Magneto-Hygrothermal Deformation of FG Nanocomposite Annular Sandwich Nanoplates with Porous Core Using the DQM

Abstract

This study introduces a novel numerical approach to analyze the axisymmetric bending behavior of functionally graded (FG) graphene platelet (GPL)-reinforced annular sandwich nanoplates featuring a porous core. The nanostructures are exposed to coupled magnetic and hygrothermal environments. The porosity distribution and GPL weight fraction are modeled as nonlinear functions through the thickness, capturing realistic gradation effects. The governing equations are derived using the virtual displacement principle, taking into account the Lorentz force and the interaction with an elastic foundation. To address the size-dependent behavior and thickness-stretching effects, the model employs the nonlocal strain gradient theory (NSGT) integrated with a modified version of Shimpi’s quasi-3D higher-order shear deformation theory (Q3HSDT). The differential quadrature method (DQM) is applied to obtain numerical solutions for the displacement and stress fields. A detailed parametric study is conducted to investigate the influence of various physical and geometric parameters, including the nonlocal parameter, strain gradient length scale, magnetic field strength, thermal effects, foundation stiffness, core thickness, and radius-to-thickness ratio. The findings support the development of smart, lightweight, and thermally adaptive nano-electromechanical systems (NEMS) and provide valuable insights into the mechanical performance of FG-GPL sandwich nanoplates. These findings have potential applications in transducers, nanosensors, and stealth technologies designed for ultrasound and radar detection.

1. Introduction

The mechanical properties of nanotechnology and foam-enhanced polymers represent a fascinating topic in engineering research and mechanics [1]. The complicated interactions between different components and their varying architectures make studying the mechanical characteristics of these materials a considerable task [2]. In this regard, the investigation and study of how composite materials respond to centripetal deformation, as well as the associated effects on their mechanical and magnetic properties, has been carried out.

Aside from the fundamental structure, the materials used in the cores and face sheets significantly impact how sandwich structures behave mechanically. In particular, the face sheets of sandwich structures may be made of functionally graded material (FGM), a kind of advanced heterogeneous composite material. In particular, the composition and nanostructure of FGMs vary continuously across the material [3,4], and the general features of FGMs vary depending on the specific component. This distinguishing property enables FGMs to demonstrate superior qualities, compared to typical isotropic materials. Therefore, FGMs have been the subject of significant research and have been used in various applications [5].

Graphene—a single layer of carbon atoms organized into a two-dimensional honeycomb lattice—has sparked widespread interest due to its remarkable mechanical, electrical, thermal, and optical capabilities. Since its discovery by Novoselov et al. [6], the incorporation of graphene into FGMs has been demonstrated to improve their performance in a variety of applications. Sahmani et al. [7] investigated the nonlinear bending behavior of porous nanobeams reinforced with FG graphene. Furthermore, Amir et al. [8] studied the free vibration of porous, three-layered annular and circular microplates reinforced with FG carbon nanotubes (CNTs). Furthermore, Barati and Zenkour [9] investigated the post-buckling of a porous nanocomposite beam reinforced with GPLs that have geometrical imperfections. Recently, the authors of [10] demonstrated that nanocomposite coatings, including ceramic-reinforced phases, may strengthen wear resistance and corrosion prevention, according to elevated radiation fluxes.

Nanostructures are studied using the theory of nonlocal elasticity, which is a concept in the mechanics of materials that refers to the relationship between stress (force acting on a material) and strain (change in shape) in an elastic material [11]. In particular, this theory states that the strain at any point in the material relies on the stress at all other points in the material, despite the distance between them. This means that a change in shape at one point affects the entire material. This concept is used to study the strains and stresses in elastic materials, and helps to understand the behavior of materials under loading and various engineering applications, such as in structural design and elastic materials. Despite nonlocal elasticity theory continuing to be widely utilized to predict the dynamics of nanostructures, it has several drawbacks. It solely discusses the impact of stiffness softening and does not account for the stiffness hardening impact seen in experimental and theoretical studies. The gradient elasticity theories extend classical elasticity equations by including higher-order strain gradient elements, employing the assumption that materials cannot be modeled as collections of points but must be considered as atoms with higher-order deformation mechanisms at the micro- or nanoscale. In particular, strain gradient theory can potentially be used to model the influence of stiffness hardening. Mindlin [12] pioneered the strain gradient theory, which included five distinct, higher-order material variables with first-order strain gradients [13]. Consequently, nonlocal elasticity theory and strain gradient theory deal with various aspects of nanoscale structural behaviors. The term “nonlocal strain gradient theory” (NSGT) refers to the generalized combination of strain gradient theory and nonlocal elasticity theory, introduced by Lim et al. [14] to explain the entire small-scale influence of such nanomaterials on mechanical characteristics. As it includes a nonlocal component and a material length-scale parameter, NSGT can reflect both stiffness hardening and softening.

Acoustic or noise elimination applications may be employed in micro- or nano-electromechanical devices (NEMS) in noisy conditions. Moreover, they may be used for military purposes such as in personal shielding, objective confusion, and shock dampening. Furthermore, a key stage in producing various NEMS devices, including oscillators, clocks, and sensor devices, is understanding their nanostructure’s vibration and bending behavior. An overview of the significance and modeling of the vibration behavior of various nanostructures was provided by Gibson et al. [15]. The nonlocal elasticity theory proposed by Eringen [16,17] has been suggested for use in continuum models for accurate vibration behavior prediction. In [18], the author reviewed several nonlocal theories related to simply supported beams and their bending deflections, buckling loads, and natural frequencies. Nanoplates have outstanding mechanical characteristics comparable to CNTs [18,19]. However, there has been little research on the vibration analysis of 2D nanoplates when compared to single-dimensional structures. Li and Hu [20] examined the buckling of an Euler–Bernoulli nanobeam using NSGT. In order to analyze the wave dispersion characteristics in homogeneous nanoplates and rotating piezoelectromagnetic nanoplates, respectively, Ebrahimi et al. [21] and Ebrahimi and Dabbagh [22] used NSGT. Furthermore, Abazid [23] used both sinusoidal shear deformation plate theory and NSGT to study thermomechanical buckling, free vibration, and wave dispersion in intelligent piezo-electromagnetic nanoplates in a humid medium. Liu et al. [24] explored the size-dependent generalized thermoelasticity model to assess the thermoelastic vibrations of Euler–Bernoulli nanobeams by including the impact of size on the constitutive and conduction of heat relations using NSGT and a dual-phase-lag (DPL) heat model. In Hong et al. [25], the authors examined the size-dependent wave propagation properties of the micro/nanostructures using an analytical technique. In particular, they provided size-dependent static bending and wave propagation analysis of FG magneto-electro-elastic (MEE) porous microbeams based on Timoshenko beam theory and modified-couple stress theory (MCsT). The static and transient response of FG MEE microplates with complicated geometries was presented by Wang et al. [26] using isogeometric analysis technique. Numerous recent studies have examined nonlinear analysis using various numerical methods [27,28,29,30,31,32].

According to the aforementioned survey, and to the best of the author’s knowledge, no article has yet addressed the bending analysis of GPL sandwich annular nanoplates with an FG porous core resting on an elastic substrate under magnetic field effects in a humid environment. Thus, the novelty of the present paper can be summarized in the following items: (a) The investigation of the hygrothermal bending analysis of FG-GPL-reinforced annular sandwich nanoplates with an FG porous core. (b) In addition, the effects of in-plane magnetic field on the bending behavior of a sandwich annular nanoplate are investigated for the first time. (c) The displacement field of the annular plate is modeled using a novel quasi-3D refined plate theory, which accounts for both transverse shear and normal strains, enhancing the accuracy of the analysis. DQM is an effective numerical technique that has been applied in several publications. This method’s main objective is to solve equations at discrete grid points by applying Lagrange interpolation polynomials to their fields’ coefficients, whereby utilizing numerous grid points might result in higher accuracy. Due to its efficiency and great accuracy, DQM was used to solve equilibrium equations. The collected results are compared to those published in the literature to validate the existing formulations. Furthermore, several numerical examples are provided to explain the influences of the following factors: graphene weight fraction, porosity, moisture, temperature, elastic foundation stiffness, core thickness, magnetic field, nonlocal coefficient, outer radius-to-thickness ratio, inner-to-outer radius ratio, and strain gradient coefficient on the deflection of the current model.

2. Formulation

Figure 1 displays a sandwich annular nanoplate constructed of three layers of inner radius $a_0$, outer radius $a_1$, and total thickness $\mathcal{H}$. It is assumed that the proposed nanoplate is composed of nanocomposite FG graphene platelet face sheets with a metal foam core resting on a Pasternak foundation while exposed to a magnetic field. The face sheets are considered to be completely connected to the core. The upper and bottom layers have the same thickness ${\mathcal{H}}_F$, while the core layer has a thickness of ${\mathcal{H}}_C$, resulting in an overall plate thickness of $\mathcal{H}=2{\mathcal{H}}_F+{\mathcal{H}}_C$. To extract the mathematical formulas, the plate shape and dimensions were established in a perpendicular cylindrical coordinate system $(r,\vartheta ,z)$; the coordinate system’s origin is determined in the middle plane at the sandwich plate’s center. The current method addresses axially symmetric deformation. Furthermore, the coordinate of the bottom surface is ${\mathcal{H}}_0=-\mathcal{H}/2$, the two internal surfaces between the layers are ${\mathcal{H}}_1=-{\mathcal{H}}_C/2$ and ${\mathcal{H}}_2={\mathcal{H}}_C/2$, and the top surface is ${\mathcal{H}}_3=\mathcal{H}/2$.

2.1. GPLs-Reinforced Face Layers

The characteristics of the face layers are supposed to change with the thickness of the layer. Accordingly, the Halpin–Tsai framework and mixture rule were used to derive the effective Young modulus $E^{\left(i\right)}$, Poisson’s ratio ${\nu }^{\left(i\right)}$, thermal expansion coefficient ${\alpha }^{\left(i\right)}$, and moisture expansion coefficient ${\beta }^{\left(i\right)}$ of the upper and lower nanocomposite layers, which can be described by [33]

$$\begin{array}{cc}\hfill E^{\left(i\right)}\left(z\right)=& {\displaystyle \frac{1}{8}}[3E_1^{\left(i\right)}\left(z\right)+5E_2^{\left(i\right)}\left(z\right)],\hfill \\ \hfill E_K^{\left(i\right)}\left(z\right)=& {\displaystyle \frac{1+{\mathrm{\Xi }}_K^g{\lambda }_K{F}_g^{\left(i\right)}}{1-{\lambda }_K{F}_g^{\left(i\right)}}}{E}_m,\hfill \\ \hfill {\lambda }_K=& {\displaystyle \frac{E_g-E_m}{E_g+2{\mathrm{\Xi }}_K^g{E}_m}},\hfill \\ \hfill {\nu }^{\left(i\right)}\left(z\right)=& F_m^{\left(i\right)}\left(z\right){\nu }_m+F_g^{\left(i\right)}\left(z\right){\nu }_g,\hfill \\ \hfill {\alpha }^{\left(i\right)}\left(z\right)=& F_m^{\left(i\right)}\left(z\right){\alpha }_m+F_g^{\left(i\right)}\left(z\right){\alpha }_g,\hfill \\ \hfill {\beta }^{\left(i\right)}\left(z\right)=& F_m^{\left(i\right)}\left(z\right){\beta }_m+F_g^{\left(i\right)}\left(z\right){\beta }_g,F_m^{\left(i\right)}\left(z\right)=1-F_g^{\left(i\right)}\left(z\right),K=1,2,i=1,3,\hfill \end{array}$$

(1)

in which the subscripts g and m correspond to the characteristics of the GPLs and metal, respectively; E, $\nu$, $\alpha$, and $\beta$ define Young’s modulus, Poisson’s ratio, the thermal expansion coefficient, and the moisture expansion coefficient, respectively; ${\mathrm{\Xi }}_1^g=l^g/{\mathcal{H}}^g$, and ${\mathrm{\Xi }}_2^g=w^g/{\mathcal{H}}^g$, where $l^g$, $w^g$, and ${\mathcal{H}}^g$ represent the length, width, and thickness of the GPLs, respectively. The thickness of the face layers influences the GPLs volume fraction, $F_g\left(z\right)$, using a modified cosine law. Based on the way the graphene platelets are dispersed throughout the thickness of the plate, three distinct patterns of GPLs distribution can be introduced in the current analysis as follows (see Figure 2):

2.1.1. Pattern I

In this type, the GPLs are scattered uniformly throughout the thickness of the sandwich faces; as a result, the volume fraction of the ith layer can take the following expression [34,35]:

$$\begin{array}{c}\hfill F_g^{\left(i\right)}\left(z\right)=V_g={\displaystyle \frac{{\rho }_m{w}_G}{{\rho }_m{w}_G+{\rho }_g(1-w_G)}},{\mathcal{H}}_0\le z\le {\mathcal{H}}_1,{\mathcal{H}}_2\le z\le {\mathcal{H}}_3,i=1,3,\mathrm{UD}\end{array}$$

(2)

in which ${\rho }_g$, ${\rho }_m$, and $w_G$ refer, respectively, to the densities of the GPLs and metal and the weight fraction of the GPLs.

2.1.2. Pattern II

The graphene variation in the current pattern achieves its highest value at the top and bottom surfaces of the plate (i.e., at ${\mathcal{H}}_3$ and ${\mathcal{H}}_0$), after which it steadily drops to its lowest value at the interfaces of the plate (${\mathcal{H}}_1$ and ${\mathcal{H}}_2$). As a result, one may write [36]:

$$\begin{array}{cc}\hfill & F_g^{\left(1\right)}\left(z\right)=V_{g}cos({\displaystyle \frac{(2z-{\mathcal{H}}_0-{\mathcal{H}}_1)\pi }{4({\mathcal{H}}_0-{\mathcal{H}}_1)}}-{\displaystyle \frac{\pi }{4}}),{\mathcal{H}}_0\le z\le {\mathcal{H}}_1,\hfill \\ & F_g^{\left(3\right)}\left(z\right)=V_{g}cos({\displaystyle \frac{(2z-{\mathcal{H}}_3-{\mathcal{H}}_2)\pi }{4({\mathcal{H}}_3-{\mathcal{H}}_2)}}-{\displaystyle \frac{\pi }{4}}),{\mathcal{H}}_2\le z\le {\mathcal{H}}_3.\hfill \end{array}$$

(3)

2.1.3. Pattern III

In contrast to Pattern II, the starting distribution of graphene is barely noticeable at the top and bottom of the plate and progressively grows to its greatest value near the interfaces (${\mathcal{H}}_1$ and ${\mathcal{H}}_2$); consequently, the volume fractions of the higher and lower layers, $F_g^{\left(i\right)}\left(z\right)$, can be stated as follows [36]:

$$\begin{array}{cc}\hfill & F_g^{\left(1\right)}\left(z\right)=V_{g}cos({\displaystyle \frac{(2z-{\mathcal{H}}_0-{\mathcal{H}}_1)\pi }{4({\mathcal{H}}_0-{\mathcal{H}}_1)}}+{\displaystyle \frac{\pi }{4}}),{\mathcal{H}}_0\le z\le {\mathcal{H}}_1,\hfill \\ & F_g^{\left(3\right)}\left(z\right)=V_{g}cos({\displaystyle \frac{(2z-{\mathcal{H}}_3-{\mathcal{H}}_2)\pi }{4({\mathcal{H}}_3-{\mathcal{H}}_2)}}+{\displaystyle \frac{\pi }{4}}),{\mathcal{H}}_2\le z\le {\mathcal{H}}_3.\hfill \end{array}$$

(4)

It is important to emphasize that the different patterns of GPLs distribution have a key influence on defining the efficiency of the plates. For instance, Pattern I could increase the mechanical properties of the plates more than the other varieties. In contrast, Pattern II has only metal interfaces between the layers and is distinguished by a strong link between the two face layers and the core layer. Therefore, it is assumed that the third pattern would be less efficient than the other two patterns.

2.2. Porous Core Layer

The metal foam forming the core layer is assumed to be constructed of aluminum, with porosities distributed either uniformly or FG across the thickness of the core (see Figure 3). The current research considers three alternative porosity variations as follows:

2.2.1. Porous-A

In this situation, the porosities are distributed evenly over the core thickness $\left(\mathrm{DE}\right)$, as follows [37,38]:

$$\begin{array}{cc}\hfill & P_{Eff}^{\left(2\right)}\left(z\right)=P_m(1-{\delta }_0\eta ),{\mathcal{H}}_1\le z\le {\mathcal{H}}_2,\hfill \end{array}$$

(5)

in which $P_{Eff}$ specifies the effective mechanical properties, including Young’s modulus, E, shear modulus, G, $\alpha$, and $\beta$. In addition, ${\delta }_0$ implies the porosity coefficient, where $0\le {\delta }_0\le 1$, and

$$\begin{array}{c}\hfill \eta ={\displaystyle \frac{1}{{\delta }_0}}-{\displaystyle \frac{1}{{\delta }_0}}[{\displaystyle \frac{2}{\pi }}\sqrt{1-{\delta }_0\eta }-{\displaystyle \frac{2}{\pi }}+1{]}^2.\end{array}$$

(6)

2.2.2. Porous-B

The pores are distributed as thinly as possible at the top and bottom interfaces of the core, and then they progressively become thicker until they reach the center of the core, where they are at their thickest $\left(\mathrm{O}\text{-}\mathrm{FG}\right)$. Consequently, the minimum mechanical characteristics will be found in the middle plane of the core; one can have the following expressions [36]:

$$\begin{array}{cc}\hfill & P_{Eff}^{\left(2\right)}\left(z\right)=P_m\{1-{\delta }_0[1-{\left({\displaystyle \frac{2z-{\mathcal{H}}_1-{\mathcal{H}}_2}{{\mathcal{H}}_1-{\mathcal{H}}_2}}\right)}^2\left]\right\},{\mathcal{H}}_1\le z\le {\mathcal{H}}_2.\hfill \end{array}$$

(7)

2.2.3. Porous-C

Here, we have the opposite of Porous-B, with maximal pore distribution at the top and bottom of the core interfaces and minimal pore distribution at the center, $\left(\mathrm{X}\text{-}\mathrm{FG}\right)$. Thus, we may write [36,38]

$$\begin{array}{cc}\hfill & P_{Eff}^{\left(2\right)}\left(z\right)=P_m[1-{\delta }_0{\left({\displaystyle \frac{2z-{\mathcal{H}}_1-{\mathcal{H}}_2}{{\mathcal{H}}_1-{\mathcal{H}}_2}}\right)}^2],{\mathcal{H}}_1\le z\le {\mathcal{H}}_2.\hfill \end{array}$$

(8)

For the aforementioned three distinct variations, the Poisson’s ratio of the FG porous core may be approximated as [39]

$$\begin{array}{c}\hfill {\nu }^{\left(2\right)}\left(z\right)={\nu }_m[0.342{(1-{\displaystyle \frac{{\rho }^{\left(2\right)}\left(z\right)}{{\rho }_m}})}^2-1.21(1-{\displaystyle \frac{{\rho }^{\left(2\right)}\left(z\right)}{{\rho }_m}})+1]+0.221(1-{\displaystyle \frac{{\rho }^{\left(2\right)}\left(z\right)}{{\rho }_m}}).\end{array}$$

(9)

Nevertheless, the porosity distribution type substantially impacts the plate’s response properties. Porous-B’s upper and bottom surfaces are pore-free, strengthening the connection between the core and face layers. Meanwhile, Porous-C may have poorer thermal conductivity at the interfaces, resulting in less heat transfer to the core layer. Consequently, the latter might be suitable for high-temperature situations. Further details on the effects of different types of GPLs distribution and porosity on the thermal bending of FG sandwich annular nanoplates will be provided in Section 4.

2.3. Elastic Foundation

In the investigation of FG-GPLs-reinforced annular sandwich nanoplates, the elastic foundation represents the underlying support structure. In particular, this type of foundation model incorporates the interaction of the structure with the foundation. Therefore, the transverse reaction of the Pasternak foundation, $\mathcal{J}$, can be expressed as [40,41,42]

$$\begin{array}{c}\hfill \mathcal{J}=K_w{u}_z{|}_{z=-{\displaystyle \frac{\mathcal{H}}{2}}}-K_P({\displaystyle \frac{{\partial }^2{u}_z}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial u_z}{\partial r}}{)}_{z=-{\displaystyle \frac{\mathcal{H}}{2}}},\end{array}$$

(10)

in which $K_w$, $u_z$, and $K_P$ refer, respectively, to Winkler springs stiffness, transverse displacement, and shear (Pasternak) layer stiffness.

2.4. Hygrothermal Conditions

Temperature considerably affects the mechanical, electrical, and structural performance, as well as the magnetic characteristics, of composite materials. Furthermore, temperature variations can cause distortions in the crystalline structure of materials, reducing their durability and hardness. Moisture, for its part, can have an impact on annular sandwich nanoplate qualities by inducing corrosion and mechanical changes. Understanding how materials react to moisture is, thus, critical to ensuring the structure’s long-term viability and dependability. In the present analysis, the proposed sandwich nanoplate is considered to be subjected to a temperature field $\mathcal{T}(r,\vartheta ,z)$ and moisture concentration $\mathcal{C}(r,\vartheta ,z)$, as described by the sinusoidal rule [43]:

$$\begin{array}{cc}\hfill & \mathcal{T}(r,\vartheta ,z)=\widehat{\mathcal{T}}\left(z\right)sin\left[{\displaystyle \frac{\pi }{a_1-a_0}}\right(r-a_0\left)\right]sin\left(\mathrm{{\rm Y}}\vartheta \right),\hfill \\ & \mathcal{C}(r,\vartheta ,z)=\widehat{\mathcal{C}}\left(z\right)sin\left[{\displaystyle \frac{\pi }{a_1-a_0}}\right(r-a_0\left)\right]sin\left(\mathrm{{\rm Y}}\vartheta \right),\mathrm{{\rm Y}}=1,2,\dots \hfill \end{array}$$

(11)

Based on the power-law modifications via the plate’s thickness, the temperature and moisture of the lower and upper surfaces are ${\mathrm{\Lambda }}_B$ and ${\mathrm{\Lambda }}_t$$(\mathrm{\Lambda }=\mathcal{T},\mathcal{C})$, respectively, and are thought to fluctuate from ${\mathrm{\Lambda }}_B$ to ${\mathrm{\Lambda }}_t$. In particular, the general mechanism of $\widehat{\mathcal{T}}\left(z\right)$ and $\widehat{\mathcal{C}}\left(z\right)$ can be given as [43]

$$\begin{array}{cc}\hfill & \widehat{\mathcal{T}}\left(z\right)=({\displaystyle \frac{z}{\mathcal{H}}}+{\displaystyle \frac{1}{2}}{)}^k\mathrm{\Delta }\mathcal{T},\mathrm{\Delta }\mathcal{T}={\mathcal{T}}_t-{\mathcal{T}}_B,\hfill \\ & \widehat{\mathcal{C}}\left(z\right)=({\displaystyle \frac{z}{\mathcal{H}}}+{\displaystyle \frac{1}{2}}{)}^k\mathrm{\Delta }\mathcal{C},\mathrm{\Delta }\mathcal{C}={\mathcal{C}}_t-{\mathcal{C}}_B,\hfill \end{array}$$

(12)

where k refers to the hygrothermal exponent, $0<k<\infty .$

2.5. Displacement Fields and Strains

The current proposed model implements a new quasi-3D plate theory (Q3HSDT) that incorporates thickness-stretching influences; in particular, the theory is modeled using Shimpi’s theory [44], which implies the subsequent hypotheses:

• The transverse strains $({ϵ}_{\vartheta z},{ϵ}_{rz},{ϵ}_{zz})$ and stresses $({\sigma }_{\vartheta z},{\sigma }_{rz},{\sigma }_{zz})$ can be identified as the consequence of transverse displacement $u_z$, which contains three elements; namely, the bending deflection ${\psi }^b$, shear deflection ${\psi }^s$, and stretching ${\psi }^n\left(r\right)$. All components are functions of r only.
• The middle-plane radial displacement u, is involved in Shimpi’s two-variable plate theory [44].

Using Q3HSDT, the displacement field of the nanoplate is characterized in the form $\mathbf{u}(r,\vartheta ,z)=u_r(r,z)\widehat{r}+u_{\vartheta }(r,z)\widehat{\vartheta }+u_z(r,z)\widehat{z}$ and is written as [45]

$$\begin{array}{cc}\hfill & u_r(r,z)=u\left(r\right)-z{\displaystyle \frac{\partial {\psi }^b}{\partial r}}-\mathrm{\Psi }\left(z\right){\displaystyle \frac{\partial {\psi }^s}{\partial r}},u_{\vartheta }(r,z)=0,\hfill \\ & u_z(r,z)={\psi }^b\left(r\right)+{\psi }^s\left(r\right)+{\overline{\mathrm{\Psi }}}^{\prime }\left(z\right){\psi }^n\left(r\right),\hfill \end{array}$$

(13)

where ${\psi }^n\left(r\right)$ is an extra displacement that accounts for the thickness-stretching effects. Additionally, $\mathrm{\Psi }\left(z\right)=z-\overline{\mathrm{\Psi }}\left(z\right)$, where the shape function $\overline{\mathrm{\Psi }}\left(z\right)$ can be demonstrated by

$$\overline{\mathrm{\Psi }}\left(z\right)={\displaystyle \frac{\mathcal{H}}{2}}{sinh}^{-1}\left({\displaystyle \frac{2z}{\mathcal{H}}}\right)-{\displaystyle \frac{2\sqrt{2}{z}^3}{3{\mathcal{H}}^2}}.$$

(14)

The shape function, $\overline{\mathrm{\Psi }}\left(z\right)$, could be validated by comparing it with various theories from the literature, including third-order theory (TPT) [46] $(\overline{\mathrm{\Psi }}\left(z\right)=z-(4z^3/3{\mathcal{H}}^2))$, sinusoidal plate theory (SPT) [47] $(\overline{\mathrm{\Psi }}\left(z\right)=\mathcal{H}sin(\pi z/\mathcal{H})/\pi )$, hyperbolic plate theory (HPT) [48] $(\overline{\mathrm{\Psi }}\left(z\right)=\mathcal{H}sinh(z/\mathcal{H})-zcosh(1/2))$, and exponential plate theory (EPT) [49] $(\overline{\mathrm{\Psi }}\left(z\right)=zexp(-2z^2/{\mathcal{H}}^2))$. The overall form of the strain-displacement relationships in cylindrical coordinates is now presented as follows:

$$\begin{array}{cc}\hfill & {ϵ}_{rr}={\displaystyle \frac{\partial u_r}{\partial r}},{ϵ}_{\vartheta \vartheta }={\displaystyle \frac{1}{r}}({\displaystyle \frac{\partial u_{\vartheta }}{\partial \vartheta }}+u_r),{ϵ}_{zz}={\displaystyle \frac{\partial u_z}{\partial z}},\hfill \\ \hfill & {ϵ}_{r\vartheta }=({\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial u_r}{\partial \vartheta }}+{\displaystyle \frac{\partial u_{\vartheta }}{\partial r}}-{\displaystyle \frac{u_{\vartheta }}{r}}),\hfill \\ & {ϵ}_{\vartheta z}=({\displaystyle \frac{\partial u_{\vartheta }}{\partial z}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial u_z}{\partial \vartheta }}),{ϵ}_{rz}=({\displaystyle \frac{\partial u_r}{\partial z}}+{\displaystyle \frac{\partial u_z}{\partial r}}).\hfill \end{array}$$

(15)

By inserting Equation (13) into Equation (15), the strain components are given as

$$\begin{array}{cc}\hfill & {ϵ}_{rr}={\displaystyle \frac{\mathrm{d}u}{\mathrm{d}r}}-z{\displaystyle \frac{{\mathrm{d}}^2{\psi }^b}{\mathrm{d}{r}^2}}-\mathrm{\Psi }\left(z\right){\displaystyle \frac{{\mathrm{d}}^2{\psi }^s}{\mathrm{d}{r}^2}},{ϵ}_{\vartheta \vartheta }={\displaystyle \frac{1}{r}}(u-z{\displaystyle \frac{\mathrm{d}{\psi }^b}{\mathrm{d}r}}-\mathrm{\Psi }\left(z\right){\displaystyle \frac{\mathrm{d}{\psi }^s}{\mathrm{d}r}}),\hfill \\ & {ϵ}_{zz}={\overline{\mathrm{\Psi }}}^{″}\left(z\right){\psi }^n,{ϵ}_{r\vartheta }=0,{ϵ}_{\vartheta z}=0,{ϵ}_{rz}={\overline{\mathrm{\Psi }}}^{\prime }\left(z\right)({\displaystyle \frac{\mathrm{d}{\psi }^s}{\mathrm{d}r}}+{\displaystyle \frac{\mathrm{d}{\psi }^n}{\mathrm{d}r}}).\hfill \end{array}$$

(16)

2.6. Nonlocal Strain Gradient Theory

NSGT was initially developed by Aifantis [16,50] and was subsequently expanded upon by Lim et al. [14]. In accordance with this theory, the total stress tensor ${\tau }_{ij}$ is written as

$${\tau }_{ij}={\sigma }_{ij}^{\left(0\right)}-\nabla ·{\sigma }_{ij}^{\left(1\right)},$$

(17)

where ∇ is the gradient operator, and ${\sigma }_{ij}^{\left(0\right)}$ and ${\sigma }_{ij}^{\left(1\right)}$ are nonlocal stress tensors and high-order nonlocal stress tensors that differ from Aifantis gradient theory, respectively; they are provided as

$$\begin{array}{cc}\hfill & {\sigma }_{ij}^{\left(0\right)}=C_{ijmn}{\int }_V{\kappa }_0\left(\right|\omega -{\omega }^{\prime }|,e_0\iota ){ϵ}_{mn}\mathrm{d}V,\hfill \\ & {\sigma }_{ij}^{\left(1\right)}={\zeta }^2{C}_{ijmn}{\int }_V{\kappa }_1\left(\right|\omega -{\omega }^{\prime }|,e_1\iota )\nabla {ϵ}_{mn}\mathrm{d}V,\hfill \end{array}$$

(18)

where $C_{ijmn}$ denotes the elastic coefficients, $\zeta$ stands for the strain-gradient length-scale parameter, and $e_0\iota$ and $e_1\iota$ are the nonlocal parameters. The nonlocal functions ${\kappa }_0\left(\right|\omega -{\omega }^{\prime }|,e_0\iota )$ and ${\kappa }_1\left(\right|\omega -{\omega }^{\prime }|,e_1\iota )$ accompany the strain tensor ${ϵ}_{mn}$ and the first-order strain gradient $\nabla {ϵ}_{mn}$, respectively, and meet Eringen’s nonlocal theory criteria. Further, the quantity $|\omega -{\omega }^{\prime }|$ refers to the Euclidean metric. According to Eringen’s nonlocal elasticity theory [17,51], and with $e_0=e_1=e$, the nonlocal stresses ${\sigma }_{ij}^{\left(0\right)}$ and ${\sigma }_{ij}^{\left(1\right)}$ in Equation (18) can be represented as

$$\begin{array}{cc}\hfill & {\mathrm{\Delta }}_N{\sigma }_{ij}^{\left(0\right)}=C_{ijmn}{ϵ}_{mn},\hfill \\ & {\mathrm{\Delta }}_N{\sigma }_{ij}^{\left(1\right)}={\zeta }^2{C}_{ijmn}\nabla {ϵ}_{mn},{\mathrm{\Delta }}_N=\left(1-{\left(e\iota \right)}^2{\nabla }^2\right).\hfill \end{array}$$

(19)

By applying the nonlocal differential operator ${\mathrm{\Delta }}_N$ to both sides of Equation (17) and then substituting Equation (19), the general constitutive relations can be stated as

$$\begin{array}{c}\hfill {\mathrm{\Delta }}_N{\tau }_{ij}={\mathrm{\Delta }}_S{C}_{ijmn}{ϵ}_{mn},{\mathrm{\Delta }}_S=\left(1-{\zeta }^2{\nabla }^2\right),\end{array}$$

(20)

in which ${\nabla }^2$ represents the Laplacian operator in the cylindrical coordinates, which can be represented in our study as follows:

$${\nabla }^2={\displaystyle \frac{{\partial }^2}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}.$$

(21)

The constitutive Equation (20) of the sandwich structure, considering the influence of hygrothermal loading, can be formulated as follows [14]:

$$\begin{array}{cc}\hfill & {\mathrm{\Delta }}_N{\tau }_{rr}^{\left(i\right)}={\mathrm{\Delta }}_S{\overline{\sigma }}_{rr}^{\left(i\right)}={\mathrm{\Delta }}_S\left(C_{11}^{\left(i\right)}{ϵ}_{rr}+C_{12}^{\left(i\right)}{ϵ}_{\vartheta \vartheta }+C_{13}^{\left(i\right)}{ϵ}_{zz}-{\overline{\alpha }}^{\left(i\right)}\mathcal{T}-{\overline{\beta }}^{\left(i\right)}\mathcal{C}\right),\hfill \\ & {\mathrm{\Delta }}_N{\tau }_{\vartheta \vartheta }^{\left(i\right)}={\mathrm{\Delta }}_S{\overline{\sigma }}_{\vartheta \vartheta }^{\left(i\right)}={\mathrm{\Delta }}_S\left(C_{12}^{\left(i\right)}{ϵ}_{rr}+C_{22}^{\left(i\right)}{ϵ}_{\vartheta \vartheta }+C_{23}^{\left(i\right)}{ϵ}_{zz}-{\overline{\alpha }}^{\left(i\right)}\mathcal{T}-{\overline{\beta }}^{\left(i\right)}\mathcal{C}\right),\hfill \\ & {\mathrm{\Delta }}_N{\tau }_{zz}^{\left(i\right)}={\mathrm{\Delta }}_S{\overline{\sigma }}_{zz}^{\left(i\right)}={\mathrm{\Delta }}_S\left(C_{13}^{\left(i\right)}{ϵ}_{rr}+C_{23}^{\left(i\right)}{ϵ}_{\vartheta \vartheta }+C_{33}^{\left(i\right)}{ϵ}_{zz}-{\overline{\alpha }}^{\left(i\right)}\mathcal{T}-{\overline{\beta }}^{\left(i\right)}\mathcal{C}\right),\hfill \\ & {\mathrm{\Delta }}_N{\tau }_{rz}^{\left(i\right)}={\mathrm{\Delta }}_S{\overline{\sigma }}_{rz}^{\left(i\right)}={\mathrm{\Delta }}_S{C}_{55}^{\left(i\right)}{ϵ}_{rz},\hfill \end{array}$$

(22)

where

$$\begin{array}{cc}\hfill & C_{11}^{\left(i\right)}=C_{22}^{\left(i\right)}=C_{33}^{\left(i\right)}={\displaystyle \frac{[1-{\nu }^{\left(i\right)}]E^{\left(i\right)}}{[1+{\nu }^{\left(i\right)}][1-2{\nu }^{\left(i\right)}]}},\hfill \\ & C_{12}^{\left(i\right)}=C_{13}^{\left(i\right)}=C_{23}^{\left(i\right)}={\displaystyle \frac{{\nu }^{\left(i\right)}{E}^{\left(i\right)}}{[1+{\nu }^{\left(i\right)}][1-2{\nu }^{\left(i\right)}]}},\hfill \\ & C_{55}^{\left(i\right)}={\displaystyle \frac{E^{\left(i\right)}}{2[1+{\nu }^{\left(i\right)}]}},{\overline{\alpha }}^{\left(i\right)}=(C_{11}^{\left(i\right)}+C_{12}^{\left(i\right)}+C_{13}^{\left(i\right)}){\alpha }^{\left(i\right)},\hfill \\ & {\overline{\beta }}^{\left(i\right)}=(C_{11}^{\left(i\right)}+C_{12}^{\left(i\right)}+C_{13}^{\left(i\right)}){\beta }^{\left(i\right)},i=1,2,3.\hfill \end{array}$$

(23)

2.7. Lorentz Magnetic Force

As stated before, the present annular sandwich nanoplate is presumed to be exposed to a uniform in-plane magnetic field, $\mathcal{M}=({\mathcal{M}}_r,0,0)$; in particular, the in-plane magnetic field acts only in the r direction. With respect to the typical Maxwell equation found in Refs. [52,53], the magnetic field induces a body force, referred to as the Lorentz force $\mathcal{F}=(f_r,f_{\vartheta },f_z)$, on all of the nanoplate’s particles, which can be represented as [53]

$$\mathcal{F}=\gamma (\widehat{\mathbf{J}}\times \mathcal{M}),\widehat{\mathbf{J}}=\nabla \times \widehat{\mathbf{g}},\widehat{\mathbf{g}}=\nabla \times (\mathbf{u}\times \mathcal{M}),$$

(24)

where $\gamma$, $\widehat{\mathbf{J}}$, and $\widehat{\mathbf{g}}$ denote, respectively, the permeability of the magnetic field, the current density, and the disturbing vector of the magnetic field. Incorporating the displacement vector, $\mathbf{u}$, that was defined in Equation (13) into the third equation of Equation (24) leads to the magnetic field distributing vector, $\widehat{\mathbf{g}}$, denoted as

$$\widehat{\mathbf{g}}={\mathcal{M}}_r\left[\right(-{\displaystyle \frac{\partial u_z}{\partial z}})\widehat{r}+({\displaystyle \frac{u_z}{r}}+{\displaystyle \frac{\partial u_z}{\partial r}}\left)\widehat{z}\right].$$

(25)

By inserting Equation (25) into the second equation of Equation (24), the current density can be given in the following form:

$$\widehat{\mathbf{J}}=-{\mathcal{M}}_r({\displaystyle \frac{{\partial }^2{u}_z}{\partial r^2}}+{\displaystyle \frac{{\partial }^2{u}_z}{\partial z^2}}+{\displaystyle \frac{\partial u_z}{r\partial r}}-{\displaystyle \frac{u_z}{r^2}})\widehat{\vartheta }.$$

(26)

Further, incorporating Equation (26) into Equation (24) yields the elements of the Lorentz magnetic force:

$$f_r=0,f_{\vartheta }=0,f_z=\gamma {\mathcal{M}}_r^2\left({\displaystyle \frac{{\partial }^2{u}_z}{\partial r^2}}+{\displaystyle \frac{{\partial }^2{u}_z}{\partial z^2}}+{\displaystyle \frac{\partial u_z}{r\partial r}}-{\displaystyle \frac{u_z}{r^2}}\right).$$

(27)

3. Governing Equations

We determine the governing equations of the equivalent nanoplate through employing the virtual work concept as follows:

$$\delta {\pi }_s-\delta {\pi }_f=0,$$

(28)

where $\delta {\pi }_s$ and $\delta {\pi }_f$ indicate, respectively, the variation in strain energy and the work conducted by the magnetic body force, external load, and elastic foundation, which can be expressed as

$$\begin{array}{cc}\hfill & \delta {\pi }_s=\sum _{i=1}^3{\int }_0^{2\pi }{\int }_{a_0}^{a_1}{\int }_{{\mathcal{H}}_{i-1}}^{{\mathcal{H}}_i}{\tau }_{ij}^{\left(i\right)}\delta {ϵ}_{ij}r\mathrm{d}z\mathrm{d}r\mathrm{d}\vartheta ,\hfill \\ \hfill & \delta {\pi }_f={\int }_0^{2\pi }{\int }_{a_0}^{a_1}{\int }_{-{\displaystyle \frac{\mathcal{H}}{2}}}^{{\displaystyle \frac{\mathcal{H}}{2}}}\mathcal{F}·\delta \mathbf{u}r\mathrm{d}z\mathrm{d}r\mathrm{d}\vartheta +{\int }_0^{2\pi }{\int }_{a_0}^{a_1}\left(q\delta u_z{{|}_{z={\displaystyle \frac{\mathcal{H}}{2}}}-\mathcal{J}\delta u_z|}_{z=-{\displaystyle \frac{\mathcal{H}}{2}}}\right)r\mathrm{d}r\mathrm{d}\vartheta ,\hfill \end{array}$$

(29)

where q specifies the external transverse load.

By incorporating Equation (29) into Equation (28) and implementing the integration in parts, the governing equations can be obtained by setting the coefficients of $\delta u$, $\delta {\psi }^b$, $\delta {\psi }^s$, and $\delta {\psi }^n$ to zero; thus, one can obtain

$$\begin{array}{cc}\hfill & \delta u:{\displaystyle \frac{\mathrm{d}{N}_{rr}}{\mathrm{d}r}}+{\displaystyle \frac{N_{rr}-N_{\vartheta \vartheta }}{r}}=0,\hfill \\ \hfill & \delta {\psi }^b:{\displaystyle \frac{{\mathrm{d}}^2{M}_{rr}}{\mathrm{d}{r}^2}}+{\displaystyle \frac{2}{r}}{\displaystyle \frac{\mathrm{d}{M}_{rr}}{\mathrm{d}r}}-{\displaystyle \frac{1}{r}}{\displaystyle \frac{\mathrm{d}{M}_{\vartheta \vartheta }}{\mathrm{d}r}}+\left(\gamma {\mathcal{M}}_r^2\mathcal{H}+K_p\right){\displaystyle \frac{{\mathrm{d}}^2{\psi }^b}{\mathrm{d}{r}^2}}+\left(\gamma {\mathcal{M}}_r^2\mathcal{H}+K_p\right){\displaystyle \frac{{\mathrm{d}}^2{\psi }^s}{\mathrm{d}{r}^2}}\hfill \\ & +A_1\gamma {\mathcal{M}}_r^2{\displaystyle \frac{{\mathrm{d}}^2{\psi }^n}{\mathrm{d}{r}^2}}+\left(\gamma {\mathcal{M}}_r^2\mathcal{H}+K_p\right){\displaystyle \frac{\mathrm{d}{\psi }^b}{r\mathrm{d}r}}+\left(\gamma {\mathcal{M}}_r^2\mathcal{H}+K_p\right){\displaystyle \frac{\mathrm{d}{\psi }^s}{r\mathrm{d}r}}+A_1\gamma {\mathcal{M}}_r^2{\displaystyle \frac{\mathrm{d}{\psi }^n}{r\mathrm{d}r}}\hfill \\ & -\left(\gamma {\mathcal{M}}_r^2\mathcal{H}+K_w\right){\displaystyle \frac{{\psi }^b}{r}}-\left(\gamma {\mathcal{M}}_r^2\mathcal{H}+K_w\right){\displaystyle \frac{{\psi }^s}{r}}-\left(A_1-A_2\right)\gamma {\mathcal{M}}_r^2{\displaystyle \frac{{\psi }^n}{r}}+q=0,\hfill \\ & \delta {\psi }^s:{\displaystyle \frac{{\mathrm{d}}^2{S}_{rr}}{\mathrm{d}{r}^2}}+{\displaystyle \frac{2}{r}}{\displaystyle \frac{\mathrm{d}{S}_{rr}}{\mathrm{d}r}}-{\displaystyle \frac{1}{r}}{\displaystyle \frac{\mathrm{d}{S}_{\vartheta \vartheta }}{\mathrm{d}r}}+{\displaystyle \frac{{\overline{R}}_{rz}}{r}}+{\displaystyle \frac{\mathrm{d}{\overline{R}}_{rz}}{\mathrm{d}r}}-\left(\gamma {\mathcal{M}}_r^2\mathcal{H}+K_p\right)\left({\displaystyle \frac{{\mathrm{d}}^2{\psi }^b}{\mathrm{d}{r}^2}}+{\displaystyle \frac{{\mathrm{d}}^2{\psi }^s}{\mathrm{d}{r}^2}}\right)\hfill \\ & -A_1\gamma {\mathcal{M}}_r^2{\displaystyle \frac{{\mathrm{d}}^2{\psi }^n}{\mathrm{d}{r}^2}}-\left(\gamma {\mathcal{M}}_r^2\mathcal{H}+K_p\right){\displaystyle \frac{\mathrm{d}{\psi }^b}{r\mathrm{d}r}}-\left(\gamma {\mathcal{M}}_r^2\mathcal{H}+K_p\right){\displaystyle \frac{\mathrm{d}{\psi }^s}{r\mathrm{d}r}}\hfill \\ & -A_1\gamma {\mathcal{M}}_r^2{\displaystyle \frac{\mathrm{d}{\psi }^n}{r\mathrm{d}r}}+\left(\gamma {\mathcal{M}}_r^2\mathcal{H}+K_w\right)\left({\displaystyle \frac{{\psi }^b}{r}}+{\displaystyle \frac{{\psi }^s}{r}}\right)+(A_1-A_2)\gamma {\mathcal{M}}_r^2{\displaystyle \frac{{\psi }^n}{r}}-q=0,\hfill \\ & \delta {\psi }^n:-R_{zz}+{\displaystyle \frac{{\overline{R}}_{rz}}{r}}+{\displaystyle \frac{\mathrm{d}{\overline{R}}_{rz}}{\mathrm{d}r}}-\gamma {\mathcal{M}}_r^2[A_1{\displaystyle \frac{{\mathrm{d}}^2{\psi }^b}{\mathrm{d}{r}^2}}+A_1{\displaystyle \frac{{\mathrm{d}}^2{\psi }^s}{\mathrm{d}{r}^2}}+A_3{\displaystyle \frac{{\mathrm{d}}^2{\psi }^n}{\mathrm{d}{r}^2}}+A_1{\displaystyle \frac{\mathrm{d}{\psi }^b}{r\mathrm{d}r}}+A_1{\displaystyle \frac{\mathrm{d}{\psi }^s}{r\mathrm{d}r}}\hfill \\ & +A_3{\displaystyle \frac{\mathrm{d}{\psi }^n}{r\mathrm{d}r}}-A_1{\displaystyle \frac{{\psi }^b}{r^2}}-A_1{\displaystyle \frac{{\psi }^s}{r^2}}+(A_4-{\displaystyle \frac{A_3}{r^2}}){\psi }^n]=0,\hfill \end{array}$$

(30)

where

$$\begin{array}{cc}\hfill & N_{rr}=\sum _{i=1}^3{\int }_{{\mathcal{H}}_{i-1}}^{{\mathcal{H}}_i}{\tau }_{rr}^{\left(i\right)}\mathrm{d}z,M_{rr}=\sum _{i=1}^3{\int }_{{\mathcal{H}}_{i-1}}^{{\mathcal{H}}_i}{\tau }_{rr}^{\left(i\right)}z\mathrm{d}z,S_{rr}=\sum _{i=1}^3{\int }_{{\mathcal{H}}_{i-1}}^{{\mathcal{H}}_i}{\tau }_{rr}^{\left(i\right)}\mathrm{\Psi }\left(z\right)\mathrm{d}z,\hfill \\ & N_{\vartheta \vartheta }=\sum _{i=1}^3{\int }_{{\mathcal{H}}_{i-1}}^{{\mathcal{H}}_i}{\tau }_{\vartheta \vartheta }^{\left(i\right)}\mathrm{d}z,M_{\vartheta \vartheta }=\sum _{i=1}^3{\int }_{{\mathcal{H}}_{i-1}}^{{\mathcal{H}}_i}{\tau }_{\vartheta \vartheta }^{\left(i\right)}z\mathrm{d}z,S_{\vartheta \vartheta }=\sum _{i=1}^3{\int }_{{\mathcal{H}}_{i-1}}^{{\mathcal{H}}_i}{\tau }_{\vartheta \vartheta }^{\left(i\right)}\mathrm{\Psi }\left(z\right)\mathrm{d}z,\hfill \\ & R_{zz}=\sum _{i=1}^3{\int }_{{\mathcal{H}}_{i-1}}^{{\mathcal{H}}_i}{\tau }_{zz}^{\left(i\right)}{\overline{\mathrm{\Psi }}}^{″}\left(z\right)\mathrm{d}z,{\overline{R}}_{rz}=\sum _{i=1}^3{\int }_{{\mathcal{H}}_{i-1}}^{{\mathcal{H}}_i}{\tau }_{rz}^{\left(i\right)}{\overline{\mathrm{\Psi }}}^{\prime }\left(z\right)\mathrm{d}z,\hfill \end{array}$$

(31)

and

$$\begin{array}{cc}\hfill & A_1={\int }_{{\displaystyle \frac{-\mathcal{H}}{2}}}^{{\displaystyle \frac{\mathcal{H}}{2}}}{\overline{\mathrm{\Psi }}}^{\prime }\left(z\right)\mathrm{d}z,A_2={\int }_{{\displaystyle \frac{-\mathcal{H}}{2}}}^{{\displaystyle \frac{\mathcal{H}}{2}}}{\overline{\mathrm{\Psi }}}^{‴}\left(z\right)\mathrm{d}z,\hfill \\ & A_3={\int }_{{\displaystyle \frac{-\mathcal{H}}{2}}}^{{\displaystyle \frac{\mathcal{H}}{2}}}({\overline{\mathrm{\Psi }}}^{\prime }\left(z\right){)}^2\mathrm{d}z,A_4={\int }_{{\displaystyle \frac{-\mathcal{H}}{2}}}^{{\displaystyle \frac{\mathcal{H}}{2}}}{\overline{\mathrm{\Psi }}}^{\prime }\left(z\right){\overline{\mathrm{\Psi }}}^{‴}\left(z\right)\mathrm{d}z.\hfill \end{array}$$

(32)

By implementing the nonlocal operator, ${\mathrm{\Delta }}_N$, into Equation (31) and then substituting Equation (22) into the resulting equations with the help of Equations (13) and (16), one can deduce

$$\begin{array}{cc}\hfill & {\mathrm{\Delta }}_N{N}_{rr}={\mathrm{\Delta }}_S[B_1{\displaystyle \frac{\mathrm{d}u}{\mathrm{d}r}}-B_2{\displaystyle \frac{{\mathrm{d}}^2{\psi }^b}{\mathrm{d}{r}^2}}-B_3{\displaystyle \frac{{\mathrm{d}}^2{\psi }^s}{\mathrm{d}{r}^2}}+B_4{\displaystyle \frac{u}{r}}-B_5{\displaystyle \frac{\mathrm{d}{\psi }^b}{r\mathrm{d}r}}-B_6{\displaystyle \frac{\mathrm{d}{\psi }^s}{r\mathrm{d}r}}+B_7{\psi }^n\hfill \\ & -B_8^{\mathcal{T}}sin\left({\displaystyle \frac{\pi }{a_1-a_0}}(r-a_0)\right)-B_9^{\mathcal{C}}sin\left({\displaystyle \frac{\pi }{a_1-a_0}}(r-a_0)\right)],\hfill \\ & {\mathrm{\Delta }}_N{M}_{rr}={\mathrm{\Delta }}_S[B_2{\displaystyle \frac{\mathrm{d}u}{\mathrm{d}r}}-B_{10}{\displaystyle \frac{{\mathrm{d}}^2{\psi }^b}{\mathrm{d}{r}^2}}-B_{11}{\displaystyle \frac{{\mathrm{d}}^2{\psi }^s}{\mathrm{d}{r}^2}}+B_5{\displaystyle \frac{u}{r}}-B_{12}{\displaystyle \frac{\mathrm{d}{\psi }^b}{r\mathrm{d}r}}-B_{13}{\displaystyle \frac{\mathrm{d}{\psi }^s}{r\mathrm{d}r}}+B_{14}{\psi }^n\hfill \\ & -B_{15}^{\mathcal{T}}sin\left({\displaystyle \frac{\pi }{a_1-a_0}}(r-a_0)\right)-B_{16}^{\mathcal{C}}sin\left({\displaystyle \frac{\pi }{a_1-a_0}}(r-a_0)\right)],\hfill \\ & {\mathrm{\Delta }}_N{S}_{rr}={\mathrm{\Delta }}_S[B_3{\displaystyle \frac{\mathrm{d}u}{\mathrm{d}r}}-B_{11}{\displaystyle \frac{{\mathrm{d}}^2{\psi }^b}{\mathrm{d}{r}^2}}-B_{17}{\displaystyle \frac{{\mathrm{d}}^2{\psi }^s}{\mathrm{d}{r}^2}}+B_6{\displaystyle \frac{u}{r}}-B_{13}{\displaystyle \frac{\mathrm{d}{\psi }^b}{r\mathrm{d}r}}-B_{18}{\displaystyle \frac{\mathrm{d}{\psi }^s}{r\mathrm{d}r}}+B_{19}{\psi }^n\hfill \\ & -B_{20}^{\mathcal{T}}sin\left({\displaystyle \frac{\pi }{a_1-a_0}}(r-a_0)\right)-B_{21}^{\mathcal{C}}sin\left({\displaystyle \frac{\pi }{a_1-a_0}}(r-a_0)\right)],\hfill \\ & {\mathrm{\Delta }}_N{N}_{\vartheta \vartheta }={\mathrm{\Delta }}_S[B_4{\displaystyle \frac{\mathrm{d}u}{\mathrm{d}r}}-B_5{\displaystyle \frac{{\mathrm{d}}^2{\psi }^b}{\mathrm{d}{r}^2}}-B_6{\displaystyle \frac{{\mathrm{d}}^2{\psi }^s}{\mathrm{d}{r}^2}}+B_1{\displaystyle \frac{u}{r}}-B_2{\displaystyle \frac{\mathrm{d}{\psi }^b}{r\mathrm{d}r}}-B_3{\displaystyle \frac{\mathrm{d}{\psi }^s}{r\mathrm{d}r}}+B_7{\psi }^n\hfill \\ & -B_8^{\mathcal{T}}sin\left({\displaystyle \frac{\pi }{a_1-a_0}}(r-a_0)\right)-B_9^{\mathcal{C}}sin\left({\displaystyle \frac{\pi }{a_1-a_0}}(r-a_0)\right)],\hfill \\ & {\mathrm{\Delta }}_N{M}_{\vartheta \vartheta }={\mathrm{\Delta }}_S[B_5{\displaystyle \frac{\mathrm{d}u}{\mathrm{d}r}}-B_{12}{\displaystyle \frac{{\mathrm{d}}^2{\psi }^b}{\mathrm{d}{r}^2}}-B_{13}{\displaystyle \frac{{\mathrm{d}}^2{\psi }^s}{\mathrm{d}{r}^2}}+B_2{\displaystyle \frac{u}{r}}-B_{10}{\displaystyle \frac{\mathrm{d}{\psi }^b}{r\mathrm{d}r}}-B_{11}{\displaystyle \frac{\mathrm{d}{\psi }^s}{r\mathrm{d}r}}+B_{14}{\psi }^n\hfill \\ & -B_{15}^{\mathcal{T}}sin\left({\displaystyle \frac{\pi }{a_1-a_0}}(r-a_0)\right)-B_{16}^{\mathcal{C}}sin\left({\displaystyle \frac{\pi }{a_1-a_0}}(r-a_0)\right)],\hfill \\ & {\mathrm{\Delta }}_N{S}_{\vartheta \vartheta }={\mathrm{\Delta }}_S[B_6{\displaystyle \frac{\mathrm{d}u}{\mathrm{d}r}}-B_{13}{\displaystyle \frac{{\mathrm{d}}^2{\psi }^b}{\mathrm{d}{r}^2}}-B_{18}{\displaystyle \frac{{\mathrm{d}}^2{\psi }^s}{\mathrm{d}{r}^2}}+B_3{\displaystyle \frac{u}{r}}-B_{11}{\displaystyle \frac{\mathrm{d}{\psi }^b}{r\mathrm{d}r}}-B_{17}{\displaystyle \frac{\mathrm{d}{\psi }^s}{r\mathrm{d}r}}+B_{19}{\psi }^n\hfill \\ & -B_{20}^{\mathcal{T}}sin\left({\displaystyle \frac{\pi }{a_1-a_0}}(r-a_0)\right)-B_{21}^{\mathcal{C}}sin\left({\displaystyle \frac{\pi }{a_1-a_0}}(r-a_0)\right)],\hfill \\ & {\mathrm{\Delta }}_N{R}_{zz}={\mathrm{\Delta }}_S[B_7\left({\displaystyle \frac{\mathrm{d}u}{\mathrm{d}r}}+{\displaystyle \frac{u}{r}}\right)-B_{14}\left({\displaystyle \frac{{\mathrm{d}}^2{\psi }^b}{\mathrm{d}{r}^2}}+{\displaystyle \frac{\mathrm{d}{\psi }^b}{r\mathrm{d}r}}\right)-B_{19}\left({\displaystyle \frac{{\mathrm{d}}^2{\psi }^s}{\mathrm{d}{r}^2}}+{\displaystyle \frac{\mathrm{d}{\psi }^s}{r\mathrm{d}r}}\right)+B_{22}{\psi }^n\hfill \\ & -B_{23}^{\mathcal{T}}sin\left({\displaystyle \frac{\pi }{a_1-a_0}}(r-a_0)\right)-B_{24}^{\mathcal{C}}sin\left({\displaystyle \frac{\pi }{a_1-a_0}}(r-a_0)\right)],\hfill \\ & {\mathrm{\Delta }}_N{\overline{R}}_{rz}={\mathrm{\Delta }}_S[B_{25}{\displaystyle \frac{\mathrm{d}{\psi }^s}{\mathrm{d}r}}+B_{25}{\displaystyle \frac{\mathrm{d}{\psi }^n}{\mathrm{d}r}}],\hfill \end{array}$$

(33)

where the coefficients $B_i$ are defined in Appendix A.

In order to construct the governing Equation (30) in terms of the displacement elements, we can apply the nonlocal operator ${\mathrm{\Delta }}_N$ into Equation (30) with the aid of Equation (33) to conclude as follows:

$$\begin{array}{c}\hfill \left[\overline{A}\right]{\left\{\mathrm{\Gamma }\right\}}^{\left(6\right)}+\left[\overline{B}\right]{\left\{\mathrm{\Gamma }\right\}}^{\left(5\right)}+\left[\overline{C}\right]{\left\{\mathrm{\Gamma }\right\}}^{\left(4\right)}+\left[\overline{D}\right]{\left\{\mathrm{\Gamma }\right\}}^{‴}+\left[\overline{E}\right]{\left\{\mathrm{\Gamma }\right\}}^{″}+\left[\overline{G}\right]{\left\{\mathrm{\Gamma }\right\}}^{\prime }+\left[\overline{J}\right]\left\{\mathrm{\Gamma }\right\}=\left\{\overline{F}\right\},\end{array}$$

(34)

where $\left\{\mathrm{\Gamma }\right\}={\left\{u{\psi }^b{\psi }^s{\psi }^n\right\}}^T$, while the matrices $\left[\overline{A}\right],\left[\overline{B}\right],\left[\overline{C}\right],\left[\overline{D}\right]$, $\left[\overline{E}\right]$, $\left[\overline{G}\right]$, and $\left[\overline{J}\right]$ and the vector $\left\{\overline{F}\right\}$ are defined in Appendix A.

Furthermore, in the current analysis, the boundary conditions can be introduced as follows:

Simply supported (SS):

$${\psi }^b={\psi }^s={\psi }^n={\mathrm{\Delta }}_N{N}_{rr}={\mathrm{\Delta }}_N{M}_{rr}={\mathrm{\Delta }}_N{S}_{rr}=0,\mathrm{at}r=a_1\mathrm{and}r=a_0.$$

(35)

Clamped (C):

$$u={\psi }^b={\psi }^s={\psi }^n=0,\mathrm{at}r=a_1\mathrm{and}r=a_0.$$

(36)

4. Numerical Solution

In this study, we implemented DQM to solve the equilibrium Equation (34) along the radial direction. It is worth noting that DQM is a numerical strategy that estimates the partial derivative of a smooth function by weighting the sum of functional values at identified grid points in the computational domain. Consequently, the annular proposed nanoplate is discretized by $\overline{n}$ discrete grid points in the domain $(a_0\le r\le a_1)$. In agreement with the Gauss–Chebyshev–Lobatto technique, the distributed mesh grid points $r_i$ are presented as follows [54]:

$$\begin{array}{c}\hfill r_i=a_0+{\displaystyle \frac{a_1-a_0}{2}}\left[1-cos\left(\pi {\displaystyle \frac{i-1}{\overline{n}-1}}\right)\right].\end{array}$$

(37)

Consequently, based on DQM, the discretizing $\overline{q}$th-order partial derivatives with respect to r of the displacement components can be written as [54]

$$\begin{array}{cc}\hfill & {\left.{\displaystyle \frac{{\mathrm{d}}^{\overline{q}}u}{\mathrm{d}{r}^{\overline{q}}}}\right|}_{r=r_i}=\sum _{j=1}^{\overline{n}}{\mathcal{A}}_{ij}^{\left(\overline{q}\right)}{u}_j,u_j=u\left(r_j\right),\hfill \\ & {\left.{\displaystyle \frac{{\mathrm{d}}^{\overline{q}}{\psi }^b}{\mathrm{d}{r}^{\overline{q}}}}\right|}_{r=r_i}=\sum _{j=1}^{\overline{n}}{\mathcal{A}}_{ij}^{\left(\overline{q}\right)}{\psi }_j^b,{\psi }_j^b={\psi }^b\left(r_j\right),\hfill \\ & {\left.{\displaystyle \frac{{\mathrm{d}}^{\overline{q}}{\psi }^s}{\mathrm{d}{r}^{\overline{q}}}}\right|}_{r=r_i}=\sum _{j=1}^{\overline{n}}{\mathcal{A}}_{ij}^{\left(\overline{q}\right)}{\psi }_j^s,{\psi }_j^s={\psi }^s\left(r_j\right),\hfill \\ & {\left.{\displaystyle \frac{{\mathrm{d}}^{\overline{q}}{\psi }^n}{\mathrm{d}{r}^{\overline{q}}}}\right|}_{r=r_i}=\sum _{j=1}^{\overline{n}}{\mathcal{A}}_{ij}^{\left(\overline{q}\right)}{\psi }_j^n,{\psi }_j^n={\psi }^n\left(r_j\right),i=1,2,\dots ,\overline{n},\hfill \end{array}$$

(38)

in which ${\mathcal{A}}_{ij}^{\left(\overline{q}\right)}$ represents the weighting coefficients analogous to the $\overline{q}$th-order derivative. Consequently, they are given as [54]

$$\begin{array}{cc}\hfill & {\mathcal{A}}_{ij}^{\left(1\right)}={\displaystyle \frac{\mathcal{K}\left(r_i\right)}{(r_i-r_j)\mathcal{K}\left(r_j\right)}},i,j=1,2,\dots ,\overline{n};i\ne j,\hfill \\ & {\mathcal{A}}_{ii}^{\left(1\right)}=-\sum _{l=1}^{\overline{n}}{\mathcal{A}}_{li}^{\left(1\right)},i=1,2,\dots ,\overline{n};l\ne i,\hfill \\ & \mathcal{K}\left(r_i\right)=\prod _{j=1}^{\overline{n}}(r_i-r_j),i\ne j.\hfill \end{array}$$

(39)

In addition, the weighting coefficients ${\mathcal{A}}_{ij}^{\left(\overline{q}\right)},(\overline{q}>1)$ for the higher-order derivatives can be calculated as [54]

$$\begin{array}{c}\hfill {\mathcal{A}}_{ij}^{\left(\overline{q}\right)}=\sum _{l=1}^{\overline{n}}{\mathcal{A}}_{il}^{\left(1\right)}{\mathcal{A}}_{lj}^{(\overline{q}-1)},i,j=1,2,\dots ,\overline{n}.\end{array}$$

(40)

The governing equations (Equation (34)) can be discretized using Equation (38), resulting in $(\overline{n}-2)$ linear algebraic equations that can be easily solved with boundary conditions to produce the displacements.

5. Numerical Results and Discussion

This section presents numerical examples to demonstrate the effects of various parameters on the deflection and stresses of a GPLs-reinforced annular plate with an FG porous core. The parameters are the nonlocal coefficient $e\iota$, the strain gradient length-scale parameter $\zeta$, the porosity factor ${\delta }_0$, porosity distribution patterns, boundary conditions, GPLs weight fraction $w_G$, parameters of temperature $\mathrm{\Delta }\mathcal{T}$ and moisture $\mathrm{\Delta }\mathcal{C}$, GPLs distribution types, the outer-radius-to-thickness ratio $a_1/\mathcal{H}$, the inner-to-outer radius ratio $a_0/a_1$, magnetic field ${\overline{\mathcal{M}}}_r$, core thickness ${\mathcal{H}}_c/\mathcal{H}$, and elastic substrate parameters $g_1$ and $g_2$.

According to earlier found solutions, numerous parameters were investigated to observe how they influence the deflections and stresses of a GPLs-reinforced annular nanoplate with an FG porous core embedded in Pasternak foundation based on Q3HSDT; the nonlocal strain gradient model was analyzed using DQM. In addition, to validate the solution approach, the current numerical results were compared with the results of references.

In order to evaluate and demonstrate the numerical findings, the dimensionless implemented parameters are defined by

$$\begin{array}{cc}\hfill & u_z^{*}=u_z\left({\displaystyle \frac{a_0+a_1}{2}},0\right){\displaystyle \frac{{10}^{-1}{E}_m}{q_0}},{\sigma }_{rr}^{*}={\overline{\sigma }}_{rr}\left({\displaystyle \frac{a_0+a_1}{2}},z\right){\displaystyle \frac{{10}^{-4}}{E_m{q}_0}},\hfill \\ & {\sigma }_{zz}^{*}=-{\overline{\sigma }}_{zz}\left({\displaystyle \frac{a_0+a_1}{2}},z\right){\displaystyle \frac{{10}^{-4}}{E_m{q}_0}},{\sigma }_{rz}^{*}={\overline{\sigma }}_{rz}\left({\displaystyle \frac{a_0+a_1}{2}},z\right){\displaystyle \frac{{10}^{-3}{a}_1}{E_m{q}_0}},\hfill \\ & {\overline{\mathcal{M}}}_r={\displaystyle \frac{\gamma \mathcal{H}{a}_1^2{\mathcal{M}}_r^2}{D_m}},g_1={\displaystyle \frac{K_w{a}_1^4}{D_m}},g_2={\displaystyle \frac{K_P{a}_1^2}{D_m}},D_m={\displaystyle \frac{E_m{\mathcal{H}}^3}{12(1-{\nu }_m^2)}}.\hfill \end{array}$$

(41)

The annular nanoplate is composed of aluminum (metal): $E_m=70{\mathrm{E}}_i$Pa, ${\nu }_m=0.3$, $\rho =2700$ kg/m³, ${\alpha }_m=16.7\times {10}^{-6}/$K, ${\beta }_m=0.44/\mathrm{wt}.\%{\mathrm{H}}_2\mathrm{O}$, and GPLs: $E_g=1010{\mathrm{E}}_i$Pa, ${\nu }_g=0.186$, $\rho =1060$ kg/m³, ${\alpha }_g=5\times {10}^{-6}/$K, ${\beta }_g=0.00026/\mathrm{wt}.\%{\mathrm{H}}_2\mathrm{O}$ where ${\mathrm{E}}_i={10}^9$. The following data were used in this study (unless mentioned otherwise): $\zeta =e\iota =1\times {10}^{-9}$ m, $w_G=0.1$%, $k=1$, $q_0=10$ Pa, ${\overline{\mathcal{M}}}_r=10$, ${\mathcal{H}}_c/\mathcal{H}=0.5$, $g_1=100$, $g_2=10$, ${\delta }_0=0.1$, $\mathrm{\Delta }\mathcal{T}=100$ K, $\mathrm{\Delta }\mathcal{C}=0.1$%, $a_0/a_1=0.1$, $a_1/\mathcal{H}=10$, $\mathcal{H}=5$ nm, $l^g=15$ nm, $w^g=9$ nm, ${\mathcal{H}}^g=0.188$ nm.

First, a convergence evaluation was undertaken to guarantee the independence of the current results of the FG porous sandwich annular nanoplate reinforced by GPLs from the precise number of mesh elements. Accordingly, the convergent solution of DQM requires an appropriate number of discrete points.

Table 1. Convergence of the results of FG porous nanocomposite sandwich annular plates.

$\overline{\mathit{n}}$	${\mathit{u}}_{\mathit{z}}^{*}$	Error	${\mathit{\sigma }}_{\mathit{rr}}^{*}$	Error	${\mathit{\sigma }}_{\mathit{zz}}^{*}$	Error	${\mathit{\sigma }}_{\mathit{rz}}^{*}$	Error
5	$3.85693$	-	$-1.79120$	-	$1.31261$	-	$-17.80773$	-
7	$-0.35939$	$4.21632$	$-0.29916$	$1.49204$	$1.31341$	$0.0008$	$-5.33885$	$12.46888$
9	$0.19221$	$0.5516$	$-0.17648$	$0.12268$	$1.30889$	$0.00452$	$-2.23510$	$3.10384$
11	$0.23392$	$0.04171$	$-0.18813$	$0.01165$	$1.30951$	$0.00062$	$-1.15908$	$1.07593$
13	$0.23461$	$0.00069$	$-0.18771$	$0.00042$	$1.30958$	$0.00007$	$-1.16427$	$0.00519$
15	$0.23533$	$0.00072$	$-0.18770$	$0.00001$	$1.30958$	0	$-1.16804$	$0.00377$
17	$0.23683$	$0.0015$	$-0.18769$	$0.00001$	$1.30958$	0	$-1.17171$	$0.00367$
19	$0.23697$	$0.00014$	$-0.18769$	0	$1.30958$	0	$-1.17014$	$0.00157$

displays the resulting convergence investigation gained from the present study, finding around 17 grid points.

For validation purposes and to indicate the precision of the current results, the results were compared with those depicted by Reddy et al. [55] employing the first-order shear deformation theory (FSDT) and Yun et al. [56] using the finite element method (FEM) as shown in

Table 2. Comparison between the present maximum deflection of an FGM circular plate and those reported by Reddy et al. [55].

BC	s	$\mathcal{H}/{\mathit{a}}_1=0.05$		$\mathcal{H}/{\mathit{a}}_1=0.1$		$\mathcal{H}/{\mathit{a}}_1=0.15$		$\mathcal{H}/{\mathit{a}}_1=0.2$
		Ref. [55]	Present	Ref. [55]	Present	Ref. [55]	Present	Ref. [55]	Present
simply	0	10.396	10.396	10.481	10.480	10.623	10.621	10.822	10.818
	2	5.714	5.713	5.756	5.754	5.826	5.822	5.925	5.918
	4	5.223	5.222	5.261	5.259	5.325	5.320	5.414	5.406
	6	4.970	4.970	5.007	5.005	5.069	5.065	5.155	5.147
	8	4.812	4.812	4.848	4.847	4.909	4.905	4.993	4.987
	10	4.704	4.703	4.739	4.738	4.799	4.796	4.882	4.876
	15	4.538	4.538	4.573	4.572	4.632	4.630	4.714	4.710
	20	4.446	4.445	4.480	4.480	4.538	4.536	4.619	4.616
Clamped	0	$2.554$	$2.561$	$2.639$	$2.667$	$2.781$	$2.844$	$2.979$	$3.093$
	2	$1.402$	$1.405$	$1.444$	$1.457$	$1.515$	$1.542$	$1.613$	$1.661$
	4	$1.282$	$1.248$	$1.320$	$1.330$	$1.384$	$1.407$	$1.473$	$1.514$
	6	$1.220$	$1.222$	$1.257$	$1.267$	$1.318$	$1.340$	$1.404$	$1.444$
	8	$1.181$	$1.184$	$1.217$	$1.227$	$1.278$	$1.300$	$1.362$	$1.401$
	10	$1.155$	$1.157$	$1.190$	$1.200$	$1.250$	$1.272$	$1.333$	$1.373$
	15	$1.114$	$1.117$	$1.149$	$1.160$	$1.208$	$1.231$	$1.289$	$1.331$
	20	$1.092$	$1.094$	$1.126$	$1.137$	$1.184$	$1.208$	$1.265$	$1.307$

and

Table 3. Comparison between the present maximum deflection of an FGM circular plate and those reported by Yun et al. [56] using the FEM.

BC	s	$\mathcal{H}/{\mathit{a}}_1=0.05$		$\mathcal{H}/{\mathit{a}}_1=0.1$		$\mathcal{H}/{\mathit{a}}_1=0.15$		$\mathcal{H}/{\mathit{a}}_1=0.2$
		FEM [56]	Present	FEM [56]	Present	FEM [56]	Present	FEM [56]	Present
simply	0	10.390	10.396	10.459	10.480	10.574	10.621	10.736	10.818
	2	5.709	5.713	5.744	5.754	5.803	5.822	5.886	5.918
	4	5.217	5.222	5.248	5.259	5.300	5.320	5.372	5.406
	6	4.965	4.970	4.994	5.005	5.043	5.065	5.112	5.147
	8	4.806	4.812	4.835	4.847	4.883	4.905	4.951	4.987
	10	4.697	4.703	4.726	4.738	4.773	4.796	4.840	4.876
	15	4.531	4.538	4.559	4.572	4.606	4.630	4.672	4.710
	20	4.437	4.445	4.465	4.480	4.512	4.536	4.578	4.616
Clamped	0	$2.549$	$2.561$	$2.629$	$2.667$	$2.765$	$2.844$	$2.956$	$3.093$
	2	$1.399$	$1.405$	$1.437$	$1.457$	$1.503$	$1.542$	$1.596$	$1.661$
	4	$1.278$	$1.248$	$1.313$	$1.330$	$1.372$	$1.407$	$1.455$	$1.514$
	6	$1.216$	$1.222$	$1.250$	$1.267$	$1.307$	$1.340$	$1.387$	$1.444$
	8	$1.178$	$1.184$	$1.211$	$1.227$	$1.267$	$1.300$	$1.346$	$1.401$
	10	$1.151$	$1.157$	$1.184$	$1.200$	$1.240$	$1.272$	$1.318$	$1.373$
	15	$1.111$	$1.117$	$1.143$	$1.160$	$1.198$	$1.231$	$1.276$	$1.331$
	20	$1.088$	$1.094$	$1.120$	$1.137$	$1.175$	$1.208$	$1.252$	$1.307$

, respectively. The results reported in these tables are calculated for various values of the thickness-to-outer radius ratio $(\mathcal{H}/a_1=0.05,0.1,0.15,0.2)$ and the power-law index $(s=0,2,4,6,8,10,15,20)$. In this case, the plate consists of FG ceramic(c)/metal(m) components with the following properties: ${\nu }_c=0.288,E_c=380$ GPa. The following law and dimensionless parameters were employed in this comparison:

$$\begin{array}{cc}\hfill & E_{eff}=E_c+(E_m-E_c){\left(0.5-{\displaystyle \frac{z}{\mathcal{H}}}\right)}^s,E_m/E_c=0.396,\hfill \\ \hfill & {\overline{u}}_z=u_z(0,0){\displaystyle \frac{64D_c}{q_0{a}_1^4}},D_c={\displaystyle \frac{E_c{\mathcal{H}}^3}{12(1-{\nu }_c^2)}}.\hfill \end{array}$$

(42)

In accordance with the data presented in Table 2 and Table 3, the maximum deflection, ${\overline{u}}_z$, of an FG circular plate was obtained for simply supported and clamped boundary conditions and for different values of the power-law index, s. It is declared that close agreement is noted among the results, especially for the thin plate ($\mathcal{H}/a_1=0.05$). However, a slight deviation is noted for the thick plate ($\mathcal{H}/a_1=0.2$) because the present theory provides a more accurate representation of transverse shear deformation, which is particularly important for thick plates where shear deformation effects are significant. Therefore, the proposed model and solution can determine the deflection of the proposed nanoplate.

To explain the effects of various parameters on the central deflection and stresses of a GPLs-reinforced annular plate with a porous core, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18 and Figure 19 are given. Figure 4 represents the changes in the central deflection, $u_z^{*}$, of sandwich annular nanoplates with a porous core (Porous-A) for various values of the graphene weight fraction $w_G$ versus the vertical load $q_0$. According to the literature review, increasing the amount of graphene in the structures significantly enhances their mechanical characteristics and stiffness. The central deflection, $u_z^{*}$, decreases as the weight fraction $w_G$ increases. It is also noted that with increasing the vertical load $q_0$, the effect of the weight fraction $w_G$ on the deflection decreases.

In Figure 5, we note that as the weight fraction $w_G$ evolves, the maximum stress ${\sigma }_{rr}^{*}$ progressively increases because the plate stiffness increases as the weight fraction $w_G$ increases. In addition, since graphene is sensitive to temperature and moisture, the maximum values of the stress ${\sigma }_{rr}^{*}$ occur in the top layer. It is also observed that the normal stress in the upper and lower layers of the plate of Patterns II and III exhibits extremums at large values of $w_G$. Note that a positive sign denotes tensile stress, whereas a negative sign denotes compressive stress.

In Figure 6, the curves represent how the transverse stress ${\sigma }_{zz}^{*}$ is distributed through the thickness of the nanoplate for different values of $w_G$. It is noted that increasing $w_G$ leads to a noticeable increment in maximum stress, ${\sigma }_{zz}^{*}$, in the upper and lower layers of the nanoplate in Pattern II and Pattern III. In Pattern I, the stress no longer increases as $w_G$ increases. Moreover, the stress profiles in Pattern I look more symmetric, indicating a more uniform distribution of graphene. In general, adding more graphene makes the nanoplate stronger and stiffer, which raises the transverse stresses.

Furthermore, as seen in Figure 7, Pattern II displays that with a height of $w_G$, the maximum tensile shear stress ${\sigma }_{rz}^{*}$ increases monotonically. Further, the compressive stress ${\sigma }_{rz}^{*}$ increases with increasing $w_G$ in Pattern III; in contrast, this trend is reversed for Pattern I, because graphene is distributed evenly through the thickness, so the whole nanoplate becomes uniformly stiffer. In addition, it is observed that shear stresses are maximum at or near the interfaces and reduce toward the surfaces, which is typical for sandwich/laminated plates.

Figure 8, Figure 9 and Figure 10 depict, respectively, the deflection $u_z^{*}$, normal stress ${\sigma }_{rr}^{*}$, and transverse normal stress ${\sigma }_{zz}^{*}$ of the sandwich annular nanoplate for various values of the porosity coefficient ${\delta }_0$, as well as for three different porosity distribution types. It can be observed that increasing the porosity minimizes the hygrothermal impact on the proposed nanoplate, because the thermal conductivity of the porosities is less than that of the solid material. Consequently, as the porosity factor rises, there is an obvious decline in deflection. Furthermore, the porosity factor ${\delta }_0$ in Figure 8 has a greater impact on the central deflection $u_z^{*}$ when $q_0$ is low. Note that the uniform distribution of the pores in the core results in a fewer deflection than other types.

It is obvious that, for all porosity distribution patterns, the influence of the coefficient ${\delta }_0$ on stress generally depends greatly on the porosity pattern. In Figure 9, the normal stress, ${\sigma }_{rr}^{*}$, in the upper and lower layers uniformly decreases as the porosity coefficient ${\delta }_0$ increases because these layers do not contain pores, while it varies differently in the core layer depending on the type of pore distribution throughout the thickness. Since the temperature is applied to the top surface of the plate, the effects of the porosity coefficient ${\delta }_0$ on the upper part of the plate are more than on the lower part.

In Figure 10, the maximum transverse normal stress ${\sigma }_{zz}^{*}$, that occurs at the upper surface of the plate, increases as the porosity factor increases. Moreover, the effects of ${\delta }_0$ on the transverse normal stress ${\sigma }_{zz}^{*}$ are less pronounced at the bottom surface of the plate for (porosity distribution) Porous-B and Porous-C, as shown in Figure 9b,c and Figure 10b,c, respectively, because the temperature is applied to the upper surface of the plate. Furthermore, it is noted that the stress ${\sigma }_{zz}^{*}$ gradually decreases with an increase in the porosity coefficient ${\delta }_0$ due to the uniform distribution of pores (Porous-A) through the thickness. While, for Porous-B, it varies parabolically through the thickness of the core because the volume fraction of pores is maximum at the mid-plane of the core. In contrast, for Porous-C, the volume fraction of pores is maximum at the upper and lower surfaces of the core; therefore, the effect of the porosity factor on ${\sigma }_{zz}^{*}$ is more pronounced at those surfaces.

Figure 11 and Figure 12 illustrate, respectively, fluctuations in the changes in the moisture $\mathrm{\Delta }\mathcal{C}$ and temperature $\mathrm{\Delta }\mathcal{T}$ parameters regarding the central deflection $u_z^{*}$, normal stress ${\sigma }_{rr}^{*}$, transverse normal stress ${\sigma }_{zz}^{*}$, and transverse shear stress ${\sigma }_{rz}^{*}$, of FG porous core sandwich annular nanoplates. As temperature and moisture levels rise, the results increase in a consistent manner as both $\mathrm{\Delta }\mathcal{C}$ and $\mathrm{\Delta }\mathcal{T}$ increase. This is because it is widely acknowledged that rising temperatures and moisture degrade structures stiffness. Furthermore, it can be observed that the sensitivity of central deflection, $u_z^{*}$, to variations in the temperature parameter $\mathrm{\Delta }\mathcal{T}$ is hardly noticeable compared to the variations with respect to the moisture concentration.

Figure 13 displays the variation in central deflection $u_z^{*}$ against vertical load $q_0$; it also shows the change in ${\sigma }_{rr}^{*}$ with the thickness of GPLs-reinforced sandwich annular nanoplates with an FG porous core for various values of the elastic foundation stiffness parameters $(g_1,g_2)$. It is noted that the elastic foundation stiffness parameters $(g_1,g_2)$ have a hardening influence because the elastic foundation supports and enhances the plate strength. Consequently, deflection increases as these parameters rise. Further, the maximum normal stress ${\sigma }_{rr}^{*}$, which occurs in the upper part of the plate, decreases when the elastic foundation coefficients $(g_1,g_2)$ rise, whereas this trend is reversed in the lower part of the plate.

Figure 14 shows how the core-to-thickness ratio ${\mathcal{H}}_c/\mathcal{H}$ affects the following parameters for FG porous core sandwich annular nanoplates: (a) central deflection $u_z^{*}$, (b) normal stress ${\sigma }_{rr}^{*}$, (c) transverse normal stress ${\sigma }_{zz}^{*}$, and (d) transverse shear stress ${\sigma }_{rz}^{*}$. Since the core layer contains pores, the increment in the core thickness leads to a decrement in the plate stiffness. Therefore, the central deflection $u_z^{*}$ and normal stress ${\sigma }_{rr}^{*}$ increase as the ratio ${\mathcal{H}}_c/\mathcal{H}$ rises. In contrast, the transverse stresses ${\sigma }_{zz}^{*}$ and ${\sigma }_{rz}^{*}$ decline as the ratio ${\mathcal{H}}_c/\mathcal{H}$ increases.

To explain the influence of the magnetic field on the obtained results, the central deflection $u_z^{*}$ of GPLs-reinforced sandwich annular nanoplates with an FG porous core is plotted against vertical load $q_0$ and the variation in ${\sigma }_{rz}^{*}$ through nanoplate thickness; this can be seen in Figure 15 for various values of the magnetic field parameter ${\overline{\mathcal{M}}}_r$. It is evident from this figure that the magnetic field parameter ${\overline{\mathcal{M}}}_r$ has a hardening impact on the proposed plate. Accordingly, a reduction in deflection and transverse shear stress occurs as ${\overline{\mathcal{M}}}_r$ rises. In particular, it can be noted that ${\sigma }_{rz}^{*}$ transfers from being compressive to tensile for higher values of ${\overline{\mathcal{M}}}_r$.

Figure 16 displays fluctuations in central deflection $u_z^{*}$ versus vertical load $q_0$, as well as changes in ${\sigma }_{rr}^{*}$ and ${\sigma }_{rz}^{*}$ via the thickness of GPLs-reinforced sandwich annular nanoplates with an FG porous core; this is shown for various values of the nonlocal coefficient, $e\iota$. It is clear that the nonlocal coefficient $e\iota$ has a slight impact on deflection. Furthermore, the inclusion of nonlocal parameters leads to an increment in the deflection of FG nanoplates, which leads, in turn, to a reduction in the stiffness of the nanoplate. Meanwhile, a significant decrement in the stress ${\sigma }_{rz}^{*}$ occurs with increases in $e\iota$. Furthermore, there is a monotonic rise in the maximum normal stress, ${\sigma }_{rr}^{*}$, which occurs at the bottom of the plate.

Figure 17 and Figure 18, respectively, display the effects of the outer radius-to-thickness ratio $a_1/\mathcal{H}$ and the inner-to-outer radius ratio $a_0/a_1$ on the central deflection, $u_z^{*}$, against vertical load $q_0$; the figures also show variations in the normal stress ${\sigma }_{rr}^{*}$ and transverse normal stress ${\sigma }_{zz}^{*}$, with respect to the thickness of the sandwich annular nanoplate. It is noticed that the increase of the ratios $a_1/\mathcal{H}$ and $a_0/a_1$ leads to a reduction in the plate strength; therefore, as the ratios $a_1/\mathcal{H}$ and $a_0/a_1$ increase, the transverse deflection and stress increase, whereas the in-plane normal stress decreases.

Figure 19 illustrates the effects of the strain gradient coefficient $\zeta$ on the central deflection $u_z^{*}$ and transverse normal stress ${\sigma }_{zz}^{*}$, via the thickness of the GPLs-reinforced sandwich annular nanoplates with an FG porous core. It can be observed that the strain gradient coefficient, $\zeta$, has a hardening impact on $u_z^{*}$; accordingly, it intensifies the stiffness of the plate. Additionally, this figure shows that ${\sigma }_{zz}^{*}$ rises as $\zeta$ increases.

6. Conclusions

As a result of this work, the axisymmetric bending response of FG-GPLs-reinforced sandwich annular nanoplates with an FG porous core was investigated. The present model is embedded on an elastic medium and subjected to a radial magnetic field, hygrothermal field, and transverse external load. Q3HSDT and NSGT were used as the foundation for the development of calculation formulas. The volume proportion of GPLs varies with thickness according to three graphene distribution patterns. The governing equations, which include the magnetic body force (Lorentz force), were displayed and numerically solved by utilizing DQM, which is employed in the radial direction for clamped and simply supported boundary conditions. The numerical outcomes indicate the following:

• The increase of the elastic foundation parameters and graphene components increases plate stiffness, leading to a significant reduction in deflection.
• Because the moisture and temperature reduce plate stiffness, the central deflection and stresses increase as moisture and temperature rise; the sensitivity to temperature fluctuations is minimal when compared to moisture concentration changes.
• Increasing the porosity coefficient decreases the deflection and stresses because the porosities dampen the hygrothermal effects on the proposed plate.
• Raising the magnetic field parameter yields a reduction in the deflection and transverse shear stress.
• The presence of the nonlocal parameter diminishes plate strength, whereas this trend is reversed when increasing the strain gradient coefficient.
• It is anticipated that the current work may have potential uses in the design and development of nanoscale devices, including circular-gate transistors.