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Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

39289400
70210
10.1038/s41598-024-70210-2
Article
Dynamic response of sandwich functionally graded nanoplate under thermal environments and elastic foundations using dynamic stiffness method
Rai Saurabh
Gupta Ankit ankit.gupta1@snu.edu.in

https://ror.org/02k949197 grid.449504.8 0000 0004 1766 2457 School of Engineering, Shiv Nadar Institution of Eminence Deemed to be University, Gautam Buddha Nagar, U.P 201314 India
17 9 2024
17 9 2024
2024
14 216899 4 2024
13 8 2024
© The Author(s) 2024
2024
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The present paper introduces the development of dynamic stiffness method for analyzing small-scale sandwich functionally graded nanoplates resting on elastic foundation in thermal environments. The mathematical formulation is based on classical plate theory in conjunction with nonlocal elasticity theory. The governing equation is derived using Hamilton’s principle. The dynamic stiffness matrix is obtained through the application of the Levy displacement approach and assembled to form the global stiffness matrix. The final matrix is solved for natural frequency of the plates using the Wittrick–Williams algorithm. The proposed methodology is validated against existing literature, demonstrating a strong agreement. Various parametric studies explore the effects of thermal environments, volume fraction index, sandwich configurations, elastic foundation characteristics, nonlocal parameter and boundary conditions. The results show the versatility of the proposed approach in addressing small scaled complex engineering structures. This research significantly contributes to the understanding and analysis of sandwich functionally graded nanoplates, providing valuable insights for applications in aerospace, structural systems, sensors, actuators, and energy harvesting devices.

Keywords

Nonlocal elasticity theory
Free vibration
Plate theory
Dynamic stiffness method
Wittrick–Williams algorithm
Sandwich functionally graded nanoplates (S-FG-nP)
Non-dimensional frequency parameter (NDFP)
Subject terms

Engineering
Nanoscience and technology
issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

In recent years, there have been significant advancements in the development of nanostructured products1. Nanostructured materials exhibit various advantageous properties, including electronic, physical, chemical, and mechanical properties as reported by Arshid et al.2. Functionally graded composite structures with viscoelastic polymers combine excellent performance and high damping, making them valuable for aerospace and mechanical engineering applications3. Gupta et al.4 investigated the vibrations of fixed-free single-walled carbon nanotubes with bacteria or viruses attached to their tips to assess their potential as detection devices. Nano-sized functionally graded materials (FGMs) have found extensive applications in nanoscale devices, including thin films and shape memory alloy films, as highlighted in the reports5,6. In the dynamic fields of energy harvesting and sensor technology, Sezer and Koc7 showcased the versatile applications of nano-sized functionally graded material (FGM) sandwich plates. To capture scale effects, researchers have proposed diverse methodologies such as nonlocal elasticity theory by Eringen and Edelen8, strain gradient theory9, surface elasticity theory10, and modified couple stress theories11. Nonlocal elasticity is widely accepted in the research community for nanostructure analysis as reported by Fatima et al.12.

A dynamic analytical model using the first-order ZIG-ZAG model and Rayleigh-Ritz method was developed to determine the natural frequency and loss factor of co-cured composite structures with varying viscoelastic damping membranes, verified against FEM results13. Similarly, a model based on higher-order shear deformation theory and Rayleigh-Ritz method was used to predict the free vibration of FG-CNTRCDS, validated with existing literature, showing the effects of structural parameters on the first-order natural frequency and loss factor14. Natarajan et al.15 investigated the free vibration of functionally graded material (FGM) nanoplates using the Mori-Tanaka homogenization scheme for material properties, while Jung and Han16 analyzed the bending and vibration behavior of sigmoid functionally graded material nanoscale plates with sigmoid-function-varied material properties. Safarpour et al. have made several contributions17, they presented a three-dimensional static and free vibration analysis of functionally graded graphene platelet-reinforced composite (FG-GPLRC) truncated conical shells, cylindrical shells, and annular plates using elasticity theory, highlighting novel formulations and showing optimal static and vibration responses through strategic graphene platelet distribution. They also investigated the vibration and bending response of FG-GPLRC rectangular plates under various substrates and thermal conditions using higher-order shear deformation theory18. Furthermore, Safarpour and Alibeigloo19 conducted a high-accuracy analysis of bending and frequency responses of sandwich cylindrical shells with FG carbon nanotube reinforced composite (FG-CNTRC) face-sheets and a polymeric core, employing three-dimensional elasticity theory and the state-space based differential quadrature method to assess the impact of initial axial stress, mechanical loadings, and boundary conditions. Bai et al.20 analyzed the coupled vibrations of nanobeams with axial and spinning motions under complex environmental changes using the nonlocal strain gradient theory (NSGT). Al-Furjan et al.21 presented a non-polynomial framework for analyzing the bending responses of FG-GPLRC disks using three-dimensional refined higher-order shear deformation theory and the differential quadrature method, revealing insights into shear stress distributions and bending behavior under various conditions. Lastly, Rahimi et al.22 investigated the bending and free vibration behavior of FG-GPLRC porous cylindrical shells using elasticity theory and various analytical methods, exploring the effects of graphene platelet distribution, porosity patterns, boundary conditions, and geometric parameters, thus providing valuable insights for designing composite structures. The study highlighted the impact of nonlocal parameters, power law index, aspect ratio, elastic modulus ratio, side-to-thickness ratio, and loading type on the mechanical response. Nami et al.23 investigated the thermal buckling behavior of FGM nanoplates using the third-order shear deformation theory (TSDT). Natarajan et al.15 employed finite element formulation in conjunction with Eringen’s theory to analyze the dynamic response of the FGM nanoplate. The author used first-order shear deformation and provided insights into the influence of the nonlocal parameter on the vibration frequency of the FGM nanoplate. Hosseini-Hashemi et al. (2013) developed an analytical formulation for free vibration using Mindlin and nonlocal elasticity. This formulation was tailored specifically for circular/annular FGM nanoplates, with a comprehensive exploration of the influence of various boundary conditions. Additionally, the study addressed the effect of an elastic foundation with added mass on the vibration behavior of FGM plates24–26. Panyatong et al. (2016) discussed the free vibration of FG nanoplates placed on an Elastic Foundation (EF) using the Second-order Shear Deformation Theory (SSDT). From the literature, it was evident that nonlocal elasticity theory had been widely adopted for the analysis of nanostructures due to its simplicity, convenience, and effectiveness in determining the bulk behavior of these structures.

As the geometric nonlinearity also affects the dynamic characteristics of the plate, several studies have been reported in the literature. In this context, the stability and vibration of carbon nanotube-reinforced composite plates under hygro-thermal environments using HSDT has been studied by Forooghi et al.27. Nonlinear numerical and experimental analyses of hyperelastic beams are conducted using Timoshenko’s beam theory28. The free vibration behavior of carbon nanotube-reinforced composite plates on a Kerr foundation is analyzed in with HSDT and nonlocal strain gradient theory29. The dynamics of axially graded beams30 with spinning and axial motions are explored in , while Forooghi et al.31 investigates FG porous plates under thermal environments using HSDT and nonlocal elasticity theory. Nonlinearity is introduced by Wu et al.32 with a model combining third-order shear theory and von Karman nonlinear theory for metamaterial arches. The dynamic behavior of carbon nanotube-reinforced composite plates is studied by Huang et al.33, extending classical laminate theory and incorporating nonlinearity. Analytical models predicting thermal expansion coefficients are presented by Huang et al.34 and Wu et al.35, emphasizing the importance of nonlinearity and unique material properties in structural analysis.

The dynamic stiffness method (DSM) is distinguished as a precise analytical technique unlike methodologies primarily relying on Finite Element Method (FEM) and analytical solutions. Banerjee36 proposes a general theory for developing the dynamic stiffness matrix of structural elements, advocating for the use of explicit analytical expressions derived through symbolic computation to optimize computational efficiency. This approach is further exemplified by Banerjee37, who derived exact explicit expressions for the coupled bending-torsional dynamic stiffness matrix of a uniform beam element, facilitating vibration analysis of coupled systems within space frame computer programs. Subsequently, Boscolo et al.38 apply the DSM to investigate the inplane free vibration behavior of plates. Kolarevic et al.39 extend DSM’s application by integrating it with a higher-order shear deformation theory (HSDT) to assess the vibrational behavior of isotropic plates. Furthermore, Kumar et al.40 utilized DSM for vibration analysis of Functionally Graded Material (FGM) plates, revealing correlations between higher modes and frequencies. Ali and Azam41 leverage DSM and CPT to conduct dynamic analysis on an FGM plate with sigmoid gradation under Levy boundary conditions.

Scope and novelty

The current literature lacks comprehensive investigations of  the small-scale effects on the vibration characteristics of S-FG-nP in thermal environments, employing nonlocal elasticity theory and the dynamic stiffness method for thin plates and small amplitude vibrations. Addressing this research gap, the paper offers a complete formulation for DSM applied to nano-sized sandwich plates in a thermal environment, resting on an elastic foundation. The material properties have been considered as temperature-dependent and follow a power-law behavior. The governing equation is derived using classical plate theory in conjunction with nonlocal theory, applying Hamilton’s principle. The derived equations are solved by taking levy-type displacement boundary conditions, while natural frequencies are computed using the Wittrick-Williams algorithm. The study reports the influence of nonlocal parameters, elastic foundation, thermal environment, and boundary conditions on the nondimensional frequency parameter of the S-FG-nP.

Geometry and material property

S-FG-nP

Sandwich functionally graded nanoplate (S-FG-nP) is shown in Fig. 1a. The sandwich nanoplates are mostly composed of three layers, consisting of the top face at z = h/2 while the bottom is at z = − h/2. S-FG-nP-A and S-FG-nP-B of the most commonly used FGM nanoplates, S-FG-nP-A has a ceramic core with an FGM face sheet and S-FG-nP-B has a core as FGM and a homogenous face sheet. The material properties of sandwich nanoplate Pn of layer n (where n=1,2,3), including the modulus of elasticity and density, are defined by Eq. (1) as defined in by Reddy42.1 Pn(z)=Pm+(Pc-Pm)Vk(z)

were, Pc and Pm represent the Young’s modulus or density of the ceramic and metal, respectively. Vk denotes the volume fraction of the ceramic, where k is the volume fraction index, and z is the coordinate along the thickness.

S-FG-nP-A

The S-FG-nP-A consists of three layers: a ceramic core and two functionally graded sheets that transition from metal to ceramic. The proportion of the nanoplate’s face sheets changes along the thickness direction following a power-law Eq. (2) as shown in Fig. 1a.2 V1(z)=z-h0h1-h0k,h0≤z≤h1V2(z)=1,h1≤z≤h2V3(z)=z-h3h2-h3k,h2≤h3

In the given equation, V1(z), V2(z), and V3(z) represent functions of the coordinate z for a sandwich plate. The regions defined by h0, h1, h2, and h3 partition the z-axis into intervals, specifying different behaviors for the functions within each region:h0 represents the lower boundary of the first region (V1(z))

h1 is the upper boundary of the first region and the lower boundary of the second region (V2(z))

h2 is the upper boundary of the second region and the lower boundary of the third region (V3(z))

h3 is the upper boundary of the third region.

For the first region, h0≤z≤h1, the function V1(z) is set to zero. Similarly, in the second region (h1≤z≤h2), V2(z) follows a power-law expression. In the third region (h2≤z≤h3), V3(z) remains constant at 1.

These parameters (h0, h1, h2, and h3) delineate the boundaries between regions, indicating distinct expressions or behaviors for the functions V1(z), V2(z), and V3(z) within each segment of the sandwich plate along the z-axis.

S-FG-nP-B

The S-FG-nP-B sandwich nanoplate consists of three layers: a FGM core bottom layer is ceramic and top layer is metal. The proportion of the nanoplate’s face sheets changes along the thickness direction following a power-law function as represented in Eq. (3) as shown in Figure 1b.3 V1(z)=0,h0≤z≤h1,V2(z)=z-h1h2-h1k,h1≤z≤h2,V3(z)=1,h2≤h3

Figure 1 Schematic of S-FG-nP resting on elastic foundation in thermal environment (a) S-FG-nP-A and (b) S-FG-nP-B.

Thermal environment

The temperature field is assumed to vary linearly across the sandwich surface, with specific temperatures specified at the top and bottom surfaces of the sandwich. The influence of the thermal environment on Young’s modulus along the thickness direction of S-FG-nP-A and S-FG-nP-B is illustrated in Fig. 2. The material properties are temperature-dependent and vary along the thickness direction as described by Eq. (4).The coefficients P0, P-1, P1, P2, and P3 in the temperature function T(K) are detailed in Table 543,44. The properties also vary with z as shown in Eqs. (2) and (3). The temperature Tb is the ambient temperature, and the temperature difference ΔT is given by:ΔT=ttzh+12

where tt is the temperature difference between the top and bottom.

Equation (4) represents the temperature-dependent variation of the material property P for both ceramic and metal:4 P=P0P-1(Tb+ΔT)-1+1+P1(Tb+ΔT)+P2(Tb+ΔT)2+P3(Tb+ΔT)3

Figure 2 Young’s modulus variation with thickness z for linear temperature variation.

Elastic foundation formulation

The S-FG-nP is assumed to rest on a Winkler-Pasternak type elastic foundation45,46, characterized by Winkler stiffness kw and shear stiffness kp, as depicted in Fig. 1. This formulation effectively captures the free vibration behavior resulting from the dynamic interaction between the plate and its elastic foundation.The reaction force of the foundation is described by Eq. (5).5 qw=kww,qp=kww-kp∇2w

Displacement field

The classical plate theory (CPT)47 is expressed as given in Eq. (6). In these equations, u(x,y,z), v(x,y,z), and w(x,y,z) denote the displacements in the x-direction, y-direction, and out-of-plane direction, respectively. The in-plane displacements u0(x,y) and v0(x,y) are influenced by the in-plane coordinates (x,y), while w0(x,y) represents the primary out-of-plane deformation.6 u(x,y,z,t)=u0(x,y,t)-zdwdxv(x,y,z,t)=v0(x,y,t)-zdwdyw(x,y,z,t)=w0(x,y,t)

Stress–strain relationship

The strain energy is evaluated using the stress–strain relationship, as expressed in Eq. (7). Furthermore, the strain is affected by the thermal environment, leading to the final constitutive relation described by Eq. (10). This comprehensive approach integrates these energy considerations to formulate the governing equation for the S-FG-nP.48.7 εxx=∂u0∂x-z∂2w∂x2,εyy=∂v0∂y-z∂2w∂y2,εxy=-2∂2w∂x∂y

8 5ε=εxxεyyεxy=ε0+zε1

9 ε0=u0,xv0,yu0,y+v0,x;ε1=w,xxw,yy2w,xy

10 σxxσyyτxy=q11q120q21q22000q66εxx-α(z)T(z)εyy-α(z)T(z)γxy

where:σxx,σyy,τxy-- Stresses\,in\, the\, plateεxx,εyy,γxy-- Strains\, in\,the\, plateq11,q12,q21,q22,q66-- Constitutive\, matrix\,coefficientsE(z,T),α(z),T(z)-- Temperature\,variation (Linear\,function\,of\,the\,z)ν-- Poisson's\,ratio

Force and moment resultant

The constitutive equation as shown in Eq. (10) can be utilized to evaluate the total force and bending force acting on the plate given in Eq. (11). The following equations to calculate stress (N) and moment resultants (M). In the Eq. (13), where D=E(z,t)/(1-ν)2 and A,B and Deff can be calculated.11 {Nij;Mij}=∫-h/2h/2σij{1;z}dz;ij=x,y

12 {Nxx;Nyy;Nxy}T=Aε0+Bε1-NT{Mxx;Myy;Mxy}T=Bε0+Cε1-NT

13 (A,B,Deff)=∫-h/2h/2D(1,z,z2)dz

14 {NT,MT}=G0=∫-h/2h/2E(z)α(z)T(z)dz

Development of governing equation

The final governing equation for a plate resting on an elastic foundation and exposed to a thermal environment can be derived using Hamilton’s variation principle44. The formulated equation is presented in Eq. (15).15 ∫0t(δU+δV-δW-δT)dt=0

Where δU, δV, and δT are the variations of the strain energy of the plate, the energy stored in the plate due to strain deformation, deformed Elastic Foundation, and the kinetic energy, respectively. The variation of the strain energy is given by Eq. (16,17).16 δU=∫S∫-h/2h/2σxxδεxx+σyyδεyy+σxyδεxydzdS

17 =∫SNxx∂δu0∂x-Mxx∂2δw∂2x+Nyy∂δv0∂x-Myy∂2δw∂2y+Nxy∂δu0∂y+∂δv0∂x-2Mxy∂2δw∂x∂ydS

The variation of the energy stored due to elastic foundation is determined by Eq. (18)44. Where R=kww-kp∇2w,∇2=(∂2∂x+∂2∂y)18 δV=∫SRδwdS

The variation of the work done by the thermal is given by the Eq. (19)19 δw=∫SNxxT∂2wδw∂2x+NyyT∂2wδw∂2y

where NxxT=NyyT=NT ,20 δw=∫SNT∂2wδw∂2x+∂2wδw∂2ydS

The kinetic energy can be evaluated as given into Eqs. (21) and (22).21 δT=∫S∫-h/2h/2ρ(z)u˙1δu˙1+v˙2δv˙2dzdS

22 =∫SI0[u˙0δu˙0+v˙0δv˙0+w˙δw˙]-I1[u˙0δw˙,x+v˙0δw˙,x+w˙δv˙0+w˙δu˙0]+I2[w˙,xδw˙,x+w˙,xδw˙,x]dS

where (I0,I1,I2)=∫-h/2h/2(1,z,z2)ρ(z)dz23 δu0:Nxx,x+Nxy,y=I0u¨0-I1w¨,x,δv0:Nxy,x+Nyy,y=I0v¨0-I1w¨,y

24 δw0:Mxx,xx+2Mxy,xy+Myy,yy+kww-kp∇2w+G0∇2w+I0w¨-I2∇2w¨

Nonlocality on structure

In classical plate theory, the stress at a given location is typically determined by the strain tensor at that specific point. However, in nonlocal continuum theory49, the stress at a particular point is influenced by the strain tensor throughout the entire continuum body. The nonlocal elasticity mimics the behavior of the nanosized plate by having the assumption that stress on a point is the function of strain at its neighboring points. The motion assumption considers the effect of the small-scale and atomic force. Small-scale forces like atomic force become considerable for small-size objects. Eringen proposed a constitutive model as shown in Eq. (25)49.25 tij=∫Vαx˙-xσij(x˙)dv(x˙)

Eq. (1) represents the Hookean stress, defined asσij=Cijklεkl, kernel function (αx˙-x) normalized over the volume of the body, i.e.∫vα(x˙)dv=1. The kernel function is obtained by matching the lattice dynamics with nonlocal results as shown in Eq. (26).26 α(x)=12πl2τ2k0|x|lτ,τ=e0al

Where, a, l are the modified bessel function with internal and external characteristic lengths. The material parameter(e0) can be evaluated experimentally. Non-local elasticity deals with spatial integrals with weighted averages, which include the contribution of all strain in the continuous body to stress at that point. The modified equation of motion considers nonlocal linear elasticity as mentioned in Eq. (27).27 tij,j+fi=ρu¨i

Where, i and j represent the meaning of x, and y components, fi, ρ, ui are the force, density, and displacement vectors of the body. Eq. (25) is substituted in Eq. (28) to get the integral from the nonlocal constitutive equation. The integrals are computationally expensive equations hence, Eringen50 proposed the differential form of the constitutive equation as given Eq. (29).28 σij,j+λ(fi-ρu¨i)=0

29 λ=1-μ∇2,μ=(e0a)2

Substituting the equation in the above differential equation gets constitutive nonlocal50 relation as given in Eq. (30).30 [1-μ∇2]tij=σij

The Eq. (30) is modified for constitutive expression for this nonlocal relationship considering thermal strain is provided by Eq. (31).31 σxxσyyτxy-μ∇2σxxσyyτxy=q11q120q21q22000q66εxx-α(z)T(z)εyy-α(z)T(z)γxy

32 (1-μ∇2){NijNL;MijNL}={Nij;Mij}

33 Mxx,xx+2Mxy,xy+Myy,yy=(1-μ∇2)(-kww+kp∇2w-G0∇2w+I0w..+I2∇2w..)

Final governing equation

The final partial differential equation for the S-FG-nP is given by Eq. (34).34 Deff∂4w∂x4+2∂4w∂x2∂y2+∂4w∂y4+I0∂2w∂t2+kww-kp-G0∂2w∂x2+∂2w∂y2-I2∂4w∂x2∂t2+∂4w∂y2∂t2-μ(ϕ1+ϕ2)=0ϕ1=∂2∂x2I0∂2w∂t2-I2∂4w∂x2∂t2+∂4w∂y2∂t2ϕ2=∂2∂y2I0∂2w∂t2-I2∂4w∂x2∂t2+∂4w∂y2∂t2

The present study considers the Levy-type boundary condition with two opposite sides simply supported for vibration analysis of the S-FG-nP. The displacement assumption, as depicted in Eq. (35)51, is substituted into Eq. (34) to derive Eq. (36). In the the equation ω is frequency, αm is half sine-wave where αm=mπ/L35 w0(x,y,t)=∑m=1m=∞wm(x)eiωtsin(αmy)

36 (Deff-I2μω2)d4wmdx4+(-2Deffα2-DeffkpL2+G0+I0μω2+I22α2μω2+I22ω2+I2α2μω2)d2wmdx2+(Deffα4+Deffα2kpL2+DeffkwL4-G0α2-I0α2μω2+I0ω2-I2α4μω2-I2α2ω2)wm=0

The given fourth-order differential Eq. (36) can be solved for the function w. To find the solution, first consider the complementary function (CF) part of the equation, which yields four roots. The variables r1 and r2 are introduced to simplify the representation of these roots. Two scenarios arise: either all four roots are real and positive, or two roots are real and positive while the other two roots are as given in Fig. 3. The solutions obtained from the differential equation are used to derive the displacement as shown in the Eqs. (38) and (40).Figure 3 Roots of equation.

Case 1:r1>r237 r1m=±2r1+r2,r2m=±2r1-r2

38 wm(x)=Amcosh(r1mx)+Bmsinh(r1mx)+Cmcosh(r2mx)+Dmsinh(r2mx)

Case 2:r1<r239 r1m=±2r1+r2,r2m=±2ir2-r1

40 wm(x)=Amcosh(r1mx)+Bmcos(r2mx)+Cmsinh(r1mx)+Dmsin(r2mx)

Development of dynamic stiffness (DS) matrix

In the DS matrix considering case 1, a unified matrix approach is employed to address the eigenvalue problem. This method utilizes a single matrix, consolidating the rotation-displacement relationship (X=AC) and the shear force-bending moment relationship (F=SC) using by extracting coefficients of Am, Bm, Cm, and Dm natural and force boundary condition as shown in the Eqs. (41) and (43). The displacemnt-rotation matrix as shown in Eq. (42) and shear force and bending moment matrix as shown in Eq. (44). Here, the rotation-displacement matrix is treated as the displacement matrix, while the force-bending moment matrix is considered the force matrix. By adopting the framework of F=kx, the dynamic stiffness matrix (DSM) is derived as DSM=S·A-1.41 x=0,wm=wa,φym=φya,x=b,wm=wb,φym=φyb

42 waφyawbφyb=10100-r1m0-r2mCh1Sh1Ch2Ch2-r1mSh1-r1mCh1-r2mSh2-r2mCh2AmBmCmDm

where Chi=cosh(rib),Shi=sinh(rib),Ci=cos(rimb),Si=sin(rimb),i=1,243 Vx=[-Deff(∂3w∂x3+(2-ν)∂3w∂x∂y2)+(kp-G0)∂w∂x]δw,Mxx=[-Deff(∂2w∂x2+ν∂2w∂y2)]δϕyx=0,Vxm=-va,Mxxm=-ma,x=b,vxm=-vb,Mxxm=-mb

44 vamavbmb=1010L10L10-R1Sh1-R1Ch1-R2Ch2-R2Ch2-L1Ch1-L1Sh1-L2Ch2-L2Sh2AmBmCmDm

where Ri=Deff(r3-(2-υ)α2rim)+rimI2ω2,Li=Deff(r2-υα2),i=1,245 DSM=SvvSvnfvvfvnSmn-fvmfmmsysSvv-SvmSmm

DS matrix assembly process

The dynamic stiffness matrix, as given in Eq. (45), serves as the fundamental component for accurately computing the natural frequencies of S-FG-nP with at least two opposite sides simply supported. For individual plates, a single dynamic stiffness (DS) element might suffice to achieve the desired accuracy in natural frequency calculations. However, for assemblies of plates and for higher natural frequencies, it is necessary to assemble the DS element matrices into a global DS matrix, similar to the process in finite element methods (FEM). Figure 4a illustrates the assembly procedure. Unlike FEM, where point nodes are used, the DS plate elements use line nodes for each strip. The assembled dynamic stiffness matrix is typically banded, akin to the FEM global stiffness matrix38. The boundary conditions are applied in DSM by penalizing the required leading diagonal element of the global DS matrices. The required element gives the penalty with a very high numerical value to this element. The Free (F) edge boundary condition is not a penalty, simply-supported (s) boundary condition, the displacement is a penalty, and in clamped (C) boundary conditions, displacement and rotation are penalties47Figure 4 schematic of (a) assembly procedure and (b) William Writtrick Algorithm.

Wittrick–Williams algorithm

The natural frequencies of the FGM plate are obtained from the global DS matrix using the well-known Wittrick–Williams (W–W) algorithm38. This algorithm is widely employed to solve transcendental characteristic equations, ensuring that no frequencies of the structure are overlooked. The steps involved are shown in a flow chart in Fig. 4b. A reliable and efficient solution technique to extract eigenvalues of natural frequencies from the analytical deterministic Dynamic stiffness formulations of a structure is the powerful Wittrick-Williams (WW) algorithm . This algorithm ensures that no eigenvalue is missed by monitoring the Sturm sequence of the ensuring matrix. According to the WW algorithm, the number of eigenvalues between 0 and a trial frequency ω∗ (mode count J) of the final structure isJ=J0+s{Kf}

where J0 is the mode count of an element with all nodes clamped, and s(Kf) is the sign count (negative inertia) of the final structure Kf evaluated at the trial frequency. For convenience, the natural frequencies of any member with both ends clamped are denoted by ωc and are called member clamped-clamped frequencies, with the corresponding modes being called local ones. And J0=∑Jm, where Jm is the number of ωc of a member subject to ωc≤ω∗, and the summation is over all members. By applying the bisection method, the eigenvalues where the mode count J shifts can be determined. It is worth emphasizing that the WW algorithm which has been used in many DS formulations52. Once the natural frequencies are computed, the mode shapes are obtained using the overall DS matrix of the structure.

Result and discussion

In this section, the natural frequencies of S-FG-nP are computed using the DSM formulation. A Python program has been developed to assess the natural frequencies. The results are examined under various Levy boundary conditions, including SSSS, SFSF, SSSF, SCSC, and SSSC, where ’C,’ ’F,’ and ’S’ signify clamped, free, and simply supported sides of the FG-nP. Additionally, a comparative analysis is conducted to compare the natural frequencies obtained through DSM with those available in published literature. Moreover, this study incorporates an assessment of the impact of design parameters such as the volume fraction index, Winkler and Pasternak elastic moduli, nonlocal parameter, and thermal environment on the natural frequencies, and it presents these findings using tables and graphs. To ensure the efficiency and accuracy of the developed methodology, a comparative study have been carried out.

Validation

Example 1

In this study, the effectiveness of a newly proposed methodology for predicting the natural frequency of nonoplate at different nonlocal parameter is evaluated. A comparison is made with the outcomes of Aghababaei and Reddy53, authors applied shear deformation theory and the CPT, incorporating Eringen’s nonlocal linear elasticity theory. The validation is performed on a square plate with specific material and geometrical properties, including length of a=10, Young’s modulus E=30×106, and Poisson’s ratio ν=0.3. The plate is subjected to simply supported boundary conditions, and NDFP ( ω11=ω×h×ρ/G). The results presented in Table 1 demonstrate the accuracy of the developed Dynamic Stiffness Matrix (DSM) formulation in predicting the natural frequency behavior of nanoplate.Table 1 Comparison of NDFP for square nanoplates: DSM versus FSDT and CPT Approaches.

a/h	μ	CPT	FSDT	DSM	
10	0	0.0963	0.0930	0.0958	
1	0.0880	0.0850	0.0875	
2	0.0816	0.0788	0.0812	
3	0.0763	0.0737	0.0758	
4	0.072	0.0696	0.0714	
5	0.0683	0.0660	0.0680	
20	0	0.0241	0.0239	0.024093	
1	0.022	0.0218	0.022019	
2	0.0204	0.0202	0.020311	
3	0.0191	0.0189	0.019091	
4	0.018	0.0178	0.017993	
5	0.0171	0.0169	0.017017	

Example 2

In this numerical validation, the current methodology is confirmed through a comparison with the outcomes provided by Sobhy and Radwan54 regarding the natural frequency of FG-nP. This FG-nP is made of SUS304/Si3N4 and exhibits varying volume fractions denoted by the index (k) and nonlocal parameters (μ). The effective material properties are determined using a power-law function whereas the mechanical properties are given as:  For Si3N4: Ec=348.43GPa, ρc=2370kg/m3, α¯c=3.3×10-6∘K-1, νc=0.24. For SUS304: Em=201.04GPa, ρm=8166kg/m3, α¯m=17.3×10-6∘K-1, νm=0. The square plates have dimensions of 10nm in length and 1nm in thickness. The comparison has been carried out while considering nondimensional parameter as (NDFP) ω11=ω×10×h×ρm/Em., In the referred paper, the results are computed using the simplified Higher-Order Shear Deformation Theory (HSDT) and quasi-3D theory 54. The results, as shown in Table 2, affirm that the proposed methodology closely aligns with the findings of Sobhy and Radwan, thus confirming its accuracy.Table 2 Comparison of NDFP for square nanoplates, as reported by Sobhy and Radwan54, considering various nonlocal parameters, volume fraction indices (k), and aspect ratio (a/h).

a/h	μ	k = 0	k = 1	k = 5	
Ref.54	Present	diff.%	Ref.54	Present	diff.%	Ref.54	Present	diff.%	
20	0	0.3555	0.3454	2.8	0.2108	0.2100	0.3	0.1707	0.1711	0.2	
	0.5	0.3471	0.3366	3.0	0.2058	0.2047	0.50	0.1667	0.1668	0.1	
	1	0.3249	0.3156	2.8	0.1926	0.1920	0.3	0.1560	0.1564	0.3	
	1.5	0.2958	0.2876	2.7	0.1754	0.1749	0.2	0.1421	0.1425	0.3	
	2	0.2658	0.2579	2.9	0.157	0.1569	0.4	0.1276	0.1278	0.1	
50	0	0.0573	0.0552	3.5	0.0339	0.0336	1.0	0.0275	0.0273	0.5	
	0.5	0.0559	0.0538	3.6	0.0331	0.0327	1.2	0.0268	0.0266	0.7	
	1	0.05236	0.0505	3.5	0.0310	0.0307	1.0	0.0251	0.0250	0.5	
	1.5	0.0476	0.0460	3.4	0.0282	0.0279	0.9	0.0229	0.0228	0.4	
	2	0.0428	0.0412	3.6	0.0254	0.0251	1.1	0.0205	0.0204	0.6	
diff.%=( Ref.-Present)×100Ref..  

Example 3

In the third validation, the accuracy of the method is assessed by comparing it to the work of Daikh et al.55, which deals with S-FG-nP. The authors incorporated Eringen’s nonlocal elasticity model to account for small size effects. They investigated two types of sandwich nanoplates: one with functionally graded material face layers and a homogeneous core, and another with homogeneous face layers and a functionally graded material core. The material properties are given in Table 5 with NDFP (ω11=ω×a2/h×(1-ν2)ρ0/E0). The study considers the case with ΔT=0, k=2, and simply supported boundary conditions with a/h = 10. The outcomes of this investigation are then compared to the results obtained using the present methodology, demonstrating a strong level of agreement between the two, thus validating the method’s accuracy. Table 3 shows a comparison of Daikh et al.’s results with the present results for different nonlocal parameters and different sandwich configurations.Table 3 Comparison of NDFP for S-FG-nP-A and S-FG-nP-B under various nonlocal parameters and configurations is conducted with DSM as detailed by Daikh et al.55.

μ	S-FG-nP-A	S-FG-nP-B	
Ref.55	Present	Ref.55	Present	Ref.55	Present	Ref.55	Present	Ref.55	Present	Ref.55	Present	
1-1-1	1-2-1	2-2-1	1-1-1	1-2-1	2-2-1	
0	6.2516	6.3672	6.5358	6.6858	6.3955	6.4881	6.4700	6.6006	6.4459	6.5342	6.3490	6.4582	
0.5	5.9404	6.0680	6.2075	6.3716	6.0763	6.2156	6.1526	6.2920	6.1308	6.2287	6.0405	6.1563	
1	5.6673	5.8020	5.9194	6.0923	5.7962	5.9432	5.8743	6.0177	5.8547	5.9572	5.7703	5.8879	
1.5	5.4251	5.5692	5.6638	5.8479	5.5477	5.7048	5.6277	5.7777	5.6099	5.7196	5.5309	5.6530	
2	5.2082	5.3697	5.4347	5.6384	5.3252	5.5004	5.4070	5.5719	5.3910	5.5159	5.3168	5.4517	

Example 4

This validation aims to assess the method’s capability to incorporate thermal effects. The results are compared with those given by  Gulshan Taj et al.56. The authors investigated the free vibration of functionally graded material (FGM) skew plates in a thermal environment. The kinematic equations were derived from Reddy’s higher-order shear deformation theory, and a nine-noded isoparametric Lagrangian element was employed to discretize the plate geometry. The results are obtained for the square plate with material properties as discussed in Table 5 and a/h = 10 under simply supported boundary conditions. The NDFP are calculated using ω11=ω×a2/h×(1-ν)ρ0/E0. The results obtained using the present method are then compared to Taj et al.’s findings, revealing a high degree of agreement between them and confirming the method’s competence in considering thermal effects. Furthermore, good agreement is observed for higher modes, as shown in Table 4.Table 4 Comparison of NDFP for various thermal environments as reported by Gulshan Taj et al.56, along with DSM results.

ΔT(K)	Ref.56	DSM	diff.%	Ref.56	DSM	% diff	Ref.56	DSM	diff.%	
	mode 1	mode 2	mode 4	
100	6.6238	6.672858	0.7	16.0508	16.703538	4.0	24.7913	26.71533	7.7	
300	6.2144	6.1209	1.5	15.6369	15.779071	0.9	24.3646	25.410372	4.2	
600	5.5438	5.465086	1.4	14.9944	14.829336	1.1	23.7107	24.141088	1.8	
900	4.7800	4.910289	2.7	14.3232	14.168592	1.0	23.0370	23.328927	1.2	

Parametric study

In this section, the authors conducted a comprehensive analysis of the free vibration characteristics of a sandwich Functionally graded nanoplate (S-FG-nP) with varying geometric and material properties. The study examined the influence of nonlocal effects on the vibration frequencies of the sandwich nanoplate, considering both S-FG-nP-A and S-FG-nP-B materials. For S-FG-nP Ti6Al4V has been considered as metal and ZrO2 as ceramic with property given in the Table 5. Additionally, temperature-dependent material properties were incorporated, and investigated the impact of thermal environment on the vibration response of the nanoplates. The S-FG-nP's length (L) is considered as 10nm, the breadth is 10nm, and the thickness is 1nm. The unit of the nonlocal parameter μ is taken as nm2. The configurations of a sandwich structure are characterized by the distribution of layer thicknesses, denoted as 111, 121, and 221. In the 111 configuration, all layers within the sandwich structure possess equal thickness. The 121 configuration indicates that the top and bottom layers share the same thickness, while the middle (core) layer has a thickness that is twice that of the outer layers. Conversely, the 221 configuration features a bottom layer with half the thickness of the top and core layers, where the top and core layers share the same thickness, and this shared thickness is twice that of the bottom layer. These numerical representations succinctly convey the specific layer thickness relationships within each configuration, providing a comprehensive understanding of the structural makeup of the sandwich assembly. Configuration had a significant impact on the material’s stiffness, and this effect differed between S-FG-nP-A and S-FG-nP-B materials, consequently affecting the natural frequency of the sandwich nanoplate. Furthermore, the study explored the impact of Wrinkles and a Pasternak foundation on the behavior of the sandwich nanoplate. The NDFP is given by the expressionNDFP=ωh·pm·21+ν/Em0,

and the parameters kw and kp used for the nondimensionalization arekw=kw·L4Deff

andkp=kp·L2Deff.

The ambient temperature is taken to be 300K and ΔT represents the temperature increase, which is assumed to be linear. The parameters pm and Em0 denote the density of the metal and the Young’s modulus of the metal at 300K, respectively.Table 5 Material properties of S-FG-nP with Ti6Al4V and ZrO256.

	ZrO2	Ti-6Al-4V	
E (Pa)	α(1/K)	E (Pa)	α(1/K)	
P0	244.27×109	12.766×10-6	122.56×109	7.5788×10-6	
P-1	0	0	0	0	
P1	-1.371×10-3	-1.491×10-3	-4.586×10-4	6.638×10-4	
P2	1.214×10-6	1.006×10-5	0	-3.147×10-6	
P2	-3.681×10-10	-6.778×10-11	0	0	
P3	168.063×109	18.591×10-6	105.698×109	6.941×10-6	

Effect of the elastic foundation and thermal environment on vibration anlysis of S-FG-nP

In this study, the influence of the elastic foundation and temperature on NDPF of square S-FG-nP-A and S-FG-nP-B with 111 configurations on SSSS boundary conditions was investigated. The material properties presented in Table 5 were utilized for the analysis. Density values were assigned as 3000 kg/m3 for ZrO2 and 4426 kg/m3 for Ti-6Al-4V. Additionally, the volumetric fraction index k was set to 1. From the given Table 6, it is evident that S-FG-nP-A-111 generally has higher NDPF values compared to S-FG-nP-B-111 under the same conditions. It can also be concluded that as kw increases to 100 for k=0 and ΔT=0, the highest change in NDPF is 33%. However, the increase in NDPF due to an increase in kw to 100 lies between 12% to 13% for both S-FG-nP-A and S-FG-nP-B when compared with (kw=0,kp=0) for the other volume fraction index (k).

Furthermore, it is observed that as kp increases from 0 to 100, the increase in NDPF is between 146% to 166%. The percentage change increases with temperature change from 0 to 300K. The inclusion of the elastic foundation leads to an increase in NDPF as it increases effective bending stiffness. The influence of the elastic foundation is more pronounced at higher temperature. It is also concluded from the results that S-FG-nP-A-111 has better performance in terms of natural frequency. As the temperature increases, the NDPF values decrease for both S-FG-nP-A-111 and S-FG-nP-B-111. This indicates that higher temperatures lead to reduced natural frequencies, which can impact structural stability. Increasing the volume fraction index (k) tends to decrease the NDPF values, indicating that a higher volume fraction leads to higher metallic content with a lesser effective Young’s modulus.Table 6 Effect of volume fraction index, elastic foundation, and thermal on NDFP of S-FG-nP (S-FG-nP-A-111 and S-FG-nP-B-111).

k	NDFP	
ΔT(K)	kw=0,kp=0	kw=100,kp=0	kw=0,kp=100	kw=100,kp=100	
S-FG-nP-A-111	
 0	0	0.1449	0.3569	0.1624	0.3644	
 1		0.1178	0.2901	0.1320	0.2962	
 5		0.1051	0.1178	0.2643		
 0	100	0.1378	0.3467	0.1553	0.3540	
 1		0.1129	0.2835	0.1272	0.2895	
 5		0.1015	0.2542	0.1142	0.2595	
 0	300	0.1259	0.3308	0.1435	0.3378	
  1		0.1040	0.2719	0.1184	0.2778	
 5		0.0945	0.2450	0.1074	0.2502	
S-FG-nP-B-111	
 0	0	0.1220	0.3004	0.1367	0.3067	
 1		0.1175	0.2895	0.1318	0.2955	
 5		0.1149	0.2830	0.1288	0.2889	
 0	100	0.1151	0.2907	0.1299	0.2968	
 1		0.1117	0.2804	0.1258	0.2863	
 5		0.1097	0.2744	0.1235	0.2802	
 0	300	0.1036	0.2758	0.1185	0.2817	
 1		0.1020	0.2667	0.1161	0.2724	
 5		0.1013	0.2614	0.1149	0.2669	

Figure 5 Effect of nonlocal parameter(μ) and configuration on NDFP of S-FG-nP.

Figure 6 Mode shapes for the square FGM nanoplate with SSSS boundary.

Table 7 Effect of configuration, volume fraction index, and temperature on NDFP of S-FG-nP.

k	ΔT(K)	μ=0	μ=1	
		111	121	221	111	121	221	
S-FG-nP-B	
 0	0	0.1220	0.1253	0.1199	0.1115	0.1145	0.1096	
 1	0	0.1175	0.1175	0.1148	0.1074	0.1074	0.1049	
 5	0	0.1149	0.1133	0.1111	0.1050	0.1035	0.1015	
 0	100	0.1151	0.1183	0.1133	0.1052	0.1081	0.1035	
 1	100	0.1117	0.1117	0.1092	0.1021	0.1021	0.0998	
 5	100	0.1097	0.1082	0.1063	0.1003	0.0989	0.0971	
 0	300	0.1036	0.1065	0.1020	0.0947	0.0973	0.0933	
 1	300	0.1020	0.1020	0.1000	0.0933	0.0933	0.0914	
 5	300	0.1013	0.1000	0.0984	0.0926	0.0914	0.0899	
S-FG-nP-A				
 0	0	0.1449	0.1449	0.1449	0.1324	0.1324	0.1324	
 1	0	0.1178	0.1227	0.1203	0.1076	0.1121	0.1099	
 5	0	0.1051	0.1112	0.1085	0.0961	0.1016	0.0991	
 0	100	0.1378	0.1378	0.1378	0.1260	0.1260	0.1260	
 1	100	0.1129	0.1175	0.1149	0.1032	0.1073	0.1050	
 5	100	0.1015	0.1071	0.1044	0.0927	0.0979	0.0954	
 0	300	0.1259	0.1259	0.1259	0.1151	0.1151	0.1151	
 1	300	0.1040	0.1081	0.1055	0.0951	0.0988	0.0964	
 5	300	0.0945	0.0995	0.0968	0.0864	0.0909	0.0885	

Table 8 Higher Modes of Vibration for S-FG-nP configurations (111, 121) under different thermal boundary conditions.

Configuration	ΔT(K)	m	n	NDFP	
S-FG-nP-A	S-FG-nP-B	
k = 0,μ = 0	k = 0,μ = 1	k = 1,μ = 0	k = 1,μ = 1	k = 0,μ = 0	k = 0,μ = 1	k = 1,μ = 0	k = 1,μ = 1	
111	0	1	2	0.3623	0.2964	0.2945	0.2410	0.3049	0.2495	0.2938	0.2404	
111	0	1	1	0.1449	0.1324	0.1178	0.1076	0.1220	0.1115	0.1175	0.1074	
111	0	2	2	0.5727	0.4281	0.4648	0.3475	0.4817	0.3601	0.4645	0.3472	
111	100	1	2	0.3498	0.2863	0.2862	0.2342	0.2930	0.2398	0.2831	0.2317	
111	100	1	1	0.1378	0.1260	0.1129	0.1032	0.1151	0.1052	0.1117	0.1021	
111	100	2	2	0.5551	0.4149	0.4534	0.3389	0.4649	0.3476	0.4491	0.3357	
111	300	1	2	0.3300	0.2700	0.2716	0.2223	0.2742	0.2244	0.2664	0.2180	
111	300	1	1	0.1259	0.1151	0.1040	0.0951	0.1036	0.0947	0.1020	0.0933	
111	300	2	2	0.5274	0.3943	0.4332	0.3238	0.4390	0.3282	0.4254	0.3180	
121	0	1	2	0.3623	0.2964	0.3067	0.2510	0.3133	0.2564	0.2938	0.2404	
121	0	1	1	0.1449	0.1324	0.1227	0.1121	0.1253	0.1145	0.1175	0.1074	
121	0	2	2	0.5727	0.4281	0.4841	0.3619	0.4948	0.3699	0.4645	0.3472	
121	100	1	2	0.3498	0.2863	0.2978	0.2437	0.3011	0.2464	0.2831	0.2317	
121	100	1	1	0.1378	0.1260	0.1175	0.1073	0.1183	0.1081	0.1117	0.1021	
121	100	2	2	0.5551	0.4149	0.4717	0.3526	0.4777	0.3571	0.4491	0.3357	
121	300	1	2	0.3300	0.2700	0.2823	0.2310	0.2818	0.2306	0.2664	0.2180	
121	300	1	1	0.1259	0.1151	0.1081	0.0988	0.1065	0.0973	0.1020	0.0933	
121	300	2	2	0.5274	0.3943	0.4502	0.3365	0.4509	0.3371	0.4255	0.3181	

Effect of geometric configuration on NDFP of S-FG-nP

In this section, the influence of geometric configuration, thermal environment and volume fraction index (k) is demonstrated. As depicted in Fig. 5, it becomes evident that the thermal environment, nonlocal parameter, and k exhibit a more pronounced effect on S-FG-nP-A compared to S-FG-nP-B. Additionally, it can be concluded that as the k increases from 0 to 5, the NDFP decreases for both types of S-FG-nP. However, the decrement is more pronounced in S-FG-nP-A compared to S-FG-nP-B. Furthermore, S-FG-nP-B is observed to be less sensitive to changes in temperature and k compared to S-FG-nP-A.It is found that the NDFP is the same for 111, 121, and 221 configurations when k=0 for the S-FGM-A type. This is due to the fact that, for k=0, the plate becomes ceramic. In the S-FGM-B type, the NDFP for k=1 is the same for the 111 and 121 configurations. This is because the effective material properties (Deff,I0,I2,G0) become identical, as calculated from Eq. (3). The table presented in Table 7 offers a comprehensive examination of the impact of different configurations (111, 121, and 221) on the NDFP of S-FG-nP. It becomes evident that the configuration labeled S-FG-nP-B exhibits a noteworthy variation in NDFP when comparing 121 with 111, amounting to a maximum change of approximately 2%. Conversely, for S-FG-nP-A, this difference is notably higher, standing at around 5%.

Furthermore, the comparison between 111 and 221, it is notable that for S-FG-nP-B, the NDFP demonstrates an increase of up to 3%. However, a contrasting trend emerges for S-FG-nP-A, where the NDFP experiences a decline, with a maximum decrease of approximately 3%. Similar trend is observed for the μ=1 with lower values. The observed phenomenon is due to the stiffness softening caused by the occurrence of nonlocality in the structures.

Effect of higher modes and nonlocal parameter on NDFP of S-FG-nP

Figure 6 illustrates a plot displaying various mode shapes under the SSSS boundary condition. In this plot, each mode shape is represented by symbols and is characterized by the values of m and n. The symbol m indicates the number of half-sine waves present in the x-direction, while n represents the specific lower frequency associated with a given m value. This plot provides a visual representation of the different mode shapes that can be observed under the SSSS boundary condition. The influence of thermal conditions and configurations on the five vibration modes, has been thoroughly investigated for volume fractions of 0 and 1, as well as nonlocal parameters (μ) of 0 and 1. It is evident from the analysis presented in Table 8 that an increase in temperature leads to a decrease in the NDFP across all configurations. This trend can be attributed to the decrease in the elastic properties of the material with rising temperatures (ΔT). Additionally, it is observed that as the nonlocal parameter μ transitions from 0 to 1, the NDFP decreases, indicating a significant influence of μ on the vibrational behavior. This phenomenon is attributed to stiffness-softening effects that occur due to the nonlocality in the structure.

Moreover, a comparison between the 111 and 121 configurations reveals interesting trends. In the case of S-FG-nP-A, the 121 configuration exhibits higher frequencies than the 111 configuration for k = 1, while yielding similar results for k = 0 and μ=0. However, in the context of S-FG-nP-B, it is noted that the NDFP remains consistent for k = 1 between the two configurations, irrespective of μ. Conversely, the 121 configuration demonstrates higher NDFP values for k = 0 compared to the 111 configuration, regardless of the value of μ. These discrepancies in behavior between S-FG-nP-B and S-FG-nP-A can be attributed to variations in material properties.

Conclusion

This research paper presents a robust methodology for analyzing the non-dimensional frequency parameter (NDFP) of functionally graded sandwich nanoplates, integrating nonlocal theory and classical plate theory (CPT) to account for various influencing factors. The DSM incorporates an elastic foundation, volume fraction index, and thermal conditions, providing a comprehensive framework for understanding the behavior of these nanoplates. One key capability of methodology is its ability to assess the impact of an elastic foundation on the effective bending stiffness of the nanoplates, leading to a noticeable increase in the NDFP. Additionally, the paper explore the influence of the nonlocal parameter on the NDFP, observing a stiffness-softening phenomenon where an increase in the nonlocal parameter results in a reduction of the NDFP because of stiffness-softening effects that occur due to the nonlocality in the structure. Another important aspect is the consideration of thermal effects, revealing how changes in temperature affect the NDFP, reflecting the thermal stability of these structures. Furthermore, our methodology enables the comparison of different plate configurations (111, 121, 221), highlighting the 121 configuration as consistently exhibiting a higher NDFP under similar conditions. By emphasizing methodology and its capabilities, this research contributes valuable insights into the design and optimization of functionally graded nanoplates, advancing their understanding and application in engineering contexts.

Appendix

The following expressions describe various aspects of the vibration and stability analysis of plates. Each term in the expressions is defined as follows:Deff: Effective bending stiffness of the plate.

I0 and I2: Moments of inertia.

b: Breadth (width) of the plate.

α: Half sine wave parameter.

G0: Thermal stress coefficient.

r1 and r2: Roots of the fourth-order differential equation.

Svv=Deff·r1·r2·(r12-r22)·(r1sinh(br1)cosh(br2)-r2sinh(br2)cosh(br1))r12sinh(br1)sinh(br2)-2r1r2cosh(br1)cosh(br2)+2r1r2+r22sinh(br1)sinh(br2)Svm=-r1r1sinh(br1)sinh(br2)-r2cosh(br1)cosh(br2)+r2(Deff·α12ν-2·Deff·α12+Deff·r12+G0+I2·ω2-kp)r12sinh(br1)sinh(br2)-2r1r2cosh(br1)cosh(br2)+2r1r2+r22sinh(br1)sinh(br2)-r2-r1cosh(br1)cosh(br2)+r1+r2sinh(br1)sinh(br2)(Deff·α12ν-2·Deff·α12+Deff·r22+G0+I2·ω2-kp)r12sinh(br1)sinh(br2)-2r1r2cosh(br1)cosh(br2)+2r1r2+r22sinh(br1)sinh(br2)Smm=-Deff·(r12-r22)·(r1sinh(br2)cosh(br1)-r2sinh(br1)cosh(br2))r12sinh(br1)sinh(br2)-2r1r2cosh(br1)cosh(br2)+2r1r2+r22sinh(br1)sinh(br2)Fvv=-Deff·r1·r2·(r12-r22)·(r1sinh(br1)-r2sinh(br2))r12sinh(br1)sinh(br2)-2r1r2cosh(br1)cosh(br2)+2r1r2+r22sinh(br1)sinh(br2)Fvm=-Deff·r1·r2·(r12-r22)·(cosh(br1)-cosh(br2))r12sinh(br1)sinh(br2)-2r1r2cosh(br1)cosh(br2)+2r1r2+r22sinh(br1)sinh(br2)Fmm=-Deff·(r13sinh(br2)-r12r2sinh(br1)-r1r22sinh(br2)+r23sinh(br1))r12sinh(br1)sinh(br2)-2r1r2cosh(br1)cosh(br2)+2r1r2+r22sinh(br1)sinh(br2)

Author contributions

A.G. provided supervision, conceptualization, and methodology for the paper. Saurabh Rai contributed to data curation, investigation, and the writing of the original draft of the paper. All authors reviewed the manuscript.

Data availibility

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.

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References

1. Rai S Kumar S Singh R Gupta A Effect of porosity inclusions on the natural frequencies of the FGM plates using dynamic stiffness method Int. J. Interact. Des. Manuf. 2023 17 2723 2730
Rai, S., Kumar, S., Singh, R. & Gupta, A. Effect of porosity inclusions on the natural frequencies of the FGM plates using dynamic stiffness method. Int. J. Interact. Des. Manuf. 17, 2723–2730 (2023).
2. Arshid E Amir S Loghman A Bending and buckling behaviors of heterogeneous temperature-dependent micro annular/circular porous sandwich plates integrated by FGPEM nano-composite layers J. Sandwich Struct. Mater. 2021 23 3836 3877
Arshid, E., Amir, S. & Loghman, A. Bending and buckling behaviors of heterogeneous temperature-dependent micro annular/circular porous sandwich plates integrated by FGPEM nano-composite layers. J. Sandwich Struct. Mater. 23, 3836–3877. 10.1177/1099636220955027 (2021).
3. Wang S Song Y Qiao Y Shao S Wang W Dynamic performance of functionally graded composite structures with viscoelastic polymers Sci. Rep. 2024 14 7613 38556537
Wang, S., Song, Y., Qiao, Y., Shao, S. & Wang, W. Dynamic performance of functionally graded composite structures with viscoelastic polymers. Sci. Rep. 14, 7613 (2024).38556537
4. Gupta A Joshi A Sharma SC Harsha SP Dynamic analysis of fixed-free single-walled carbon nanotube-based bio-sensors because of various viruses IET Nanobiotechnol. 2012 6 115 121 22894536
Gupta, A., Joshi, A., Sharma, S. C. & Harsha, S. P. Dynamic analysis of fixed-free single-walled carbon nanotube-based bio-sensors because of various viruses. IET Nanobiotechnol. 6, 115–121 (2012).22894536
5. Craciunescu CM Wuttig M New ferromagnetic and functionally graded shape memory alloys ChemInform 2003
Craciunescu, C. M. & Wuttig, M. New ferromagnetic and functionally graded shape memory alloys. ChemInform10.1002/chin.200339234 (2003).
6. Fu Y Du H Zhang S Functionally graded TiN/TiNi shape memory alloy films Mater. Lett. 2003 57 2995 2999
Fu, Y., Du, H. & Zhang, S. Functionally graded TiN/TiNi shape memory alloy films. Mater. Lett. 57, 2995–2999. 10.1016/S0167-577X(02)01419-2 (2003).
7. Sezer N Koç M A comprehensive review on the state-of-the-art of piezoelectric energy harvesting Nano Energy 2021 80 105567
Sezer, N. & Koç, M. A comprehensive review on the state-of-the-art of piezoelectric energy harvesting. Nano Energy 80, 105567. 10.1016/j.nanoen.2020.105567 (2021).
8. Eringen A Edelen D On nonlocal elasticity Int. J. Eng. Sci. 1972 10 233 248
Eringen, A. & Edelen, D. On nonlocal elasticity. Int. J. Eng. Sci. 10, 233–248. 10.1016/0020-7225(72)90039-0 (1972).
9. Lim CW Zhang G Reddy JN A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation J. Mech. Phys. Solids 2015 78 298 313
Lim, C. W., Zhang, G. & Reddy, J. N. A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J. Mech. Phys. Solids 78, 298–313. 10.1016/j.jmps.2015.02.001 (2015).
10. Ansari R Ashrafi M Pourashraf T Sahmani S Vibration and buckling characteristics of functionally graded nanoplates subjected to thermal loading based on surface elasticity theory Acta Astronaut. 2015 109 42 51
Ansari, R., Ashrafi, M., Pourashraf, T. & Sahmani, S. Vibration and buckling characteristics of functionally graded nanoplates subjected to thermal loading based on surface elasticity theory. Acta Astronaut. 109, 42–51. 10.1016/j.actaastro.2014.12.015 (2015).
11. Salehipour H Shahidi A Nahvi H Modified nonlocal elasticity theory for functionally graded materials Int. J. Eng. Sci. 2015 90 44 57
Salehipour, H., Shahidi, A. & Nahvi, H. Modified nonlocal elasticity theory for functionally graded materials. Int. J. Eng. Sci. 90, 44–57 (2015).
12. Bayones FS Mondal S Abo-Dahab SM Kilany AA Effect of moving heat source on a magneto-thermoelastic rod in the context of Eringen’s nonlocal theory under three-phase lag with a memory dependent derivative Mech. Based Des. Struct. Mach. 2023 51 2501 2516
Bayones, F. S., Mondal, S., Abo-Dahab, S. M. & Kilany, A. A. Effect of moving heat source on a magneto-thermoelastic rod in the context of Eringen’s nonlocal theory under three-phase lag with a memory dependent derivative. Mech. Based Des. Struct. Mach. 51, 2501–2516. 10.1080/15397734.2021.1901735 (2023).
13. Wang S Zhang J Li Q Su J Liang S Free vibration of co-cured composite structures with different numbers of viscoelastic damping membranes Compos. Struct. 2020 247 112434
Wang, S., Zhang, J., Li, Q., Su, J. & Liang, S. Free vibration of co-cured composite structures with different numbers of viscoelastic damping membranes. Compos. Struct. 247, 112434 (2020).
14. Wang S Free vibration of functionally graded carbon nanotube-reinforced composite damping structure based on the higher-order shear deformation theory Polym. Compos. 2023 44 873 885
Wang, S. et al. Free vibration of functionally graded carbon nanotube-reinforced composite damping structure based on the higher-order shear deformation theory. Polym. Compos. 44, 873–885 (2023).
15. Natarajan S Chakraborty S Thangavel M Bordas S Rabczuk T Size-dependent free flexural vibration behavior of functionally graded nanoplates Comput. Mater. Sci. 2012 65 74 80
Natarajan, S., Chakraborty, S., Thangavel, M., Bordas, S. & Rabczuk, T. Size-dependent free flexural vibration behavior of functionally graded nanoplates. Comput. Mater. Sci. 65, 74–80. 10.1016/j.commatsci.2012.06.031 (2012).
16. Jung W-Y Han S-C Analysis of sigmoid functionally graded material (S-FGM) nanoscale plates using the nonlocal elasticity theory Math. Probl. Eng. 2013 2013 1 10
Jung, W.-Y. & Han, S.-C. Analysis of sigmoid functionally graded material (S-FGM) nanoscale plates using the nonlocal elasticity theory. Math. Probl. Eng. 2013, 1–10. 10.1155/2013/476131 (2013).
17. Safarpour M Rahimi A Alibeigloo A Static and free vibration analysis of graphene platelets reinforced composite truncated conical shell, cylindrical shell, and annular plate using theory of elasticity and dqm Mech. Based Des. Struct. Mach. 2020 48 496 524
Safarpour, M., Rahimi, A. & Alibeigloo, A. Static and free vibration analysis of graphene platelets reinforced composite truncated conical shell, cylindrical shell, and annular plate using theory of elasticity and dqm. Mech. Based Des. Struct. Mach. 48, 496–524 (2020).
18. Safarpour M Forooghi A Dimitri R Tornabene F Theoretical and numerical solution for the bending and frequency response of graphene reinforced nanocomposite rectangular plates Appl. Sci. 2021 11 6331
Safarpour, M., Forooghi, A., Dimitri, R. & Tornabene, F. Theoretical and numerical solution for the bending and frequency response of graphene reinforced nanocomposite rectangular plates. Appl. Sci. 11, 6331 (2021).
19. Safarpour M Alibeigloo A Elasticity solution for bending and frequency behavior of sandwich cylindrical shell with FG-CNTRC face-sheets and polymer core under initial stresses Int. J. Appl. Mech. 2021 13 2150020
Safarpour, M. & Alibeigloo, A. Elasticity solution for bending and frequency behavior of sandwich cylindrical shell with FG-CNTRC face-sheets and polymer core under initial stresses. Int. J. Appl. Mech. 13, 2150020 (2021).
20. Bai Y Suhatril M Cao Y Forooghi A Assilzadeh H Hygro-thermo-magnetically induced vibration of nanobeams with simultaneous axial and spinning motions based on nonlocal strain gradient theory Eng. Comput. 2022 38 2509 2526
Bai, Y., Suhatril, M., Cao, Y., Forooghi, A. & Assilzadeh, H. Hygro-thermo-magnetically induced vibration of nanobeams with simultaneous axial and spinning motions based on nonlocal strain gradient theory. Eng. Comput. 38, 2509–2526 (2022).
21. Al-Furjan M Non-polynomial framework for stress and strain response of the FG-GPLRC disk using three-dimensional refined higher-order theory Eng. Struct. 2021 228 111496
Al-Furjan, M. et al. Non-polynomial framework for stress and strain response of the FG-GPLRC disk using three-dimensional refined higher-order theory. Eng. Struct. 228, 111496 (2021).
22. Rahimi A Alibeigloo A Safarpour M Three-dimensional static and free vibration analysis of graphene platelet-reinforced porous composite cylindrical shell J. Vib. Control 2020 26 1627 1645
Rahimi, A., Alibeigloo, A. & Safarpour, M. Three-dimensional static and free vibration analysis of graphene platelet-reinforced porous composite cylindrical shell. J. Vib. Control 26, 1627–1645 (2020).
23. Nami MR Janghorban M Damadam M Thermal buckling analysis of functionally graded rectangular nanoplates based on nonlocal third-order shear deformation theory Aerosp. Sci. Technol. 2015 41 7 15
Nami, M. R., Janghorban, M. & Damadam, M. Thermal buckling analysis of functionally graded rectangular nanoplates based on nonlocal third-order shear deformation theory. Aerosp. Sci. Technol. 41, 7–15. 10.1016/j.ast.2014.12.001 (2015).
24. Tran TT Pham Q-H Nguyen-Thoi T Dynamic analysis of functionally graded porous plates resting on elastic foundation taking into mass subjected to moving loads using an edge-based smoothed finite element method Shock. Vib. 2020 2020 1 19
Tran, T. T., Pham, Q.-H. & Nguyen-Thoi, T. Dynamic analysis of functionally graded porous plates resting on elastic foundation taking into mass subjected to moving loads using an edge-based smoothed finite element method. Shock. Vib. 2020, 1–19. 10.1155/2020/8853920 (2020).
25. Tran TT Pham Q-H Nguyen-Thoi T An edge-based smoothed finite element for free vibration analysis of functionally graded porous (FGP) plates on elastic foundation taking into mass (EFTIM) Math. Probl. Eng. 2020 2020 1 17
Tran, T. T., Pham, Q.-H. & Nguyen-Thoi, T. An edge-based smoothed finite element for free vibration analysis of functionally graded porous (FGP) plates on elastic foundation taking into mass (EFTIM). Math. Probl. Eng. 2020, 1–17. 10.1155/2020/8278743 (2020).
26. Banichuk N Analysis and optimization against buckling of beams interacting with elastic foundation Mech. Based Des. Struct. Mach. 2018 46 615 633
Banichuk, N. et al. Analysis and optimization against buckling of beams interacting with elastic foundation. Mech. Based Des. Struct. Mach. 46, 615–633. 10.1080/15397734.2017.1377619 (2018).
27. Forooghi A Fallahi N Alibeigloo A Forooghi H Rezaey S Static and thermal instability analysis of embedded functionally graded carbon nanotube-reinforced composite plates based on HSDT via GDQM and validated modeling by neural network Mech. Based Des. Struct. Mach. 2023 51 7149 7182
Forooghi, A., Fallahi, N., Alibeigloo, A., Forooghi, H. & Rezaey, S. Static and thermal instability analysis of embedded functionally graded carbon nanotube-reinforced composite plates based on HSDT via GDQM and validated modeling by neural network. Mech. Based Des. Struct. Mach. 51, 7149–7182 (2023).
28. Azarniya O Rahimi G Forooghi A Large deformation analysis of a hyperplastic beam using experimental/fem/meshless collocation method Waves Random Compl. Media 2023
Azarniya, O., Rahimi, G. & Forooghi, A. Large deformation analysis of a hyperplastic beam using experimental/fem/meshless collocation method. Waves Random Compl. Media10.1080/17455030.2023.2184645 (2023).
29. Forooghi A Alibeigloo A Hygro-thermo-magnetically induced vibration of FG-CNTRC small-scale plate incorporating nonlocality and strain gradient size dependency Waves Random Compl. Media 2022
Forooghi, A. & Alibeigloo, A. Hygro-thermo-magnetically induced vibration of FG-CNTRC small-scale plate incorporating nonlocality and strain gradient size dependency. Waves Random Compl. Media10.1080/17455030.2022.2037784 (2022).
30. Ebrahimi-Mamaghani A Forooghi A Sarparast H Alibeigloo A Friswell M Vibration of viscoelastic axially graded beams with simultaneous axial and spinning motions under an axial load Appl. Math. Model. 2021 90 131 150
Ebrahimi-Mamaghani, A., Forooghi, A., Sarparast, H., Alibeigloo, A. & Friswell, M. Vibration of viscoelastic axially graded beams with simultaneous axial and spinning motions under an axial load. Appl. Math. Model. 90, 131–150 (2021).
31. Forooghi A Rezaey S Haghighi SM Zenkour AM Thermal instability analysis of nanoscale fg porous plates embedded on kerr foundation coupled with fluid flow Eng. Comput. 2022 38 2953 2973
Forooghi, A., Rezaey, S., Haghighi, S. M. & Zenkour, A. M. Thermal instability analysis of nanoscale fg porous plates embedded on kerr foundation coupled with fluid flow. Eng. Comput. 38, 2953–2973 (2022).
32. Wu M-J Huang X-H Azim I Zhu J Chen H Nonlinear dynamic and vibration characteristics of metamaterial shallow arches Eur. J. Mech. A Solids 2023 102 105084
Wu, M.-J., Huang, X.-H., Azim, I., Zhu, J. & Chen, H. Nonlinear dynamic and vibration characteristics of metamaterial shallow arches. Eur. J. Mech. A Solids 102, 105084 (2023).
33. Huang X-H Yang J Wang X-E Azim I Combined analytical and numerical approach for auxetic FG-CNTRC plate subjected to a sudden load Eng. Comput. 2022 38 55 70
Huang, X.-H., Yang, J., Wang, X.-E. & Azim, I. Combined analytical and numerical approach for auxetic FG-CNTRC plate subjected to a sudden load. Eng. Comput. 38, 55–70 (2022).
34. Huang X-H Yu N-T Azim I Zhu J Wu M-J A comparative analysis of thermos-mechanical behavior of CNT-reinforced composite plates: Capturing the effects of thermal shrinkage Case Stud. Therm. Eng. 2022 38 102347
Huang, X.-H., Yu, N.-T., Azim, I., Zhu, J. & Wu, M.-J. A comparative analysis of thermos-mechanical behavior of CNT-reinforced composite plates: Capturing the effects of thermal shrinkage. Case Stud. Therm. Eng. 38, 102347 (2022).
35. Wu M-J Zhao S-Y Azim I Zhu J Huang X-H Design and thermo-mechanical analysis of sandwich structures with negative thermal expansion Int. J. Mech. Mater. Des. 2022 18 807 822
Wu, M.-J., Zhao, S.-Y., Azim, I., Zhu, J. & Huang, X.-H. Design and thermo-mechanical analysis of sandwich structures with negative thermal expansion. Int. J. Mech. Mater. Des. 18, 807–822 (2022).
36. Banerjee JR Dynamic stiffness formulation for structural elements: A general approach Comput. Struct. 1997 63 101 103
Banerjee, J. R. Dynamic stiffness formulation for structural elements: A general approach. Comput. Struct. 63, 101–103 (1997).
37. Banerjee JR Coupled bending-torsional dynamic stiffness matrix for beam elements Int. J. Numer. Methods Eng. 1989 28 1283 1298
Banerjee, J. R. Coupled bending-torsional dynamic stiffness matrix for beam elements. Int. J. Numer. Methods Eng. 28, 1283–1298. 10.1002/nme.1620280605 (1989).
38. Boscolo M Banerjee JR Dynamic stiffness elements and their applications for plates using first order shear deformation theory Comput. Struct. 2011 89 395 410
Boscolo, M. & Banerjee, J. R. Dynamic stiffness elements and their applications for plates using first order shear deformation theory. Comput. Struct. 89, 395–410. 10.1016/j.compstruc.2010.11.005 (2011).
39. Kolarevic N Nefovska-Danilovic M Petronijevic M Dynamic stiffness elements for free vibration analysis of rectangular Mindlin plate assemblies J. Sound Vib. 2015 359 84 106
Kolarevic, N., Nefovska-Danilovic, M. & Petronijevic, M. Dynamic stiffness elements for free vibration analysis of rectangular Mindlin plate assemblies. J. Sound Vib. 359, 84–106. 10.1016/j.jsv.2015.06.031 (2015).
40. Kumar S Ranjan V Jana P Free vibration analysis of thin functionally graded rectangular plates using the dynamic stiffness method Compos. Struct. 2018 197 39 53
Kumar, S., Ranjan, V. & Jana, P. Free vibration analysis of thin functionally graded rectangular plates using the dynamic stiffness method. Compos. Struct. 197, 39–53. 10.1016/j.compstruct.2018.04.085 (2018).
41. Ali I Azam MS Exact solution by dynamic stiffness method for the natural vibration of porous functionally graded plate considering neutral surface J. Sandwich Struct. Mater. 2021 235 7 1585 1603
Ali, I. & Azam, M. S. Exact solution by dynamic stiffness method for the natural vibration of porous functionally graded plate considering neutral surface. J. Sandwich Struct. Mater. 235(7), 1585–1603 (2021).
42. Reddy J Microstructure-dependent couple stress theories of functionally graded beams J. Mech. Phys. Solids 2011 59 2382 2399
Reddy, J. Microstructure-dependent couple stress theories of functionally graded beams. J. Mech. Phys. Solids 59, 2382–2399 (2011).
43. Zare M Nazemnezhad R Hosseini-Hashemi S Natural frequency analysis of functionally graded rectangular nanoplates with different boundary conditions via an analytical method Meccanica 2015 50 2391 2408
Zare, M., Nazemnezhad, R. & Hosseini-Hashemi, S. Natural frequency analysis of functionally graded rectangular nanoplates with different boundary conditions via an analytical method. Meccanica 50, 2391–2408. 10.1007/s11012-015-0161-9 (2015).
44. Pham Q-H A nonlocal quasi-3D theory for thermal free vibration analysis of functionally graded material nanoplates resting on elastic foundation Case Stud. Therm. Eng. 2021 26 101170
Pham, Q.-H. et al. A nonlocal quasi-3D theory for thermal free vibration analysis of functionally graded material nanoplates resting on elastic foundation. Case Stud. Therm. Eng. 26, 101170. 10.1016/j.csite.2021.101170 (2021).
45. Pasternak, P. L. On a new method of analysis of an elastic foundation by means of two foundation constants. Gos. Izd. Lit. po Strait i Arkh (1954).
46. Winkler, E. Theory of elasticity and strength. Dominicus Prague 36 (1867).
47. Kumar S Ranjan V Jana P Free vibration analysis of thin functionally graded rectangular plates using the dynamic stiffness method Compos. Struct. 2018 197 39 53
Kumar, S., Ranjan, V. & Jana, P. Free vibration analysis of thin functionally graded rectangular plates using the dynamic stiffness method. Compos. Struct. 197, 39–53. 10.1016/j.compstruct.2018.04.085 (2018).
48. Chauhan M Dwivedi S Jha R Ranjan V Sathujoda P Sigmoid functionally graded plates embedded on Winkler–Pasternak foundation: Free vibration analysis by dynamic stiffness method Compos. Struct. 2022 288 115400
Chauhan, M., Dwivedi, S., Jha, R., Ranjan, V. & Sathujoda, P. Sigmoid functionally graded plates embedded on Winkler–Pasternak foundation: Free vibration analysis by dynamic stiffness method. Compos. Struct. 288, 115400. 10.1016/j.compstruct.2022.115400 (2022).
49. Eringen AC On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves J. Appl. Phys. 1983 54 4703 4710
Eringen, A. C. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710 (1983).
50. Eringen A Wegner J Nonlocal continuum field theories Appl. Mech. Rev. 2003 56 B20 B22
Eringen, A. & Wegner, J. Nonlocal continuum field theories. Appl. Mech. Rev. 56, B20–B22. 10.1115/1.1553434 (2003).
51. Kumar S Jana P Accurate solution for free vibration behaviour of stepped FGM plates implementing the dynamic stiffness method Structures 2022 45 1971 1989
Kumar, S. & Jana, P. Accurate solution for free vibration behaviour of stepped FGM plates implementing the dynamic stiffness method. Structures 45, 1971–1989. 10.1016/j.istruc.2022.10.035 (2022).
52. Boscolo M Banerjee JR Layer-wise dynamic stiffness solution for free vibration analysis of laminated composite plates J. Sound Vib. 2014 333 200 227
Boscolo, M. & Banerjee, J. R. Layer-wise dynamic stiffness solution for free vibration analysis of laminated composite plates. J. Sound Vib. 333, 200–227. 10.1016/j.jsv.2013.08.031 (2014).
53. Aghababaei R Reddy J Nonlocal third-order shear deformation plate theory with application to bending and vibration of plates J. Sound Vib. 2009 326 277 289
Aghababaei, R. & Reddy, J. Nonlocal third-order shear deformation plate theory with application to bending and vibration of plates. J. Sound Vib. 326, 277–289. 10.1016/j.jsv.2009.04.044 (2009).
54. Sobhy M Radwan AF A new quasi 3D nonlocal plate theory for vibration and buckling of FGM nanoplates Int. J. Appl. Mech. 2017 09 1750008
Sobhy, M. & Radwan, A. F. A new quasi 3D nonlocal plate theory for vibration and buckling of FGM nanoplates. Int. J. Appl. Mech. 09, 1750008. 10.1142/S1758825117500089 (2017).
55. Daikh AA Drai A Bensaid I Houari MSA Tounsi A On vibration of functionally graded sandwich nanoplates in the thermal environment J. Sandwich Struct. Mater. 2021 23 2217 2244
Daikh, A. A., Drai, A., Bensaid, I., Houari, M. S. A. & Tounsi, A. On vibration of functionally graded sandwich nanoplates in the thermal environment. J. Sandwich Struct. Mater. 23, 2217–2244 (2021).
56. Gulshan Taj MN Chakrabarti A Prakash V Vibration characteristics of functionally graded material skew plate in thermal environment Compos. Part B 2014 8 142 153
Gulshan Taj, M. N., Chakrabarti, A. & Prakash, V. Vibration characteristics of functionally graded material skew plate in thermal environment. Compos. Part B 8, 142–153. 10.5281/zenodo.1090741 (2014).
