Refined Plate Theory for Critical Buckling Analysis of FG Sandwich Nanoplates Considering Neutral Surface Concept and Piezoelectric Surface Effects
Subject Areas : Mechanics of SolidsAmir Hossein Soltan Arani 1 , Ali Ghorbanpour Arani 2 * , Zahra Khoddami Maraghi 3
1 - Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran
2 - Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran---Institute of Nanoscience & Nanotechnology, University of Kashan, Kashan, Iran
3 - Faculty of Engineering, Mahallat Institute of Higher Education, Mahallat, Iran
Keywords: Critical Buckling Load, Neutral Surface Concept, Nonlocal Strain Gradient Theory, Surface Effect, Refined Plate Theory,
Abstract :
The nonlocal quasi-3D static stability analysis of the sandwich simply supported nanoplates embedded in an orthotropic Pasternak foundation placed in an electrical environment by considering the surface effects based on a five-variable refined plate theory by taking into account the stretching effects is investigated in the current study. The core of the structure is functionally graded along its thickness using a power law model. The concept of neutral surface position is applied to achieve symmetry in the distribution of material properties across the thickness. The piezoelectric face sheets are actuators and sensors for the functionally graded layer based on the surface piezoelasticity theory. According to the nonlocal strain gradient theory, the higher-order shear deformation theory is utilized to develop the linear equilibrium equations of motion based on the principle of minimum potential energy. Eventually, a Navier-type solution is applied to obtain the analytical results of a three-layered nano-plate subjected to the electric field. Evaluation of the accuracy and efficiency of the current approach demonstrates a good agreement between the obtained results from this model and those published in the reviewed literature. Eventually, a comprehensive study is conducted to examine the influences of various parameters on the critical buckling load of the functionally graded sandwich structure in detail. Numerical results indicate significant influences of residual surface stress and neutral surface position on the critical buckling load, particularly in thick nanoplates. These findings are expected to aid in designing micro/nano-electro-mechanical system components based on smart nanostructures.
[1] Feri M., Krommer M., Alibeigloo A., 2021, Three-dimensional static analysis of a viscoelastic rectangular functionally graded material plate embedded between piezoelectric sensor and actuator layers, Mechanics Based Design of Structures and Machines 51(7): 3843-3867.
[2] Marzavan S., Nastasescu V., 2023, Free vibration analysis of a functionally graded plate by finite element method, Ain Shams Engineering Journal 14(8): 102024.
[3] Frahlia H., Bennai R., Nebab M., Ait Atmane H., Tounsi A., 2022, Assessing effects of parameters of viscoelastic foundation on the dynamic response of functionally graded plates using a novel HSDT theory, Mechanics of Advanced Materials and Structures 30(13): 2765-2779.
[4] Zargar Ershadi M., Faraji Oskouie M., Ansari R., 2022, Nonlinear vibration analysis of functionally graded porous circular plates under hygro-thermal loading, Mechanics Based Design of Structures and Machines 52(2): 1042-1059.
[5] Xie K., Chen H., Wang Y., Li L., Jin F., 2024, Nonlinear dynamic analysis of a geometrically imperfect sandwich beam with functionally graded material facets and auxetic honeycomb core in thermal environment, Aerospace Science and Technology 144: 108794.
[6] Arefi M., Soltan Arani A.H., 2018, Higher order shear deformation bending results of a magnetoelectrothermoelastic functionally graded nanobeam in thermal, mechanical, electrical, and magnetic environments, Mechanics Based Design of Structures and Machines 46(6): 669-692.
[7] Ghorbanpour Arani A., Kolahdouzan F., Abdollahian M., 2018, Nonlocal buckling of embedded magnetoelectroelastic sandwich nanoplate using refined zigzag theory, Applied Mathematics and Mechanics 39(4): 529-546
[8] Singh A.K., Koley S, Negi A., 2022. Remarks on the scattering phenomena of love-type wave propagation in a layered porous piezoelectric structure containing surface irregularity, Mechanics of Advanced Materials and Structures 30(12): 2398-2429.
[9] Dhua S., Mondal S., Maji A., 2024, Surface effects on wave propagation in piezoelectric–piezomagnetic loosely bonded bilayer system using nonlocal theory of elasticity, thin wall structure 197: 111612.
[10] Biswas M., Sahu S.A., 2022, Surface wave dispersion in imperfectly bonded flexoelectric-piezoelectric/FGPM bi-composite in contact of Newtonian liquid, Mechanics of Advanced Materials and Structures 30(14): 2995-3012.
[11] Mirzaei M., 2022, Vibration characteristics of sandwich plates with GPLRC core and piezoelectric face sheets with various electrical and mechanical boundary conditions, Mechanics Based Design of Structures and Machines 52(2): 990-1013.
[12] Hai T., Al-Masoudy M.M., Alsulamy S., El Ouni M.H., Ayvazyan A., Kumar A., 2023, Size-dependent free vibration analysis of honeycomb sandwich microplates integrated with piezoelectric actuators based on the modified strain gradient theory, Composite Structures 305: p.116555.
[13] Ghorbanpour Arani A., Jamali M., Mosayyebi M., Kolahchi R., 2016, Wave propagation in FG-CNT-reinforced piezoelectric composite micro plates using viscoelastic quasi-3D sinusoidal shear deformation theory, Composites Part B: Engineering 95: 209-224.
[14] Cao T.N.T., Reddy J.N., Lieu Q.X., Nguyen X.V., Luong V.H., 2021, A multi-layer moving plate method for dynamic analysis of viscoelastically connected double-plate systems subjected to moving loads, advanced structural engineering 24(9): 1798-1813.
[15] Ragb O., Matbulyb M.S., 2021, Nonlinear vibration analysis of elastically supported multi-layer composite plates using efficient quadrature techniques, International Journal for Computational Methods in Engineering Science and Mechanics 23(2): 129-146.
[16] Taghizadeh S.A., Naghdinasab M., Madadi H., Farrokhabadi A., 2021, Investigation of novel multi-layer sandwich panels under quasi-static indentation loading using experimental and numerical analyses, Thin wall structure 160: 107326.
[17] Amoozgar M., Fazelzadeh S.A., Ghavanloo E., Ajaj R.M., 2022, Free vibration analysis of curved lattice sandwich beams, Mechanics of Advanced Materials and Structures 31(2): 343-355
[18] Sahoo B., Sharma N., Sahoo B., Ramteke P.M., Panda S.K., Mahmoud S.R., 2022, Nonlinear vibration analysis of FGM sandwich structure under thermal loadings, Structures 44: 1392-1402.
[19] Derikvand M., Farhatnia F., Hodges H., 2021, Functionally graded thick sandwich beams with porous core: buckling analysis via differential transform method, Mechanics Based Design of Structures and Machines 51(7): 3650-3677.
[20] Li F., Yuan W., Zhang C., 2021, Free vibration and sound insulation of functionally graded honeycomb sandwich plates, Journal of Sandwich Structures and Materials 24(1): 565-600.
[21] Ghorbanpour Arani A., Shahraki M.E., Haghparast E., 2022, Instability analysis of axially moving sandwich plates with a magnetorheological elastomer core and GNP-reinforced face sheets, journal of brazilian society of mechanical sciences and engineering 44(4):150
[22] Li J., Kardomateas G., Liu L., 2023, Vibration analysis of thick-section sandwich structures in thermal environments, international journal of mechanical sciences 241: 107937.
[23] Liu S., Wang K., Wang B., 2023, Buckling and vibration characteristic of anisotropic sandwich plates with negative Poisson’s ratio based on isogeometric analysis, Mechanics Based Design of Structures and Machines 9:1-16.
[24] Ren H., Zhuang X., Oterkus E., Zhu H., Rabczuk T., 2021, Nonlocal strong forms of thin plate, gradient elasticity, magneto-electro-elasticity and phase-field fracture by nonlocal operator method, Engineering Computations 39: 23–44.
[25] Pham Q.H., Nguyen P.C., Tran T.T., 2022, Dynamic response of porous functionally graded sandwich nanoplates using nonlocal higher-order isogeometric analysis, Composite Structure 290: 115565,
[26] Nguyen N.V., Phan D.H., 2023, A refined quasi-3D isogeometric model for dynamic instability of graphene nanoplatelets-reinforced porous sandwich plates, aerospace science and technology 142: 108595.
[27] Hung P.T., Phung-Van P., Thai C.H., 2023, Small scale thermal analysis of piezoelectric–piezomagnetic FG microplates using modified strain gradient theory, International Journal of Mechanics and Materials in Design 19(4): 739-761.
[28] Jin Q., 2021, Interlaminar stress analysis of functionally graded graphene reinforced composite laminated plates based on a refined plate theory, Mechanics of Advanced Materials and Structures 29(25): 4138-4150.
[29] Tharwan M.Y., Daikh A.A., Assie A.E., Alnujaie A., Eltaher M.A., 2023, Refined quasi-3D shear deformation theory for buckling analysis of functionally graded curved nanobeam rested on Winkler/Pasternak/Kerr foundation, Mechanics Based Design of Structures and Machines 1-24.
[30] Hai Van N.T., Hong N.T., 2023, Novel finite element modeling for free vibration and buckling analysis of non-uniform thickness 2D-FG sandwich porous plates using refined Quasi 3D theory, Mechanics Based Design of Structures and Machines 1-27.
[31] Shahmohammadi M.A., Azhari M., Salehipour H., Thai H.T., 2023, Buckling of multilayered CNT/GPL/fibre/polymer hybrid composite plates resting on elastic support using modified nonlocal first-order plate theory, Mechanics Based Design of Structures and Machines 52(3): 1785-1810.
[32] Ghandourah E.E., Daikh A.A., Alhawsawi A.M, Fallatah O.A., EltaherM.A., 2022, Bending and buckling of FG-GRNC laminated plates via quasi-3D nonlocal strain gradient theory, Mathematics 10(8): 1321
[33] Hung D.X., Van Long N., Tu T.M., Trung D.X., 2024, Bending analysis of FGSP nanoplate resting on elastic foundation by using nonlocal quasi-3D theory, Thin Wall Structure 196: 111510.
[34] Daikh A.A., Belarbi M.O., Khechai A., Li L., Ahmed H.M., Eltaher M.A., 2023, Buckling of bi-coated functionally graded porous nanoplates via a nonlocal strain gradient quasi-3D theory, Acta Mechanica 234(8): 3397-3420.
[35] Shahzad M.A., Sahmani S., Safaei B., Basingab M.S., Hameed A.Z., 2023, Nonlocal strain gradient-based meshless collocation model for nonlinear dynamics of time-dependent actuated beam-type energy harvesters at nanoscale, Mechanics Based Design of Structures and Machines 1-35.
[36] Soleimani A., Zamani F., Haghshenas Gorgani H., 2022, Buckling analysis of three-dimensional functionally graded Euler-Bernoulli nanobeams based on the nonlocal strain gradient theory, Journal of Applied and Computational Mechanics 53(1): 24-40.
[37] Li L., Hu Y., Ling L., 2015, Flexural wave propagation in small-scaled functionally graded beams via a nonlocal strain gradient theory, Composite Structure 133: 1079-1092.
[38] Jalaei M.H., Civalek Ö.M.E.R., 2019, A nonlocal strain gradient refined plate theory for dynamic instability of embedded graphene sheet including thermal effects, Composite Structure 220: 209-220.
[39] Ebrahimi F., Dabbagh A., 2017, On flexural wave propagation responses of smart FG magneto-electro-elastic nanoplates via nonlocal strain gradient theory, Composite Structure 162: 281-293.
[40] Ma L.H., Ke L., Reddy J.N., Yang J., Kitipornchai S., Wang Y.S., 2018, Wave propagation characteristics in magneto-electro-elastic nanoshells using nonlocal strain gradient theory, Composite Structure 199: 10-23.
[41] Sahmani S., Aghdam M.M., Rabczuk T., 2018, Nonlinear bending of functionally graded porous micro/nano-beams reinforced with graphene platelets based upon nonlocal strain gradient theory, Composite Structure 186: 68-78.
[42] Jamali M., Shojaee T., Mohammadi B., 2020, Analytical buckling and post-buckling characteristics of Mindlin micro composite plate with central opening by use of nonlocal elasticity theory, Journal of Applied and Computational Mechanics 51(1): 231-238.
[43] Barati M.R., Shahverdi H., 2017, An analytical solution for thermal vibration of compositionally graded nanoplates with arbitrary boundary conditions based on physical neutral surface position, Mechanics of Advanced Materials and Structures 24(10): 840-853.
[44] Bellifa H., Benrahou K.H., Hadji L., Houari M.S.A., Tounsi A., 2016, Bending and free vibration analysis of functionally graded plates using a simple shear deformation theory and the concept the neutral surface position, Journal of the Brazilian Society of Mechanical Sciences and Engineering 38(1): 265-275.
[45] Farzam-Rad S.A., Hassani B., Karamodin A., 2017, Isogeometric analysis of functionally graded plates using a new quasi-3D shear deformation theory based on physical neutral surface, Composites Part B: Engineering 108: 174-189.
[46] Barati A., Norouzi S., 2020, Nonlocal elasticity theory for static torsion of the bi-directional functionally graded microtube under magnetic field, Journal of Applied and Computational Mechanics 51(1): 30-36.
[47] Ghorbanpour-Arani A.A., Khoddami Maraghi Z., Ghorbanpour Arani A., 2023, The Frequency Response of Intelligent Composite Sandwich Plate Under Biaxial In-Plane Forces, Journal of Solid Mechanic 15(1): 1-18.
[48] Mahmoudi A., Benyoucef S., Tounsi A., Benachour A., Adda Bedia E.A., Mahmoud, S.R., 2019, A refined quasi-3D shear deformation theory for thermo-mechanical behavior of functionally graded sandwich plates on elastic foundations, Journal of Sandwich Structures and Materials 21(6): 1906-1929.
[49] Meziane M.A.A., Abdelaziz H.H., Tounsi A., 2014, An efficient and simple refined theory for buckling and free vibration of exponentially graded sandwich plates under various boundary conditions, Journal of Sandwich Structures and Materials 16(3): 293-318.
[50] El Meiche N., Tounsi A., Ziane N., Mechab I., 2011, A new hyperbolic shear deformation theory for buckling and vibration of functionally graded sandwich plate, International Journal of Mechanical Sciences 53(4): 237-247.
[51] Ghorbanpour Arani A., Zarei H.B.A., Haghparast E., 2018, Vibration response of viscoelastic sandwich plate with magnetorheological fluid core and functionally graded-piezoelectric nanocomposite face sheets, Journal of Vibration and Control 24(21): 5169-5185.
[52] Ghorbanpour Arani A., Haghparast E., Zarei H.B.A., 2016, Nonlocal vibration of axially moving graphene sheet resting on orthotropic visco-Pasternak foundation under longitudinal magnetic field, Physica B: Condensed Matter 495: 35-49.
[53] Alipour M.M., Shariyat M., 2011, A power series solution for free vibration of variable thickness Mindlin circular plates with two-directional material heterogeneity and elastic foundations, Journal of Solid Mechanics 3(2): 183-197.
[54] Najafizadeh M.M., Raki M., Yousefi P., 2018, Vibration analysis of FG nanoplate based on third-order shear deformation theory (TSDT) and nonlocal elasticity, Journal of Solid Mechanics 10(3):464-475.
[55] Molla-Alipour M., Shariyat M., Shaban M., 2020, Free vibration analysis of bidirectional functionally graded conical/cylindrical shells and annular plates on nonlinear elastic foundations, based on a unified differential transform analytical formulation, Journal of Solid Mechanics 12(2): 385-400.
[56] Ghorbanpour Arani A., Haghparast E., Zarei, H.B.A., 2016, Vibration of axially moving 3-phase CNTFPC plate resting on orthotropic foundation, Structural Engineering and Mechanics 57(1): 105-126.
[57] Thai H.T., Vo T.P., 2013. A new sinusoidal shear deformation theory for bending, buckling, and vibration of functionally graded plates. Applied mathematical modelling 37(5): 3269-3281.
[58] Shufrin I., Eisenberger M., 2005, Stability and vibration of shear deformable plates––first order and higher order analyses. International Journal of Solids and Structures 42(3-4): 1225-1251.
[59] Haghparast E., Ghorbanpour-Arani A., Ghorbanpour Arani A., 2020. Effect of fluid–structure interaction on vibration of moving sandwich plate with Balsa wood core and nanocomposite face sheets. International Journal of Applied Mechanics 12(07): 2050078.
Journal of Solid Mechanics Vol. 16, No. 1 (2024) pp. 97-128 DOI: 10.60664/jsm.2024.3091790 |
Research Paper Refined Plate Theory for Critical Buckling Analysis of FG Sandwich Nanoplates Considering Neutral Surface Concept and Piezoelectric Surface Effects |
A.H. Soltan Arani 1, A. Ghorbanpour Arani 1,21 , Z. Khoddami Maraghi 3 | |
1 Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran 2 Institute of Nanoscience & Nanotechnology, University of Kashan, Kashan, Iran 3 Faculty of Engineering, Mahallat Institute of Higher Education, Mahallat, Iran | |
Received 11 September 2023; Received in revised form 20 June 2024; Accepted 24 June 2024 | |
| ABSTRACT |
| The nonlocal quasi-3D static stability analysis of the sandwich simply supported nanoplates embedded in an orthotropic Pasternak foundation placed in an electrical environment by considering the surface effects based on a five-variable refined plate theory by taking into account the stretching effects is investigated in the current study. The core of the structure is functionally graded along its thickness using a power law model. The concept of neutral surface position is applied to achieve symmetry in the distribution of material properties across the thickness. The piezoelectric face sheets are actuators and sensors for the functionally graded layer based on the surface piezoelasticity theory. According to the nonlocal strain gradient theory, the higher-order shear deformation theory is utilized to develop the linear equilibrium equations of motion based on the principle of minimum potential energy. Eventually, a Navier-type solution is applied to obtain the analytical results of a three-layered nano-plate subjected to the electric field. Evaluation of the accuracy and efficiency of the current approach demonstrates a good agreement between the obtained results from this model and those published in the reviewed literature. Eventually, a comprehensive study is conducted to examine the influences of various parameters on the critical buckling load of the functionally graded sandwich structure in detail. Numerical results indicate significant influences of residual surface stress and neutral surface position on the critical buckling load, particularly in thick nanoplates. These findings are expected to aid in designing micro/nano-electro-mechanical system components based on smart nanostructures. |
| Keywords: Critical Buckling Load; Neutral Surface Concept; Nonlocal Strain Gradient Theory; Surface Effect, Refined Plate Theory.
|
1 INTRODUCTION
D
A multi-layered plate is a special form of a sandwich structure comprising a combination of different laminates that are bonded to each other so that its properties are considered as the properties of an integrated structure. The primary advantage of multi-layered plates is very high stiffness-to-weight and high bending strength-to-weight ratio. Lightweight and stiff laminated panels are vital elements of many modern civil, aircraft, and spacecraft designs. Subsequently, researchers started to investigate the behavior of the multi-layered structures in the last few years.
Cao et al. [14] studied dynamic analysis of viscoelastically subjected to moving loads using the multi-layer moving plate method. They extracted the governing equations of the connected double-plate system by using the Reissner-Mindlin plate theory. Ragb and Matbuly [15] introduced different numerical schemes to formulate and solve nonlinear vibration analysis of elastically supported multilayer composite plates resting on the Winkler-Pasternak foundation by a first-order shear deformation theory (FSDT). The obtained results show that the used method is an accurate efficient model in the dynamic analysis of discontinuity structure resting on a nonlinear elastic foundation. Taghizadeh et al. [16] investigated the mechanical behavior of novel multi-layer sandwich panels subjected to indentation of a spherical indenter load experimentally and numerically. Amoozgar et al. [17] employed a combining a two-dimensional a one-dimensional nonlinear beam analysis to study the influences of initial curvature and lattice core shape on the vibration of sandwich beams. They used a time-space scheme to obtain nonlinear governing equations of the sandwich beam. Their results show that the lattice unit cell shape affects both in-plane and out-of-plane stiffness and results in changes in the dynamic behavior of the beam. Sahoo et al. [18] predicted nonlinear vibration analysis of FGM sandwich structure under linear and nonlinear temperature distributions numerically using the higher-order shear deformation theory (HSDT). A parametric study on the buckling behavior of a sandwich beam consisting of a porous ceramic core including the effects of length-to-thickness ratio, the volume fraction of FGM, and various porosity patterns based on third-order shear deformation theory (TSDT) was presented by Derikvand et al. [19]. The governing equilibrium equations were solved for different end conditions using the differential transform method and the physical neutral axis of the beam. Li et al. [20] used hyperbolic tangent shear deformation theory for the analysis of free vibration of FG honeycomb sandwich plates with negative Poisson’s ratio. They solved the derived governing dynamic equations by applying Navier’s method and fluid-solid interface conditions. The corresponding results display that the FG honeycomb core with negative Poisson’s ratio can yield much lower frequencies. Instability analysis of axially moving sandwich plates with magnetorheological core and polymeric face sheets reinforced with graphene nanoplatelets by using FSDT was studied by Ghorbanpour Arani et al. [21]. The Halpin–Tsai model and the rule of mixture are utilized to estimate the effective mechanical properties. A novel unified model for vibration analysis of a thick-section sandwich structure was presented based on the variational asymptotic method by Li et al. [22]. They studied the effects of temperature gradients in the thickness direction, core thickness, and boundary conditions by a detailed parametric study. Liu et al. [23] analyzed the buckling and vibration studies of the sandwich plates based on the isogeometric analysis in conjunction with the refined shear deformation theory.
Many researchers have studied the mechanical behavior of structures over the past centuries. Today, with the advancement of technology and the development of industries, achieving the exact results requires the use of new models and methods. Laminated structures are used in many engineering industries. The different theories are used to simulate and obtain analytical results the most common of these theories are Classical plate theory (CPT), FSDT, and HSDT. As the thickness of the sheet increases, the accuracy of these theories decreases. Size effects play a significant role in predicting mechanical behavior when the structure is being studied on a small scale. The best alternative approach to studying the mechanical behavior of materials is the use of the continuum mechanics relationships. The effect of size is not taken into account in classical continuum mechanics theory. For this reason, this theory cannot predict the mechanical behavior of nanostructures and microstructures well. Various theories including Strain gradient theory (SGT), Modified strain gradient theory (MSGT), Couple stress theory (CST), and Modified couple stress theory (MCST), are proposed to eliminate this defect. Layer-wise (LW) and zig-zag (ZZ) theories provide sufficiently accurate responses for relatively thick laminated structures. These theories can capture the inter-laminar stress fields near the edges. Refined plate theories (RPT) are theories that assume the uniaxial and lateral displacements have bending and shear components. In them, the bending and shear components do not contribute toward shear forces and bending moments, respectively. The most interesting feature of these theories is that they have high accuracy for a quadratic variation of the transverse shear strains across the thickness and also satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. Several models of RPT with different form functions by dividing the transverse displacement into bending and shear parts for plate structures are proposed. Quasi-three-dimensional and Three-dimensional (3D) are the new contributions of the proposed theories which are compatible with the numerical method and naturally taken into account in the thickness direction. Examples of the use of these theories in the published articles are expressed as follows.
Ren et al. [24] derived the nonlocal strong forms for various physical models in traditional methods. They derived the nonlocal forms of electro-magneto-elasticity thin plate and phase-field fracture method based on the nonlocal operator method by using the variational principle/weighted residual method. Pham et al. [25] studied the nonlocal dynamic response of sandwich nanoplates with a porous FG core using higher-order isogeometric theory. They extracted the governing equations of motion of sandwich nanoplates by the Hamilton principle and solved them by the Newmark method. Dynamic instability behavior of graphene nanoplatelets-reinforced porous sandwich plates subjected to periodic in-plane compressive loads based on a four-variable refined quasi-3D plate theory was investigated by Nguyen and Phan [26]. They used Bolotin's method to solve the Mathieu–Hill equation. Their results show that the thickness stretching effect should be carefully evaluated for moderate to thick plate structures, such as sandwich plates. Free vibration and buckling analyses of piezoelectric–piezomagnetic FG microplates in the thermal environment using MSGT were investigated by Hung et al. [27]. They derived the equilibrium equations by using Hamilton’s principle. They reported the effect of the electric voltage, power index, magnetic potential, length scale parameters, and geometrical parameters on the dimensionless frequencies and critical buckling loads of microplates. Jin [28] used a refined plate theory to examine the Interlaminar stress analysis of composite laminated plates reinforced with FG graphene particles. Tharwan et al. [29] utilized a novel refined three-variable quasi-3D shear deformation theory to study the buckling behavior of multi-directional FG curved nanobeam rested on an elastic foundation. They used a novel solution to effectively address a range of boundary conditions. Quasi 3D free vibration and buckling analysis of non-uniform thickness sandwich porous plates in a hygro-thermal environment utilizing a refined plate theory and novel finite element model were provided by Hai Van and Hong [30]. They considered the non-uniform thickness sandwich porous plates as bi-directional FGM. Their results reveal that the novel porosity patterns and the boundary conditions have a substantial impact on the mechanical behaviors of sandwich porous plates. Shahmohammadi et al. [31] extended the modified nonlocal FSDT to study buckling analysis of multilayered composite plates reinforced with FG carbon nanotube or FG graphene platelets resting on elastic foundations. A novel quasi-3D hyperbolic HSDT in association with nonlocal MSGD was considered by Ghandourah et al. [32] to analyze the bending and buckling behaviors of FG graphene-reinforced nanocomposite plates. The modified model of Halpin–Tsai and the rule of mixture were employed to compute the effective Young’s modulus, Poisson’s ratio, and mass density of FG graphene-reinforced nanocomposite plates. The inclusion of thickness stretching, nonlocal parameters, and length-scale parameters has a significant effect on the response of the GRNC plate. Hung et al. [33] employed a quasi-3D HSDT to study the bending response of FG-saturated porous nanoplate resting on an elastic foundation. According to their findings, the deflection and stresses increase by increasing the values of the nonlocal parameter. Daikh et al. [34] proposed a Quasi-3D HSDT to examine the buckling behavior of bilayer FG porous plates based on nonlocal strain gradient theory (NSGT). They developed the equilibrium equations using the virtual work principle and solved them utilizing the Galerkin method to cover various boundary conditions. Shahzad et al. [35] analyzed the size-dependent nonlinear dynamic of piezoelectric nanobeam subjected to a time-dependent mechanical uniform load. They formulated the NSGT based on a quasi-3D beam theory to take into account the size dependency.
Sandwich structures are one of the most advanced and modern structures that are utilized for strengthening based on the materials used in their construction. Fixed and mobile refrigerated warehouses, metal industries, spatial structures, and industrial and semi-industrial cold stores are examples of the use of sandwich structures in different industries. Piezoelectric materials have been widely employed as sensors and actuators in microelectromechanical systems (MEMS) and nanoelectromechanical systems (NEMS). In addition, various piezoelectric materials have been considered for applications in energy harvesting, biomedical engineering, and additive manufacturing. Therefore, due to the existence of piezoelectric layers, piezoelectric nano-sandwiches have many applications in the medical industries including drug delivery, cartilage, nerve, skin, tendon, and muscle regeneration as well as military industries.
Many researchers have investigated the behavior of structures over the past centuries. Today, based on the growth and development of industries and the increasing progress of technology, it is necessary to achieve accurate and reliable results using new models and methods. The use of piezoelectric face sheets as sensors and actuators as well as protection and prevention of damage to FGMs is of great importance considering the cost of construction and the production process of these materials. According to the comprehensive literature survey and the best of the authors’ knowledge, there has been no attempt concerning the study of the bi-axial buckling analysis of FG nano-plate covered with piezoelectric face-sheets by considering elastic foundations. Motivated by these considerations, the current paper is the first attempt to present the exact solution for size-dependent quasi-3D buckling analysis of a three-layer FG nanoplate integrated with piezoelectric layers supported by orthotropic Pasternak medium subjected to electric field and in-plane forces. The surface effect responsible for size-dependent characteristics can become distinctly important for piezoelectric nanomaterials in which large surface-to-volume ratio. Also, understanding the buckling behavior of sandwich nano-systems could be a key point for the application in electromechanical resonators, hence, for the first time surface effect, neutral surface position of FGMs and thickness stretching effects together with NSGT are applied to sandwich piezoelectric nanoplate. Eventually, one of the innovations of the presented research is the presentation of comparative results in different models for the critical buckling load of the nano-plate.
2 THEORETICAL FORMULATIONS
2.1 Basic Assumptions
The following assumptions based on the equations of the quasi-three-dimensional theory are presented in this paper to accurately simulate the behavior of the desired sandwich structure with a close approximation of the actual material properties. This theory reliably approximates the actual behavior of thick plates in the thickness direction, considering that 
is significantly smaller compared to 
and 
, except at the edges of the structure. The assumptions underlying the current theory and the used materials are outlined as follows:
· It is assumed that there are no slip conditions between the core and the face sheets, ensuring complete continuity and integration between all layers.
· The FGM is modeled as a linear elastic material in the pre-yield condition.
· The origin of the Cartesian coordinate system is placed at the neutral surface of the FG plate.
· The displacements are relatively small compared to the plate thickness, leading to infinitesimal strains.
· The displacements 
in the 
-direction and 
in the 
-direction are composed of extension, bending, and shear components.
· The transverse displacement 
comprises three parts: bending (
) and shear (
) and normal stress (
) (stretching effect). These components are solely functions of the coordinates 
and time (
).
2.2 Theoretical Formulations
Consider a rectangular sandwich nanoplate with FGM core and piezoelectric face-sheets at the top and bottom of the core via length a, width b, and total thickness h according to Fig. 1. The thickness of the core and bonded layers are 
and 
, respectively. The displacement field of the current formulation is obtained based on the above assumptions. In this research, a higher-order quasi-3D theory according to the four-variable plate theory is developed by considering the thickness stretching parameter. Based on the given assumptions, the considered displacement field is capable of satisfying the transverse shear stresses related to shear strains at the uppermost and lowermost surfaces of the sandwich structure. Hence, the displacement field at any point of the three-layered nanoplate can be expressed as below [35,36];
| (1) |
| (2) |
| (3) |
The bending components 
and 
are considered analogous to the displacements described by classical plate theory. Hence, the expressions for and can be formulated as follows [25,37]:
| (4) |
|
Fig. 1 Geometry of three-layered FG nano-plate integrated with piezoelectric face-sheets. |
The shear components 
and 
together with 
create parabolic variations in shear strains 
and 
. This, in turn, affects the shear stresses 
and 
throughout the plate's thickness, ensuring that and are zero at the top and bottom faces of the plate. As a result, the expressions for and can be defined as follows [26,37,38]:
| (5) |
On the other hand, 
and 
represent the displacements in the 
and 
directions at a corresponding point on the reference surface and also 
and 
are the bending and shear components of the transverse displacement, respectively.
represents a shape function that estimates the distribution of shear stress across the plate's thickness, eliminating the need for any shear correction factor. This shape function can originate from different types of functions, including trigonometric, polynomial, and hyperbolic forms. A polynomial function is selected in this study, similar to the utilized methodology in the hybrid-type quasi-3D shear deformation theory. The shape functions 
and 
can be expressed as [26,39];
| (6) |
| (7) |
The linear strain-displacement relations based on the quasi-3D displacement field can be written as:
| (8) |
| (9) |
2.3 Piezoelectric Materials
Piezoelectric materials have the unique capacity to regulate electrical phases. These materials can produce various physical and chemical responses to electrical phase changes, such as pressure or electric fields. Their key features include high precision and sensitivity, stability and durability, and rapid response times. These distinctive capabilities play an essential role in advancing modern technologies. With the advancement of nanotechnology and composite materials, the applications of piezoelectric materials are expected to expand into new domains, driving the development of more intelligent devices and systems. The constitutive piezo-elasticity relations for a piezoelectric material based on the continuum mechanics approach can be formulated using stress and strain components, electric displacement relations, and the field strength matrix as follows [40]:
| (10) |
| (11) |
in which 
are stiffness matrix components, 
are piezo-electric coefficients, and 
are dielectric permeability coefficients. 
represent electric displacements and 
are electric field components. The electric fields must be chosen to satisfy Maxwell’s relation and can be represented as follows [40-41]:
| (12) |
| (13) |
where 
represents the externally applied electric voltage between the top and bottom of the piezoelectric layers. Additionally, 
denotes the spatial variation of the electric potential in two-dimensional directions. Consequently, the electric components in the three spatial directions are as follow [38,41]:
| (14) |
| (15) |
| (16) |
The transformed coordinates for the upper and lower layers, relative to the mid-plane of the piezoelectric face-sheets can be expressed as follows.
| (17) |
| (18) |
2.4 FGM Properties
FGMs are smart materials typically composed of ceramic and metal and the effective properties of these materials continuously vary along the thickness direction according to the specific relationship. In this paper, the material properties of the FG plate change based on the power-law distribution. Hence, the effective non-homogeneous properties of FG nanoplate utilizing the rule of mixture can be expressed by [40-44]:
| (19) |
In this Eq., subscripts 
and 
represent the properties of ceramic and metallic materials, respectively. Additionally, the parameter 
indicates the gradient index distribution of properties along the thickness direction of the plate. Due to the asymmetric distribution of properties along the functional grading of these materials, the position of the neutral surface does not coincide with the mid-plane in the graded direction. The inherent asymmetry in material properties of FG plates relative to the mid-plane causes the stretching and bending equations to be coupled. Thus, by appropriately selecting the origin of the coordinates in the direction of property variations, the coupling between stretching and bending can be neglected. The symmetry of properties along the direction of variation significantly simplifies the analysis of these materials. Hence, two different planes 
and 
are considered for the measurement of z from the middle surface and the neutral surface of the plate, respectively, to specify the position of the neutral surface of FG plates, as shown in Fig. 2.
| (20) |
| (21) |
| (22) |
Notably, the material properties at the top and bottom surfaces of the FG plate are pure ceramic and pure metal, respectively. 
are Young’s modulus and mass density of FG nanoplate that are assumed to vary according to a power law distribution in terms of the volume fractions of the constituents.
|
Fig. 2 Neutral surface position of FG nano-plate. |
The distance of the neutral surface from the mid-plane is represented by 
in Eq. (2) and this distance can be defined as follows [42]:
| (23) |
The linear constitutive relations of a FG plate can be written as [40]:
| (24) |
Using the material properties defined in Eq. (3), stiffness coefficients, 
can be expressed as [40]:
| (25) |
2.5 Non-local Strain Gradient Theory
In this paper, the NSGT is employed to analyze the size-dependent behavior of sandwich nanoplate. A new non-conventional continuum theory of elasticity called NSGT is introduced to simultaneously consider the opposite characteristics of small-scale effects. In this theory, the coupled physical influences of nonlocal and strain gradient size effects are simultaneously considered. By ignoring the effect of body force, the general constitutive relations with the framework of NSGT are written as [45-51]:
| (26) |
| (27) |
| (28) |
in which 
represents the internal material length scale parameter, 
denotes the nonlocal parameter, and 
is the Laplacian operator.
2.6 Surface Piezoelectricity Effects
In nano-scale structures, the energy fraction stored in the surface layers is considerably higher than that in the bulk material, and surface or near-surface atoms are typically subject to a range of environmental influences distinct from those affecting bulk atoms. Indeed, a key characteristic of nanostructures is their high surface-to-volume ratio. Consequently, surface elasticity theory is combined with non-classical continuum theory to examine these significant effects. The primary equations for stresses and electric displacements on the surface of piezoelectric materials based on NSGT can be formulated as [49-51]:
| (29) |
| (30) |
where 
and 
are the surface elastic constants, surface piezoelectric constants, surface dielectric constants, and the residual surface stress tensor. Also, 
.
2.6 Equations of motion
The equilibrium governing equations for the nonlocal sandwich FG nanoplate are formulated using Hamilton’s principle. According to the Hamilton’s principle, one can get that [52-56];
| (31) |
where 
and 
are the first variation of strain energy and work done by external forces leading foundation and applied electrical field, respectively. The total potential energy of sandwich nanoplate with FG core and piezoelectric face sheets including both the bulk part and two surface layers can be expressed as [54]:
| (32) |
where 
and 
relates to upper surface layers and lower surface layers, respectively.
| (33) |
The converted coordinate based on the neutral axis is 
. The above equation, 
and 
implies the force, moment and transverse shear stress resultants that can be described as:
| (34) |
| (35) |
| (36) |
| (37) |
| (38) |
The external works can be extracted into two parts. One part is an orthotropic Pasternak medium and the other part is a two-dimensional electric field applied to the piezoelectric face sheets [53, 57-60].
| (39) |
In contrast to other models, this foundation can simulate both normal and shear loads in any given direction. The applied force induced by the orthotropic Pasternak foundation can be defined as [57,58]:
| (40) |
in which 
describes linear spring coefficient and shear layers in two arbitrary directions, respectively. Also, the angle 
specifies the orientation of the local -direction of the orthotropic foundation with respect to the global -axis of the system. The applied forces from the electric field can be calculated as [45,51,60-61]:
|
| ||
[1] Corresponding author. Tel.: +98 31 55913434; Fax: +98 31 55913444.
E-mail address: aghorabn@kashanu.ac.ir (A. Ghorbanpour Arani)