta \phi_{y} }}{\partial x} + M_{xy}^{{S_{2} }} \frac{{\partial \delta \psi_{x} }}{\partial y} + M_{xy}^{{S_{2} }} \frac{{\partial \delta \psi_{y} }}{\partial x} \\&\quad+ Q_{xz}^{1} \left[ {\delta \phi_{x} + \frac{{\partial \delta \phi_{z} }}{\partial x}} \right] + Q_{xz}^{2} \left[ {\delta \psi_{x} + \frac{{\partial \delta \psi_{z} }}{\partial x}} \right]\\&\quad + Q_{yz}^{1} \left[ {\delta \phi_{y} + \frac{{\partial \delta \phi_{z} }}{\partial y}} \right] + Q_{yz}^{2} \left[ {\delta \psi_{y} + \frac{{\partial \delta \psi_{z} }}{\partial y}} \right]{\text{d}}y\,{\text{d}}x \\&\quad + \int\limits_{0}^{a} {} \int\limits_{0}^{b} {} \left[ \begin{aligned} &q + k_{w} \left( {w_{0} + f_{1}^{\prime} (z)\phi_{z} + f_{2}^{\prime} (z)\psi_{z} } \right)\\&\quad - k_{p1} \frac{{\partial^{2} }}{{\partial x^{2} }}\left( {w_{0} + f_{1}^{\prime} (z)\phi_{z} + f_{2}^{\prime} (z)\psi_{z} } \right) \\&\quad - k_{p2} \frac{{\partial^{2} }}{{\partial y^{2} }}\left( {w_{0} + f_{1}^{\prime} (z)\phi_{z} + f_{2}^{\prime} (z)\psi_{z} } \right) \end{aligned} \right]\delta w_{0} {\text{d}}x {\text{d}}y \\&\quad + \int\limits_{0}^{a} {} \int\limits_{0}^{b} {} \left[ \begin{aligned}& q + k_{w} \left( {w_{0} + f_{1}^{\prime} (z)\phi_{z} + f_{2}^{\prime} (z)\psi_{z} } \right)\\&\quad - k_{p1} \frac{{\partial^{2} }}{{\partial x^{2} }}\left( {w_{0} + f_{1}^{\prime} (z)\phi_{z} + f_{2}^{\prime} (z)\psi_{z} } \right) \\&\quad - k_{p2} \frac{{\partial^{2} }}{{\partial y^{2} }}\left( {w_{0} + f_{1}^{\prime} (z)\phi_{z} + f_{2}^{\prime} (z)\psi_{z} } \right) \\& \end{aligned} \right]\delta \phi_{z} {\text{d}}x {\text{d}}y \\&\quad + \int\limits_{0}^{a} {} \int\limits_{0}^{b} {} \left[ \begin{aligned}& q + k_{w} \left( {w_{0} + f_{1}^{\prime} (z)\phi_{z} + f_{2}^{\prime} (z)\psi_{z} } \right)\\&\quad - k_{p1} \frac{{\partial^{2} }}{{\partial x^{2} }}\left( {w_{0} + f_{1}^{\prime} (z)\phi_{z} + f_{2}^{\prime} (z)\psi_{z} } \right) \\&\quad - k_{p2} \frac{{\partial^{2} }}{{\partial y^{2} }}\left( {w_{0} + f_{1}^{\prime} (z)\phi_{z} + f_{2}^{\prime} (z)\psi_{z} } \right) \end{aligned} \right]\delta \psi_{z} \\&\quad\times\,{\text{d}}x {\text{d}}y = 0, \end{aligned} $$
(10)
where
$$ \begin{aligned} \left( {N_{x} ,M_{x}^{b} ,M_{x}^{{S_{1} }} ,M_{x}^{{S_{2} }} } \right) = \int\limits_{ - h/2}^{ + h/2} {\sigma_{x } \left[ {1,z,f_{1} (z),f_{2} (z)} \right]} {\text{d}}z, \hfill \\ \left( {N_{y} ,M_{y}^{b} ,M_{y}^{{S_{1} }} ,M_{y}^{{S_{2} }} } \right) = \int\limits_{ - h/2}^{ + h/2} {\sigma_{y } \left[ {1,z,f_{1} (z),f_{2} (z)} \right]} {\text{d}}z, \hfill \\ \left( {N_{xy} ,M_{xy}^{b} M_{xy}^{1} ,M_{xy}^{2} } \right) = \int\limits_{ - h/2}^{ + h/2} {\tau_{xy} \left[ {1,z,f_{1} (z),f_{2} (z)} \right]} {\text{d}}z, \hfill \\ \left( {Q_{xz}^{1} ,Q_{xz}^{2} } \right) = \int\limits_{ - h/2}^{ + h/2} {\tau_{xz } \left[ {f_{1}^{\prime} (z),f_{2}^{\prime} (z)} \right]} {\text{d}}z, \hfill \\ \left( {Q_{yz}^{1} ,Q_{yz}^{2} } \right) = \int\limits_{ - h/2}^{ + h/2} {\tau_{yz } \left[ {f_{1}^{\prime} (z),f_{2}^{\prime} (z)} \right]} {\text{d}}z, \hfill \\ Q_{z}^{1} = \int\limits_{ - h/2}^{ + h/2} {\sigma_{z } f_{1}^{\prime \prime} (z)} {\text{d}}z, \hfill \\ Q_{z}^{2} = \int\limits_{ - h/2}^{ + h/2} {\sigma_{z } f_{2}^{\prime \prime} (z)} {\text{d}}z. \hfill \\ \end{aligned} $$
(11)
Here, \( N_{x} ,N_{y} ,N_{xy} \) = resultant in-plane forces, \( M_{x}^{b} ,M_{y}^{b} ,M_{xy}^{b} \) = resultant moments analogous to classical plate theory, \( M_{x}^{{S_{1} }} ,M_{x}^{{S_{2} }} ,M_{xy}^{{S_{1} }} ,M_{xy}^{{S_{2} }} ,M_{y}^{{S_{1} }} ,M_{y}^{S2} \) = resultant moments due to shear deformation, \( Q_{xz}^{1} ,Q_{xz}^{2} ,Q_{z}^{1} ,Q_{z}^{2} ,Q_{yz}^{1} , Q_{yz}^{2} ,Q_{z}^{{S_{1} }} , Q_{z}^{{S_{2} }} \) = resultant shear forces.The governing differential equations of the present theory are obtained by integrating Eq. (10) by parts; collecting the coefficients of \( \delta u_{0} ,\delta v_{0} , \)\( \delta w_{0} ,\delta \phi_{x} ,\delta \phi_{y} ,\delta \psi_{x} ,\delta \psi_{y} ,\delta \phi_{z} ,\delta \phi_{z} \) and setting them equal to zero.
$$ \delta u_{0} :\frac{{\partial N_{x} }}{\partial x} + \frac{{\partial N_{xy} }}{\partial y} = 0 $$
(12)
$$ \delta v_{0} :\frac{{\partial N_{y} }}{\partial y} + \frac{{\partial N_{xy} }}{\partial x} = 0 $$
(13)
$$ \begin{aligned} &\delta w_{0} :\frac{{\partial^{2} M_{x}^{b} }}{{\partial x^{2} }} + \frac{{\partial^{2} M_{y}^{b} }}{{\partial y^{2} }} + 2\frac{{\partial^{2} M_{xy}^{b} }}{\partial x\partial y} \\&\quad+ q + k_{w} \left( {w_{0} + f_{1}^{\prime} (z)\phi_{z} + f_{2}^{\prime} (z)\psi_{z} } \right) \\&\quad- k_{p1} \frac{{\partial^{2} }}{{\partial x^{2} }}\left( {w_{0} + f_{1}^{\prime} (z)\phi_{z} + f_{2}^{\prime} (z)\psi_{z} } \right) \\&\quad - k_{p2} \frac{{\partial^{2} }}{{\partial y^{2} }}\left( {w_{0} + f_{1}^{\prime} (z)\phi_{z} + f_{2}^{\prime} (z)\psi_{z} } \right) = 0 \hfill \\ \end{aligned} $$
(14)
$$ \delta \phi_{x} :\frac{{\partial M_{x}^{{S_{1} }} }}{\partial x} + \frac{{\partial M_{xy}^{{S_{1} }} }}{\partial y} - Q_{{xz^{1} }} = 0 $$
(15)
$$ \delta \psi_{x} : \frac{{\partial M_{x}^{{S_{2} }} }}{\partial x} + \frac{{\partial M_{xy}^{{S_{2} }} }}{\partial y} - Q_{{xz^{2} }} = 0 $$
(16)
$$ \delta \phi_{y} : \frac{{\partial M_{y}^{{S_{1} }} }}{\partial y} + \frac{{\partial M_{xy}^{{S_{1} }} }}{\partial x} - Q_{{yz^{1} }} = 0 $$
(17)
$$ \delta \psi_{y} : \frac{{\partial M_{y}^{{S_{2} }} }}{\partial y} + \frac{{\partial M_{xy}^{{S_{2} }} }}{\partial x} - Q_{{yz^{2} }} = 0 $$
(18)
$$ \begin{aligned} &\delta \phi_{z} : \frac{{\partial Q_{xz}^{1} }}{\partial x} + \frac{{\partial Q_{yz}^{1} }}{\partial y} - Q_{z}^{1} \\ &\quad+ q + k_{w} \left( {w_{0} + f_{1}^{\prime} (z)\phi_{z} + f_{2}^{\prime} (z)\psi_{z} } \right)\\ &\quad - k_{p1} \frac{{\partial^{2} }}{{\partial x^{2} }}\left( {w_{0} + f_{1}^{\prime} (z)\phi_{z} + f_{2}^{\prime} (z)\psi_{z} } \right) \hfill \\&\quad - k_{p2} \frac{{\partial^{2} }}{{\partial y^{2} }}\left( {w_{0} + f_{1}^{\prime} (z)\phi_{z} + f_{2}^{\prime} (z)\psi_{z} } \right) = 0 \hfill \\ \end{aligned}$$
(19)
$$ \begin{aligned} &\delta \psi_{z} : \frac{{\partial Q_{xz}^{2} }}{\partial x} + \frac{{\partial Q_{yz}^{2} }}{\partial y} - Q_{z}^{2} \\ &\quad + q + k_{w} \left( {w_{0} + f_{1}^{\prime} (z)\phi_{z} + f_{2}^{\prime} (z)\psi_{z} } \right) \\ &\quad- k_{p1} \frac{{\partial^{2} }}{{\partial x^{2} }}\left( {w_{0} + f_{1}^{\prime} (z)\phi_{z} + f_{2}^{\prime} (z)\psi_{z} } \right) \hfill \\&\quad - k_{p2} \frac{{\partial^{2} }}{{\partial y^{2} }}\left( {w_{0} + f_{1}^{\prime} (z)\phi_{z} + f_{2}^{\prime} (z)\psi_{z} } \right) = 0 \hfill \\ \end{aligned} $$
(20)

3.4 Navier’s solution

A FG plate simply supported at its edges is considered for the analytical solutions. Analytical solutions are obtained using Navier solution technique. Following are the boundary conditions at the four edges of the simply supported plate.

at edges x = 0 and x = a;
$$ v_{0} = 0, w_{0} = 0, \phi_{y} = 0, \psi_{y} = 0, \phi_{z} = 0, \psi_{z} = 0, M_{x}^{b} = 0,M_{x}^{{s_{1} }} = 0,M_{x}^{{s_{2} }} = 0, N_{x} = 0 $$
(21)
at edges y = 0 and y = b;
$$ u_{0} = 0, w_{0} = 0, \phi_{x} = 0, \psi_{x} = 0, \phi_{z} = 0, \psi_{z} = 0, M_{y}^{b} = 0,M_{y}^{{s_{1} }} = 0,M_{y}^{{s_{2} }} = 0, N_{y} = 0 $$
(22)
The displacement variables are assumed in double trigonometric series as follows:
$$ \begin{aligned} \left( {u_{0} ,\phi_{x} ,\psi_{x} } \right) = \left( {u_{1} ,\phi_{1} ,\psi_{1} } \right)\cos \alpha x\sin \beta y , \hfill \\ \left( {v_{0} , \phi_{y} ,\psi_{y} } \right) = \left( {v_{1} ,\phi_{2} ,\psi_{2} } \right)\sin \alpha x\cos \beta y , \hfill \\ \left( {w_{0} ,\phi_{z} ,\psi_{z} } \right) = \left( {w_{1} ,\phi_{3} ,\psi_{3} } \right)\sin \alpha x\sin \beta y, \hfill \\ \end{aligned} $$
(23)
where \( \alpha = \pi /a, \beta = \pi /b \) and \( u_{1,} w_{1,} v_{1,} \phi_{1} , \phi_{2} , \phi_{3} ,\psi_{1} ,\psi_{2} , \psi_{3} \) are the unknown coefficients to be determined. The mechanical, thermal and moisture loads are also expanded in double trigonometric form.
$$ \begin{aligned} \left( {q,T_{0} ,C_{0} } \right) = \left( {q_{0} ,t_{0} ,c_{0} } \right)\sin \alpha x\sin \beta y, \hfill \\ \left( {T_{1} ,C_{1} } \right) = \left( {t_{1} ,c_{1} } \right)\sin \alpha x\sin \beta y, \hfill \\ \left( {T_{2} ,C_{2} } \right) = \left( {t_{2} ,c_{2} } \right)\sin \alpha x\sin \beta y, \hfill \\ \left( {T_{3} ,C_{3} } \right)) = \left( {t_{3} ,c_{3} } \right)\sin \alpha x\sin \beta y \hfill \\ \end{aligned} $$
(24)
Substitution of Eqs. (23) and (24) into the governing Eqs. (12)–(20) leads to the equation:
$$ \left[ K \right]\left\{ \Delta \right\} = \left\{ f \right\}, $$
(25)
where [K] is the stiffness matrix, \( \left\{ f \right\} \) is the force vector and \( \left\{ \Delta \right\} \) is the vector of unknowns. The elements of stiffness matrix is as follows:
$$ \begin{aligned}& K_{11} = - (A_{11} \alpha^{2} + A_{66} \beta^{2} ), K_{12} = - (A_{12} + A_{66} )\alpha \beta , K_{13} = B_{11} \alpha^{3} + B_{12} \alpha \beta^{2} + 2B_{66} \alpha \beta^{2} , \\& K_{14} = - (C_{11} \alpha^{2} + C_{66} \beta^{2} ), K_{15} = - (D_{11} \alpha^{2} + D_{66} \beta^{2} ), K_{16} = - (C_{12} + C_{66} )\alpha \beta , \\& K_{17} = - (D_{12} + D_{66} )\alpha \beta , K_{18} = I_{13} \alpha , K_{19} = J_{13} \alpha , K_{22} = - (A_{66} \alpha^{2} + A_{12} \beta^{2} ), \\& K_{23} = B_{12} \alpha^{2} \beta + 2B_{66} \alpha^{2} \beta + B_{22} \beta^{3} , K_{24} = - (C_{12} + C_{66} )\alpha .\beta , K_{25} = - (D_{12} + D_{66} )\alpha \beta , \\& K_{26} = - (C_{22} \beta^{2} + C_{66} \alpha^{2} ), K_{27} = - (D_{22} \beta^{2} + D_{66} \alpha^{2} ), K_{28} = I_{23} \beta , K_{29} = J_{23} \beta , \\& K_{33} = - (A_{S11} \alpha^{4} + 2A_{S12} \alpha^{2} \beta^{2} + A_{S22} \beta^{4} + 4A_{s66} \alpha^{2} \beta^{2} ) - (k_{w} + k_{{p_{1} }} \alpha^{2} + k_{{p_{2} }} \beta^{2} ) \\ &K_{34} = C_{S11} \alpha^{3}+C_{S12} \alpha\beta^{2}+2C_{S66}\alpha\beta^{2}, \\ &K_{35} = D_{S11} \alpha^{3}+D_{S12} \alpha\beta^{2}+2D_{s66}\alpha\beta^{2}, \\ & K_{36} = C_{S12} \alpha^{2} \beta + C_{S22} \beta^{3} + 2C_{S66} \alpha^{2} \beta ,\, K_{37} = D_{S12} \alpha^{2} \beta + D_{S22} \beta^{3} + 2D_{S66} \alpha^{2} \beta , \\& K_{38} = - (I_{S13} \alpha^{2} + I_{S23} \beta^{2} ) - (k_{w} + k_{{p_{1} }} \alpha^{2} + k_{{p_{2} }} \beta^{2} ) , \\& K_{39} = - (J_{S13} \alpha^{2} + J_{S23} \beta^{2} ) - (k_{w} + k_{{p_{1} }} \alpha^{2} + k_{{p_{2} }} \beta^{2} ), K_{41} = - (C_{11} \alpha^{2} + C_{66} \beta^{2} ), \\& K_{44} = - (C_{SS111} \alpha^{2} + C_{SS166} \beta^{2} + C_{SSS144} ), K_{45} = - (C_{SS211} \alpha^{2} + C_{SS266} \beta^{2} + C_{SSS244} ), \\& K_{46} = - (C_{SS112} + C_{SS166} )\alpha \beta , K_{47} = - (C_{SS212} + C_{SS266} )\alpha \beta , K_{48} = (I_{SS113} - C_{SSS144} )\alpha , \\& K_{49} = (J_{SS113} - D_{SSS144} )\alpha , K_{55} = - (D_{SS211} \alpha^{2} + D_{SS266} \beta^{2} + D_{SSS244} ), K_{56} = - (C_{SS212} + C_{SS266} )\alpha \beta , \\& K_{57} = - (D_{SS212} + D_{SS266} )\alpha .\beta , K_{58} = (I_{SS213} - C_{SSS244} )\alpha , K_{59} = (J_{SS213} - D_{SSS244} )\alpha , \\& K_{66} = - (C_{SS122} \beta^{2} + C_{SS166} \alpha^{2} + C_{SSS155} ), K_{67} = - (C_{SS222} \beta^{2} + C_{SS266} \alpha^{2} + C_{SSS255} ), \\& K_{68} = (I_{SS123} - C_{SSS155} )\beta , K_{69} = (J_{SS123} - C_{SSS255} )\beta , K_{77} = - (D_{SS222} \beta^{2} + D_{SS266} \alpha^{2} + D_{SSS255} ), \\& K_{78} = (I_{SS233} - C_{SSS255} )\beta , K_{79} = (J_{SS233} - D_{SSS255} )\beta , \\& K_{88} = - (C_{SSS144} \alpha^{2} + C_{SS155} \beta^{2} + I_{SSS133} ) - (k_{w} f_{1}^{\prime} (z) + k_{{p_{1} }} f_{1}^{\prime} (z) \alpha^{2} + k_{{p_{2} }} f_{1}^{\prime} (z) \beta^{2} ), \\& K_{89} = - (C_{SSS244} \alpha^{2} + C_{SS255} \beta^{2} + I_{SSS233} ) - (k_{w} f_{1}^{\prime} (z) + k_{{p_{1} }} f_{1}^{\prime} (z) \alpha^{2} + k_{{p_{2} }} f_{1}^{\prime} (z) \beta^{2} ), \\& K_{99} = - (D_{SSS244} \alpha^{2} + D_{SSS255} \beta^{2} - J_{SSS233} ) - (k_{w} f_{2}^{\prime} (z) + k_{{p_{1} }} f_{2}^{\prime} (z) \alpha^{2} + k_{{p_{2} }} f_{2}^{\prime} (z) \beta^{2} ), \\& {\text{symmetrical}}\, {\text{elements}} \\& K_{21} = K_{12} , K_{31} = K_{13} , K_{41} = K_{14} , K_{51} = K_{15} , K_{61} = K_{16} , K_{71} = K_{17} , K_{81} = K_{18} , K_{91} = K_{19} , \\& K_{32} = K_{23} , K_{42} = K_{24} , K_{52} = K_{25} , K_{62} = K_{26} , K_{72} = K_{27} , K_{82} = K_{28} , K_{92} = K_{29} , K_{43} = K_{34} , \\& K_{53} = K_{35} , K_{63} = K_{36} , K_{73} = K_{37} , K_{83} = K_{38} , K_{93} = K_{39} , K_{54} = K_{45} , K_{64} = K_{46} , K_{74} = K_{47} , \\& K_{84} = K_{48} , K_{94} = K_{49} , K_{65} = K_{56} , K_{75} = K_{57} , K_{85} = K_{58} , K_{95} = K_{59} , K_{76} = K_{67} , K_{86} = K_{68} , \\& K_{96} = K_{69} , K_{87} = K_{78} , K_{97} = K_{79} , K_{98} = K_{89} \\& \end{aligned} $$
(26)
Vector of unknown is
$$ \left\{ \Delta \right\} = \left\{ {u_{mn} , v_{mn} , w_{mn} , \phi_{xmn} , \psi_{xmn} , \phi_{ymn} , \psi_{ymn} , \phi_{zmn} , \psi_{zmn} } \right\}^{T} . $$
(27)
The elements of force vector {f} are as follows, stated in Eqs. (28)–(36)
$$ \begin{aligned} f_{1}& = - (A_{11}^{T} + A_{12}^{T} + A_{13}^{T} )t_{0} \alpha - (B_{11}^{T} + B_{12}^{T} + B_{13}^{T} )t_{1} \alpha\\&\quad - (C_{11}^{T} + C_{12}^{T} + C_{13}^{T} )t_{2} \alpha - (D_{11}^{T} + D_{12}^{T} + D_{13}^{T} )t_{3} \alpha \\&\quad - (A_{11}^{C} + A_{12}^{C} + A_{13}^{C} )c_{0} \alpha - (B_{11}^{C} + B_{12}^{C} + B_{13}^{C} )c_{1} \alpha \\&\quad- (C_{11}^{C} + C_{12}^{C} + C_{13}^{C} )c_{2} \alpha - (D_{11}^{C} + D_{12}^{C} + D_{13}^{C} )c_{3} \alpha \hfill \\ \end{aligned} $$
(28)
$$ \begin{aligned} f_{2}& = - (A_{12}^{T} + A_{22}^{T} + A_{23}^{T} )t_{0} \beta - (B_{12}^{T} + B_{22}^{T} + B_{23}^{T} )t_{1} \beta\\&\quad - (C_{12}^{T} + C_{22}^{T} + C_{23}^{T} )t_{2} \beta - (D_{12}^{T} + D_{22}^{T} + D_{23}^{T} )t_{3} \beta \\&\quad - (A_{12}^{C} + A_{22}^{C} + A_{23}^{C} )c_{0} \beta - (B_{12}^{C} + B_{22}^{C} + B_{23}^{C} )c_{1} \beta\\&\quad - (C_{12}^{C} + C_{22}^{C} + C_{23}^{C} )c_{2} \beta - (D_{12}^{C} + D_{22}^{C} + D_{23}^{C} )c_{3} \beta \hfill \\ \end{aligned} $$
(29)
$$ \begin{aligned} f_{3}& = - (B_{11}^{T} + B_{12}^{T} + B_{13}^{T} )t_{0} \alpha^{2} - (AS_{11}^{T} + AS_{12}^{T} + AS_{13}^{T} )t_{1} \alpha^{2}\\&\quad - (CS_{11}^{T} + CS_{12}^{T} + CS_{13}^{T} )t_{2} \alpha^{2} - (DS_{11}^{T} + DS_{12}^{T} + DS_{13}^{T} )t_{3} \alpha^{2} \\&\quad - (B_{12}^{T} + B_{22}^{T} + B_{23}^{T} )t_{0} \beta^{2} - (AS_{12}^{T} + AS_{22}^{T} + AS_{23}^{T} )t_{1} \beta^{2}\\&\quad - (CS_{12}^{T} + CS_{22}^{T} + CS_{23}^{T} )t_{2} \beta^{2} - (DS_{12}^{T} + DS_{22}^{T} + DS_{23}^{T} )t_{3} \beta^{2} \\&\quad - (B_{11}^{C} + B_{12}^{C} + B_{13}^{C} )c_{0} \alpha^{2} - (AS_{11}^{C} + AS_{12}^{C} + AS_{13}^{C} )c_{1} \alpha^{2}\\&\quad - (CS_{11}^{C} + CS_{12}^{C} + CS_{13}^{C} )c_{2} \alpha^{2} - (DS_{12}^{C} + DS_{22}^{C} + DS_{23}^{C} )c_{3} \alpha^{2} \\&\quad - (B_{12}^{C} + B_{22}^{C} + B_{23}^{C} )c_{0} \beta^{2} - (AS_{12}^{C} + AS_{22}^{C} + AS_{23}^{C} )c_{1} \beta^{2}\\&\quad - (CS_{12}^{C} + CS_{22}^{C} + CS_{23}^{C} )c_{2} \beta^{2}\\&\quad - (DS_{12}^{C} + DS_{22}^{C} + DS_{23}^{C} )c_{3} \beta^{2} - q \end{aligned} $$
(30)
$$ \begin{aligned} f_{4}& = - (C_{11}^{T} + C_{12}^{T} + C_{13}^{T} )t_{0} \alpha - (CS_{11}^{T} + CS_{12}^{T} + CS_{13}^{T} )t_{1} \alpha \\&\quad- (CSS_{111}^{T} + CSS_{112}^{T} + CSS_{113}^{T} )t_{2} \alpha - (CSS_{211}^{T} + CSS_{212}^{T}\\&\quad + CSS_{213}^{T} )t_{3} \alpha - (C_{11}^{C} + C_{12}^{C} + C_{33}^{C} )c_{0} \alpha - (CS_{11}^{C} + CS_{12}^{C}\\&\quad + CS_{13}^{C} )c_{1} \alpha - (CSS_{111}^{C} + CSS_{112}^{C} + CSS_{113}^{C} )c_{2} \alpha\\&\quad - (CSS_{211}^{C} + CSS_{212}^{C} + CSS_{213}^{C} )c_{3} \alpha \end{aligned} $$
(31)
$$ \begin{aligned} f_{5}& = - (D_{11}^{T} + D_{12}^{T} + D_{13}^{T} )t_{0} \alpha - (DS_{11}^{T} + DS_{12}^{T} + DS_{13}^{T} )t_{1} \alpha \\&\quad- (CSS_{211}^{T} + CSS_{212}^{T} + CSS_{213}^{T} )t_{2} \alpha - (DSS_{211}^{T} + DSS_{212}^{T} \\&\quad+ DSS_{213}^{T} )t_{3} \alpha - (D_{11}^{C} + D_{12}^{C} + D_{13}^{C} )c_{0} \alpha - (DS_{11}^{C} + DS_{12}^{C} \\&\quad+ DS_{13}^{C} )c_{1} \alpha - (CSS_{211}^{C} + CSS_{212}^{C} + CSS_{213}^{C} )c_{2} \alpha - (DSS_{211}^{C}\\&\quad + DSS_{212}^{C} + DSS_{213}^{C} )c_{3} \alpha \end{aligned} $$
(32)
$$ \begin{aligned} f_{6} &= - (C_{12}^{T} + C_{22}^{T} + C_{23}^{T} )t_{0} \beta - (CS_{12}^{T} + CS_{22}^{T} + CS_{23}^{T} )t_{1} \beta\\&\quad - (CSS_{112}^{T} + CSS_{122}^{T} + CSS_{123}^{T} )t_{2} \beta - (CSS_{212}^{T} + CSS_{222}^{T}\\&\quad + CSS_{223}^{T} )t_{3} \beta - (C_{12}^{C} + C_{22}^{C} + C_{23}^{C} )c_{0} \beta - (CS_{12}^{C}\\&\quad + CS_{22}^{C} + CS_{23}^{C} )c_{1} \beta - (CSS_{112}^{C} + CSS_{122}^{C} + CSS_{123}^{C} )c_{2} \beta \\&\quad- (CSS_{212}^{C} + CSS_{222}^{C} + CSS_{223}^{C} )c_{3} \beta \end{aligned} $$
(33)
$$ \begin{aligned} f_{7}& = - (D_{12}^{T} + D_{22}^{T} + D_{23}^{T} )t_{0} \beta - (DS_{12}^{T} + DS_{22}^{T} + DS_{23}^{T} )t_{1} \beta\\&\quad - (CSS_{212}^{T} + CSS_{222}^{T} + CSS_{223}^{T} )t_{2} \beta - (DSS_{212}^{T} + DSS_{222}^{T} \\&\quad+ DSS_{223}^{T} )t_{3} \beta - (D_{12}^{C} + D_{22}^{C} + D_{23}^{C} )c_{0} \beta - (DS_{12}^{C} + DS_{22}^{C}\\&\quad + DS_{23}^{C} )c_{1} \beta - (CSS_{212}^{C} + CSS_{222}^{C} + CSS_{223}^{C} )c_{2} \beta\\&\quad - (DSS_{212}^{C} + DSS_{222}^{C} + DSS_{223}^{C} )c_{3} \beta \end{aligned} $$
(34)
$$ \begin{aligned} f_{8} &= - (I_{13}^{T} + I_{23}^{T} + I_{33}^{T} )t_{0} - (IS_{13}^{T} + IS_{23}^{T} + IS_{33}^{T} )t_{1} - (ISS_{113}^{T} \\&\quad+ ISS_{123}^{T} + ISS_{133}^{T} )t_{2} - (ISS_{213}^{T} + ISS_{223}^{T} + ISS_{233}^{T} )t_{3}\\&\quad - (I_{13}^{C} + I_{23}^{C} + I_{33}^{C} )c_{0} - (IS_{13}^{C} + IS_{23}^{C} + IS_{33}^{C} )c_{1} \\&\quad - (ISS_{113}^{C} + ISS_{123}^{C} + ISS_{133}^{C} )c_{2} - (ISS_{213}^{C} \\&\quad+ ISS_{223}^{C} + ISS_{233}^{C} )c_{3} - f_{1}^{\prime} \left( z \right)q \end{aligned} $$
(35)
$$ \begin{aligned} f_{9}& = - (J_{13}^{T} + J_{23}^{T} + J_{33}^{T} )t_{0} - (JS_{13}^{T} + JS_{23}^{T} + JS_{33}^{T} )t_{1} - (JSS_{113}^{T}\\&\quad + JSS_{123}^{T} + JSS_{133}^{T} )t_{2} - (JSS_{213}^{T} + JSS_{223}^{T} + JSS_{233}^{T} )t_{3} \\&\quad- (J_{13}^{C} + J_{23}^{C} + J_{33}^{C} )c_{0} - (JS_{13}^{C} + JS_{23}^{C} + JS_{33}^{C} )c_{1}\\&\quad - (JSS_{113}^{C} + JSS_{123}^{C} + JSS_{133}^{C} )c_{2} \\&\quad- (JSS_{213}^{C} + JSS_{223}^{C} + JSS_{233}^{C} )c_{3} - f_{2}^{\prime} \left( z \right)q \end{aligned} $$
(36)

4 Illustrative examples and discussion

To illustrate the accuracy and efficiency of the present theory, the authors have presented the results with and without considering thickness stretching effect which is mentioned below through numerical examples 1 through 3. The functionally graded plate made up of Titanium (metal) and Zerconia (ceramic) materials are considered for the analysis.

Properties of these materials are as follows:
$$ \begin{aligned} &{\text{Titanium }}\left( {{\text{Ti}} - 6 {\text{Al}} - 4 {\text{V}}} \right) :E_{\text{m}}\, = \, 6 6. 2 {\text{ GPa}},\mu \, = \,0. 3 3,\\&\quad\alpha \, = \, 10. 3\times \left( { 10^{ - 6} /{^\circ }{\text{C}}} \right),\beta = 0.33 \\& {\text{Zirconia }}\left( {{\text{ZrO}}_{ 2} } \right): \, E_{\text{c}}\, = \, 1 1 7 {\text{ GPa}}, \, \mu \, = \,0. 3 3,\\&\quad\alpha \, = \, 7. 1 1 { } \times \left( { 10^{ - 6} /{^\circ }{\text{C}}} \right),\beta = 0 \end{aligned} $$
Following numerical examples are solved in the present study to predict hygro-thermo-mechanical response of FG plate:
  1. 1.

    FG plate resting on elastic foundation and subjected to mechanical load (T = C=0).

     
  2. 2.

    FG plate resting on elastic foundation and subjected to linear hygro-thermo-mechanical load (T1 = 10, C1 = 100).

     
  3. 3.

    FG plate resting on elastic foundation and subjected to non-linear hygro-thermo-mechanical load (T1 = T2 = 10, C1 = C2 = 100).

     
The results are presented in the non-dimensional form, shown as follows:
$$ \begin{aligned}& \bar{w} \left( {\frac{a}{2},\frac{b}{2},\frac{z}{h}} \right) = \frac{{100E_{c} h^{3} }}{{q_{0} a^{4} }}(w),\\&(\bar{\sigma }_{x} ,\bar{\sigma }_{y} ) \left( {\frac{a}{2},\frac{b}{2},\frac{z}{h}} \right), = \frac{h}{{q_{0} a}} (\sigma_{x,} \sigma_{y,} ), \\& (\bar{\tau }_{xy} ) \left( {0,0,\frac{z}{h}} \right), = \frac{h}{{q_{0} a}}(\tau_{xy} ),\\&(\bar{\tau }_{xz} )\left( {\frac{a}{2},0,\frac{z}{h}} \right) = \frac{h}{{q_{0} a}} (\tau_{xz} ), \\& (\bar{\tau }_{yz} ) \left( {0,\frac{b}{2},\frac{z}{h}} \right) = \frac{h}{{q_{0} a}}(\tau_{yz} ),\\&k_{0} = \frac{{k_{w} D}}{{a^{4} }}, k_{P1} = \frac{{k_{1} D}}{{a^{2} }}, \\& k_{P2} = \frac{{k_{2} D}}{{b^{2} }}, D = \frac{{h^{3} E_{C} }}{{12(1 - \mu^{2} )}},\\&AR = a/h. \\& \end{aligned} $$
(37)
Numerical results are tabulated in Tables 1 through 5 and graphically plotted in Figs. 4 and 5. Validation of the present theory for mechanical analysis is presented in Table 1, whereas hygro-thermo-mechanical analysis is presented in Tables 2 through 5. Plate theories developed and presented by Kirchhoff [1], Mindlin [2], Zidi et al. [39], Reddy [43] and Zenkour [44] are used for the comparison purpose. The room temperature and initial moisture concentration are taken by 25 °C and 0%, respectively. Table 1 shows comparison of displacements and stresses for FG rectangular plate (b = 3a) subjected to sinusoidal mechanical load. From Table 1, it is observed that the transverse deflection and stresses are decreasing with increase in stiffness of the elastic foundation, i.e. the inclusion of Pasternak foundation parameters yields lowest value of deflection and stresses. Also, as the power law index increases, the deflection will increase and stresses will decrease. This is due increase in flexibility of the plate with increase in power law index. It is important to observe that the stresses for a fully ceramic plate are not the same as that for a fully metal plate with elastic foundations. Table 2 shows effects of power law index and side-to-thickness ratio on the non-dimensional transverse displacement of square (b = a) FG plates resting on elastic foundation and subjected to linear and non-linear hygro-thermo-mechanical loadings. Examination of Table 2 reveals that non-dimensional deflection increases with inclusion of non-linear hygro-thermal effect in the plate. This is due to the considerable effect of temperature and moisture on the extensional behavior of the plate comparatively the flexural behavior. It is also observed that the transverse deflection decreases with increases in side-to-thickness ratio. In addition to this, it is also pointed out that the transverse deflections are observed to decrease with the increase in stiffness of the elastic foundations. Effect of non-linear hygro-thermo-mechanical load on deflection of FG rectangular plate (b = 3a) is presented in Table 3. From Table 3 it is point out that the transverse deflection values are more for rectangular plate compared to square plate. Other variations are similar to square plate. Table 4 represents the effect of power law index (p) on the non-dimensional displacements and stresses of on FG rectangular plates subjected to linear hygro-thermo-mechanical loadings. Table shows the comparison between present fifth-order shear and normal deformation theory (FOSNDT) and different higher order theories, first order shear deformation theory (FSDT) and classical plate theory (CPT). Table 4 shows that the present results are in good agreement with other higher order theories. It is observed that the deflection and stresses are decreasing from without elastic foundation to two-parameter elastic foundation. The inclusion of Pasternak foundation parameters yields lowest value of deflection and stresses. As the power law index increases (p), the deflection is increases and stresses are decreases. This is due increase in flexibility of the plate with increase in power law index. Table 5 represents the effect of power law index on the non-dimensional displacements and stresses of on FG rectangular plates resting on elastic foundation and subjected to non-linear hygro-thermo-mechanical loadings. Figures 4 and 5 show the variation of non-dimensional stresses with respect to thickness coordinate (z/h) for rectangular FG plate resting on elastic foundation.
Table 1

Effect of power law index on the non-dimensional displacements and stresses of an FG rectangular plates (b = 3a) resting on elastic foundation and subjected to mechanical loadings with (\( \varepsilon_{z} \ne 0 \)) and without (\( \varepsilon_{z} = 0 \)) considering the effect of thickness stretching (a/h = 10, q0 = 100, T = 0, C = 0)

P

k0

k1

k2

Theory

Model

\( \bar{w} \)

\( \bar{\sigma }_{\varvec{x}} \)

\( \bar{\tau }_{xy} \)

\( \bar{\tau }_{{\varvec{xz}}} \)

0

0

0

0

Present (\( \varepsilon_{z} \ne 0 \))

FOSNDT

0.85890

0.51230

0.72570

0.42920

    

Present (\( \varepsilon_{z} = 0 \))

FOSNDT

0.85540

0.51115

0.72880

0.42920

    

Zidi et al. [39]

FVRPT

0.85891

0.51545

0.72797

0.42956

    

Zenkour [44]

SSDT

0.85891

0.51545

0.72797

0.42956

    

Reddy [43]

TSDT

0.85887

0.51362

0.72784

0.44327

    

Mindlin [2]

FSDT

0.85892

0.51065

0.72949

0.34377

    

Kirchhoff [1]

CPT

0.83156

0.51065

0.72949

 

100

0

0

Present (\( \varepsilon_{z} \ne 0 \))

FOSNDT

0.46240

0.27710

0.39230

0.23220

    

Present (\( \varepsilon_{z} = 0 \))

FOSNDT

0.46200

0.27620

0.39240

0.23090

    

Zidi et al. [39]

FVRPT

0.46206

0.27620

0.39162

0.23109

    

Zenkour [44]

SSDT

0.46206

0.27620

0.39162

0.23100

    

Reddy [43]

TSDT

0.46205

0.27631

0.39158

0.23847

    

Mindlin [2]

FSDT

0.46206

0.27471

0.39246

0.18493

    

Kirchhoff [1]

CPT

0.45402

0.27879

0.39830

 

100

100

100

Present (\( \varepsilon_{z} \ne 0 \))

FOSNDT

0.08270

0.04960

0.07020

0.04150

    

Present (\( \varepsilon_{z} = 0 \))

FOSNDT

0.08230

0.04920

0.06990

0.04110

    

Zidi et al. [39]

FVRPT

0.08228

0.04919

0.06972

0.04116

    

Zenkour [44]

SSDT

0.08224

0.04919

0.06972

0.04246

    

Reddy [43]

TSDT

0.08224

0.04919

0.06972

0.04246

    

Mindlin [2]

FSDT

0.08228

0.04890

0.06986

0.03293

    

Kirchhoff [1]

CPT

0.08201

0.05035

0.07193

1

100

100

100

Present (\( \varepsilon_{z} \ne 0 \))

FOSNDT

0.08460

0.04580

0.05060

0.03260

    

Present (\( \varepsilon_{z} = 0 \))

FOSNDT

0.08400

0.04540

0.05200

0.03229

    

Zidi et al. [39]

FVRPT

0.08418

0.04574

0.05190

0.03214

    

Zenkour [44]

SSDT

0.08417

0.04574

0.05190

0.03318

    

Reddy [43]

TSDT

0.08417

0.04574

0.05191

0.03318

    

Mindlin [2]

FSDT

0.08418

004546

0.05198

0.02572

    

Kirchhoff [1]

CPT

0.08399

0.04683

0.05353

5

100

100

100

Present (\( \varepsilon_{z} \ne 0 \))

FOSNDT

0.08530

0.04660

0.04480

0.02830

    

Present (\( \varepsilon_{z} = 0 \))

FOSNDT

0.08476

0.04620

0.04560

0.02790

    

Zidi et al. [39]

FVRPT

0.08491

0.04652

0.04549

0.02735

    

Zenkour [44]

SSDT

0.08491

0.04652

0.04549

0.02832

    

Reddy [43]

TSDT

0.08492

0.04656

0.04549

0.02832

    

Mindlin [2]

FSDT

0.08491

0.04630

0.04568

0.02085

    

Kirchhoff [1]

CPT

0.08471

0.04780

0.04714

Metal

100

100

100

Present (\( \varepsilon_{z} \ne 0 \))

FOSNDT

0.08630

0.02930

0.04130

0.02450

    

Present (\( \varepsilon_{z} = 0 \))

FOSNDT

0.08580

0.02900

0.04150

0.02430

    

Zidi et al. [39]

FVRPT

0.08584

0.02905

0.04115

0.02428

    

Zenkour [44]

SSDT

0.08584

0.02906

0.04116

0.02507

    

Reddy [43]

TSDT

0.08584

0.02906

0.04116

0.02428

    

Mindlin [2]

FSDT

0.08584

0.02888

0.04126

0.01943

    

Kirchhoff [1]

CPT

0.08569

0.02978

0.04250

Fig. 4

Variation of non-dimensional transverse shear stress (\( \bar{\tau }_{xz} \)) through the thickness of rectangular FG plate resting on elastic foundation and subjected to non-linear hygro-thermo-mechanical load

Fig. 5

Variation of non-dimensional in-plane shear stress (\( \bar{\tau }_{xy} \)) through the thickness of rectangular FG plate resting on elastic foundation and subjected to non-linear hygro-thermo-mechanical load

Table 2

Effect of power law index on the non-dimensional transverse displacement of FG square plates (b = a) resting on elastic foundation and subjected to hygro-thermo-mechanical loadings

P

k0

k1

k2

q0

t1

t2

t3

C1

C2

C3

a/h

5

10

20

50

100

0

100

0

0

100

0

0

0

0

0

0

0.2381

0.2130

0.2065

0.2046

0.2043

    

100

10

0

0

100

0

0

1.9795

0.6345

0.3109

0.2213

0.2085

    

100

10

10

0

100

100

0

3.5061

0.9816

0.3960

0.2347

0.2119

    

100

10

10

10

100

100

100

6.2323

1.7254

0.5854

0.2653

0.2195

 

0

100

100

100

0

0

0

0

0

0

0.0448

0.0431

0.0425

0.0423

0.0423

    

100

10

0

0

100

0

0

0.3728

0.1282

0.0640

0.0458

0.0432

    

100

10

10

0

100

100

0

0.6603

0.1984

0.0814

0.0486

0.0439

    

100

10

10

10

100

100

100

1.1737

0.3487

0.1205

0.0549

0.0455

 

100

100

100

100

0

0

0

0

0

0

0.0430

0.0413

0.0408

0.0406

0.0406

    

100

10

0

0

100

0

0

0.3573

0.1230

0.0614

0.0439

0.0414

    

100

10

10

0

100

100

0

0.6328

0.1903

0.0781

0.0466

0.0421

    

100

10

10

10

100

100

100

1.1249

0.3345

0.1156

0.0527

0.0436

1

100

0

0

100

0

0

0

0

0

0

0.2889

0.2604

0.2528

0.2506

0.2503

    

100

10

0

0

100

0

0

2.6316

0.8316

0.3946

0.2734

0.2560

    

100

10

10

0

100

100

0

4.6811

1.3030

0.5100

0.2919

0.2609

    

100

10

10

10

100

100

100

8.3717

2.3170

0.7693

0.3336

0.2713

 

0

100

100

100

0

0

0

0

0

0

0.0448

0.442

0.0440

0.0440

0.0440

    

100

10

0

0

100

0

0

0.4220

0.1427

0.0689

0.0480

0.0450

    

100

10

10

0

100

100

0

0.7503

0.2235

0.0890

0.0512

0.0458

    

100

10

10

10

100

100

100

1.3419

0.3973

0.1343

0.0585

0.0476

 

100

100

100

100

0

0

0

0

0

0

0.0429

0.0423

0.0421

0.0421

0.0421

    

100

10

0

0

100

0

0

0.4038

0.1366

0.0660

0.0459

0.0431

    

100

10

10

0

100

100

0

0.7181

0.2140

0.0853

0.0490

0.0439

    

100

10

10

10

100

100

100

1.2842

0.3805

0.1286

0.0561

0.0456

2

100

0

0

100

0

0

0

0

0

0

0.3041

0.2729

0.2646

0.2622

0.2618

    

100

10

0

0

100

0

0

2.7412

0.8698

0.4129

0.2860

0.2678

    

100

10

10

0

100

100

0

4.9113

1.3630

0.5336

0.3051

0.2726

    

100

10

10

10

100

100

100

8.7203

2.4241

0.8053

0.3489

0.2835

 

0

100

100

100

0

0

0

0

0

0

0.0467

0.0450

0.0445

0.0443

0.0443

    

100

10

0

0

100

0

0

0.4207

0.1435

0.0694

0.0483

0.0453

    

100

10

10

0

100

100

0

0.7537

0.2252

0.0897

0.0516

0.0461

    

100

10

10

10

100

100

100

1.3383

0.3999

0.1354

0.0590

0.0480

 

100

100

100

100

0

0

0

0

0

0

0.0432

0.0426

0.0425

0.0424

0.0424

    

100

10

0

0

100

0

0

0.4025

0.1374

0.0665

0.0463

0.0434

    

100

10

10

0

100

100

0

0.7212

0.2156

0.0859

0.0494

0.0442

    

100

10

10

10

100

100

100

1.2805

0.3828

0.1297

0.0565

0.0459

5

100

0

0

100

0

0

0

0

0

0

0.3202

0.2859

0.2767

0.2741

0.2737

    

100

10

0

0

100

0

0

2.8569

0.9085

0.4315

0.2989

0.2799

    

100

10

10

0

100

100

0

5.1346

1.4224

0.5563

0.3187

0.2849

    

100

10

10

10

100

100

100

9.2104

2.5558

0.8471

0.3656

0.2966

 

0

100

100

100

0

0

0

0

0

0

0.0470

0.0454

0.0448

0.0447

0.0446

    

100

10

0

0

100

0

0

0.4194

0.1441

0.0699

0.0487

0.0456

    

100

10

10

0

100

100

0

0.7539

0.2257

0.0901

0.0519

0.0464

    

100

10

10

10

100

100

100

1.3522

0.4054

0.1372

0.0595

0.0484

 

100

100

100

100

0

0

0

0

0

0

0.0450

0.0434

0.0429

0.0427

0.0427

    

100

10

0

0

100

0

0

0.4012

0.1379

0.0669

0.0466

0.0437

    

100

10

10

0

100

100

0

0.7211

0.2160

0.0863

0.0497

0.0445

    

100

10

10

10

100

100

100

1.2934

0.3880

0.1313

0.0570

0.0463

Metal

100

0

0

100

0

0

0

0

0

0

0.3573

0.3238

0.3148

0.3122

0.3118

    

100

10

0

0

100

0

0

3.2192

1.0256

0.4892

0.3402

0.3188

    

100

10

10

0

100

100

0

5.7314

1.6043

0.6307

0.3626

0.3244

    

100

10

10

10

100

100

100

10.451

2.8994

0.9619

0.4160

0.3378

 

0

100

100

100

0

0

0

0

0

0

0.0478

0.0462

0.0457

0.0456

0.0455

    

100

10

0

0

100

0

0

0.4307

0.1464

0.0711

0.0496

0.0466

    

100

10

10

0

100

100

0

0.7668

0.2290

0.0916

0.0529

0.0474

    

100

10

10

10

100

100

100

1.3983

0.4139

0.1397

0.0607

0.0493

 

100

100

100

100

0

0

0

0

0

0

0.0457

0.0442

0.0437

0.0436

0.0436

    

100

10

0

0

100

0

0

0.4117

0.1400

0.0480

0.0474

0.0445

    

100

10

10

0

100

100

0

0.7330

0.2190

0.0876

0.0506

0.0453

    

100

10

10

10

100

100

100

1.3366

0.3958

0.1336

0.0580

0.0472

Table 3

Effect of power law exponent on the non-dimensional transverse displacement of FG rectangular plates (b = 3a) resting on elastic foundation and subjected to hygro-thermo-mechanical loadings

P

k0

k1

k2

q0

t1

t2

t3

C1

C2

C3

a/h

    

5

10

20

50

100

0

100

0

0

100

10

0

0

100

0

0

2.5514

0.9774

0.5848

0.4750

0.4593

    

100

10

10

0

100

100

0

4.2987

1.4021

0.6886

0.4916

0.4634

    

100

10

10

10

100

100

100

7.7704

2.3224

0.9236

0.5293

0.4729

 

0

100

100

100

10

0

0

100

0

0

0.4838

0.1905

0.1148

0.0934

0.0904

    

100

10

10

0

100

100

0

0.8151

0.2745

0.1352

0.0967

0.0912

    

100

10

10

10

100

100

100

1.4733

0.4526

0.1813

0.1041

0.0930

 

100

100

100

100

10

0

0

100

0

0

0.4438

0.1748

0.1054

0.0858

0.0829

    

100

10

10

0

100

100

0

0.7477

0.2449

0.1242

0.0887

0.0837

    

100

10

10

10

100

100

100

1.3515

0.4154

0.1664

0.0956

0.0854

1

100

0

0

100

10

0

0

100

0

0

3.1559

1.1806

0.6849

0.5461

0.5262

    

100

10

10

0

100

100

0

5.3562

1.7132

0.8169

0.5671

0.5314

    

100

10

10

10

100

100

100

9.7567

2.8914

1.1165

0.6153

0.5435

 

0

100

100

100

10

0

0

100

0

0

0.5381

0.2063

0.1204

0.0962

0.0927

    

100

10

10

0

100

100

0

0.9133

0.3020

0.1436

0.0999

0.0936

    

100

10

10

10

100

100

100

1.6637

0.5052

0.1963

0.1084

0.0957

 

100

100

100

100

10

0

0

100

0

0

0.4927

0.1852

0.1094

0.0881

0.0849

    

100

10

10

0

100

100

0

0.8362

0.2687

0.1316

0.0915

0.0858

    

100

10

10

10

100

100

100

1.5233

0.4627

0.1798

0.0933

0.0877

2

100

0

0

100

10

0

0

100

0

0

3.2348

1.2146

0.7049

0.5619

0.5414

    

100

10

10

0

100

100

0

5.5142

1.7647

0.8411

0.5836

0.5469

    

100

10

10

10

100

100

100

10.005

2.9771

1.1500

0.6333

0.5593

 

0

100

100

100

10

0

0

100

0

0

0.5378

0.2071

0.1210

0.0967

0.0932

    

100

10

10

0

100

100

0

0.9167

0.3010

0.1444

0.1004

0.0941

    

100

10

10

10

100

100

100

1.6634

0.5078

0.1975

0.1089

0.0962

 

100

100

100

100

10

0

0

100

0

0

0.4922

0.1868

0.1108

0.0885

0.0883

    

100

10

10

0

100

100

0

0.8390

0.2659

0.1322

0.0919

0.0862

    

100

10

10

10

100

100

100

1.5224

0.4649

0.1808

0.0998

0.0881

5

100

0

0

100

10

0

0

100

0

0

3.3117

1.2473

0.7245

0.5776

0.5566

    

100

10

10

0

100

100

0

5.6439

1.8072

0.8629

0.5597

0.5621

    

100

10

10

10

100

100

100

10.358

3.0826

1.1881

0.6520

0.5752

 

0

100

100

100

10

0

0

100

0

0

0.5369

0.2078

0.1215

0.0971

0.0936

    

100

10

10

0

100

100

0

0.9151

0.3010

0.1448

0.1008

0.0945

    

100

10

10

10

100

100

100

1.6795

0.5135

0.1975

0.1096

0.0967

 

100

100

100

100

10

0

0

100

0

0

0.4912

0.1873

0.1113

0.0889

0.0857

    

100

10

10

0

100

100

0

0.8371

0.2662

0.1325

0.0923

0.0878

    

100

10

10

10

100

100

100

1.5364

0.4700

0.1825

0.1003

0.0885

Metal

100

0

0

100

10

0

0

100

0

0

3.5575

1.3407

0.7815

0.6246

0.6022

    

100

10

10

0

100

100

0

6.1516

1.9500

0.9306

0.6084

0.6081

    

100

10

10

10

100

100

100

11.199

3.3250

1.2823

0.7050

0.7050

 

0

100

100

100

10

0

0

100

0

0

0.4974

0.1916

0.1123

0.0899

0.0867

    

100

10

10

0

100

100

0

0.3035

0.9237

0.1462

0.1020

0.0958

    

100

10

10

10

100

100

100

1.7133

0.5196

0.2015

0.1109

0.0980

 

100

100

100

100

10

0

0

100

0

0

0.4974

0.1887

0.1123

0.0899

0.0867

    

100

10

10

0

100

100

0

0.8442

0.2678

0.1337

0.0933

0.0875

    

100

10

10

10

100

100

100

1.5659

0.4751

0.1842

0.1014

0.0896

Table 4

Effect of power law index on the non-dimensional displacements and stresses of an FGM rectangular plates (b = 3a) resting on elastic foundation and subjected to linear hygro-thermo-mechanical loadings (a/h = 10, q0 = 100, T1 = 10, C1 = 100)

P

k0

k1

k2

Theory

Model

\( \bar{w} \)

\( \bar{\sigma }_{\varvec{x}} \)

\( \bar{\tau }_{{\varvec{xy}}} \)

\( \bar{\tau }_{{\varvec{xz}}} \)

0

100

100

100

Present (\( \varepsilon_{z} \ne 0 \))

FOSNDT

0.17480

0.50560

0.16530

0.38630

    

Present (\( \varepsilon_{z} = 0 \))

FOSNDT

0.17300

0.50410

0.17560

0.38750

    

Zidi et al. [39].

FVRPT

0.17309

0.50498

0.17490

0.38766

    

Zenkour [44]

SSDT

0.17309

0.50498

0.17490

0.38766

    

Reddy [43]

TSDT

0.17309

0.50498

0.17490

0.38766

    

Mindlin [2]

FSDT

0.17309

0.50245

0.17351

0.31204

1

100

100

100

Present (\( \varepsilon_{z} \ne 0 \))

FOSNDT

0.18520

0.51540

0.15530

0.44540

    

Present (\( \varepsilon_{z} = 0 \))

FOSNDT

0.18510

051407

0.15710

0.44720

    

Zidi et al. [39].

FVRPT

0.18504

0.51450

0.15631

0.44545

    

Zenkour [44]

SSDT

0.18504

0.51450

0.15631

0.44545

    

Reddy [43]

TSDT

0.18504

0.51450

0.15631

0.44545

    

Mindlin [2]

FSDT

0.18505

0.51084

0.15694

0.35542

2

100

100

100

Present (\( \varepsilon_{z} \ne 0 \))

FOSNDT

0.18680

0.50470

0.13450

0.43790

    

Present (\( \varepsilon_{z} = 0 \))

FOSNDT

0.18570

0.50300

0.13530

0.43870

    

Zidi et al. [39].

FVRPT

0.18560

0.50336

0.13451

0.43831

    

Zenkour [44]

SSDT

0.18560

0.50336

0.13451

0.43831

    

Reddy [43]

TSDT

0.18560

0.50336

0.13451

0.43831

    

Mindlin [2]

FSDT

0.18567

0.49980

0.13244

0.33958

5

100

100

100

Present (\( \varepsilon_{z} \ne 0 \))

FOSNDT

0.18730

0.48970

0.12410

0.43670

    

Present (\( \varepsilon_{z} = 0 \))

FOSNDT

0.18700

0.48710

0.12500

0.43610

    

Zidi et al. [39].

FVRPT

0.18696

0.48940

0.12417

0.43754

    

Zenkour [44]

SSDT

0.18696

0.48940

0.12417

0.43754

    

Reddy [43]

TSDT

0.18696

0.48940

0.12417

0.43754

    

Mindlin [2]

FSDT

0.18712

0.48601

0.12125

0.32978

Metal

100

100

100

Present (\( \varepsilon_{z} \ne 0 \))

FOSNDT

0.18870

0.43400

0.12020

0.45960

    

Present (\( \varepsilon_{z} = 0 \))

FOSNDT

0.18835

0.43160

0.12190

0.44210

    

Zidi et al. [39].

FVRPT

0.18840

0.43095

0.12087

0.45993

    

Zenkour [44]

SSDT

0.18840

0.43117

0.12092

0.47465

    

Reddy [43]

TSDT

0.18840

0.43095

0.12087

0.45993

    

Mindlin [2]

FSDT

0.18840

0.42794

0.11921

0.36808

Table 5

Effect of power law index on the non-dimensional displacements and stresses of an FGM rectangular plates (b = 3a) resting on elastic foundation and subjected to non-linear hygro-thermo-mechanical loadings (a/h = 10, q0 = 100, T1 = T2 = 10, and C1 = C2 = 100)

P

k0

k1

k2

Theory

Model

\( \bar{w} \)

\( \bar{\sigma }_{\varvec{x}} \)

\( \bar{\tau }_{{\varvec{xy}}} \)

\( \bar{\tau }_{{\varvec{xz}}} \)

0

0

0

0

Present (\( \varepsilon_{z} \ne 0 \))

FOSNDT

2.54060

0.52680

2.20470

0.42380

    

Present (\( \varepsilon_{z} = 0 \))

FOSNDT

2.55068

0.52526

2.23290

0.41370

    

Zidi et al. [39]

FVRPT

2.54076

0.52522

2.20374

0.42454

    

Zenkour [44]

SSDT

2.54068

0.52522

2.20366

0.43753

    

Reddy [43]

TSDT

2.54076

0.52522

2.20374

0.42454

 

100

0

0

Present (\( \varepsilon_{z} \ne 0 \))

FOSNDT

1.36930

0.17790

1.20740

0.16410

    

Present (\( \varepsilon_{z} = 0 \))

FOSNDT

1.37550

0.17645

1.22570

0.15970

    

Zidi et al. [39]

FVRPT

1.36682

0.17643

1.20877

0.16257

    

Zenkour [44]

SSDT

1.36680

0.17649

1.20881

0.16834

    

Reddy [43]

TSDT

1.36682

0.17643

1.20877

0.16257

 

100

100

100

Present (\( \varepsilon_{z} \ne 0 \))

FOSNDT

0.24490

0.84910

0.25630

0.72770

    

Present (\( \varepsilon_{z} = 0 \))

FOSNDT

0.24490

0.84802

0.26010

0.72410

    

Zidi et al. [39]

FVRPT

0.24366

0.84804

0.25658

0.72442

    

Zenkour [44]

SSDT

0.24337

0.84835

0.25670

0.74816

    

Reddy [43]

TSDT

0.24366

0.84804

0.25658

0.72442

1

100

100

100

Present (\( \varepsilon_{z} \ne 0 \))

FOSNDT

0.26870

0.86380

0.23720

0.84720

    

Present (\( \varepsilon_{z} = 0 \))

FOSNDT

0.26610

0.86263

0.24160

0.84430

    

Zidi et al. [39]

FVRPT

0.26330

0.86205

0.23762

0.82148

    

Zenkour [44]

SSDT

0.26330

0.86252

0.23772

0.84866

    

Reddy [44]

TSDT

0.26330

0.86205

0.23762

0.82148

2

100

100

100

Present (\( \varepsilon_{z} \ne 0 \))

FOSNDT

0.26590

0.84210

0.20290

0.82490

    

Present (\( \varepsilon_{z} = 0 \))

FOSNDT

0.26393

0.84138

0.20510

0.82360

    

Zidi et al. [39]

FVRPT

0.26396

0.84135

0.20133

0.80652

    

Zenkour [44]

SSDT

0.26395

0.84185

0.20154

0.83484

    

Reddy [43]

TSDT

0.26396

0.84138

0.20133

0.80652

5

100

100

100

Present (\( \varepsilon_{z} \ne 0 \))

FOSNDT

0.26620

0.81730

0.18550

0.82900

    

Present (\( \varepsilon_{z} = 0 \))

FOSNDT

0.26050

0.81700

0.18810

0.82390

    

Zidi et al. [39]

FVRPT

0.26601

0.81745

0.18459

0.80297

    

Zenkour [44]

SSDT

0.26600

0.81792

0.18486

0.83230

    

Reddy [43]

TSDT

0.26601

0.81745

0.18459

0.80297

Metal

100

100

100

Present (\( \varepsilon_{z} \ne 0 \))

FOSNDT

0.26780

0.71660

0.18300

0.85850

    

Present (\( \varepsilon_{z} = 0 \))

FOSNDT

0.26430

0.71650

0.18402

0.85860

    

Zidi et al. [39]

FVRPT

0.26774

0.71656

0.18286

0.84003

    

Zenkour [44]

SSDT

0.26775

0.71694

0.18298

0.86748

    

Reddy [43]

TSDT

0.26774

0.71656

0.18286

0.84003

5 Conclusions

This study deals with the effect of non-linear hygro-thermo-mechanical loadings on the bending response of FG plates resting on elastic foundation using a new fifth-order shear and normal deformation theory (FOSNDT). The theory involves the effects of transverse shear and normal deformations, i.e. thickness stretching. Observations of the numerical results obtained by the present theory are excellent in agreement with previously published results reported in the literature. Hence, it is recommended that the present theory is accurate and efficient in predicting the bending responses of FG plates subjected to non-linear hygro-thermo-mechanical loading.

Notes

Compliance with ethical standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

References

  1. 1.
    G.R. Kirchhoff, Uber das gleichgewicht und die bewegung einer elastischen Scheibe. J. Reine. Angew. Math. (Crelle’s J) 40, 51–88 (1850)Google Scholar
  2. 2.
    R.D. Mindlin, Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. ASME J. Appl. Mech. 18, 31–38 (1951)zbMATHGoogle Scholar
  3. 3.
    A.S. Sayyad, Y.M. Ghugal, On the free vibration analysis of laminated composite and sandwich plates: a review of recent literature with some numerical results. Compos. Struct. 129, 177–201 (2015)Google Scholar
  4. 4.
    A.S. Sayyad, Y.M. Ghugal, Bending, buckling and free vibration of laminated composite and sandwich beams: a critical review of literature. Compos. Struct. 171, 486–504 (2017)Google Scholar
  5. 5.
    A.S. Sayyad, Y.M. Ghugal, Modeling and analysis of functionally graded sándwich beams: a review (Adv. Mater. Struct, Mech, 2018).  https://doi.org/10.1080/15376494.2018.1447178 CrossRefGoogle Scholar
  6. 6.
    D.K. Jha, T. Kant, R.K. Singh, A critical review of recent research on functionally graded plates. Compos. Struct. 96, 833–849 (2013)Google Scholar
  7. 7.
    K. Swaminathan, D.M. Sangeetha, Thermal analysis of FGM plates—a critical review of various modeling techniques and solution methods. Compos. Struct. 160, 43–60 (2017)Google Scholar
  8. 8.
    K. Swaminathan, D.T. Naveenkumar, A.M. Zenkour, E. Carrera, Stress, vibration and buckling analyses of FGM plates-A state-of-the-art review. Compos. Struct. 120, 10–31 (2015)Google Scholar
  9. 9.
    A. Tounsi, M.S.A. Houari, S. Benyoucef, E.A.A. Bedia, A refined trigonometric shear deformation theory for thermoelastic bending of functionally graded sandwich plate. Aerosp. Sci. Technol. 24, 209–220 (2013)Google Scholar
  10. 10.
    M.N.A.G. Taj, A. Chakrabarti, A.H. Sheikh, Analysis of functionally graded plates using higher order shear deformation theory. Appl. Math. Model. 37, 8484–8494 (2013)MathSciNetzbMATHGoogle Scholar
  11. 11.
    T.S. Daouadji, A. Tounsi, E.A.A. Bedia, Analytical solution for bending analysis of functionally graded plates. Sci. Iran. 20(3), 516–523 (2013)Google Scholar
  12. 12.
    L.V. Tran, A.J.M. Ferreira, H.N. Xuan, Isogeometric analysis of functionally graded plates using higher order shear deformation theory. Compos. Part B Eng. 51, 368–383 (2013)Google Scholar
  13. 13.
    K.K. Pradhan, S. Chakraverty, Static analysis of functionally graded thin rectangular plates with various boundary supports. Arch. Civ. Mech. Eng. 15, 721–734 (2015)Google Scholar
  14. 14.
    M. Filippi, E. Carrera, A.M. Zenkour, Static analyses of FGM beams by various theories and finite elements. Compos. Part B Eng. 72, 1–9 (2015)Google Scholar
  15. 15.
    J.L. Mantari, E.M. Bonilla, C.G. Soares, A new tangential-exponential higher orders shear deformation theory for advanced composite plates. Compos. Part B Eng. 60, 319–328 (2014)Google Scholar
  16. 16.
    J.L. Mantari, I.A. Ramos, E. Carrera, M. Petrolo, Static analysis of functionally graded plates using new non polynomial displacement fields via Carrera unified formulation. Compos. Part B Eng. 89, 127–142 (2016)Google Scholar
  17. 17.
    A.S. Sayyad, Y.M. Ghugal, Thermoelastic bending analysis of laminated composite plates according to various shear deformation theories. Open Engg. 5, 18–30 (2015)Google Scholar
  18. 18.
    A.S. Sayyad, Y.M. Ghugal, A unified shear deformation theory for the bending of isotropic, functionally graded, laminated and sandwich plates Int. J. Appl. Mech. 9(1), 1–36 (2017)Google Scholar
  19. 19.
    E. Carrera, S. Brischetto, M. Cinefra, M. Soave, Effect of thickness stretching in functionally graded plates and shells. Compos. Part B-Eng. 42, 123–133 (2011)Google Scholar
  20. 20.
    E. Carrera, S. Brischetto, Modeling and analysis of functionally graded beams, plates and shells-Part I. Mech. Adv. Mater. Struct. 17(8), 585 (2010)Google Scholar
  21. 21.
    E. Carrera, S. Brischetto, Modeling and analysis of functionally graded beams, plates and shells Part-II. Mech. Adv. Mater. Struct. 18(1), 1–2 (2011)zbMATHGoogle Scholar
  22. 22.
    S.M. Ghumare, A.S. Sayyad, A new fifth-order shear and normal deformation theory for static bending and elastic buckling of P-FGM beams. Lat. Am. J. Solids Struct. 14, 1893–1911 (2017)Google Scholar
  23. 23.
    S.M. Ghumare, A.S. Sayyad, A new quasi-3D model for functionally graded plates. J. Appl. Comput. Mech. 5(2), 367–380 (2019)Google Scholar
  24. 24.
    N.S. Naik, A.S. Sayyad, 1D analysis of laminated composite and sandwich plates using new fifth-order shear and normal deformation theory. Lat. Am. J. Solids Struct. 15(1), 1–17 (2018)Google Scholar
  25. 25.
    N.S. Naik, A.S. Sayyad, 2D analysis of laminated composite and sandwich plates using new fifth order theory. Lat. Am. J. Solids Struct. 15(9), 1–27 (2018)Google Scholar
  26. 26.
    N.S. Naik, A.S. Sayyad, An accurate computational model for thermal analysis of laminated composite and sandwich plates. J. Therm. Stresses 42(5), 559–579 (2019)Google Scholar
  27. 27.
    Y.L. Chung, H.X. Chang, Mechanical behavior of rectangular plates with functionally graded coefficient of thermal expansion subjected to thermal loading. J. Therm. Stresses 31, 368–388 (2008)Google Scholar
  28. 28.
    H. Matsunaga, Stress analysis of functionally graded plates subjected to thermal and mechanical loadings. Compos. Struct. 87, 344–357 (2009)Google Scholar
  29. 29.
    M. Cinefra, E. Carrera, S. Brischetto, S. Belouettar, Thermo-mechanical analysis of functionally graded shells. J. Therm. Stresses 33, 942–963 (2010)Google Scholar
  30. 30.
    F.A. Fazzolari, E. Carrera, Thermal stability of FGM sandwich plates under various through-the-thickness temperature distributions. J. Therm. Stresses 37, 1449–1481 (2014)Google Scholar
  31. 31.
    D. Li, Z. Deng, H. Xiao, Thermomechanical bending analysis of functionally graded sandwich plates using four-variable refined plate theory. Compos. Part B Eng. 106, 107–119 (2016)Google Scholar
  32. 32.
    M. Bouazza, A. Boucheta, T. Becheri, N. Benseddiq, Thermal stability analysis of functionally graded plates using simple refined plate theory. Int. J. Auto. Mech. Eng. 14(1), 4013–4029 (2017)Google Scholar
  33. 33.
    M.D. Demirbas, Thermal stress analysis of functionally graded plates with temperature-dependent material properties using theory of elasticity. Compos. Part B Eng. 131, 100–124 (2017)Google Scholar
  34. 34.
    R. Leetsch, T. Wallmersperger, K. Kroplin, Thermo-mechanical modeling of functionally graded plates. J. Intell. Mater. Syst. Struct. 20, 1799–1813 (2009)Google Scholar
  35. 35.
    S. Brischetto, R. Leetsch, E. Carrera, T. Wallmersperger, B. Kroplin, Thermo-mechanical bending of functionally graded plate. J. Therm. Stresses 31, 286–308 (2008)Google Scholar
  36. 36.
    A.S. Sayyad, Y.M. Ghugal, A four-variable plate theory for thermoelastic bending analysis of laminated composite plates. J. Therm. Stresses 38, 904–925 (2015)Google Scholar
  37. 37.
    A.S. Sayyad, Y.M. Ghugal, Thermal stress analysis of laminated composite plates using exponential shear deformation theory. Int. J. Auto. Compos. 2(1), 23–40 (2016)Google Scholar
  38. 38.
    A.M. Zenkour, The refined sinusoidal theory for FGM plates on elastic foundations. Int. J. Mech. Sci. 51, 869–880 (2009)Google Scholar
  39. 39.
    M. Zidi, A. Taunsi, M. Hauari, E. Bedia, O.A. Beg, Bending analysis of an FGM plates under hygro-thermo-mechanical loading using a four variable refined plate theory. Aerosp. Sci. Technol. 14, 24–34 (2014)Google Scholar
  40. 40.
    T.H. Daouadji, B. Adim, R. Benferhat, Bending analysis of an imperfect FGM plates under hygro-thermo-mechanical loading with analytical validation. Adv. Mater. Res. 5(1), 35–53 (2016)Google Scholar
  41. 41.
    A.M. Zenkour, M.L. Allam, A.F. Radwan, Effects of transverse shear and normal strains on FG plates resting on elastic foundations under hygro-thermo-mechanical loading. Int. J. Appl. Mech. 6(5), 1–26 (2014)Google Scholar
  42. 42.
    A.S. Sayyad, Y.M. Ghugal, Effects of non-linear hygro-thermo-mechanical loading on bending of FGM rectangular plates resting on two-parameter elastic foundation using four-unknown plate theory. J. Therm. Stresses 42(2), 213–232 (2018)Google Scholar
  43. 43.
    J.N. Reddy, A simple higher order theory for laminated composite plates. ASME J. Appl. Mech. 51, 745–752 (1984)zbMATHGoogle Scholar
  44. 44.
    A.M. Zenkour, Generalized shear deformation theory for bending analysis of functionally graded plates. Appl. Math. Model. 30, 67–84 (2006)zbMATHGoogle Scholar
  45. 45.
    E. Winkler, Die Lehre von der Elasticitaet und Festigkeit (Prag, Dominicus, 1867)Google Scholar
  46. 46.
    P.L. Pasternak, On a new method of analysis of an elastic foundation by means of two foundation constants (Gosudarstvennoe Izdatelstvo Literaturi po Stroitelstvu i Arkhitekture, Moscow, 1954)Google Scholar
  47. 47.
    M. Arefi, G.H. Rahimi, Non linear analysis of a functionally graded square plate with two smart layers as sensor and actuator under normal pressure. Smart. Struct. System. 8(5), 433–447 (2011).  https://doi.org/10.12989/sss.2011.8.5.433 CrossRefGoogle Scholar
  48. 48.
    M. Arefi, G.H. Rahimi, Studying the nonlinear behavior of the functionally graded annular plates with piezoelectric layers as a sensor and actuator under normal pressure. Smart. Struct. Syst. 9(2), 127–143 (2012)Google Scholar
  49. 49.
    M. Arefi, G.H. Rahimi, Non-linear responses of an arbitrary FGP circular plate resting on the Winkler-Pasternak foundation. Smart. Struct. Syst. 16(1), 81–100 (2015). 10.12989/sss.2015.16.1.081 Google Scholar
  50. 50.
    M. Arefi, E. Mohammad, R. Bidgoli, A.M. Zenkour, Size-dependent free vibration and dynamic analyses of a sandwich microbeam based on higher-order sinusoidal shear deformation theory and strain gradient theory. Smart. Struct. Syst. 22(1), 27–40 (2018).  https://doi.org/10.12989/sss.2018.22.1.027 CrossRefGoogle Scholar
  51. 51.
    M. Arefi, A.H.S. Arani, Higher order shear deformation bending results of a magnetoelectrothermoelastic functionally graded nanobeam in thermal, mechanical, electrical, and magnetic environments. Mech. Based Des. Struct. Mach. 46(6), 669–692 (2018).  https://doi.org/10.1080/15397734.2018.1434002 CrossRefGoogle Scholar
  52. 52.
    M. Arefi, A.M. Zenkour, Size-dependent electro-elastic analysis of a sandwich microbeam based on higher-order sinusoidal shear deformation theory and strain gradient theory. J. Intel Mater. Syst. Struct. 29(7), 1394–1406 (2018)Google Scholar
  53. 53.
    M.Mohammadi, M. Arefi, R. Dimitri, F.Tornabene, Higher-order thermo-elastic analysis of FG-CNTR cylindrical vessels surrounded by a pasternak foundation. Nanomaterials 9(1), 79-1-21 (2019)Google Scholar
  54. 54.
    M. Arefi, A.M. Zenkour, Transient sinusoidal shear deformation formulation of a size-dependent three-layer piezo-magnetic curved nanobeam. Acta Mech. (2017).  https://doi.org/10.1007/s00707-017-1892-6 MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    M. Arefi, A.M. Zenkour, Thermal stress and deformation analysis of a size-dependent curved nanobeam based on sinusoidal shear deformation theory. Alex. Eng. J. 57(3), 1–9 (2018).  https://doi.org/10.1016/j.aej.2017.07.003 CrossRefGoogle Scholar
  56. 56.
    M. Aerfi, Size-dependent bending behavior of three-layered doubly curved shells: modified couple stress formulation. J. Sandw. Struct. Mater. (2018).  https://doi.org/10.1177/1099636218793993 CrossRefGoogle Scholar
  57. 57.
    M.A. Arefi, T. Rabczuk, A nonlocal higher order shear deformation theory for electro-elastic analysis of a piezoelectric doubly curved nano shell. Compos. Part B Eng. 168, 496–510 (2019)Google Scholar
  58. 58.
    M. Arefi, A.M. Zenkour, Effect of thermomagnetoelectro mechanical fields on the bending behaviors of a three-layered nanoplate based on sinusoidal shear deformation plate theory. J. Sandw. Struct. Mater. (2017).  https://doi.org/10.1177/1099636217697497 CrossRefGoogle Scholar
  59. 59.
    M. Arefi, E.M.R. Bidgoli, R. Dimitri, M. Bacciocchi, F. Tornabene, Application of sinusoidal shear deformation theory and physical neutral surface to analysis of functionally graded piezoelectric plate. Compos. Part B-Eng. 151, 35–50 (2018).  https://doi.org/10.1016/j.compositesb.2018.05.050 CrossRefGoogle Scholar
  60. 60.
    A.M. Zenkour, A.F. Radwan, Hygro-thermo-mechanical buckling of FGM plates resting on elastic foundations using a quasi-3D model (J. Comput. Meth. Eng. Sci. Mech, Int, 2019).  https://doi.org/10.1080/15502287.2019.1568618 CrossRefGoogle Scholar
  61. 61.
    D.S. Chilton, J.W. Wekezer, Plates on elastic foundation. J. Struct. Eng. 116(11), 3236–3241 (1990)Google Scholar
  62. 62.
    C.X. Sheng, A free rectangular plate on elastic foundation. J. Appl. Math. Mech. 13(10), 977–982 (1992)Google Scholar
  63. 63.
    F. Najafi, M.H. Shojaeefard, H.S. Googarchin, Nonlinear low-velocity impact response of functionally graded plate with nonlinear three-parameter elastic foundation in thermal field. Compos. Part B Eng. 107, 123–140 (2016)Google Scholar
  64. 64.
    A.T. Daloglu, K. Ozgan, The effective depth of soil stratum for plates resting on elastic foundation. Struct. Eng. Mech. 18(2), 1–12 (2004)Google Scholar
  65. 65.
    A.M. Zenkour, M. Sobhy, Elastic foundation analysis of uniformly loaded functionally graded viscoelastic sandwich plates. J. Mech. 28(03), 439–452 (2012)Google Scholar
  66. 66.
    Y. Kiani, A.H. Akbarzadeh, Z.T. Chen, M.R. Eslami, Static and dynamic analysis of an FGM doubly curved panel resting on the Pasternak-type elastic foundation. Compos. Struct. 94, 2474–2484 (2012)Google Scholar
  67. 67.
    A.S. Sayyad, Y.M. Ghugal, Bending of shear deformable plates resting on Winkler foundations according to trigonometric plate theory. J. Appl. Comput. Mech. 4(3), 187–201 (2018)Google Scholar
  68. 68.
    A.S. Sayyad, Y.M. Ghugal, An inverse hyperbolic theory for FG beams resting on Winkler–Pasternak elastic foundation. Adv. Aircr. Spacecr. Sci. 5(6), 671–689 (2018)Google Scholar

Copyright information

© The Korean Society of Mechanical Engineers 2019

Authors and Affiliations

  1. 1.Department of Civil Engineering, SRES’s Sanjivani College of EngineeringSavitribai Phule Pune UniversityKopargaonIndia

Personalised recommendations