Computational Linear and Nonlinear Free Vibration Analyses of Micro/Nanoscale Composite Plate-Type Structures With/Without Considering Size Dependency Effect: A Comprehensive Review

Abstract

Recently, the mechanical performance of various mechanical, electrical, and civil structures, including static and dynamic analysis, has been widely studied. Due to the neuroma's advanced technology in various engineering fields and applications, developing small-size structures has become highly demanded for several structural geometries. One of the most important is the nano/micro-plate structure. However, the essential nature of highly lightweight material with extraordinary mechanical, electrical, physical, and material characterizations makes researchers more interested in developing composite/laminated-composite-plate structures. To comprehend the dynamical behavior, precisely the linear/nonlinear-free vibrational responses, and to represent the enhancement of several parameters such as nonlocal, geometry, boundary condition parameters, etc., on the free vibrational performance at nano/micro scale size, it is revealed that to employ all various parameters into various mathematical equations and to solve the defined governing equations by analytical, numerical, high order, and mixed solutions. Thus, the presented literature review is considered the first work focused on investigating the linear/nonlinear free vibrational behavior of plates on a small scale and the impact of various parameters on both dimensional/dimensionless natural/fundamental frequency and Eigen-value. The literature is classified based on solution type and with/without considering the size dependency effect. As a key finding, most research in the literature implemented analytical or numerical solutions. The drawback of classical plate theory can be overcome by utilizing and developing the elasticity theories. The nonlocality, weight fraction of porosity, or the reinforcements, and its distribution type of elastic foundation significantly influence the frequencies.

1 Introduction

1.1 Linear and Nonlinear Definition and Theories

In the definition of linear motion, at a thickness assumed to be huge compared to the deflections, the middle surface's strains can be neglected, and this assumption is only valid in theories, while in practice, the deflection has the thickness magnitude [1]. Defining a system as linear or nonlinear is based on the essential components that generate the overall schematic. One component acting non-linearly means that the total vibrational system is NL. For example, having a nonlinear (NL) stiffness in a system will lead to a non-sinusoidal displacement even if a sinusoidal harmonic load has been applied. However, a system with only linear components is specified as a linear system. Whether the nano-plate (NP)/ micro-scale plate (MSP) structure is made of composite or functionally graded material (FGM), they are extensively utilized in various developed industrial fields. The Von-Karman theory (VKT) has been developed to study the significance of the NL effect in aerodynamic structures and investigate huge deflections in thin structures. The free vibration (FV) analysis of an NP with a single Graphene (Gr) sheet layer has been accomplished using the generalized differential quadrature Method (GDQM) combined with the first order shear deformation theory (FSDT) [2]. The FV performance of the Mindlin NPs has been investigated by implementing the refined Levy-Solution (LS) [3]. Nevertheless, the FSDT has been utilized to examine the FV of orthotropic NPs [4]. Furthermore, the thermal vibration of sandwich NPs made of functionally graded (FG) material has been studied by using the Navier’s solution (NvS) [5]. Shen et al. [6] presented various NL mechanical analyses, such as; NL vibration of FGM shells and plate structure, which were performed by using various methodologies such as; Finite element method (FEM), Galerkin, classical plate theory (CPT), etc. Van et al. [7] investigated the NL geometrical analysis of functionally graded plates (FGPs) through FEM by utilizing the CS-MIN3 elements, which are based on the C⁰- higher-order shear deformation theory (HSDT). Furthermore, Karakoti et al. [8] performed an NL transient analysis of functionally graded porous plates (FGPPs) and functionally graded porous (FGP) shell panels subjected to a blast loading in a TE. In addition, Sahmani and Aghdam [9] explored the NL vibrational responses of pre/post-buckled lipid supramolecular MNS tubules by nonlocal strain gradient (NSGT), while the analytical solution was performed by applying the improved perturbation approach combined with Galerkin methodology. Meanwhile, Zamani [10] investigated the NL vibrational behavior of piezoelectric (PE) Gr-reinforced composite laminated panels subjected to TEs. Moreover, Kee et al. [11] investigated the NL FV behavior of composite rotor blades to develop the aerodynamics characterizations. Van et al. [12] employed the Isogeometric analysis (IGA) to perform an NL transient analysis for porous FGP under a hygro-thermomechanical load. Additionally, Song et al. [13] investigated the NL FV of cracked functional graded graphene platelets reinforced nanocomposite (FG-GPLRC) nanobeam (NB) subjected to a thermal environment (TE). Furthermore, for examining the NL FV of the Gr sheet in a TE Shen et al. [14] applied the nonlocal (NLC) plated model by utilizing the CPT combined with VKT of kinematic NL. In a further study presented by the same author, Shen et al. [15] the same method was employed to determine the NL FV of a thin film plate based on elastic foundation (EF) in a TE. However, the nonlocal elasticity theory (NLET) combined with the DQ method, was used to examine the vibration of orthotropic single-layer graphene sheet (SLGS) [16]. In addition, the same model has been subjected to a pasternak foundation (PF) and tested by combining the NLET with the DQ method by Pradhan et al. [17]. Meanwhile, Thang et al. [18] utilized the NLSGT and presented an NvS to examine the FV of an FG (SUS403-Si₃N)-Carbon nanotube (CNT) reinforced composite NP. The NL vibrational behavior of a bistable asymmetric composite cross-ply laminated cantilever shell subjected to external excitation has been investigated by Ren et al. [19]. However, multi beams based on NL EF made of multiscale composite were considered by Li et al. [20] to investigating the NL vibrational analysis. In addition, Liu et al. [21] investigated the NL vibrational behavior of curved pipes made of FG-CNTRC. Karličić et al. [22] designed and analyzed an NL model for a SWCNT in an NB shape using the nonlocal continuum theory (NLCT). Nevertheless, Li et al. [23] investigated the NL FV of a unique sandwich cylindrical shell structure with a composite hexagon honeycomb and reinforced with functional graded fiber-reinforced composite (FG-FRC) by implementing various methodologies such as Hamilton’s principle (HP), HT, Galerkin, VKT, and the modified Gibson Formula. Karimiasl and Alibeigloo [24] presented an NL FV and force vibration analysis of sandwich cylindrical panels with a sandwich structure consisting of an auxetic core and two face sheets of GPLRC. Yang et al. [25] employed the FSDT to examine the NL vibrational behavior of FG-GPLRC truncated conical shell. Furthermore, Ma [26] developed a simplified form of Hamiltonian based frequency-amplitude for NL vibrational analysis.

From the literature, several authors were interested in presenting review studies concerning the mechanical analyses of small-scale structures. In this frame work, Nuhu and Safaei [27] accomplished a comprehensive review to investigate the current nonlocal continuum elasticity theories (NCET), which was implemented for stability analyses for nano/microplates. In a further comprehensive review published by the same authors, Nuhu and Safaei [28] discussed the vibrational analysis of nano/micro-scale (NMS) structure by utilizing the NCET. Moreover, the same authors published a further review study, Nuhu and Safaei [29] to investigate the NP/ microplate (MP) structure explored through implementing the NCET. Meanwhile, Yee and Ghayesh [30] presented a comprehensive review that focused on linear and NL solid mechanics analysis (theoretically and arithmetically) of GLP-reinforced composite plate, shell, and beam structures. Besides the FGM, porous material and nano/microstructure were addressed. Kong [31] presented a review that focused on the size-dependent model of MP/beam based on modified couple stress theory (MCST). Kanu et al. [32] performed a review study that covered various mechanical analyses of FGM, including innovative structures, such as vibrational, stability, and fracture issues. Besides, Jha et al. [33] performed a review study investigating FGPs and their FV, stability, and stress analyses. Wu and Yu [34] presented a review that covers the implementations of Eringen's NCET for examining various mechanical analyses of NB with rectangular shapes, single walled carbon nanotube (SWCNT), double walled carbon nanotube (DWCNT), walled carbon nanotube (MWCNT). Furthermore, Farzad et al. [35] presented a comprehensive review investigating the nanoscale structures that display PE performance and its various material properties. Meanwhile, Farajpour et al. [36] accomplished a comprehensive review study to investigate the utilization of the NCET and the NLSGET in analyzing various mechanical behaviors for numerous nanoscale (NS) structures. Additionally, İnada et al. [37] investigated the effectiveness of nanomaterial in solar energy packing systems in a novel review study. However, the applications of finite element analysis (FEA) in designing and analyzing various honeycomb sandwich structures were reviewed by Onyibo and Safaei [38]. The material applicability, physics, and arithmetic of the NLCT mechanics were reviewed comprehensively by Shaat et al. [39]. While, the NMS shallow arch structures made of multilayer of FG designed and analyzed computationally were reviewed by Hurdoganoglu et al. [40]. Besides, the static and dynamic performances were highlighted. Besides, a variety of NC structures reinforced with nano FGM and CNT which have been analyzed through various approaches were reviewed by Garg [41]. Al Mahmoud et al. [42] presented a comprehensive review highlighting the applications of various mesh-free (MF) methodologies in static and dynamic analysis for various structures in NMS as well as macroscale.

Where there are numerous methods and techniques to investigate the mechanical response of NMS structure and parameters to be considered, this review focused on covering the studies in the literature that examine the vibrational response, specifically the linear/NL FV response of the NMS CPs and laminated composite plates (LCPs) structures. The developed mathematical models, generated by various methodologies and steps, are solved by analytical, numerical, high-order, and mixed solutions. The SDE, which is classified as one of the most prominent parameters that affect dynamic behavior, especially in studying the small scale effect (SSE), is further considered and discussed. Based on the literature, researchers are seriously interested in developing effective numerical methods and exact models to examine the FV of NPs/ microplates (MPs). Based on the literature, the parameters that directly effects on the FV behavior are classified as illustrated in Fig. 1

Fig. 1

The classification of parameters effect on the FV behavior

1.2 Literature Approach

The main database that has been used as in obtaining the articles in this literature review is Scopus database. The steps of selecting the articles to be studied are as followed:

• 1. As the 1st search cord ("free vibration")) AND ((("linear" OR "nonlinear")) AND ("plate" OR "mirco-plate" OR "nano-plate" OR "microplate" OR "NP")) AND ("composite" OR "composite plate" OR "composite structure") as a search in the search cord. 511 documents emerged. Not all of the 511 documents are close to the scope of this review study; so several keywords were excluded. Due to the fact that Scopus doesn’t exclude the other types of structures and their analysis such as shear deformation, buckling, shear stress, shell, beams and girders, strain, structural panels, cylinders, bending laminates, shell panels, shear strain, nanotube and cylindrical shell from these search cord. The total, documents found by Scopus after exclude the mentioned key words is 237, though again some documents were not relevant.

• 2. For advance search, further documents were collected from additional databases such as; Google Scholar, Science Direct and Web of Science. To produce the theoretical background of the research; where the supportive articles commonly assistance in developing the contexts of the review study.

Figure 2, shows Scopus statistics of linear/NL free vibration analysis for CP structure in NP/ MSP (2010-until now).

Fig. 2

Scopus statistics of linear/NL free vibration analysis for CP structure in NP/ MSP (2010-until now)

1.3 Various Composite Nano/Microplates

The implementation of composite materials has increased in several industrial and engineering applications [43]. Besides, to accomplish structure stability under various types of load such as electrical, magnetic, thermal as well as mechanical several numerical and analytical methodologies are developed. However, one of the most important factors to consider in composite materials are normal stresses and the through-thickness distributions of transverse [44]. Furthermore, by implementing the sandwich structures the optimum material strength and modulus of the face sheet can be used, with considering the benefits of low core density to increase the thickness of component and influence the bending stiffness of the structure [45]. Certainly, with the vast progress in engineering technologies and science applications, developing novel structures in NMS has become more significant. NMS structures usually have advanced properties, specifically their mechanical and electrical characterizations and physical properties, which makes their application widely spread. Moreover, as a common example of NMS; micro-electro-mechanical systems (MEMS), actuators, energy harvesting, and biomedical engineering applications. Besides, composite nano/microstructures based on EF are widely used in engineering under various operating conditions [46]. However, Arash et al. [47] presented a comprehensive review that addressed various nano-mechanical resonators and their uses in molecular transportation as well as sensors. Moreover, one of the most favorable composite structures is the sandwich structure. Certainly, lightweights sandwich structures are currently in considerable demand in various industries [48]. Meanwhile, LCPs have a great specific stiffness and strength, making their applications expansive [49]. Besides, small-scale analysis of lightweight sandwich forms may improve structural mechanical properties, which seems essential for several industrial improvements. In this context, the first nanostructure analysis studied the mechanical behavior was accomplished by utilizing the strain gradient interpretation (SGI) of size dependency effect (SDE) [50], (class gradient method) CGM [51], and the Gradient approach relation to Eringen’s nonlocal theory (GAENLT) [52] and the obtained results of the nano/microstructure performance were investigated by utilizing the nonlocal theory (NLT) [53]. Recently, a review that focused on the enhancement of CNT as well as porosity on the vibrational performance of various NC structures has been published by Alibar et al. [54]. Besides, the NMS structures’ size-dependent continuum mechanics were reviewed comprehensively and classified into linear and NL by Roudbari et al. [55]. Meanwhile, Liu et al. [56] presented a comprehensive review study which explore the various thin walled deployable composite assemblies, with highlighting its design, theories as well as the applications. However, the energy harvesting assemblies consisting of PE and auxetic materials were extensively reviewed by Tabak et al. [57]. Likewise, Barbaros et al. [58] performed a comprehensive review that presented various applications, fabrications, and mechanical characterizations of FGP NC material.

Moreover, Bonthu et al. [59] performed FV and stability analyses of 3D-printed FG sandwich foams by utilizing numerical and experimental methods. In addition, Dastjerdi et al. [60] developed a new Active multidisciplinary sandwich plate (AMSP), made of two active PE faces and a core of advanced porous embedded with CNT. In a further work, Dastjerdi et al. [61] presented a novel smart multidisciplinary plate (SMP) made of NC porous plate with piezoceramics and the core in between made of simultaneous FG-CNT. Yang et al. [62] designed MSPs, made FG composites with variable thickness and employed the IGA to investigate the linear and NL flexural responses. In a novel study proposed by Chen et al. [63] a porous MSP consisting of FGM with various shapes of central cutout was tested to explore the size-dependent geometrical NL deflection performance through IGA. The same structure in the nanoscale has been investigated in a further study presented by Fan et al. [64] to examine the thermal post-buckling by IGA. Furthermore, Safaei et al. [65] proposed a novel NC sandwich plate structure made of an isotropic polymer material with two outer layers reinforced by FG-CNT agglomerations and investigated its forced vibration. Besides, in a further study Safaei et al. [66] employed an MF method to explore the thermoelastic performance of a sandwich plate structure consisting of a core made of porous polymer and outer layers made of CNT clusters/polymer NC layers. In addition, Safaei et al. [67] employed the FSDT combined with Mori–Tanaka (MT) to investigate the critical stability force and temperature for porous plates with a sandwich structure with outer face sheets made of FG-CNT-reinforced polymeric NC. Nevertheless, Feng et al. [68] designed and processed a sandwich of CPs for natural inspiration for vibrational destruction, and various dynamics and vibrational tests were performed. Meanwhile, Sahmani et al. [69] developed a model consist of a novel bio-NC plate with a sandwich structure consist of two facesheet made of GN-IBO thin film and a core made of n-HA-ZnO bulk with accomplishing an analytical analysis based on the shear deformation theory (SDT). Moreover, Ebrahimi and Barati [70] explored the buckling analysis of flexoelectric NPs embedded, as presented in Fig. 2 (c) in which surface enhancement has been considered. While, Fan et al. [71] studied the shear stability properties of FGM skew NPs. Rao et al. [72] designed and investigated a composite MSP made of porous FG, which had a central cutout by IGA and was molded through the coupled stress theory (CST) quasi-3D plate. Besides, Sahmani et al. [73] explored the geometrical NL bending performance of a sector and elliptical NPs made of FG composite. Additionally, Safaei et al. [74] employed the Galerkin FEM as well as unit cell boundary conditions (BCs) based on previously published models to develop an arithmetical model of platelet-reinforced composites. Meanwhile, Ma et al. [75] examined MSPs made of FGM with various thicknesses by implementing the NLSGET. Using MF methodology. In addition, Safaei et al. [76] explored the improvement of CNT clusters on the mechanical characterizations mainly the thermoelastic stress wave propagation in NC plates with sandwich structure by applying the MF methodology. Nevertheless, Karimi and Farajpour [77] examined the static behavior of NPs made of BiTiO₃-CoFe₂O₄ shown in Fig. 3 (a). While, Nikrad et al. [78] investigated the development of Gr reinforcement dispersal on the energy release rate and the vibration of CPs subjected to a pre/post-buckled. However, a multifunctional sandwich foam plates cohesive by composite faces were considered and studied by Safaei [79] to explore the frequency reliant on damped vibrations. Besides, Karimi and Shahidi [80] investigated the asynchronous bending/stability analyses of skew double layered magneto-electro-thermo-elastic (METE) NPs shown in Fig. 3 (b). Also, Sengar et al. [81] employed the IGA to investigate the post-buckled vibrational performance of skew sandwich plates (SSPs) with a core generated from metal foam and subjected to random edge compressive loads. Furthermore, Sari et al. [82] examined the stability behavior of FGM NPs subjected to thermo-mechanical loading, as illustrated in Fig. 3 (d). Else, Mousavi et al. [83] presented a novel methodology (fully modified NLC) to examine the static analyses of NPs. Moreover, Ebrahimi and Barati [84] examined the NLC thermal stability of embedded METE nonhomogeneous NPs, shown in Fig. 3 (e). Besides, Karimi and Shahidi [85] considered the METE NPs shown in Fig. 3 (f) to examine the stability behavior by utilizing the Galerkin methodology and focusing on the surface energy layers. In addition, Hughes et al. [86] presented an IGA analysis to investigate various thin-shape structures and define their filed variables. Based on NSGT and MCST which focused on enhancing the surface layers’ variations rate. However, the asymmetric/axisymmetric stability of a FGM circular/annual NPs shown in Fig. 3 (g) through the implementation of the Mindlin NCET by Bedroud et al. [87].

Fig. 3

Various NPs schematics a BiTiO₃-CoFe₂O₄ NPs subjected to thermal & in-plane loads [77], b skew double layered METE NPs [80], c flexoelectric NPs with EF [70], d FGM NPs subjected to linearly varying biaxial thermos-mechanical load [82], e embedded FGM NPs subjected to ME field [84], f METE NPs with a surface layers [85], & g FGM annular NP subjected to a uniform compressive load [87]

Assadi and Najaf [88] explored the NL bending of a single crystalline circular NPs with a cubic material anisotropy. Meanwhile, the MSCT Reddy for rectangular plates has been utilized to investigate the bending as well as biaxial buckling of double-coupled polymeric NC plates made of FG-SWCNTs and FG-SWN-BNNTs as illustrated Fig. 4 (a) by Mohammadimehr et al. [89]. Likewise, Chebakov et al. [90] employed the NLC asymptotic approach to explore thin plates, while the exponential NLC Kernel across the thickness was assumed to vary along the length. In addition, Mazari et al. [91] investigated the bending behavior of FGM NPs subjected to fully simply supported (SSSS) BCs analytically. The 3D hyperbolic shear deformation theory (HBSDT) has been utilized to define the displacement field model, while the virtual work principle has been implemented to drive the QE. Besides, Dastjerdi et al. [92] employed the NLET to explore the enhancement of vacant defects in Gr nanosheets based on EF, as shown in Fig. 4 (b), on the bending behavior. Furthermore, Aizikovich et al. [93] presented a solution to the interaction problem between an asymmetrically loaded thin FG circular NP and a support system made of EF by reducing the system into a DE of plates’ bending and dual integral formula for unknown pressure in regular contact. Additionally, Arani and Zamani [94] examined the electromechanical bending performance of sandwich NP consisting of two PE face sheets and a core made of FGP and based on silica Aerogel basics, as illustrated in Fig. 4 (c). On the other hand, Repka et al. [95] performed a numerical analysis using the moving finite element (FE) to examine the SDE on the bending behavior in nano/micro-plates (NMPs). However, Salehipour et al. [96] presented an analytical solution that focused on exploring the bending behavior of FG NMPs based on EF shown in Fig. 4 (d), through implementing MCST. Meanwhile, Kananipour et al. [97] utilized the differential quadrature method (DQM) to explore the bending and stability analysis of NPs, as shown in Fig. 4 (e), found on NLC Kirchhoff plate theory (KPT) as well as Mindlin plate theory (MPT). Nevertheless, Arefi et al. [98] employed the nonlocal elasticity and strain gradient theory (NLESGT) to examine a sandwich NP consisting of a porous core and two PM Face sheets, as shown in Fig. 5 (c). Moreover, The axisymmetric bending of an annular single-layered Gr sheet based on EF has been considered and investigated by Ahmadi and Ghassemi [99] to analyze the enhancement of EF parameters as well as the SSEs. While, Arefi and Zenkour [100] explored the thermos-electro-magneto-mechanical deflection performance of NPs with a sandwich PM structure, as shown in Fig. 4 (f). Besides, Ghobadi et al. [101] investigated the SDE thermo-electro-magnetic mechanical NL bending analysis of a flexoelectric NP subjected to a magnetic field, as shown in Fig. 5 (d). Wang and Nie [102] explored the FV of variable angle tow (VAT) composite reinforced laminated annular sector plate by implementing the DQM. Carvalho et al. [103] examined the enhancement of uncertain geometrical as well as material characterizations on various responses of the composite fiber-reinforced plate shown in Fig. 6. Furthermore, the FV of a cracked innovative plate made of MEE FGM based on PF has been explored by Pham et al. [104] by utilizing the HSDT.

Fig. 4

Schematic of various plates a double-coupled PE polymeric NC plates reinforced by FG-SWCNTs under in-plane loading and applied voltage [89], b Gr nanosheet with eccentric defects & based on Winkler-Pasternak foundation (WPF) [92], c sandwich NP consist of two PE face sheets and a core made of FGP and based on silica Aerogel basis [94], d FG NMPs based on EF [96], e double layered Gr sheet [97], f Sandwich NP founded on PF [100]

Fig. 5

a The single-layered annular Gr sheet in continuum model, b the Single-layered annular Gr sheet fixed on EF [99], c Sandwich NP integrated with two PM fache-sheet [98], d flexoelectric NP [101]

Fig. 6

The schematic structure of a composite plate consists of three-layered & reinforced with fibers [103]

Moreover, Kim et al. [105] developed a composite structure consist of PDMS reinforced with a hybrid filler and investigates the electromechanical behavior. Additionally, Rama et al. [106] developed a linear triangular shell element based on the FSDT and the equivalent single-layer method. Though, the discrete shear gap has been used to resolve the shear locking effects. However, the developed model can be used in analyzing the NL/linear geometric of LCPs and other thin walled structures. Furthermore, Sahmani et al. [107] explored the NL axial instability of FGPPs reinforced with Graphene nanoplatelet (GNP) by using NSGT and the virtual work’s principle was used to drive the NCL GE. The obtained results revealed that by increasing the porosity coefficient values at various types of porosity dispersal the size dependent critical stability load. However, Ansari et al. [108] explored the influence of surface stress on the pull-in instability of a circular NPs with developing a NCL model. Besides the pull-in hydrostatic and pull-in voltage pressure for the structure subjected to arbitrary edge BCs were predicted. Notably, this study consider as a benchmark for exploring the mechanical properties of electrostatical actuators. In addition, Sahmani et al. [109] presented a novel calibrated NLC anisotropic plate scheme which used in examining the size-dependent NL uniaxial instability of the 3D nanosheet made of metallic carbon. Although, the interlocking was considered and examined. Nonetheless, Sahmani et al. [110] developed a circular NPs model based on the HSDT with considering the impact of surface energy to examine the postbuckling, where the Gurtin–Murdoch (GM) elasticity approach was considered to define the impact of surface energy. Furthermore, Sahmani and Fattahi [111] presented a developed sufficient calibrated NLC plate model for analyzing the NL axial instability of a nanosheets made of zirconia by utilizing the molecular dynamic simulation (MDS)m with considering the SDE. However, Marinkovic and Zehn [112] presented a FEA for various smart structures with active laminated composite (LC) by using shell element found by Reissner–Mindlin kinematic approach for designing moderately thick and thin structures. Where, the smart active plate was consist of two outer face sheets of hybrid LC (PZT G1195 piezoceramic) and a core made of Gr//Ep (T300/976). The obtained results validated and compared with previous results obtained by Ritz approximation. Additionally, Milic and Marinkovic [113] accomplished modified version of IGA based on NURBS shape function to over the C0-continuity at the element boundaries. The modified algorithm was test and the analysis was accomplished and examined a plate with hole. Moreover, Kiarasi et al. [114] explored the stability performance of FGPPs under various types of loading conditions with considering three various porosity dispersal patterns through thickness. Which are uniform dispersal, NL symmetric and NL asymmetric. To perform the analysis a novel method was developed which based on GDQM and FEM and known as FE-GDQM. Nonetheless, the static analysis of a moderately thick FGPPs with considering the NLC elasticity and subjected in a hygro-TE was examined semi-analytically by implementing the polynomial methodology by Dastjerdi et al. [115]. However, Ipek et all [116] examined the stability analysis of FG-CNT moderately thick plate under in-plae biaxial compression loads subjected in elastic TE in the context of FSDT.

1.4 Free Vibration Analysis for Various Structures

The FV analysis considered as a dynamic analysis. It has recently been applied extensively for investigating various mechanical and civil structures in NMS and macroscales by employing diverse numerical, analytical, and other methods. The steps of FV analysis processes are shown in Fig. 7. The main categorization of applied methods and approaches in linear/NL FV analysis are presented in Fig. 8.

Fig. 7

Steps of FV analysis processe

Fig. 8

The main categorization of applied methods and approaches in linear/NL FV analysis

1.4.1 Shell and Cylindrical Structures

In this context, Jin et al. [117] developed an Fourier series (FS) solution with RR for analyzing the FV of conical shells with a truncated structural shape. Besides, Tornabene et al. [118] applied the radial basis function (RBF) MF methodology to examine the FV of doubly-curved (DC) laminated composite shells and panels. In addition, Tornabene [119] utilized the FSDT as well as the GDQM to inspect the FV of a thick-conical-cylindrical shell made of FGM with annular plates (APs). Alibeigloo et al. [120] employed the elasticity theory to explore the FV of a PE layer embedded with a composite cylindrical panel and reinforced with an FG-CNT. Likewise, Badarloo and Salehipour [121] accomplished an analytical solution (closed-form) for FV as well as a buckling simulation of curvy sandwich panels made of NC reinforced with (GPL and CNT) face sheets and a core made of porous metal foam. Meanwhile, Cho [122] explored the NL FV behavior of FG GPLRC conical panels laid on EF, through the 2D planar MF method, the od-based NL arithmetical approach. Furthermore, Mouthanna et al. [123] performed arithmetical, experimental as well as analytical examinations to explore the FV of a single-phase FGP cylindrical shell structure with sandwich geometry. Karimiasl and Alibeigloo [124] employed the homotopy method (HM) to study the NL vibration of a cylindrical panel with a sandwich structure made of FGM and an auxetic core. However, Zhu et al. [125] studied the FV performance of orthotropic cylindrical NSs. The enhancement of viscosity on the FV performance of a thin cylindrical NS has been examined by Sharaf et al. [126]. Nonetheless, The enhancement of thermomechanical loading as well as porosity on the FV and NL dynamical behaviors of the FG shells with a sandwich structure and double curvature by Trinh et al. [127]. She et al. [128] investigated the main NL resonance of the initially stressed DC shells made of GLPs reinforced metal foams and had geometric imperfections. The dynamic buckling and microtubes structures made of FG- Graphene platelets (GPLs) composites by Lu et al. [129]. Moreover, Le et al. [130] utilized the IGA to perform a 3D solution for FV and stability for various structures, such as conical and annular plate cylinders and cylindrical shells made of FGP cellular materials. Chen et al. [131] reexamined the linear and NL FV of a porous cylindrical shell sandwich structure reinforced with GPLs. Li et al. [132] explored the NL vibrational performance of cylindrical shells made of fiber reinforcement polymer (FRP) and based on EF subjected to temperature gradient conditions. Besides, the NL, vibration, and NL static bending of a porous sandwich panel GPLRC have been investigated by Shen and Li [133]. A fluid-conveying pipe made of FG-CNT subjected in a TE has been considered and investigated by Chen et al. [134]. However, the FV and static analyses of an LC cylindrical shell are accomplished by implementing the zig-zag theory (ZZT) and Kirchhoff quadrilateral element by Dagade and Kulkarni [135]. Besides, Kulkarni and Walunjkar [136] employed the developed four-node discrete Kirchhoff quadrilateral element based on Reddy's third-order (DKTOT) to investigate the FV behavior of isotropic cylindrical shells. Li et al. [137] employed the MF Chebyshev RPIM to analyze the FV of rotating cross-ply laminated combined cylindrical-conical shells in a heated environment. In addition, Gupta and Pradyumna [138] investigated the NL vibrational behavior of sandwich structure shell panpanel consisting of a core made of an auxetic honeycomb and curvilinear fiber-reinforced Face sheets. Meanwhile, the enhancement of surface stress on the NL FV of FG composite nanoshells structure in modal interaction presence by Li et al. [139]. Furthermore, Yi et al. [140] studied the NL FV behavior of nanoshells made of FGP by employing the closed-cell Gaussian-Random field and GM of the surface approach of elasticity. Hashemi Kachap [141] performed a NL FV and buckling analysis of a nanoshells structure consist of piezo-harmo-electrostatic, based on modified strain gradient theory (MSGT) and surface/interface enhancements. However, Sobhani et al. [142] investigated the FV of a hybrid matrix porous NC-joined hemispherical-cylindrical-conical shells by implementing the FSDT. Civalek [143] explored the FV behavior of rotating shells with various material features by utilizing the discrete singular convolution method (DSCM). Besides, Rout and Hota [144] investigated the NL flexural FV of a sandwich hyper-shell structure founded in a TE utilizing HSDT and the NL Green–Lagrange theory. Pakravan et al. [145] explored the FV and the stability behaviors of conical shells reinforced with FG-CNT in various distribution patterns, subjected to a Haar wavelet technique, and found in TEs. Additionally, Melaibari et al. [146] studied the FV performance of a composited laminated shell and plate structures reinforced with FG-SWCNTs in four dissimilar dispersal patterns: V, O, X, and UD distributions. Zhang et al. [147] employed the refined ZZT to analyze the stability and FV of composite laminated shells. Babaei [148] investigated the FV as well as the snap during the instability of shallow arches made of FG-CNTRC and founded on an NL EF. Moreover, the composite and sandwich structure of conical shell panels, cylindrical as well as spherical, were considered and examined by Dagade and Kulkarni [149] to explore the natural fiber (NF), deflection, and stress by developing a quadrilateral flat-shell element. Besides, Shamloofard et al. [150] presented a novel super-element to investigate composite spherical shells' FV and deformation behaviors. Yang et al. [151] implemented the FSDT to investigate the FV and stability of cylindrical shells consisting of eccentric rotating FG-GPLRC. In addition, Farsadi et al. [152] employed the Genetic algorithm (GA) optimization methodology to optimize the fundamental natural frequency (FNF) of composite skew and taper cylindrical panels. Meanwhile, Ersoy and Civalek [153] combined the DQ with the DSCM to define the differential GEs. Besides, the two diverse types of material properties were presumed to vary continuously along the thickness path according to the V_F power law as well as the general four-parameter power law distributions. It is seen that all modes changed for conical panels when Ns = 11 grid points have been implemented in the circumferential directions in results obtained by employing the DSCM. Besides, the results of frequencies and the frequency parameter were in complete agreement with the results obtained in the author's previous study. Likewise, Kapuria et al. [154] employed the ZZT and 2D-FE models to investigate the FV of a bright piezo-bonded laminated shell featuring delamination and transducer debonding. Soureshjani et al. [155] investigated thermal enhancement on the FV of joined conicals made of FG-CNTRC-conical shells. Furthermore, Ghahfarokhi et al. [156] presented novel analytical and numerical solutions with experimental work that focused on exploring and predicting the stability as well as the FV of composite lattice cylinders with a sandwich structure by implementing the vibration correlation technique (VCT). However, Azarafza et al. [157] analyzed the FV of a cylindrical shell structure consisting of a Grid-stiffened composite reinforced with Carbon nanotubes (CNTs) analytically by implementing the FSDT, and the material features were obtained through the rule of mixture (ROM). While, Sciascia et al. [158] employed the Ritz methodology to investigate FV and the linear transient analysis of a DC shell structure with variable stiffness. Fan et al. [159] employed the Walsh series methodology to investigate the FV of the FG cylindrical shell. Carminelli and Catanica [160] employed the FEM B-Spline to inspect the FV of a double-curved thin-walled shell. However, Nguyen-Thanh et al. [161] presented a developed smooth FEM for analyzing shell which considered robust and free of shear locking by membrane bending founded on Mindline-Reissner approach. Kallannavar and Kattiman [162] examined the enhancement of porosity as well as temperature on the FV performance of a DC LC sandwich shell structure generated through 3D printing and having a core made of PLA. Furthermore, Sharma et al. [163] employed the FSDT to determine the flutter caused by an aerodynamic loading in an investigation to study the NL aeroelastic flutter characteristics of laminated composites cylindrical and flat curved structures. Nonetheless, Milić et al. [164] developed a modified IGA for active LC shells with PE layer, which offered a significant tool for modeling electrical and mechanical field and their coupling in such thin walled LC structures. Besides, to guaranty that the numerical level effort is in an acceptable domain, the modified form was based on FSDT and Reissner–Mindlin. Furthermore, Kachapi [165] investigated the NL vibrational response of PE nano-sensor with cylindrical structure under harmonic load and electrostatic excitation. The GM was employed to accomplish the energy approach with considering the subjected load, while the HP and Lagrange-Euler’s and the arc-length continuation as well as the complex average methodologies were used to examine the enhancement of surface/interface features. Besides, Marinkovic [166] presented a developed and analyze a novel model of PE in adaptive structure which in used in actuator and sensors by applying the FEM by implementing a model reduction technique. Moreover, Ramezani et al. [167] explored the NL FV responses of a cylindrical shells with a sandwich structure made of FG/SMA/FG analytically by implementing the HSDT with the semi assumed natural strain functions. Nonetheless, Sobhani et al. [168] inspected the FV behavior of porous NCs cylindrical shells reinforced with Gr oxide powder to enhance the mechanical properties. The ROM and Haplin–Tsai (HT) were implemented to define the mechanical properties of the hybrid metal hybrid matrix and hybrid polymer matrix. Besides, the GEs determined by using the FSDT and HP and solved by GDQM. Furthermore, Tornabene et al. [169] investigated the FV analysis of the DC laminated shell with an advance materials by implementing the GDQM. Besides, the displacement model was generated based on higher order expression following the equivalent single layer. Nonetheless, a semi-analytical approaches has been presented by Vescovini and Fantuzzi [170], to analyze the FV responses of conical shells by implementing the Ritz integral approach with minimizing the required CPU parallel with achieving high efficiency. However, Sofiyev and Fantuzzi [171] presented an analytical solution to explore the buckling FV analysis of sandwich cylindrical shells coated with an FGM layer and subjected to clamped BCs based on SDT. Although, various shear stress functions were utilized to accomplish the analysis such as uniform shear stress function, parabolic shear stress function, and cosine-hyperbolic shear stress function.

1.4.2 Beam Structure

In the framework of exploring the dynamic responses, Larbi et al. [172] investigated the FV of an FG beam analytically by utilizing the FSDT, while the BCs and the equation of motion (EOM) were obtained using the HPs. Avcar and Mohammed [173] employed the classical beam theory to investigate the FV of an FGM beam based on WPF. Besides, Sahouane et al. [174] utilized the HSDT to numerically examine the FV behavior of an FG beam. While, AlSaid-Alwan and Avcar [175] performed an analytical study to examine the FV performance of the FG beam by implementing various beam theories, and a comparison between the results obtained from each result has been highlighted. Li and Hu [176] employed the Timoshenko beam and the Euler-Bernolli models to study the NL bending and FV performance in the NLSGT framework. Nonetheless, Civalek and Gürses [177] utilized the DSCM to investigate the FV behavior of nonhomogeneous annular membranes. In addition, Chen et al. [178] investigated the NL FV and post-buckling of an FG Gr-reinforced porous NC beam by employing the Ritz methodology as well as DIM. Also, Kitipornchai et al. [179] studied the FV and the elastic stability of FGP beams reinforced with GPLs. In contrary, Feng et al. [180] examined the NL FV of FG polymer composite beams embedded with GLPs. Ladmek et al. [181] performed an analytical solution to explore the FV performance of a novel FG-CNTRC NB with Winkler foundation (WF) created of NSGT and two variable HSDT. Furthermore, Jiang et al. [182] explored the FV as well as the forced vibration behaviors of several 2D linear elastic structures, such as beams, by developing an overlapping FEM. Nonetheless, Mehrparvar et al. [183] implemented the high order (HO) Haar wavelet methodology to examine the FV analysis of the Timoshenko beam. Moreover, in a novel study for predicting the Frequency ratio (FR) performance of constant and variable stiffness general layup composite arched beams, Manickam et al. [184] employed the HSDT of beams. Besides, the mixed-developed ZZT has been used by Sorrenti and Gherlone [185] to analyze the dynamical response of a sandwich beam with adhesive layers. Malekzadeh et al. [186] explored the FV of a tapered-shaped NB by implementing FSDT combined with CST. Additionally, Ansari et al. [187] investigated the FV of PE NB subjected to post-buckling. Aghazadeh et al. [188] implemented various beam methodologies for investigating the FV and static analysis of NMS FG beams. However, by using a multiple-scale methodology, Eipakchi and Nasrekani [189] investigated the linear/NL FV analysis of extreme-light composite beams, including a honeycomb core. Likewise, Ke et al. [190] examined the NL FV behavior of FG-CNT-reinforced composite beam structures. Reddy [191] implemented the NLC methodologies to investigate the mechanical performance of various beams. Though, Aydogdu [192] investigated the FV, stability, as well as bending of an NB by employing the NLC beam methodology. In addition, Hadji and Avcar [193] employed the hyperbolic shear deformation beam approach to examine the NLC FV of porous FG NB. Zerrouki et al. [194] investigated the FV behavior of beam structure reinforced with NL FG-CNT dispersed in a polymer matrix by employing the HSDT. Nonetheless, a curved beam made of GPLRC has been considered by Moghaddasi and Kiani [195] to explore its FV as well as force vibration performances through utilizing the FSDT. Although, Xu et al. [196] investigated the FV of a composite Timoshenko beam consisting of rotating FG-CNTRC subjected to general BCs in a TE using the FSDT. Moreover, Garg and Chalak [197] developed an HZZT for studying laminated sandwich beams' FV and static behaviors. Else, Zhang and Shi [198] employed the exact FS method to present an exact solution of the FV behavior of LC-double-beam with elastic constraints. Furthermore, Jin et al. [199] explored the NL/linear FV of imperfect symmetrically/anti-symmetrically laminated FRC beams subjected to pre post-buckling (PPB). Also, Li et al. [200] investigated the FV and thermal post-buckling of the symmetric beam with sandwich structure subjected to fully clamped (CCCC) and SSSS BCs. In addition, Ansari et al. [201] explored the FR of microbeam made of FGM with considering SDE based on the strain gradient theory (SGT) Timoshenko beam. Furthermore, Sahmani et al. [202] explored the enhancement of surface energy on the FV behavior of the postbuckled NB modeled by third shear deformation theory (TSDT) and GM elasticity theory. Besides, the HP was used to define the NCL GEs which discretized by GDQM and the eigenvalue solved by the pseudo-arc-length (PAL). Additionally, Safaei et al. [203] explored the FV of a beam structure with a honeycomb sandwich structure made of various type of materials by applying the RVE. However, the FV of the smart sandwich NB consist of FGP and PE layers subjected to electromechanical forces with considering the SDE was examined by Jankowski [204]. Furthermore, Faghidian and Tounsi [205] explored the dynamic behavior of an elastic NB by using the mixture unified gradient theory of elasticity. Besides, an analytical solution for the wave dispersion response of the structure. Moreover, Limkatanyu et al. [206] presented a novel size-dependent beam-substrate medium model for static and dynamic analysis, though the modified SGT including three NCL material constant to identify the beam-bulk material SSE. Nonetheless, Karami et al. [207] explored the FV of non-uniform NB made of FGM with considering the SDE NLESGT. The NB properties assumed to be dependent on porosity and temperature and vary along the thickness and length directions.

1.4.3 Plate Structure

Based on the diversity of CP and laminated composite plate (LCP) application, author’s shows interests in exploring the mechanical behavior. For instance, Poojary and Rajamohan [208] explored the NL FV analysis of the internal thickness-CPs with rectangular geometry shown in Fig. 9, by employing FEM, and the time response has been determined by Newton–Raphson iterative (N-RI). Besides, Shi et al. [209] employed the variational asymptotic method (VAM) to investigate the static and dynamic performances of a composite honeycomb sandwich plate. In addition, Wei et al. [210] explored the FV as well as the static analyses of thin, stiffened flexible plates by generating and utilizing a wavelet FEM. Meanwhile, Thai et al. [211] implemented the IGA to study the multi-directional FGPs’ FV behavior in TEs. However, the exact 3D solution has been employed by Vel and Batra [212] to examine the vibrational performance (FV and forced vibration) of FGPs. In contrast, Saibaba et al. [213] presented a numerical solution for the FV characterizations of sandwich panels subjected to a TE consisting of two face sheets made of GRPC and a homogeneous core made of titanium. Besides, Zamani [214] investigated the FV of DC LC panels using the HSDT. Likewise, Liu et al. [215] investigated the FV, stability, and bending behaviors of sandwich panels with a honeycomb core. Where the core sheet treated as beams, while the sandwich panel treated as a composite structure. The GEs defined by HP and solved by implementing the Galerkin combined with FS. The obtained results show that, by increasing the height core the first NF increased until it reached a specific value then it started to decrease. Besides, decreasing the face sheet thickness leads to a decrease in the plate stiffness and reduces the NF. However, as the sandwich panel becomes lighter the NF increased. Moreover, Shen and He [216] explored the FV and the large amplitude of composite DC panels made of FG-SWCNT. The EOMs were defined by using the HSDT and VKT, and solved by two-step perturbation approach. The outcome of this study shows that the NL/linear FR of FG-Λ & FG-X dispersals increased due to increasing temperature. Besides, the FG reinforcement has remarkable influence on the NL vibrational characterizations of DC panel CNTRC. Likewise, in some circumstances increasing the temperature of the TE cause the frequency amplitude curve to become softening in case of FG-V & FG-U. However, Alanby et al. [217] studied the FV of a thick quadrilateral laminated plate by implementing TSDT combined with normal deformation theory (NDT). Nonetheless, Zenkour [218] utilized the sinusoidal shear deformation theory (SSDT) for plates to study the FV and stability behaviors of FG sandwich plates subjected to SSSS BCs. Hou et al. [219] investigated the FR analysis of Mindlin plates analytically, followed by numerical examples by developing a novel DSC Ritz method. Various BCs were utilized. Moreover, Das [220] investigated the NF of folded plates made of composite laminated by implementing the FEM. Civalek [221] investigated the NL dynamic behavior of LCP of temperately thick designed by the FSDT. The Karman NL equation has been used for the plates’ formulation. Besides, the NL EF governing equations (GEs) were integrated by utilizing the DSC-DQ coupled methods.

Fig. 9

The representation of composite sandwich plate with a ply-drop configuration, a staircase arrangement, b overlapping dropped plies and c continuous plies interspersed [208]

Furthermore, Selim et al. [222] utilized the interpolating moving least square (IMLS)-Ritz MF based on Reddy’s HSDT to study the FV of CNTRC plates with PE. Recently, Yesil et al. [223] performed an FV examination of twin PE inclusions implanted in an elastic medium using the (EET. Likewise, a PE bridge-type energy harvester in micro has been considered by Alshenawy et al. [224] to examine its NL dynamic performance through strain gradient MS collocation methodology. Besides, Nguyen-Xuan et al. [225] investigated the FV, bending and buckling of Reissner–Mindlin plates by implementing t node-based smooth FEM with 3-node triangular elements. Where maintain the formulation stability and to avoid the transvers shear (TS) locking a stabilization approach combined with the discrete shear gab algorithm were used. In addition, Karličić et al. [226] explored the FV of a VE multi-NP structure. Meanwhile, Akgöz and Civalek [227] employed the sinusoidal plate theory (SPT) to examine the stability as well as the FV of scale-based geometrical plates. The axisymmetric FV, as well as the forced vibration of circular NPs, has been performed by Malekzadeh and Farajpour [228]. Nonetheless, the Chebyshev–Ritz method (CRM) has been utilized in order to investigate the FV of plates made of FGM with inconstant delamination parameters by Wang et al. [229]. Conversely, Shafiei et al. [230] employed the MCST to study the buckling and FV performances of various sheet structures. Additionally, Natarajan et al. [231] employed the IGA to investigate the FV, stability as well and flutter performances of tow-steered composite laminates. However, Eftekhari [232] presented a developed simple procedure of finite element (FE) for FV analysis of thin as well as thick plates with rectangular geometry. Cai et al. [233] employed a numerical WFm to explore the FV and statically analyze thin isotropic triangular plates subjected to different BCs and internal supports. Also, Wu and Lu [234] utilized the pb-2 Ritz methodology and Reddy’s TSDT to present and investigate the FV behavior of plates consisting of internal columns subjected to elastic edges as a support system. Beside, Shi et al. [235] employed the unified methodology to analyze the FV performance of annular circular and sector plates subjected to random BCs. In addition, Mahran et al. [236] presented a comparison study examining various FE elements for analyzing FV, stress, and aero-elastic. The results show that the quadrilateral element has been the most appropriate element for elastic FV analysis through the deformation-based shape function. Garg et al. [237] examined the stability and FV behaviors of bioinspired laminated sandwich plate structures consisting of a soft core and helicoidal/bouligand Face sheets. Moreover, Peng et al. [238] employed the moving Kriging interpolation (MKI) MF to examine the bending and FV analyses for stiffened CPs made of FGM and subjected to TE. Alternatively, the NL FV of rectangular CPs made of GPLs reinforcement was investigated by Liu[239] through implementing the VKT and HPs followed by Galerkin. Additionally, Rama et al. [106] developed a linear triangular shell element based on the FSDT and the equivalent single-layer method. Though, the discrete shear gap has been used to resolve the shear locking effects’. However, the developed model can be used in analyzing the NL/linear geometric of LC of various thin walled structure. Besides, in order to maximize the FNF for the arbitrary quadrilateral LCPs shown in Fig. 10, Wang et al. [240] employed the hybrid whale optimization algorithm (HWOA) optimization approach, as illustrated in the flowchart in Fig. 11, after utilizing the FSDT for modeling the plates. At the same time, the RR methodology has been used for defining the FNF. Additionally, an optimized design has been accomplished and analyzed by Lu et al. [241] for maximizing and investigating the fundamental frequency (FF) of the NL FV of helicoidally laminated plates’ made of CFRPC. Meanwhile, to enhance the FV and stability behaviors of a composite VAT plate structure. Fallahi [242] employed the GA optimization algorithm, as shown in Fig. 12, to maximize the first NF and the first critical stability load. Besides, the NF and the buckling load were defined by utilizing the one-dimensional Carrera’s unified formulation (1D-CUF). Similarly, Farsadi et al. [243] employed GA optimization methodologies to optimize the FF of composite skew plates with variable stiffness. Contrary, Jing et al. [244] developed a new variable stiffness optimization algorithm (VSOA) to optimize the vibration of variable stiffness CPs by utilizing a linear deviation fiber path function. The developed design phases for the VSOA are illustrated in Fig. 13. Furthermore, Feng et al. [245] used the GA optimization approach to develop an optimal design of a sandwich plate structure with a honeycomb core. In addition, a plate made of FGM material and subjected to NL thermal loading has been investigated in order to optimize it is parametric FV characterizations through implementing the FEM as well as the nature-based algorithms by Rout et al. [246]. Besides, various developed hybrid optimization theories were employed to define the most extraordinary optimal result, followed by a data validation process. Furthermore, Wei et al. [247] employed the natural element method (NEM) to determine the direct FV analysis of the power-law FG plate based on the FSDT of the plate, followed by developing and solving the NL optimization problem of FF by implementing the Nelder-Mcad simplex methodology. The results revealed that the best surrogate model with minimum sample data was achieved by employing the piecewise cubic Hermitc interpolating polynomial. Additionally, Zhuang et al. [248] presented a deep auto-encoder technique based on energy methodology for analyzing the FV, and static responses of Kirchhoff plates for minimizing the overall potential energy, which considered as a specific type of the feedforward deep neural network. The presented algorithm has the ability to identify the basic patterns of energy system such as critical buckling load (CBL) factors and NF. The basic schematic of the presented transfer learning approach is illustrated in Fig. 14.

Fig. 10

The schematic of LCPs with its coordinate system & boundary spring [240]

Fig. 11

The flowchart of HWOA optimization method steps [240]

Fig. 12

The GA flowchart combined with Carrera’s unified formulation (CUF) [242]

Fig. 13

The developed design phases for the VSOA [244]

Fig. 14

The basic of the transfer learning approach [248]

Moreover, Dang et al. [249] employed the TSDT and HPs to investigate the NL vibrational characterizations of rotating composite blades in a heated environment. Also, Ansari et al. [250] performed a systematic stepwise analysis to investigate the FV behavior of rotated FG-reinforced turbo-machinery blades. Chen et al. [251] investigated the FV of composite blades made of FG-GPLRC with cross-sectional sectional area by implementing the Rayleigh–Ritz method (RRM) to obtain the NF and HT to formulate the Young's modulus (E). However, the other material characterizations were determined by utilizing the ROM. Meanwhile, Bidzard et al. [252] investigated the FV of toroidal micro-panels consisting of multilayers of FG-GPLRC subjected in TE by developing an NL FE model based on the MSGT combined with the FSDT. Likewise, the NL FV of blades made of rotating FG-GPLs has been investigated by Wang et al. [253]. The linear/NL frequencies were obtained through the RR method, while the HT model and ROM were utilized to define the mass density, E, and the effective Poisson’s ratio. The results show that the linear frequency was raised by increasing the rotational speed, while the increment of the taper ratio increased the NL FR.

1.5 The Basic Theories

1.5.1 Basic Plates’ Theories

The implemented theories in local and NLC domain in FV analysis for NMS plate structures are illustrated in Fig. 15. The NLC is a physical methodology with a particle that produces an instantaneous impact over distant arrangements. Otherwise, NLC based on two fundamental parameters: the steering of physical statuses at a space and the local uncertainty, as recorded by Ramanathan et al. [254]. The main general EOM NLC plate theory can be written as [255]:

$$N_{i\beta ,\beta }^L = - [1 - \left( {e_0 a)^2 \nabla^2 } \right]p_i + \mathop \smallint \limits_{\frac{ - h}{2}}^\frac{h}{2} \rho {\text{\"u }}_i dx_3 - \left( {e_0 a} \right)^2 \mathop \int \limits_{\frac{ - h}{2}}^\frac{h}{2} \nabla^2 \left( {\rho {\text{\"u }}_i } \right)dx_3$$

(1)

$$M_{\alpha \beta ,\beta }^L - N_{\alpha 3}^L = \mathop \int \limits_{\frac{ - h}{2}}^\frac{h}{2} \rho {\text{\"u }}_\alpha dx_3 - \left( {e_0 a} \right)^2 \mathop \int \limits_{\frac{ - h}{2}}^\frac{h}{2} \rho {\text{\"u }}_\alpha x_3 dx_3 - \left( {e_0 a} \right)^2 \mathop \int \limits_{\frac{ - h}{2}}^\frac{h}{2} \nabla^2 \left( {\rho {\text{\"u }}_i x_3 } \right)dx_3$$

(2)

Fig. 15

The classification of implemented theories in local/NLC domain in FV analysis for NMS plate structures

While the equations of motion for the NLC Kirchhoff plate theorem can be written as [255]:

$${A}_{\alpha \beta \omega \rho }{u}_{\omega ,\rho \beta }^{0}-{B}_{\alpha \beta \omega \rho }{u}_{\omega ,\rho \beta }^{0}+[1-\left({e}_{0}a{)}^{2}{\nabla }^{2}\right]\left({p}_{\alpha }-{I}_{0}{\ddot{u}}_{\alpha }^{0}\right)=0$$

(3)

$$B_{\alpha \beta \omega \rho } u_{\omega ,\rho \alpha \beta }^0 - D_{\alpha \beta \omega \rho } u_{3,\omega \rho \alpha \beta }^0 + \left[ {1 - (e_0 a)^2 \nabla^2 } \right](p_\alpha - I_0 {\text{\"u }}_\alpha^0 ) = 0$$

(4)

The equations of motion for the NLC MPT can be written as [255]:

$$A_{\alpha \beta \omega \rho } u_{\omega ,\rho \beta }^0 + B_{\alpha \beta \omega \rho } \psi_{\omega ,\rho \beta } + \left[ {1 - (e_0 a)^2 \nabla^2 } \right](p_\alpha - I_0 {\text{\"u }}_\alpha^0 ) = 0$$

(5)

$${A}_{3\beta 3\rho }({u}_{3,\rho \beta }^{0}+{\Psi }_{\rho ,\beta })+[1-\left({e}_{0}a{)}^{2}{\nabla }^{2}\right]\left({p}_{3}-{I}_{0}{\ddot{u}}_{3}^{0}\right)=0$$

(6)

$${B}_{\alpha \beta \omega \rho }{u}_{\omega ,\rho \beta }^{0}+{D}_{\alpha \beta \omega \rho }{\Psi }_{\omega ,\rho \beta }-{A}_{3\alpha 3\rho }\left({u}_{3,\rho }^{0}+{\Psi }_{\rho }\right)-[1-\left({e}_{0}a{)}^{2}{\nabla }^{2}\right]{I}_{2}{\ddot{\Psi }}_{\alpha }=0$$

(7)

The EOM for the NLC TSDT can be written as [256]:

$${A}_{\alpha \beta \omega \rho }{u}_{\omega ,\rho B}^{0}+{B}_{\alpha \beta \omega \rho }{\varphi }_{\omega ,\rho B}-{c}_{1}{D}_{\alpha \beta \omega \rho }\left({\varphi }_{\omega ,\rho B}+{w,}_{\beta \omega \rho }^{0}\right)+\left[1-\mu {\nabla }^{2}\right]\left({F}_{\alpha }-{I}_{0}{\ddot{u}}_{\alpha }^{0}\right)=0$$

(8)

$${A}_{z\alpha z\rho }\left({\varphi }_{\rho ,\alpha }+ {w}_{,\alpha \rho }^{0}\right)-{c}_{2}{C}_{z\alpha z\rho }\left({\varphi }_{\rho ,\alpha }+ {w}_{,\alpha \rho }^{0}\right)-{c}_{2}{[c}_{z\alpha z\rho }\left({\varphi }_{\rho ,\alpha }+ {w}_{,\alpha \rho }^{0}\right)-{c}_{2}{E}_{z\alpha z\rho }\left({\varphi }_{\rho ,\alpha }+ {w}_{,\alpha \rho }^{0}\right)]+{c}_{2}{[D}_{\alpha \beta \omega \rho }{u}_{\omega ,\rho \alpha \beta }^{0}+{E}_{\alpha \beta \omega \rho }{\varphi }_{\omega ,\rho \alpha \beta }-{c}_{1}{F}_{\alpha \beta \omega \rho }\left({\varphi }_{\omega ,\rho \alpha \beta }+{w}_{,\alpha \beta \omega \rho }^{0}\right)]+[1-\mu {\nabla }^{2}]({q}_{z}-{I}_{0}{\ddot{w}}^{0}+{c}_{1}]{I}_{4}{\ddot{\varphi }}_{\alpha ,\alpha }-{c}_{1}{I}_{6}({\ddot{\varphi }}_{\alpha ,\alpha }+{\ddot{w}}_{\alpha ,\alpha })])=0$$

(9)

$${B}_{\alpha \beta \omega \rho }{u}_{\omega ,\rho B}^{0}+{c}_{\alpha \beta \omega \rho }{\varphi }_{\omega ,\rho B}-{c}_{1}{E}_{\alpha \beta \omega \rho }\left({\varphi }_{\omega ,\rho B}+{w,}_{\beta \omega \rho }^{0}\right)-{c}_{1}{[D}_{\alpha \beta \omega \rho }{u}_{\omega ,\rho \alpha \beta }^{0}+{E}_{\alpha \beta \omega \rho }{\varphi }_{\omega ,\rho B}-{c}_{1}{F}_{\alpha \beta \omega \rho }\left({\varphi }_{\omega ,\rho B}+{w}_{,\beta \omega \rho }^{0}\right)]-{A}_{z\alpha z\rho }\left({\varphi }_{\rho }+ {w}_{,\rho }^{0}\right)+{c}_{2}{c}_{z\alpha z\rho }\left({\varphi }_{\rho }+ {w}_{,\rho }^{0}\right)+{c}_{2}{[c}_{z\alpha z\rho }\left({\varphi }_{\rho }+ {w}_{,\rho }^{0}\right)-{c}_{2}{E}_{z\alpha z\rho }\left({\varphi }_{\rho }+ {w}_{,\rho }^{0}\right)-\left[1-\mu {\nabla }^{2}\right][\left({I}_{2}{\ddot{\varphi }}_{\alpha }-{c}_{1}{I}_{4}({\ddot{\varphi }}_{\alpha }+{\ddot{w}}_{,\alpha }^{0}\right)-{c}_{2}({I}_{4}{\ddot{\varphi }}_{\alpha }-{c}_{1}{I}_{6}\left({\ddot{\varphi }}_{\alpha }+{\ddot{w}}_{,\alpha }^{0}\right))]=0$$

(10)

The general formula of the NSGT constitutive relation for the stress as well as strain components can be written as [257]:

$$\left[1-{\left({e}_{1}a\right)}^{2}{\nabla }^{2}\right]\left[1-{\left({e}_{0}a\right)}^{2}{\nabla }^{2}\right]{\sigma }_{ij}={C}_{ijkl}\left[1-{{\left({e}_{1}a\right)}^{2}\nabla }^{2}\right]{\varepsilon }_{kl}-{C}_{ijkl}{l}^{2}[1-{\left({e}_{0}a\right)}^{2}{\nabla }^{2}]{\nabla }^{2}{\varepsilon }_{kl}$$

(11)

The EOM of HPs by substituting the NSGT formulas can be written as [257]:

$$\frac{\partial {N}_{xx}}{\partial x}+\frac{\partial {N}_{xy}}{\partial y}={I}_{0}{\ddot{u}}_{0}-{I}_{1}\frac{\partial {\ddot{w}}_{b}}{\partial x}-{J}_{1}\frac{\partial {\ddot{w}}_{s}}{\partial x}$$

(12)

$$\frac{\partial {N}_{xy}}{\partial x}+\frac{\partial {N}_{yy}}{\partial y}={I}_{0}{\ddot{v}}_{0}-{I}_{1}\frac{\partial {\ddot{w}}_{b}}{\partial y}-{J}_{1}\frac{\partial {\ddot{w}}_{s}}{\partial y}$$

(13)

The displacement field of the HO refined plate theory which can be used in defining the GEs written as [258]:

$${u}_{1}\left(x,y,z\right)=-z\frac{{\partial }_{wb}}{\partial x}-f(z)\frac{{\partial }_{ws}}{\partial x}$$

(14)

$${u}_{2}\left(x,y,z\right)=-z\frac{{\partial }_{wb}}{\partial y}-f(z)\frac{{\partial }_{ws}}{\partial y}$$

(15)

$${u}_{3}\left(x,y,z\right)={w}_{b}\left(x,y\right)+{w}_{s}(x,y)$$

(16)

The governing equation (GE) for the NL FV of plates based on HPs can be written as [259]:

$${\int }_{{t}_{1}}^{{t}_{2}}\delta \left(U-T\right)dt=0$$

(17)

$$t={t}_{1}, {t}_{2}: \delta {u}_{0}=\delta {v}_{0}=\delta {w}_{0}=\delta {\varphi }_{x}=\delta {\varphi }_{y}=0$$

(18)

The displacement of FV problem based on 2D-DQM can be written as [260]:

$${u}_{0}\left(x,y,t\right)={\overline{u} }_{0}(x,y){e}^{\overline{i\omega t} }$$

(19)

$${v}_{0}\left(x,y,t\right)={\overline{v} }_{0}(x,y){e}^{\overline{i\omega t} }$$

(20)

$${w}_{0}\left(x,y,t\right)={\overline{w} }_{0}(x,y){e}^{\overline{i\omega t} }$$

(21)

$${\varphi }_{x}\left(x,y,t\right)={\overline{\varphi } }_{x}(x,y){e}^{\overline{i\omega t} }$$

(22)

$${\varphi }_{y}\left(x,y,t\right)={\overline{\varphi } }_{y}(x,y){e}^{\overline{i\omega t} }$$

(23)

where the vibrational frequency is defining as $\overline{\omega }$ and ${\overline{u} }_{0}$,${\overline{w} }_{0}$,${\overline{v} }_{0}$, ${\overline{\varphi } }_{y}$ and ${\overline{\varphi } }_{x}$ are the spectral components.

The strain energy U based on the MSGT can be written as [227, 261]:

$$U=\frac{1}{2}\underset{\Omega }{\overset{-}{\int }}({\varepsilon }_{ij}+{p}_{i}{\gamma }_{i}+{\tau }_{ijk}^{(1)}{\eta }_{ijk}^{(1)}+{m}_{ij}^{s}{x}_{ij}^{s})dv$$

(24)

where the x^s defined the symmetric rotation gradient tensor, and ${x}_{ij}^{s}$,${\gamma }_{i}$,${\varepsilon }_{ij}$, and ${\eta }_{ijk}^{(1)}$ defining the components of displacement vector.

The basic of SPT defined the displacement components as [227, 262]:

$${u}_{1}\left(x,y,z,t\right)=u\left(x,y,t\right)-z\frac{\partial \omega \left(x,y,t\right)}{\partial x}+\frac{h}{\pi }\text{sin}\left(\frac{\pi z}{h}\right)\left(\frac{\partial \omega \left(x,y,t\right)}{\partial x}-{\varphi }_{1}(x,y,t)\right)$$

(25)

$${u}_{2}\left(x,y,z,t\right)=v\left(x,y,t\right)-z\frac{\partial \omega \left(x,y,t\right)}{\partial y}+\frac{h}{\pi }\text{sin}\left(\frac{\pi z}{h}\right)\left(\frac{\partial \omega \left(x,y,t\right)}{\partial y}-{\varphi }_{2}(x,y,t)\right)$$

(26)

$${u}_{3}\left(x,y,z,t\right)=\omega \left(x,y,t\right)$$

(27)

The x, y and z directional component were defined as u₁, u₂, and u₃ and the mid-plane of MPs as u,$v$ and $\omega$; hence the GEs of the MPs founded by SPT and the cross ponded BCs were presented in [227].

Based on KPT the displacement rotation and the GE can be written as [263]:

$$u=-z\frac{\partial \omega }{\partial x}{e}_{1}-z\frac{\partial \omega }{\partial y}{e}_{2}+\omega {e}_{3}$$

(28)

$$\theta =\frac{\partial \omega }{\partial y}{e}_{1}-\frac{\partial \omega }{\partial x}{e}_{2}$$

(29)

$${N}_{\alpha \beta ,\beta }={J}_{1}{\ddot{u}}_{\alpha }^{0}$$

(30)

$${M}_{\alpha \beta ,\alpha \beta }+q\left(x,t\right)={J}_{1}{\ddot{\omega }}^{0}-{J}_{3}{\ddot{\omega }}_{,\alpha \alpha }^{0}$$

(31)

Various forms of Lemma theory and Corollary approaches were presented by Furtsev and Rudoy [264] for modeling a modified KPT with soft as well as stiff interfaces. While, the classical formulation of the displacement-traction problem of KPT VKT theory NL elastic plates has been presented by Geymonat and Krasucki [265]. Hence, a various modification on the KPT has been accomplished based on type of plate aimed to analyze.

The displacement field equations and the GEs based on MPT can be written as [97]:

$${u}_{x}=u\left(x.y.z\right)+z{\psi }_{x}$$

(32)

$${u}_{y}=v\left(x.y.z\right)+z{\psi }_{y}$$

(33)

$${u}_{z}=w\left(x.y.z\right)$$

(34)

$$\frac{{\partial }^{2}{\psi }_{x}}{\partial {x}^{2}}+\frac{1-v}{2}\frac{{\partial }^{2}{\psi }_{x}}{\partial {y}^{2}}+\frac{1+v}{2}\frac{{\partial }^{2}{\psi }_{y}}{\partial x\partial y}-\frac{\kappa Gh}{D}\left(\frac{\partial w}{\partial x}+{\psi }_{x}\right)=0$$

(35)

$$\frac{{\partial }^{2}{\psi }_{y}}{\partial {y}^{2}}+\frac{1-v}{2}\frac{{\partial }^{2}{\psi }_{y}}{\partial {x}^{2}}+\frac{1+v}{2}\frac{{\partial }^{2}{\psi }_{x}}{\partial x\partial y}-\frac{\kappa Gh}{D}\left(\frac{\partial w}{\partial y}+{\psi }_{y}\right)=0$$

(36)

$${\nabla }^{2}w+\left(\frac{\partial {\psi }_{x}}{\partial x}+\frac{\partial {\psi }_{y}}{\partial y}\right)=-\frac{\left(1-{\left({e}_{0}a\right)}^{2}{\nabla }^{2}\right)}{\kappa Gh}(q+{G}_{b}{\nabla }^{2}w-{k}_{w}w)$$

(37)

The GE of NLC Mindlin plate after reduced to the NLC KPT can be written as [97]:

$$D{\nabla }^{2}{\nabla }^{2}w=(1-{\left({e}_{0}a\right)}^{2}{\nabla }^{2})(q+{G}_{b}{\nabla }^{2}w-{k}_{w}w)$$

(38)

The strain energy of a linear elastic continuum body in MCST in local theories usually define by a function of strain and curvature tensors and written as follow [266]:

$$U=\frac{1}{2}{\int }_{V}^{-}\left(\sigma :\varepsilon +m:x\right)dV$$

(39)

where: σ is the Cauchy stress, ε is the strain tensors, $m$ is a deviatoric part of the coupled stress tensor and $x$ is the symmetric curvature tensor.

The Galerkin WFm for FV and bending analysis of NPs written as follow [267]:

$${\int }_{V}^{-}{\sigma }_{ij}\delta {\varepsilon }_{ij}dV+{\int }_{V}^{-}(1-\mu {\nabla }^{2}{)\rho u}_{i}{\ddot{u}}_{i}dV={\int }_{V}^{-}{f}_{i}(\delta {u}_{i}-\mu {\nabla }^{2}\delta {u}_{i})dV+{\int }_{{\Gamma }_{g}}^{-}{\sigma }_{ij}{n}_{i}\delta {u}_{i}d{\Gamma }_{g}$$

(40)

For size dependent analysis the stress tensor founded on NLC Eringen’s approach with SSE can be written as [267, 268]:

$${t}_{ij}\left(x\right)={\int }_{V}^{-}\alpha \left(\left|{x}{\prime}-x\right|,\mu \right){\sigma }_{ij}\left({x}{\prime}\right)d{x}{\prime}$$

(41)

The displacement fields based on the refined plate theory (RPT) can be written as follow [267, 269]:

$$u={u}_{0}+z{u}_{1}+f(z){u}_{2}$$

(42)

NvS for FV, stability and bending in case of implementing the SS BCs can be expressed as [270]:

$${W}^{b}=\sum\limits_{m=1}^{\infty }\sum\limits_{n=1}^{\infty }{W}_{mn}^{b}\text{sin}(\alpha x)\text{sin}(\beta y)$$

(43)

$${W}^{s}=\sum\limits_{m=1}^{\infty }\sum\limits_{n=1}^{\infty }{W}_{mn}^{s}\text{sin}(\alpha x)\text{sin}(\beta y)$$

(44)

The GM surface elasticity theory assume that the surface thickness is equal to zero, and the constitutive law linking to the deformability of the surface is required. The GM at small deformation the linear constitutive ℑ written as [271]:

$$\mathfrak{I}=\mathcal{M}+{\mathbb{C}}s :\mathcal{E}+(\mathcal{F} - {\varvec{I}})\mathcal{M}$$

(45)

where ℳ is the residue surface stress, ℂ^s is the forth-order tensor for surface elastic moduli and ℰ is the surface strain.

Based on the MT technique the local effective material characterizations of FGM NPs such as (mass density, E, thermal extension coefficient and Poisson’s ratio) can be written as [272]:

$$\rho \left(z\right)={\rho }_{c}{V}_{c}\left(z\right)+{\rho }_{m}{V}_{m}$$

(46)

$$E\left(z\right)={E}_{c}{V}_{c}\left(z\right)+{E}_{m}{V}_{m}$$

(47)

$$\alpha \left(z\right)={\alpha }_{c}{V}_{c}\left(z\right)+{\alpha }_{m}{V}_{m}$$

(48)

$$v\left(z\right)={v}_{c}{V}_{c}\left(z\right)+v{V}_{m}$$

(49)

Moreover, the stress component σ_zz was assumed to vary linearly through the plates’ thickness and obtained as [273]:

$${\sigma }_{zz}=\frac{\left(\frac{\partial {\sigma }_{xz}^{s}}{\partial x}+\frac{\partial {\sigma }_{yz}^{s}}{\partial y}-{\rho }^{s}\frac{{\partial }^{2}w}{\partial {t}^{2}}\right)at top+\left(\frac{\partial {\sigma }_{xz}^{s}}{\partial x}+\frac{\partial {\sigma }_{yz}^{s}}{\partial y}-{\rho }^{s}\frac{{\partial }^{2}w}{\partial {t}^{2}}\right)at bottom}{2}+\frac{\left(\frac{\partial {\sigma }_{xz}^{s}}{\partial x}+\frac{\partial {\sigma }_{yz}^{s}}{\partial y}-{\rho }^{s}\frac{{\partial }^{2}w}{\partial {t}^{2}}\right)at top+\left(\frac{\partial {\sigma }_{xz}^{s}}{\partial x}+\frac{\partial {\sigma }_{yz}^{s}}{\partial y}-{\rho }^{s}\frac{{\partial }^{2}w}{\partial {t}^{2}}\right)at bottom}{h}Z$$

(50)

1.5.2 High Order Methods

The application of high order methods in dynamic system is essential when significant accuracy at a very short time is required for a system with infinite number of unknown, as well as in dynamic control system. The high order methods consist of three approaches which are; RK, Predictor corrector, and Adams. A classification of various types of high order methods are illustrated in Fig. 16. Certainly, in mathematical framework, the initial value problems which were identified after the discretization process to ordinary differential equations (ODEs), though the prominent of those initial value problems is the 4ht-RK. However, in some complex system the RK45 approach can be used. Besides, the Runge–Kutta (RK) method consider as an explicit approach. The general formula of s-stage RK function can be written as [274]:

Fig. 16

Types and classifications of high order methods

$${Y}_{i}={y}_{n}+h{\sum }_{i,j=1}^{s}{a}_{ij}f({x}_{n}+{c}_{j}h,Yj)$$

(51)

While the 4th-RK utilized in solving ODEs combined with initial value problem is written as [275]:

$$\frac{dy(t)}{dt}=f\left(t,y\left(t\right)\right),y\left(0\right)={y}_{0}$$

(52)

The 4th-RK consider as h5 and written as:

$${k}_{4}=h({y}_{n}+{k}_{3},{t}_{n}+h)$$

(53)

$${y}_{n+1}={y}_{n}+({k}_{1}+2{k}_{2}+2{k}_{3}+{k}_{4})/6$$

(54)

To accomplish significant accuracy in predicting deformation, stress and frequency responses, predictor corrector methodology have significant predicting potential. Besides, the main concept of the predictor–corrector methodology is to employ an appropriate combination of an implicit and an explicit approaches to come up with optimized approach with superior convergence features. For example, the combination of (FE with second order Adams–Moulton. Where the method consists of two steps, step one predictor and step 2 corrector, a further modified method known as weighted predictor–corrector was accomplished [276]. Notably, the concept of the modified approach is to determine the number density at the step n + 1, N_n+1 utilizing the weight parameter as:

$${N}_{n+1}={e}^{(\left(1-w\right)ln{N}_{n+1}^{p}+wln{N}_{n+1}^{c})}$$

(55)

Furthermore, the predictor–corrector methodology has been modified to solve generalized-Caputo type FDEs using the L1 (which refer to the group of integrable functions with finite absolute value on a given measure space)-based discretization concept [277]. In addition, to avoid over estimation of materials’ temperature at large time domain which may be achieved by direct implementation of the predictor–corrector methodology, Shi and Xie [278] accomplished a solution for TRT formulas by employing predictor–corrector Monte Carlo and a liner solution was accomplished. However, based on Wen [279] numerous theories verify that the predictor–corrector approach might receive the asymptotic mean square stability of beyond two original problems.

The concept of Adams’ methodologies is the approximation of integrand with polynomial within a specific interval defined as (t_n,t_n+1). For instance, the Adams’ schemes consist of implicit form which defined as Adams–Moulton approach and the explicit form is the Adams–Bashforth method which are the simple form of backward and forward of Euler approaches correspondingly. Remarkably, each method has its own advantages above the other. For example, when numerical stability is required it is recommended to apply the second order Adams–Moulton which may be a drawback for second order Adams–Bashforth. Although, second order Adams–Bashforth considered as time saving and cost saving compare with the second order Adams–Moulton which required solving NL algebraic equations at each time step. However, the second order Adams–Bashforth can be written as:

$${y}_{n+1}={y}_{n}+\frac{h}{3}(3f({y}_{n},{t}_{n}-f\left({y}_{n-1},{t}_{n-1}\right))$$

(56)

where the time step of second order Adams–Moulton which defined also as the trapezoidal rule can be written as:

$${y}_{n+1}={y}_{n}+\frac{h}{2}(f({y}_{n+1},{t}_{n+1}+f\left({y}_{n},{t}_{n}\right))$$

(57)

2 Linear Free Vibration Analysis

2.1 Linear Analysis by Analytical Method

2.1.1 Analytical Solution Without Considering the SDE

Analytical solutions have many advantages over other types of solutions one of the most important is, having fewer unknowns. However, the most used analytical solution in plate analysis is the NvS. Besides, it is employed for plants that are subjected to SSSS BCs Meanwhile, the LS can achieve significant accurate results for plates subjected to various BCs and to accomplish layer-wise (LW) benchmark solutions above the drawback of numerical solution. Furthermore, it is recommended to implement the analytical solution instead of the numerical solution whenever it is possible by Boscolo [280]. In this context many authors shows interested in implementing analytical solution for analyzing FV behavior of CPs. Chanda et al. [281] accomplished an analytical study and FEM solutions to investigate the impact of EL on the controlled and uncontrolled FV, forced vibration, and static responses of smart multi-layered laminated composite (SMLLC) plates by implementing the non-polynomial HZZPT kinematics as well as HP for driving the GE. The GEs were solved by NvS and C⁰ isoparametric finite element (IFE). Figure 17 illustrates the schematic of the plate, which deliberates the realistic parabolic behavior of the TS stresses across the thickness of the LCP. Besides, the PF has been utilized. The results show that the FS increases the FV frequency and modifies the NF. Joshan et al. [282] employed the non-polynomial shear deformation theories (NPSDTs) to investigate the thermomechanical behavior of LCPs and the NL. Besides, linear thermal gradient characteristics were examined, and the GEs were developed using the vertical work principle and solved by NvS. Furthermore, Bessaim et al. [283] developed an innovative HSDT and normal deformation theory for examining the bending as well as the FV performance of sandwich plates made of FG isotropic face sheets ( Aluminum and Alumina) and a soft/hardcore subjected to SSSS BCs. Besides, the GEs have been obtained by utilizing the HPs and solved analytically. Hence, the methods were validated, and exact and numerical solutions were presented. The results show that plates with homogeneous soft core k have less effect on NF compare with homogeneous hardcore. Additionally, the maximum values were obtained in ceramic plates and the minimum in metal plates, and by increasing the width-to-thickness ratio (b/h), the NF decreased for all k values.

Fig. 17

The schematic of a smart CP with PE actuator & PE sensor maintained on PF [281]

GhorbanpourArani et al. [284] presented an FV analysis for a VE sandwich plate with CNTRP- Polymethyl methacrylate (PMMA) and a homogenized core made of Ti-6Al-4V as shown in Fig. 18. By utilizing the FSDT and HP. The outcomes show that implementing the foundation causes a significant reduction in the frequency, although it raises the damping coefficient of the material, foundation, and the VE structure. Furthermore, the stability of the CP is highly affected by the foundations’ stiffness and damping, the distribution of FG-CNTs, volume fraction (V_F) of the CNTs, core-to-face sheet thickness ratios, and structural damping. Moreover, Amir et al. [285] employed the FSDT to study the FV of sandwich flexoelectric plates, illustrated in Fig. 19 with PF at the base. The NvS has been implemented to present an analytical solution, and the HP has been used for GE elicitation. It is observed that the flexoelectric plays a significant role, which has a significant effect on the NP sandwich structure, mainly if a fragile plate is utilized as a face sheet. In other words, face sheets with high thickness values lead to smaller values of NF. Additionally, NF is considered highly sensitive to the subjected electrical voltage on the face sheet. Besides, the NF can be increased by raising the V_F of the CNT fibers, which, as a consequence, will cause a rise in the composite core plates’ strength.

Fig. 18

The schematic of the VE sandwich plate on a VE foundation with various FG-CNTs distribution in the face sheet [284]

Fig. 19

The schematic of sandwich plates CNTRP core & core flexoelectric face sheet on PF [285]

Hadji et al. [286] investigated the NF of an imperfect FGP with a sandwich structure with EF with various porosity dispersal analytical, as shown in Fig. 20 and subjected it to SSSS BCs. The kinematic relationship of the structure has been defined by utilizing the TSDT, while the HPs have been used to define the EOM. Besides, an assumption of instantaneously variable mechanical features through the thickness direction has been considered. The obtained results show that the EF presence causes an increase in the non-dimensional natural frequencies (NDNF), although it reduces the impact of porosity V_F and varying V_F index. In addition, the porosity dispersal remarkably affects the NDNF. Also, the layup schemes significantly enhance the NDNF. Furthermore, Nguyen et al. [287] performed an FEA by implementing the IGA Bézier formulation connected with C⁰-HSDT for a CPs structure, which consists of two outer layers of PE & FG porous reinforced with GPLs as a core that is dispersed uniformly or NY in the metal matrix in various arrangements. The obtained NF in all various meshes was validated with previous analytical results, and the relative error was calculated. Mishra et al. [288] examined the FV performance of FGM plates with rectangular shapes by employing the FSDT combined with the DSM under various BC sets. It has been seen that increasing the material gradient index (GI) causes a reduction in frequency parameters. Above all, the results obtained in that study were found to be identical when compared to previous published data. Additionally, adding at least one clamped to the BCs set increases the NF in all five modes. However, the spline finite point method (SFPM) has been employed by Zhou and Li [289] to analyze the FV of various plates with a sandwich structure made of core and laminated faces. Giunta et al. [290] employed FEM to investigate the FV of CPs with a variable stiffness by using the Hierarchical modeling where the CUF has been expanded and utilized. Besides, the outcomes show that applying the Reissner mixed variational theorem (RMVT) through the CUF guaranty a remarkable potential in enhancing the accuracy as well as the efficiency of modeling the VAT plates in the FV analyses.

Fig. 20

The schematic of imperfect FGP sandwich & the porosity dispersal [286]

Tao et al. [291] noticed that the maximum linear NDFF of an MSP made of multilayers of FG-GPLs has been obtained by utilizing the FG-X dispersion. Besides, by comparing the results obtained by FG-GPLs' annular sector, MSP has significantly greater FF than the results obtained by pure epoxy. In other words, GPLs greatly enhance the MSP FF. Wang et al. [292] explored the dynamic performance (FV and forced vibration) as well as bending deformation of the plats shown in Fig. 21 and subjected to SSSS BCs, analytically through implementing pseudo-Stroh formulism (PSF) and propagation matrix method (PMM). The impact of imperfect interface parameters and stacking sequences on the static and dynamic response were considered. It has been noticed that the natural frequencies (NFs) reduced due to the increase in the imperfect interface parameter.

Fig. 21

The schematics of a N-layer QC plate with the imperfect interfaces & b the QC coated with aluminum based CP structure with the imperfect interfaces [292]

Zhai et al. [293] presented an analytical solution by utilizing the NvS to investigate the FV as well as the buckling behaviors of composite sandwich plates subjected to a TE, and the HP has been used to drive the EOM. It has been seen that increasing the aspect ratio (a/h) cause to reduce the NF subjected to various temperatures; in other words, thicker plates exceed higher FF compared with thin plates. Besides, the frequencies increased dramatically at the shear parameter (g) < 0.001. At the same time, when it reaches a value greater than 0.001, the FF reaches and maintains a steady state with high values. Furthermore, Nguyen et al. [294] employed the IGA based on the RPT to investigate the FV bending and the stability of circular NP made of FGM. Yesi and Yahnioglu [295] employed the HP and 3D exact solution by Fortran to examine the FV behavior of PE plates subjected to SSSS BCs and containing a cylindrical cavity. It has been observed that the PE effects significantly enhanced the FFs. Besides, by determining the FFs in various polled axes, it is seen that the maximum values were recorded in axes that were perpendicular to the cylindrical location planes’, although as the distance between the cavity’s top and plane top reduced, the FFs reduced too. Saidi et al. [296] investigated the FV of moderately thick FG annular plates by implementing the FSDT and five coupled partial differential equations (PDEs). At all various wave numbers, the NF decreased due to the increase in n, while the wave numbers depended on a/h, thickness, BCs, and the sector angle of the FG plates. Otherwise, the FF is reduced due to increasing the a/h. A composite bi-layered plate consisting of PE dielectric as well as PE semiconductor layers made of ZnO-BaTiO₃, and subjected to SSSS BCs has been considered by Guo et al. [297] to examine the FV analytically by utilizing various methods such as; PS theories, first ZZT, and HPs. The dimensionless natural frequencies (DLNF) shows a sharp reduction due to the increase in steady-state electron density from 10¹⁹ to 10²² m⁻³, while no significant enhancement was noticed when applying other steady-state values. The required vibrational frequency of the plate can be achieved by controlling the absorption of the majority of free electrons at a specific value by changing the donor impurity. Furthermore, Kuo and Wei [298] employed the PSF combined with the PMM to examine the FV of magneto-electro-elastic (MEE) laminated plates analytically, followed by implementing the solution to a CP structure made of BaTiO₃-CoFe₂O₄. The obtained results show that reducing the relative scaling parameters causes a reduction in the NF. Zhang et al. [299] presented a novel analytical solution for FV of orthotropic CPs, the integral kernel of a double half sinusoidal series, to generate the integral transformation pair for the novel, precise vibrational analysis. Besides, many other theories were implemented to complete the analytical solution and obtain the NF and the mode shape, such as PDEs, Fourier coefficient, and the inverse formula. Also, other than 500 inclusive analytical solutions were compared and validated with arithmetical outcomes. Moreover, Chanda and Sahoo [300] employed the non-polynomial Inverse Hyperbolic ZZT to perform an analytical simulation to investigate the static and FV behaviors of SSSS sandwich plates and the HP implemented to drive the dynamic GE. Besides, the Newmark's time integration (NTI) and NvS to define the coupled equations, it has been noted that the material properties, as well as the geometrical characterization of the structure, significantly enhanced the frequency and amplitude responses. Also, the amplitude is highly affected by the shock pulse length factor, although decreasing the a/h, which is caused by reducing the b value, leads to an increase in the frequency and amplitude. In a further study, Joshan et al. [301] employed the NPSDTs to perform an analytical investigation to explore the FV and the static responses of composite MSP. At the same time, the LSEs have been deliberated through implementing the CST, and various shear shape functions were considered to generate the TS deformation. Besides, the GEs have been formulated through utilizing the HP and solved by NvS. The results in Fig. 22 revealed that increasing the a/h, number of layers, or anisotropy ratio causes an increase in the NDFF. Additionally, the obtained results by inverse hyperbolic shear deformation theory (IHSDT) as well as ITSDT were more accurate and similar to the FEM than those of EPT, SPT as well as HPT, and the LSE also has been considered more obviously. While, Trapezon et al. [302] investigated the FV of CP with variable thickness by developing a novel analytical algorithm.

Fig. 22

a The first six vibrational modes shapes for anti-symmetric laminated [0/60/0/90] at a/h = 5, l/h = 1, b the NDFF for multi-layered anti-symmetric [0/90]_n cross-ply composite MP & c the deviation in NDFF with anisotropy ratio of two layered [0°/90°] cross-ply composite MP at a/h = 10 [301]

Alaimo et al. [303] employed the principle of virtual displacements (PVDs) combined with LW to explore the NF and the analytical damped FV behaviors of CPs embedded in VE layers. Besides, the FS has been used to define the local modes of the VE layer in the damped FV analysis. Based on this, the required FS terms are reduced with the a/h. In other words, increasing the plates' thickness required more terms of the FS. Additionally, it has been noted that the stress behavior has been affected by the evaluation frequency. This is related to the fractional derivative VE model's frequency dependence. As seen in Fig. 23, in case 2 for mode one, asymmetric vibration is revealed, and the obtained vibrational frequencies in case 1 enhanced by the damping effect of the local vibration frequencies of the significant VE layer were much lower than those of case 2. However, Li et al. [304] presented an FV and static analysis of LCPs by utilizing the IGA based on NURBS by the TSDT. However, a penalty methodology was used to enforce the essential BCs. It has been determined that the results of raising the NURBS elements’ order and improving the mesh quality congregate with previously published data. Although the NF results were slightly differences from results obtained earlier by IGA CPT. Through utilizing FS method, Amoushahi and Goodarzian [305] explored the stability and FV responses of LCPs with and without strip delamination under hygrothermal effects. The obtained results show that at various moisture concentrations and temperatures, increasing the thickness of the plates caused a rise in the NF values. Besides, increasing the reduction rate by raising the delamination area when the delamination gets closer to the mid-layer causes a reduction in the non-dimensional frequency (NDF) at SCSS and SCSF BCs. The variation in NDF of LCPs subjected to SSSS through various delamination places at elevated moisture concentrations is illustrated in Fig. 24, in which it is revealed that the NF reduced due to changing the delamination location from the edge side toward the plates’ center.

Fig. 23

The first mode shape of CPs for various cases case 2 & case 3 LW⁴_∑4,4 [303]

Fig. 24

The variation on NDF of LCPs subjected to SSSS through various delamination place at elevated moisture concentration ∆C = 0% [305]

Zhai and Liang [306] employed HP to define the equilibrium equations for three vibrational models. They solved them by NvS closed-form to investigate the linear FV of cross-ply CPs with sandwich structure, the core of which is assumed to be VE. It is seen that by increasing the a/h, the frequency has reduced dramatically at a/h up to 10, and then it keeps reducing gradually. Increasing the VE core thickness to the total plate thickness up to 0.4 caused a reduction in the frequency, but when increasing from 0.4 to 0.9, it noticed an increase in the frequency. Moreover, Chanda and Sahoo [307] developed a plate that was defined as an interlaminar transverse shear stress continuous (TSSC) plate theory based on the CPT to model the deformation behavior and utilized a trigonometric function to determine the NL behavior of the model. This study aimed to investigate the FV and stress behaviors of LCPs with EF, through FE and NvS solutions, and HP obtained the required EOM. It is noticed that the stiffness of the foundation has a remarkable impact on the NF, vibrational amplitude, stress, and deformation. Furthermore, increasing the foundations’ stiffness causes an increase in the value of the FF, while it causes a reduction in the vibrational amplitude. The enhancement of PF on the structural response was noticed to be greater than that of WF. However, Chanda and Sahoo [308] employed the trigonometric ZZT to examine the FV and the transient performances of cross-ply LCPs. Besides, the HP has been used to drive the equilibrium equation, and the analytical solutions were determined by NvS combined with the NTI scheme. As shown in Fig. 25 (a) increasing the number of layers causes an increase in FF associated with the span-thickness ratio. Besides, the maximum FF has been recorded by Gr/epoxy (Ep) and the minimum by G/Ep. The amplitude of the FV behavior after a brutal blast depended on the time the load was delayed. The NF and the HO vibrational modes were highly influenced by material characterizations and geometrical characterization, such as a/h, core thickness, density, and modular ratio. Likewise, Sciuva and Sorrenti [309] employed the refined ZZT with the C0 quadrilateral plate element to analyze various sandwiches and LCPs. The FF error has been defined analytically by utilizing the FEM.

Fig. 25

a The deviation in normalized NF associated with number of layers of the laminated CP, & b normalized NF of several LCPs [308]

Thakur et al. [310] employed the NPSDTs to study the enhancement of the hygro TE on the dynamical responses of one/two-fold folded LCPs. Besides, for studying the FV, steady-state and transient responses of the structure, the C0 FEM has been utilized, and other methods such as the Green–Lagrange strain (Green-LS), Lagrangian approach, and the NTI methodology were implemented to complete the analysis successfully. It has been revealed that the enhancement of the folding on the NF of cross-ply was higher than that of angle-ply, and it is noticed that in case of changing the crank angle from 90° to 120° in the two folded plates has a remarkable effect on several values of moisture, temperature, and NF. From Fig. 26 it can be observed that the concentrated load causes higher values of deflection amplitude compared to those subjected to UD load in both cases of BCs, and the frequency is reduced due to the reduction in the plates’ stiffness caused by increasing the temperature. Moreover, Zaoui et al. [311] employed the 2D-HSDT to develop a computational methodology that focused on investigating the FV performance of developed CPs made of FGM ZrO2/Ti-6Al-4V subjected to SSSS BCs in a TEs. Besides, the EOM has been determined HP and solved based on NvS to define the NF. The obtained results show that increasing temperature causes a decrease in the NF, which is considered a consequence of reducing the E by raising the temperature. Besides, the reduction in NF that occurred in lower modes is less than that occurred in higher vibrational mode. Likewise, Adhikari and Singh [312] employed the HSDT to investigate the FV and dynamical characteristics of FG-CNT CP based on EF. Besides, the GEs were obtained by utilizing the energy principles and solved by implementing the FEM, while the force responses were defined by using the NTI technique. As demonstrated, increasing the thickness ratio causes an increase in the NDNF. However, increasing the a/h causes a reduction in the NDNF, and FG-X recorded the maximum values. Besides, it was revealed that increasing the V_F of CNTs enhanced the NDNF. Additionally, as a remarkable outcome, it has been observed that subjecting the plates that consist of FG-X to SCSC BCs at EF parameter Ks > 120 does not impact NF.

Fig. 26

The enhancement of various BCs & harmonic load on the frequency performances of [0°/90°/0°] LCP subjected to thermal load & with material characterizations MM3; a UD load subjected to SSSS BCs, b central concentrated load subjected to SSSS BCs, c UD load subjected to CCCC BCs, & d central concentrated load subjected to CCCC BCs [310]

Adhikari and Singh [313] employed the HNP Quasi-3D approach for investigating the dynamic behavior of LCPs and the Lagrange equation has been used to determine the GEs, although the forced responses were obtained by implementing the NTI approach. It has been revealed that the dynamic behavior critically affected by the temperature variation in which, a reduction in NF has been noticed due to considering the temperature effect. However, the reduction in frequency values of thick plate was much lower than those of thin plate owing to the extra losses in thin plates’ stiffness. In addition, raising the temperature cause to reduce the frequency. Wang et al. [314] analyzed the FV of the CPs with honeycomb core made of foam based on the HSDT. The HP was used to define the GEs, which solved analytically. The obtained results shows that changing the modulus of the core causes a shifting in the initial symmetric frequency, and an improvement in the associated vibration pattern. Besides, Hadji et al. [315] performed an analytical FV analysis of the sandwich plate with a ceramic core and FGP face sheet, as shown in Fig. 51, subjected to different BCs. Also, the HPs have been employed for defining the EOM, while the kinematic relation of the FGP has been determined using the hyperbolic shear displacement theory (HySDT). The results show that by increasing the V_F of the porosity, the NDNF has decreased in all porosity distribution models. Besides, by increasing the sides-to-thickness ratio, the NDNF has increased until it reaches a specific value, then it remains constant in the various porosity distribution models (Fig. 27).

Fig. 27

The schematic of a sandwich plate consist of a ceramic core and FGP face sheet & the porosity distribution models [315]

2.1.2 Analytical Solution with Considering the SDE

Due to the significant importance of discernment of the mechanical behavior of NMS structures, the attention increases widely towards the SDE on the FV analysis and other mechanical responses. Several NLET approaches were considered and employed in the literature to explore the enhancement of Length scale parameter (LSP) or NLC parameters in NPs. Notably, Salehipour et al. [266] presented an exact closed-form analysis for FV of FG NP/MSPs subjected to SSSS BCs found on the MCST and 3D elasticity theories. Besides, the HP has been utilized to define the EOM and BCs. The enhancement of the LSP and the material gradient on NF were examined and presented numerically. The results show that the GI on shear stiffness can be neglected at minor LSP. In addition, reducing the GI or increasing the LSP causes an increase in the NDF. Cutolo [316] presented an analytical solution for FV and stability analysis of thick NPs made of FGM subjected to WPF and subjected to various BCs. The third NLET has been considered and employed, although the TSDT has defined the GEs. Besides, a sensitivity analysis has been performed to prove the efficiency of NLC parameters. It is revealed that the NLC parameter impacts the mechanical response, and a reduction in FR was highly pronounced at higher frequencies due to raising the NLC parameter. Moreover, Naghinejad and Ovesy [317] investigated the VE FV behavior of NPs throughout the FEM based on the two-phase-NLC integral theory. Also, for developing the FEM, the principle of total energy has been utilized. However, the formulation has been generated by implementing the CPT. The obtained results show that the complex NF (real and imaginary) decreased by producing the NLC parameters. While the imaginary eigenvalues have increased by producing the VE parameters, the opposite is correct for the real eigenvalues. Besides, at different values of NLC parameters, the NF shows a relative increase by increasing the cutout place length in NS. Furthermore, Pham et al. [318] investigated the FV of the FGP NNU NPs based on WF, as shown in Fig. 28 by implementing the FEM based on FSDT and by using ES-MITC3 element. Furthermore, the NLT has been used to examine the SSE. Also, the GE has been generated based on the HP. The obtained results illustrate that the MITC3 element gives less accurate results than those obtained using the ES-MITC3 element. Besides, the WF enhance the annular NPs’ stiffness. In addition, significant observation shows that increasing the NLC parameters causes a drop in the NPs’ rigidity. Another significant outcome of increasing the NLC parameters is that it reduces the frequency rapidly.

Fig. 28

The schematic of FGP-NU annular-NP based on WF [318]

Moreover, Bui et al. [319] employed the MKI meshless method combined with CPT to compute the NF of the LCPs. Besides, the enhancement of scaling factor, fiber orientations stiffness ratio, and another parameter on the NF were investigated. Besides, Le et al. [320] employed the IGA combined with MCST to examine the small size-effect analysis for linear and NL performance of an MSP structure made of porous metal foam. Besides, a seventh-order distribution function has been utilized to define the constitutive relations between the stress–strain and the modified couple stress. Additionally, the equilibrium formula was defined using the HPs. As shown in Fig. 29, the maximum frequency values were obtained by subjecting the plates to CCCC BCs with two different patterns, uniform and symmetric.

Fig. 29

The improvement of various BCs on the NF of Steel foam MSP structure a Uniform & b Symmetric [320]

Furthermore, Tran et al. [321] generated an FE formulation by utilizing four unknown SDT incorporating with NLCT to examine the FV and bending behaviors of FG NPs based on EF, as presented in Fig. 30 (a). Besides, the Hermitiain interpolation and Lagrangian functions were used to develop 4-node Quadrilateral elemnt. While the EOM was defined by HP. The obtained show that the FF was reduced due to softening the NPs by increasing the b/a while increasing EF stiffness caused an increase in the FF. Besides, it is seen from Fig. 30 (b and c) that the maximum FF has been achieved by subjecting the NPs to CCCC BCs. However, increasing the V_F causes a reduction in the FF until a specific value remains constant, and the opposite is correct regarding increasing the a/h. Singh et al. [322] performed an analytical solution by utilizing the LS support conditions through the extended Kantorovich method (EKM) combined with FS, and the HPs have been used to define the GE to investigate the VE FV analysis of the schematic shown in Fig. 31. Which consists of an orthotropic plates made of IPFG material and integrated with PE sensors. It is worth mentioning that a numerical solution has followed the analytical solution, and various examples were presented and solved. The results show that the gradation indexes enhanced the NF, although this enhancement directly depends on the mechanical support conditions utilized. However, Sahmani and Ansari [323] employed the NvS to solve the FV of FGM MSPs subjected to SSSS BCs after utilizing various methods in which the material characterizations were assumed based on the MT homogenization technique. Besides, the SGTE has been used to develop an NLC HSDT of plates with three different material LSP, and the HP has been utilized to define the GEs. The results show that the maximum NDNF was obtained by plates based on SGET, while the lowest was obtained by implementing the CPT. It is remarkable to mention that the various values of h/l in the case of utilizing the CPT have no evident influence on the FV. Moreover, Rong et al. [324] developed a novel analytical approach for analyzing the FV, forced vibration, and stability of NPs with rectangular shapes based on NCET. The analytical model has been generated by implementing the HP, KPT, and Eringen’s NLCT. Besides, the mechanical characterizations were obtained by eigenvalue analysis and the expansion of Eigen-function with the absence of trial functions. Additionally, the benchmark results were presented and validated.

Fig. 30

a The schematic of FG NPs based on two layers EF, (b & c) the variation of dimensionless fundamental frequency (DLFF) of the FG NPs at various BCs, (b) at k_w1 = k_s1 = 10 & μ = 1, (c) at a/h (n = 1) & h = 10 as a V_F power-law index (n) [321]

Fig. 31

The schematic of smart plate made of IPFG [322]

Zare et al. [325] presented a novel analytical solution that analyzed the NF of a Al/Al₂O₃ FG NPs with rectangular geometry, and subjected to various BCs. Besides, the NLET has been implemented for size dependency according to Eringen’s differential form. From the obtained results it is noticed that the maximum NDF has been recorded at SCSC BCs and the minimum at SFSF; this is since more significant constraints at the edges cause greater flexural rigidity that in consequence will lead to obtaining higher frequency response. In contrast, the NDF has the same trends in all various types of BCs. Ezzin et al. [326] employed the general stiffness matrix method (GSMM) to investigate the FV of an FG MEE CP by assuming the material features are continuously graded along the thickness direction. Furthermore, Tseng [327] presented an analytical model for modeling the FV of moderate-thick LCPs by employing the FSDT and NL optimization solver. The model has been distinguished using the Kirchhoff circuit with wave digital filters. In a novel investigation published by Khalafi and Fazilati [328] they developed an IGA approach to investigate the FV and linear flutter responses of skew LCPs and finite squares. Besides considering the enhancement of shear stresses, the FSDT has been implemented to generate the geometry of the plates. The enhancement of varying ply curvilinear fiber orientation angles subjected to various BCs and geometries was examined. Moreover, Su et al. [329] implemented the FS methodology to examine the FV of LC with four parameters FGM sector plates subjected to various BCs as presented in Fig. 32 (a). Also, the FSDT has been implemented to include the enhancement of rotary inertias and shear deformation and to solve the exact solution. The RRM has been utilized based on the energy function of the plates. The results in Fig. 32 (b) show that the non-dimensional (ND) frequency parameter increased for specific values of the elastic restraint parameter when it increased. Accordingly, the elastic edges were found by setting the appropriate values of the springs’ stiffness. Besides, increasing the sector angle caused a reduction in the plates’ NF. In addition, it has been remarkably noticed that the deviation of FGM sector plates’ FNF with CCCC, SSCC, and SCSC BCs was similar; Fig. 32 (c) illustrates the deviation at SCSC BCs examined. The flowchart proposed by Motamedi et al. [330] for investigating the stability as well as the FV analyses by a novel MF methodology is illustrated in Fig. 33.

Fig. 32

a The sector plate structure, b the enhancement of elastic resistance stiffness on the FGM sector plate frequency parameter in the second mode & c the deviation of FF of the FGM plates at various n at SCSC BCs [329]

Fig. 33

The flowchart of investigating the stability as well as the FV analyses by a novel MF methodology [330]

Hosseini-Hashemi et al. [331] utilized an exact analytical approach to examine the FV of thick circular/annular Mindlin NPs made of FGM, as shown in Fig. 34 under various BCs, using the Eringen NCET. It has been observed that at a high number of modes or when the atom interactions increased, the enhancement of the NLC parameter increased. Besides, as seen in Fig. 34, the impact of the NLC parameter on the FNFs increased due to the increase in the GI. Also, increasing the plates’ rigidity will enhance the effect of NLC parameters on NF. Moreover, Sobhy [332] employed SSDT to analyze the bending and FV behavior of NPs embedded in two parameters: thermal and mechanical loading conditions subjected to various BCs. The Eringen NLC differential constitutive relations have been implemented to reformulate the SSDT, while the analytical solution has been used to solve the EOM. Figure 35 presents the enhancement of various side-to-thickness ratios and lengths on the FV of NP; it is seen that CCCC recorded the maximum FV values at various side-to-thickness ratios. Besides, the greatest number of modes combined with increasing the length cause to rise the FV until it reaches the maximum value then it remains constant.

Fig. 34

The schematic of FG annular NP& the dimensionless frequency (DLF) in 2D/3D plots with NLC parameter for various values of GI [331]

Fig. 35

a The FV Vs. side to thickness ratio subjected to different BCs, & b FV Vs. square NP length at different number of modes [332]

Zenkour and Arefi [333] explored the thermos-electromechanical bending and vibrational analyses of an FG PE nanosheet based on Visco-PF, by utilizing the NLET combined with CPT. the nanosheet basic formulations were obtained, and by using the HP, the EOM was obtained and solved with an assumption founded by an unknown transient field utilizing the trigonometric functions. A reduction in FF was caused by increasing the NLC parameter for all various values of the nonhomogeneous index. However, increasing the nonhomogeneous index caused a reduction in the FF. This article came with a remarkable outcome: materials and nanoscale structures have greater stiffness than large-scale structures such as micro and macro. Furthermore, Van et al. [267] presented a refined NLC IGA combined with an RPT model for investigating the FV and bending behaviors of multilayer FG-GPLRC NPs with various dispersal patterns. It has been obtained that the plates' stiffness improved by the GPLs reinforcement, and several benchmark problems were obtained based on the results of the SDEs' of NPs defined by implementing Eringen's NLET. It is seen from Fig. 36 that the UD dispersal is considered an isotropic homogeneous NP. However, FG-O and FG-X are symmetrically rendered to the plate mid-surface, excluding FG-A. Besides, the NF was enhanced by adding the reinforcement NFs into the NPs, although the NF was reduced due to the increase in the length-thickness ratio (l/t) and NLC parameters. The highest and the lowest NF were obtained in FG-X and FG-O, respectively. Else, Murmu and Adhikari [334] investigated the NLC parameter effect on the vibrational behavior of bonded double-NPs connected through an enclosing elastic medium, as shown in Fig. 37, where multiple Gr sheets were dispersed in the NCs. The FV has been defined through implementing the NLET, and the NF has been obtained by analytical methodology. It has been seen that fewer SSEs during the asynchronous modes have been obtained due to increasing the stiffness of coupling springs. Besides two various types of vibration, the system has experts. The first one combined with low frequencies is the synchronous vibration, and the one combined with high frequencies is called the asynchronous vibration. However, the stiffness affects the asynchronous vibrational modes, where it causes a reduction in the SSEs. In contrast, an absence of any effect was observed in the case of synchronous vibrational modes. Nonetheless, Afshari et al. [335] explored the size dependency enhancement on FV and the stability behavior of FG-GNP MPs. The MP model has been generated based on MCST combined with SSDT, and the material characterizations were defined by employing the HT model associated with ROM. Besides, the GEs were used to define the GEs that were solved further by NvS. It is revealed that the NF is reduced by increasing the MPs’ length, which is caused by the reduction in the structures’ rigidity. Besides, the MP stiffness increases due to increasing the MP thickness. Raising the LSP causes an increase in the NF ascendingly.

Fig. 36

The E of GPLs/polymer NCs with ten layers [267]

Fig. 37

a The coupled Gr sheet system in PM environment, b Double NPs system connected through elastic medium, c mathematical representation of the system & the coupling springs presenting the elastic medium, & d NPs dispersed in polymer matrix in NCs [334]

In addition, Zhang et al. [336] employed the SGET combined with Ritz SDT to develop efficient size-dependent MSPs with EF and made of FG model as illustrated in Fig. 38 (a), and analyzed its FV and static behaviors. Besides the GEs defined by employing the HP, an analytical solution has been presented, and homogenization methodologies were used to define E based on by Balabed et al. [337]. As can be notice from Fig. 38 (b) and (c), reducing the a/h or raising the n caused lower NF and the classical model at h/l = 1 especially has less influence on NF than the influence caused by the TS deformation. Additionally, the relationship between the Winkler parameter and NDF is linear, where it is NL between the Pasternak parameter and DLF. Additional, to explore the effect of various micromechanical models such as LRVE, Reuss and Tamura on the FV on inhomogeneous NPs founded on WF and PF, Shahsavari and Karami [338] employed the Quais-3D SDT with general strain gradient theory (GSGT) for modeling the NPs model which consist of FGM, and the exact solution was obtained by NvS. Besides, it was seen increasing the V_F index cause to decrease the NDNF dramatically at all various homogenous structures. In contrast, increasing the NLC parameter reduce the NDNF values at various V_F index. A classification of linear analysis by analytical solution with/without considering the SDE based on obtained results presented in Table 1. And Table 2 present classification of linear analysis by analytical solution with/without considering the SDE based on CPs schematic.

Fig. 38

a the schematic of FG MSP based on WPF b the enhancement of n & a/h on DLFF & c the enhancement of ND EF parameters on the DLFF at various h/l values [336]

Table 1 Classification of linear analysis by analytical solution with/without considering the SDE based on obtained results

Table 2 Classification of linear analysis by analytical solution with/without considering the SDE based on CPs schematic

2.2 Linear Analysis by Numerical Method

2.2.1 Numerical Solution Without Considering the SDE

Employing the CPT in CPs analysis is insufficient; since the TS deformation is ignored. While, one of the considerable challenges in CPs is that, the ratio of in-plane E/TS modulus is moderately great due to significant variation between the elastic characterizations of the fiber filament and the matrix materials. In other words, the mathematical difficulty in modeling and analyzing the dynamical responses of CPs can’t be solved by implementing the CPT. Consequently, there is an essential for employing arithmetical solutions such as FEM and finite difference method (FDM). For example, Malekzadeh et al. [352] applied the transformed DQM to analyze the FV behavior of the schematic in Fig. 39, which consists of two PE layers with FG multi-layer NCs reinforced with GPLs. Besides, the NCs’ mechanical characterizations were defined by utilizing the modified HT. Based on the FSDT, the HPs and Maxwell’s equation were implemented to drive the GE. Also, the method was numerically validated. The results show that the frequency parameters increase due to the increase in the PE to core thickness ratio In contrast, the FF shows a reduction by increasing the eccentricity parameter. Additionally, the external electric voltage causes a monotonic reduction in the frequency parameters in all various GPL dispersal arrangements. In addition, the flowchart of the developed transformed DQM is illustrated in Fig. 40. Furthermore, Wang et al. [353] employed the p-Ritz approach to investigate the FV performance of SSPs made of core in-between two laminated face sheets, and a code has been developed. Various BCs were implemented, and the obtained results show that the maximum frequency parameters were recorded by implementing the CCCC BCs. Besides, in CCCC BCs, maintaining the skew angle (SA) constant and increasing the number of facing layers enhances the frequency parameters, and this observation has been noticed only in CCCC BCs. However, Xiang et al. [354] employed the MF global collocation approach to analyze the FV of FG sandwich plates. Furthermore, the sandwich NF with various side-to-thickness ratios, thickness ratio of each layer, BCs, and material properties were observed. Besides, nth-order SDT has been employed. Moreover, Yang et al. [355] employed the CRM to examine the stability and FV behaviors of FG porous NC plates reinforced by GLPs. Else, for defining the mechanical characterizations of the porous NCs, the modified HT micromechanics mode, open-cell metal foams’ mechanical properties, as well as the extended rule of mixture. The output of this study shows that rising the porosity coefficient caused a reduction in the FNF. Though, the GPLs enhanced the FNF and other mechanical properties. Also, the reduction in the FF caused by increasing the thickness ratio is evidence of the significant drop in the plate flexural rigidity.

Fig. 39

The schematic of the FG GPLs reinforced composite eccentric APs implanted as a core in between PE layers [352]

Fig. 40

The flowchart of the developed transformed DQM [352]

A biocomposite SSPs laminated with a face sheet made of bamboo fibers and PLA core, has been investigated by Gwande et al. [356] to examine the enhancement of temperature and moisture on its FV characterizations. The investigation was performed by utilizing the HSDT. The enhancement of SAs with various BCs has been considered. The results indicate that the CP structure has significant potential for application in enormous environmental conditions. Besides, the enhancement of SA on NDF at various BCs. It is seen that the maximum NDF values at various SAs were recorded by subjecting the CCCC BCs. Additionally, increasing the SA causes greater values of NDF. Hence, Moreira et al. [357] implemented the LW model to investigate the FV, as well as the static PE CPs. The obtained results show that the NF was marginally overestimated by using the user element subroutine, and the obtained NF values by both models of user elements were identical. Furthermore, the annular sector and the sector plates with various FGM distribution and LC cases were considered by Civalek and Baltacıoglu [358] to examine its frequency response. The modal analysis GEs were obtained and solved by utilizing the DSCM combined with harmonic differential quadrature (HDQ). Meanwhile, the FSDT and Love’s conical shell methodologies were employed to obtain the annual sector plate equations. It has been seen that increasing the angle sector causes a reduction in the FF and the frequency parameter of the plates subjected to CCCC BCs, and the same effects have been noticed by increasing the radius ratio. Nevertheless, Ansari et al. [359] employed the TSDT combined with the VDQ to investigate the FV as well as the post-buckling response of porous plates with various geometries and made of FG-GPL. Besides, the material characterizations were determined using the HT combined with closed-cell Gaussian Random. It is revealed from Fig. 41 that, increasing the porosity coefficient at a certain level of PPB will cause a reduction followed by an increase in the NDNF. Also, it is noticed that in post-buckling conditions, the enhancement of the porosity coefficient for the lower vibrational mode is much more significant than that for high modes. Additional results show that by increasing the internal porosity density, the NF reduced as a consequence of reducing the total stiffness of the porous plates. While implementing the VKT, nonlinearity caused an increase in the frequency of the porous plates with higher stiffness under post-buckling conditions.

Fig. 41

The enhancement of porosity coefficient on the first three values of frequency of the FGPPs with type 2 of GPL dispersal & porosity type 1 at Ω_GPL = 0.6%, β = 45° [359]

Additionally, Karimi et al. [360] employed the Ritz solution combined with the polynomial series expansion, TSDT, and Reddy SDT to investigate the FV of two various types of CPs (variable stiffness composite laminated and hybrid composite laminated), which were subjected to interaction with the sloshing fluid as shown in Fig. 42. The obtained results show that the overall stiffness in both plate types is reduced by increasing the thickness ratio, which consequently leads to a decrease in the FF. However, at a high thickness ratio, the amount of variation in the FF has reduced. On the contrary, it can be seen that the eigenvalue increases due to increasing the a/h.

Fig. 42

The geometrical schematic of a CPs; a coupled to the fluid, b plate mad of HCL & c plate made of VSCL [360]

Seba and Kebdani [361] presented an arithmetical simulation as well as an experimental study that focused on investigating the FV behavior of an LCP containing elastoplastic layers and subjected to various BCs through implementing the FEM. It is seen that adding more elastoplastic layers leads to a reduction in the NF, which, in consequence, will reduce the von Mises stress. Besides, the most outstanding NF values were recorded by applying CCCC and the lowest by CFFF. Also, raising the a/h, length-to-width ratio (a/b) reduces the NF whether the elastoplastic layers were contained, and increasing the thickness ratio causes the same results. Subsequently, Das and Niyogi [362] examined the FV behavior of a cross-play LC folded plate made of Gr/Ep by employing the FEM and the Mindlin isotropic plate approach. Besides, studying the enhancement of crack angles, stacking sequence, and plate thickness on the FF of a fold-folded plate subjected to thermal load shows that when the plate thickness increased, the plate stiffness increased, and the FF dropped as a result. Also, increasing the plates’ overall thickness can resist a higher thermal load. Besides, by considering the same plate structure while changing the thermal load to a hygral load, it is seen that the same mode shape was achieved compared to the results obtained by a high thermal load, and increasing the hygral load has no significant effect on changing the mode shapes. Likewise, Duc and Minh [363] investigated the FV of FG-CNTRC plates exposed to cracks using the PFT combined with Shi’s TSDT. Besides, the FEM has been used to solve the cracked equation around the cracked area. As seen from Fig. 43 (a), the frequency decreased due to decreasing the plate hardness caused by increasing the edge ratio L/H. Besides, by increasing the length of the crack, the elastic energy will be reduced, which in turn will remarkably reduce the FV frequency parameter. The hardness is noticed to be proportional to the elastic energy. Additionally, it is revealed from Fig. 43 (b), that when the thickness ratio increases, it leads to a reduction in plate stiffness, which causes lower FV frequency values. Moreover, Vidal et al. [364] employed the variable separation method (VSM) to investigate the FV of LCPs, and the FEM has been used for the in-plane description. Furthermore, Fantuzzi et al. [365] performed a simulation analysis on LCPs with various geometries by utilizing the linear approach of moderately thick plates, and the strong form of FEM as well as WFm was used for solving the problem. The results obtained of the first two NFs of circular LCPs with symmetric lamination by utilizing different types of elements revealed that fewer adequate elements were found through Quad 4 element from Straus7. Further results show that the commercial FE codes achieved more remarkable accuracy error than the substantial form FEM in symmetric LCPs, and the NF was found to reduce due to increasing the plates’ thickness at various element types.

Fig. 43

a The enhancement of cracks on the NDF & b the enhancement of cracked’ length & thickness on the first FV frequency parameters [363]

The FV analysis of CP, which has an axially moving multiscale with a thermal effect, has been examined by Marynowski [366] by generating a linear arithmetical model in the form of an equilibrium state equation. The investigation was run for (0.1–1) wt. % multiscale FRCPs in temperature range (35–200) °C. Otherwise, the HT model has been utilized to define the relationship between the tested plates' orthotropic coefficient and the fiber's V_F in multiscale composite plates. It has been revealed that raising the internal damping causes an extension of the frequency range among consecutive NFs and enlarges the critical transport speeds of all tested plates. However, increasing the fiber's V_F reduces the tested plates' critical transport speeds. The computational representation of the implemented multiscale composite mechanical properties is illustrated in Fig. 44 [367]. Furthermore, Kuma et al. [368] employed the RRM to investigate the FV and buckling performance for composite skew plates that exposed a pre-buckling. The obtained results show that raising the edge restraint and the SA caused an increase in the CBL and the pre-buckling V_F. Besides, by increasing the SA, the NDFF slightly increased, and the maximum values at various SA were obtained by applying the CCCC. However, increasing the NDL in-plane load causes a reduction in the NDFF.

Fig. 44

The computational representation of the implemented multiscale composites mechanical properties [367]

Besides, Sekhavatjou et al. [428] employed the higher ZZT combined with the bubble complex finite strip method (BCFSM) to investigate the FV and stability behaviors of LCPs, which consist of angle-ply and cross-ply laminates. The enhancement of a number of layers, fiber orientation, b/h, various dimensional ratios on the NF, and dynamic behavior were studied numerically. Furthermore, Dastjerdi et al. [369] investigated the enhancements of a/h, V_F waviness and the distribution of CNT on the FV as well as forced vibration of sandwich CPs made of FG-CNTRC with an isotropic core and based on PF, by employing the moving least square (MLS) MF and FSDT. The obtained results shows that, increasing the b/a ratio cause a reduction in frequency parameters, though, it increase by the EF. Besides, the highest values of frequency were recorded in V-CNTRC. In addition, Valizadeh et al. [370] investigated the flexural FV, bending, stability as well as supersonic flutter responses of FGM plates (Al/Al₂O₃ and Al/ZrO₂) subjected to various BCs by implementing the NURBS approach (IGA-quadratic and IGA-cubic). Besides, the material properties were computed by ROM and Mori-Tanka and the plates kinematics formulated based on the FSDT. The enhancement of GI, a/h, and the plates’ thickness on the mechanical responses were considered. The obtained NF shown to be in good agreements with previous published data by various methods such as kp-Ritz and HSDT at all various BCs. Additionally, Thai et al. [371] examined the static and FV behaviors of LCPs Reissner–Mindlin plates based on FSDT and subjected to various BCs, by implementing the NURBS-based IGA (quadratic and cubic). Besides, the shear locking in the stiffness was treated by adopting a stabilization technique. The obtained NDNF shown to be in good agreements with previous published data by various methodologies. Meanwhile, Thai et al. [372] proposed a developed inverse trigonometric shear deformation plate theory (TSDPT) for LCPs, and by utilizing the HSDT the necessities of shear correction factors were eliminated. Besides, based on the developed approach the static and FV responses were examined by implementing the IGA. The FV analysis has been accomplished for various types of plates. For instance, by analyzing a square LCP with [0°/90°/90°/0°] subjected to SSSS BCs it was seen that the values of NF obtained by 17 × 17 meshes were similar to those of 13 × 13 meshes. In contrast by analyzing the sandwich CPs by employing FSDT and HSDT, a remarkable gap between the frequency values obtained by both methods was revealed for thick plates which conclude the necessity to use the HSDT. However, increasing the face sheet thickness ration cause to reduce the NF. Likewise, the obtained results shows to be in good agreements with previous published data by analytical and numerical approaches. Thai et al. [373] examined the static and FV behaviors of LCPs based on HSDT by using the IGA (quartic, cubic and quadratic), with eliminating the shear correction factor by introducing TSDT with C¹-continuity. In the FV analysis three and four layer laminates were considered with [0°/90°/0°] and [0°/90°/90°/0°] respectively. The obtained results were compared with previous published data by exact, global and local all based on HSDT, and moving least square, RBF wavelet all based on FSDT.

2.2.2 Numerical Solution with Considering the Dependency Size Effect

The NLC theories have the ability to accomplished modeling the response of NSs effectively though, consider moderately simple in formulation. Certainly, the NLC theories, incorporating the LSP in the constitutive clarification for the purpose of capturing the SDE by Raghu et al. [374]. For instance, Wan et al. [375] developed a Q8 plate element based on the FSDT to investigate the FV and static behavior of LC Reissner-Mindline plates, and the stiffness matrix has been generated through implementing the first order of Taylor’s expansion. It has been observed that by decreasing the integration QSR with the presence of standard integration QSN in the isoparametric Q8 element, a shear-locking phenomenon was obtained. Besides, a small effect caused by the correction factor has been observed. In general, outstanding responses of the distorted meshes likewise endorse the applicative capability of this methodology for problems with complex domains. Furthermore, Wu and Hsu [376] investigated the FV and statically behaviors of SSSS FG MSPs, as shown in Fig. 45, by utilizing a 3D-WF-consistent couple stress theory (CCST)and the FS combined with Lagrange/Hermite polynomials with a further improvement of LW C0, C1 for 2–3 nodes quadratic and cubic FEM. Besides, the MCST-based on FEM has been employed, and the results were compared with CCST results. In the extensional modes, increasing the LSP caused an increase in the differences between the results obtained by both methods. However, the frequency parameter obtained by CCST was lower than those obtained by MSCT by 4.5%. In the FV analysis, both methods' frequency parameters of the obtained flexural modes were similar.

Fig. 45

a The schematic of elastic FG MSP under uniform external mechanical loads, & b local & global thickness coordinates inside the plates’ layers [376]

Benjeddou et al. [377] accomplished an FV of SSSS piezoelectric sandwich plate (PSP) adaptive plates by using FSDT quadratic non-uniform (QNU) electric potential followed by numerical validation of various hybrid plates made of graphite/epoxy cross-ply and found to be identical similar to the 3D coupled exact solution. The FV investigation of arbitrarily shaped LCPs has been accomplished using the FSDT, followed by radial basis function differential quadrature (RBF-DQ) theory [378]. The results show that employing the RBF-DQ is much more efficient than the traditional DQ approach, specifically in excellent subdomain dimensions in composite material. Moreover, Guo et al. [379] employed the IMLS-Ritz meshless methodology to investigate the FV of laminated composite quadrilateral plates reinforced with GPLs. Besides, the FSDT has been utilized to define the functional energy formula. In addition, all required material characterizations were defined by the modified HT model associated with ROM. Various BCs were subjected and examined, and various GPL distribution patterns, the enhancement of GPLs size and geometry were presented in the form of a/b defined as l_GPL/ w_GPL as well as a/h defined as l_GPL/ h_GPL. The obtained results shows that by rising the l_GPL/ h_GPL up to a 1000, the NF has increased while further increasing has no significant effect on the NF. On the other hand, raising the l_GPL/ w_GPL up to 10 caused a reduction in the NF, as illustrated in Fig. 46. Once the number of layers is sufficiently remarkable, the NF values turn to be constant. Besides, Chiker et al. [380] explored the enhancement of NF distribution on the FV performance of composite laminated plates reinforced with a multilayered FG-CNT. The authors determined the NF by employing the FEM, while the ROM was utilized for driving the material features, and the FSDT combined with the Lagrange equation for obtaining the GE. The obtained results show that increasing the number of layers to greater than 20 will not cause any modifications in NDF for bath cases of plats subjected to SSSS as well as CCCC BCs. Besides, by examining the enhancement of the orientation angle of CNT, it is seen that up to 45° the NDF raised but when increasing the orientation angle from 45° up to 75° the NDF reduced.

Fig. 46

The first order NF of the quadrilateral plates reinforced with GPLs with various geometries ratio [379]

Chen et al. [381] examined the FV and linear bending behaviors of a CP made of an FG-CNT and based on EF by implementing various MF methods mainly based on the improved Reddy type TSDT. Besides, the HPs and the SMKI were utilized to drive the EOM and discretization, respectively. However, the benchmark example has been used to study the accuracy and effectiveness of the proposed method. The obtained results show that the plates’ stiffness gradually increased due to the foundation coefficient and the CNTs’ V_F rise. Moreover, Fang et al. [382] investigated the vibrational as well as the thermal stability analysis of Reddy MSPs, which are made of a rotating porous FG–GPLs reinforcement NC. The GE of motion by the MCST has been defined by utilizing the TSDT of Reddy as well as Lagrange’s equation. Besides, the material features of the MSP were defined by implementing the HT and ROM. In contrast, the Galerkin has been adopted as the arithmetical model. It has been observed that the CCCC BCs enhanced the FF and the critical stability temperature. Also, the rotating motion may have a negative effect. Additionally, the centrifugal stiffening and the SSE have a significant impact on the stability and vibrational behaviors. Furthermore, Kablia et al. [383] investigated the FV behavior of two types of FGPPs with sandwich structure and deviation in porosity dispersal by utilizing a modified mixing law. HP have defined the EOM. Also, the authors presented an original HSDT for the FV of sandwich plates. The results were validated with previously published data from other HO theories and three-dimensional theory of elasticity (3D-ET). It has been realized that the deviation in porosity dispersal is directly associated with the inhomogeneity factor, and the NF has an inverse relation with the inhomogeneity factor. Additionally, the NF is highly influenced by the variation of porosity dispersal rate and the increase in the a/h a/b. Besides, Lazar et al. [384] performed an FV analysis of CP structures made of FGM-CNTs by utilizing the exponential function approach, and the ROM has been used to define the CNTs’ mechanical properties. The exponential law has been employed to implement the FSDT in the FEA formulation. Figure 47 shows the first six vibration modes of composite FGM-CNTs plats subjected to CCCC BCs with various reinforcement distributions FG-X, FG-O, and FG-A. It has been observed that the enhancement of the exponential law on NF values is much higher than the enhancement of the n. Besides, raising the V_F of CNTs leads to a rise in the NF values and a decrease or increase in NF values can be obtained by decreasing or increasing the width or length of the plate, respectively.

Fig. 47

The first 6 modes of FGM-CNTs plates with various reinforcement distribution FG-X, FG-O and FG-A at N_L = 20, width-to-height ratio (w/h) = 10, L/W = 1, & f^*_r = 0.11 [384]

Cho et al. [385] employed the MF natural element method (NEM) and the linear ROM to investigate the mechanical behavior of an FG-CNT plate structure numerically. It has been observed that increasing the b/h causes an increase in the NF. Besides, the maximum NF was obtained by implementing the FG-X distribution, while the minimum NF was by FG-O distribution. However, the trends for NF and stability behaviors were reversed entirely. Recently, a mathematical model of bio-inspired CPs subjected to SSSS and CCCC BCs has been developed by Mohamed [260] to explore the dynamical response in which the FSDT has been utilized to represent the geometrical kinematic relations of displacement. Besides, the 2D-DQM solves the integro-DE. The results show that as long as the a/h increases, the NF increases in plates subjected to both cases of BCs. Also, the highest values of NF were obtained by implementing the UD distribution at a/b < 1. However, Civalek [386] investigated the FV behavior of symmetrically LCPs subjected to various BCs by implementing the DSCM combined with FSDT. The results obtained show that increasing the plate thickness/side ratio will cause a reduction in the frequency parameters, although it was raised due to increasing the a/h. Furthermore, Civalek [387] investigated the FV performance of LC panels as well as plates with a curved shape, which has various dispersal patterns of FGM composite. Besides, the FSDT combined with Love’s shell theory was utilized to obtain the GEs. By comparing the obtained frequency results through utilizing both methods, it is noticed that Love’s shell theory guarantees more significant results than FSDT. Additionally, increasing the number of modes causes more significant frequency parameters. However, increasing the semi-vertex angle causes a reduction in the obtained frequencies. Additionally, Kallannavar et al. [388] explored the impact of temperature and moisture on the FV performance of an SLCS, consisting of two face sheets of GrEp composite and a core-made CNTRC. Besides, the FSDT has been utilized in order to formulate the coupling relationship between the thermos-elastic and hygro-elastic, although the NL strain–displacement relationship has been implemented to develop the initial stress stiffness matrix to represent the nonmechanical stiffness matrices. Likewise, Chiker et al. [389] employed the FSDT and the FEM to investigate the FV and the NDNF of hybrid laminated plates consisting of FG-CNTRC. The results show that the FF increased due to the number of plies, and the BCs strongly affect the FF, although the maximum FF values were recorded by subjected the CCCC BCs. Besides, increasing the a/b as well as the b/h causes to increase and decrease correspondingly in the FF values. Additionally, in the relationship between NDFF and the number of layers under various distribution patterns of composites, it is seen that at UD, there is an absence of the effect of several layers, owing to the reason of homogeneous distribution. However, in plates with non-uniform distribution, it is noticed that an increasing number of layers will cause an increase in the NDFF for FG-X distribution and a reduction in the case of implementing the FG-O and FG-Λ distribution patterns. Nevertheless, Kallannavar et al. [390] explored the enhancement of moisture and temperature on the FV performance of a hybrid skew LC as well as sandwich plates with a VE softcore by utilizing the FSDT. It has been observed that the NF is reduced at low a/h values due to the increase in a/h. In contrast, the opposite trend in deviation is noticed at high a/h values, as presented in Fig. 48. Besides, increasing the SA caused an increase in the NF. Meanwhile, Chiker et al. [391] employed the LW formulation model to examine the FV of multilayer FG-CNTRC and FG-GPLRC polymer NC plates reinforced with various distributions of nanofillers. The FSDT has been utilized to determine the GEs, while the ROM has been used to define the E of CNTRC, contrary to the HT used for the GPLRC. It has been observed that increasing the n causes an increase in the FF, as shown in Fig. 49, and the BCs, dispersal patterns, and weight fraction influence the plates’ rigidity.

Fig. 48

The enhancement of SA on NF of five-layered SSSS hybrid skew LCP with 0.25% of moisture with several a/h (a) a/b = 0.5, (b) a/b = 1..0, (c) a/b = 1.5 & (d) a/b = 2.0 [390]

Fig. 49

The enhancement of n on the FF change % of GPLRC plates at a/b = 1; (a) & (b) b/h = 10, (c) & (d) b/h = 20 & (f) b/h = 50 [391]

Ragb et al. [392] employed the sinc differential quadrature method (SDQM) combined with discrete singular convolution differential quadrature method (DSCDQM) to investigate the FV of a PE CP shown in Fig. 50 with NL/linear EF. The GE has been driven through implementing the 3D-elasticity approach. The results show that changing the NL WF parameter has no significant effect on changing the NF. However, it converges at a more excellent value of the linear WF parameter. Besides, the NDFF of the PZT-4 is smaller than those of BaTiO₃. In addition, increasing the thickness of the composite layer causes the NT values to remain constant, or reducing the thickness of the piezo layer causes the NF. Khouzestani et al. [393] employed the FSDT to analyze the axisymmetric FV and stress behaviors of composite annular plats consisting of saturated porous. Besides, the EOM has been obtained by HP and variational formulation and solved by GDQM. It has been revealed that increasing the porosity cause a decrease in the first NF and reduces the stiffness as well as the density. While, increasing the skeleton coefficient causes the rise of the first NF. Furthermore, Gholamia et al. [394] employed GDQM combined with the PAL scheme to investigate the FV performance of FG-CNT annular plates exposed to PPB. Besides, the material properties were estimated by implementing the equivalent continuum approach based on modified ROM, and for formulating the vibrational problem, the FSDT associated with VKT nonlinearity was utilized. It has been seen that increasing the radial load in the pre-buckled state caused a reduction in the FF. The main reason was that the plate's stiffness was reduced by raising the compressive radial loads. In addition, the plates did not experience any vibrational behavior at a specific point, and FF = 0.0; the reason for this phenomenon is that the stiffness matrix = 0.0 is due to raising the compressive radial load to a new high value.

Fig. 50

The schematic of PE composite the actuator made of Ba₂NaNb₅O₁₅ & PZT-4 sensor founded on NL EF [392]

Moreover, Nor Hafizah et al. [395] utilized the FSDT to analyze the FV performance of antisymmetric angle-ply composite, and the cubic splines method was employed to determine the solution. It has been seen that the FV performance was influenced by varying ply angles as well as several layers, although the angular frequency obtained with ply angle 30°/-30° was higher than those obtained by 45°/-45°. Likewise, Qin et al. [396] developed an MF methodology based on the FSDT in order to examine the FV as well as bending analyses of circular stiffened CPs, and the MLS approach was used to determine the shape functions and to generate the flat circular plates’ displacement fields. By validating the obtained results, it has been revealed that the presented approach can analyze the FV and the bending of the structure effectively with high stability and quickly converge with low relative error values. Additionally, high flexibility has been achieved as a consequence of the high capability that the developed method guarantees, which is to locate the stiffener in any position, and this significant flexibility can be implemented to obtain an optimum position for the stiffeners. Moreover, Chattopadhyay et al. [397] employed the linear elasticity theory (LET) and the normal TSDT of plate to investigate the model localization in FV as well as the forced vibration of monolithic and composite rectangular plates, which consist of internal constrained points subjected to SSSS and CCCC BCs. It has been revealed that the normal TSDT of plate imposing interior constraints has no impact on the first 100 values of frequency, while it affected the following mode shape significantly: 2, 4, 5, 10 as well as 20 and the vibrational regions in the mode shapes founds to vary. Besides, it is noticed that the eccentricity highly influenced the mode localization factor, and the first 52 vibrational modes were observed to be nearly undeformed in regions 1 and 2 with mode localization factor = 0 or 1. A reverse relationship between the mode localization factor and several modes can also be observed. However, Hoang and Thanh [398] presented a novel TSDPT to explore the FV analysis of various types and thicknesses of FGM plates made of Al-Al₂O₃ and Al-ZrO₂ with the Kerr EF model. The HP has been employed to generate the GEs for the CPs, while the Galerkin methodology has been utilized to define the NFs. Besides, the results obtained were compared and validated with previous data that were achieved using the CPT, FSDT, and further shear theories. Additionally, the results indicate that the NF values recorded by implementing the Al-Al₂O₃ were remarkably greater than those of Al-ZrO₂ and had superior properties. Nonetheless, the enhancement of the upper and lower elastic layers and six different BCs on NFs were further examined. Thai et al. [399] applied the MKI MF to investigate the free vibration of NP with and without a hole. The outcomes show that in the plate with a complicated hole, the frequencies increase with a rise of l/t in addition to the value of V ∗ _CNT. Again, the FGX and FGO patterns are given the most extensive and minor NDNF. To study the free vibration behavior of a SLGS. Else, Zhang et al. [400] implement the Kp-Ritz method with the NLCT as a MF computational framework. In addition, the Eringen NLC constitutive formula with the CPT was combined by the NLCT, and it was revealed to have the capability to take the SSEs into account. Furthermore, Rajabi and Hashemi [401] implemented the NSGT to perform the size-dependent FV of FSD orthotropic NPs. It has been observed that the NLC parameters highly influence the frequency parameters, while the gradient parameters have a higher influence on the frequency parameter than the NLC parameters. Yet, the solution has been accomplished by implementing Kantorovich's methodology combined with GDQM. However, the bi-axially compressed double-layered Gr sheets shown in Fig. 51, with various types of the bilayer systems’ motion based on NSGT were considered by Ebrahimi and Barati [258] to explore the vibrational analysis. Where the GEs have been generated by HPs and solved using the Galerkin approach.

Fig. 51

The schematic of the Gr sheets with EF based & various types of the bilayer Gr sheets systems’ motion [258]

Li et al. [402] employed the NLC MPT to explore the stability and FV performances of a magneto-electro-elastic NP founded on PF. The Maxwell equations have been implemented, and the HP has been used to define the GEs, solved by Cramer’s rule. It has been seen that raising the NLC parameter caused a reduction in the FV frequency, although it increased the shear coefficients as well as the normalized spring, which, in consequence, reduced the structures’ stiffness. Additionally, the FV frequency reduced due to increasing the electrical potential and increased due to increasing the magnetic potential. Moreover, Kumar and Gupta [403] employed the NLC continuum model combined with MSDT to explore the enhancement of randomness in material properties such as (V_F, Poisson’s ratio, and E) on the vibrational behavior of FG NPs. Besides, a practical stochastic FE expression has been acknowledged for capturing the first/second-order NFs’ statistics. However, the ROM has been utilized to obtain efficient material properties and combine the inclusion of porosity. It is revealed that Poisson’s ratio has a significantly lower effect on the stochastic properties than that of E. Additionally, the frequency variation coefficient increased due to changing the FG structure from a macroscale to a nanoscale and raising the NLC parameter from 0 to 1, causing a remarkable decline in NF. Higher values were obtained by subjecting CCCC BCs than those of SSSS BCs. Furthermore, Qu et al. [404] presented a dynamic analysis to examine the microstructure enhancement and Mindlin’s high-order theory was used to define the SDE. Besides, the FSDT, CST and MSCT were implemented. It is revealed that the CST has a remarkable impact on mechanical responses. The FV of the MPs was obtained by the Double FS. In addition, Dastjerdi and Behdinan [405] explored the temperature effect on the FV of SMP sandwich structure made of polymer Gr NC and piezoceramics by using the MLS MF and TSDT. While the mechanical properties were defined by employing the HT model. It has been revealed that, inserting of pores inside the core will enhance the NF of the plate due to the significant reduction of structure weight. Also, increasing the quantity of graphene leads to enhance the NF. Moreover, Ansari et al. [406] explored the influence of surface stress on the vibrational behavior of circular NPs subjected to different BCs. The NCL GE and BCs were define by using HP, where the surface energy was included through GM elasticity theory. Besides, the GDQM was employed to discretize the GE. It is seen that, the impact of surface stress vary with changing BCs, mode numbers, surface elastic constant and plate thickness. In addition, Ansari et al. [407] investigated the influence of surface stress on the FV as well as postbuckling of axisymmetric circular NPs under various BCs. The GM elasticity theory combined with the Mindlin’s plate approach were employed to drive the NL EOM incorporating the geometric nonlinearity as well as the surface stress impact. Besides, the GDQM and the NCL differential GE were used to discretize the EOM which solved further by using the PAL continuation methodology. The obtained results reveals that, by reducing the thickness magnitude the influence of surface stress became significantly prominent. Although, for NPs with small thickness the influence of surface stress on the FV in the vicinity of the postbuckling has increased. Furthermore, a NCL high-order shear deformable circular NPs model considering the surface effect has been developed and investigated by Sahmani et al.[408]. The developed model was accomplished by using the GM elasticity theory which includes all various surface effects. Meanwhile, the VKT NL approach was used to define geometrical nonlinearity, and the HP was used to define the NCL differential GEs which further discretized by GDQM and solved by PAL. It is remarkably noticed that by rising the plate thickness the NL vibrational performance predicted by the modified NCL plate model became similar to that obtained with the CPT. Furthermore, Belarbi et al. [409] utilized the LW approach to examine the FV behavior of NPs made of FGM with considering the SDE by using the Eringen’s NCET. Besides, the classical HP was used to define the GEs. It was seen that the NLC parameter enhanced the DLFF, and the maximum effect of NLC on NT was obtain by applying the CCCC BCs while the lowest by FSSF BCs. However, the FF increased by increasing the number of clamped at the edges. In addition, it was revealed that the nonlocality cause a stiffness-softening effect as by increasing the NLC parameter the FF reduces. Karami et al. [410] explored the FV and forced vibration of FGM poroelastic thick MPs with considering the SDE by employing the MCST, quasi-3D model, power-law scheme. Besides, the linear-displacement relationship was utilized to model the infinitesimal deformation of the MPs, although the virtual work principle was used to define the GEs. Meanwhile, the trigonometric expressions was used to discretize the PDs of the fourfold coupled (axial-transverse-rotation-stretching). It was revealed that a significant NDFF obtained by increasing the LSP. In contrast, by increasing the l/t the differences between fundamental frequencies becomes less. Nonetheless, Bacciocchi et al. [411] presented a novel FEM to explore the FV and stability analysis of LCPs subjected in hygro-TE modeled based on SGT and Kirchhoff. Besides, the laminated CPT was used to generate the kinematic model. The obtained results revealed that the NF increased by increasing the NLC parameter. However, the plates subjected to CCCC BCs observed to be more stable and it require significant initial temperature to reach the critical temperature which define the stiffness’ losses. Furthermore, Bacciocchi et al. [412] investigated the FV and bending analysis of LCPs containing holes with considering the SDE by SGT subjected to various BCs which cannot solve analytically. The mechanical responses were evaluated by using a FEM based on higher-order Hermite interpolating. The obtained results reveal that NF increased due to increasing the NLC parameter. In addition, Natarajan et al. [413] implemented the IGA approach to explore the flexural FV responses of FGM NPs with considering the SDE based on Eringen’s NCET with considering the material feature to vary along the plates’ thickness. Besides, the MT approach was used to compute the effective properties of the NPs and the impact of characteristic internal length, GI, a/h, BCs and plates’ thickness on the mechanical responses were investigated. It was reveal that the fundamental frequencies reduced due to increasing the characteristic internal length and GI. The classification of linear analysis by numerical solution with/without considering the size depending effect show in Table 3.

Table 3 Classification of linear analysis by numerical solution with/without considering the size depending effect

2.3 Linear Analysis by High Order Theory

2.3.1 High Order Solution Without Considering the SDE

In a mathematical context, after defining the GEs and discretizing to an ODEs. This ODEs with primary constraints identifies the unidentified function of a specified point in a sphere, defined as an initial value problem. Besides, the high-order methodologies allow mesh and polynomial refinement strategies. While, the most prominent of all initial value problems is the 4th-RK, though, in some systems when the solution rapidly oscillates’ an embedded method can be used such as the RK45 methodology. The necessity of high order solution is based on demanding a significant accuracy for functions with infinite unknown with large deflection such as when deflection as f (t) is required and in dynamic control applications. In this frame work some studies that performed based on high-order solutions are presented. For instance, Neves et al. [428] utilized the RBF to resolve Eigen-problems and develop a HO-SDT that consider the extensibility in the thickness direction during modeling FGPs. The obtained results shows that when ϵzz ≠ 0, excellent correlation with exact theories was obtained, and the convergence solutions were found. Besides, the CUF virtual displacements principle was used to derive the explicit GEs and BCs. Besides, Khalili and Mohammadi [429] presented a novel high-order sandwich plate theory for analyzing the FV behavior of a sandwich plate made of FG face sheets and soft/hardcore in a TE. Besides, the HPs have been utilized for defining the GEs. The results show that, at different n, the NF decreased by increasing the temperature; when the sheet-to-core thickness ratio became more remarkable, the NF parameters increased. Also, the NF decreased and increased for hard and softcore, respectively. However, Jin and Yao [430] employed accurate mixed ZZT to investigate the FV response of LC and sandwich plates. It has been obtained that the NF and the higher ones are greatly enhanced by the TS stresses derived by the RMVT involved in the equilibrium EOM at various thicknesses and material characterizations. The high values of NF could not be predicted by previous methods in the literature, such as ZZT and local–global models. Furthermore, Thakur et al. [431] employed the higher-order nonpolynomial shear deformation theory (HNPSDT) to develop a C0 FE modeling to investigate the dynamic performance of a folded LCP. The IHSDT combined with nine-nodded Lagrange isoparametric FE, while the HP has been utilized to drive the GEs, and the subspace iteration method (SIM) has been implemented to solve the FV referred to as the eigenvalue problem. The results show that employing various crank angles in the case of a fold isotropic shows that, there is no significant diversion in the first values of NDNF. However, there are remarkable differences in NDNF values in the case of flat plates. Furthermore, in the case of the two-fold, remarkable differences in NDNF were obtained and caused by diverse fold angles.

Moreover, Van Do et al. [432] employed the hybrid HSDT associated with local Bézier extraction based IGA for the purpose of investigating the FV and dynamical response of CNTRC plates. Moreover, for simulating the resonance phenomena the first NF assumed to be equal to the excitation frequency. It has been revealed that the lowest NF values recorded by using the FG-O dispersal patterns, while the highest by FG-X dispersal patterns. Besides, the sensitivity of the NF to b/h became greater with the plate constraint existence improved. The combination of those two methods found to be efficient and accurate numerical methodology for forecasting the dynamical performance. Likewise, Singh and Singh [433] developed two novel HSDT for the purpose of investigating the FV as well as stability analysis of braided and LCPs. The obtained results show to be in good agreement with previous published data obtained either by exact or different numerical methodologies and both novel methods were accurate and recommended for analyzing 3D braided and LCPs in further future works. However, in a novel study, Habibi et al. [434] explored the vibrational properties of FG-GPLRC in the context of HSDT. The HP has been implemented to define the GEs, and the KV was used to define the VE properties. Besides, the 4th-order RK has been employed to solve the deflection as f (t). Also, the GDQM was used to present the numerical solution. The outcomes of this study show that significant stability and NF can be accomplished by utilizing the FG-X dispersal patterns. However, the FG-A, FG-UD, and FG-V have identical impacts on the NF. Furthermore, the maximum values of NF were obtained by implementing more square shapes near the bottom and top surfaces. Nonetheless, Safarpour et al. [435] employed the GDQM to explore the frequency and FV performance of thick annular circular plates made of FG-GPLRC VE. Besides the 4th-order RK was used to solve the deflection as f (t). Where, the formulation was valid in forecasting the extraordinary frequencies. Meanwhile, Bai et al. [436] performed a semi-numerical mimic to analyze the vibration behavior of an imperfect VE reinforced with GPLs circle plates consisting of an Al honeycomb core. The material properties were defined by employing HT and ROM, through the GEs, defined using HP, and solved by GDQM. However, the KV was used to define the VE properties. Besides, the 4th-order RK was utilized to resolve the deflections as f (t). It has been revealed from the obtained results, that to understand the dependency of the number of grins in GDQM minimum 10 grid points must be selected. The CPs with FG-X dispersal experienced the highest NF values, while the lower NF values were achieved by FX-O dispersal. Besides, the CPs’ stability can be improved by reducing the fiber angle at CCCC and CSCS BCs. Additionally, the NF increased by increasing the t/l. Furthermore, Ngak et al. [437] investigated the static and FV of multilayered FGM elastic plates with imperfect interfaces shown in Fig. 52 and subjected to SSSS BCs. The imperfection between the layers was modeled by using the spring-type layer model, while a semi-analytical solution combining the state-space approach with 4th-order RK was employed in each layer. Besides, the propagator matrix methodology was used to propagate the predicted solution in each layer in order from the bottom to the top layer considering the transfer matrix at the imperfect interface. The results have been validated and compared with results obtained previously by various methods such as pseudo-Stroh formalism, FEM, LW, discrete layer, etc. Also, the amount of error at all vibrational modes was small and accepted except for the fourth vibrational mode. However, the main disadvantage of this method was that the frequency equation order increased due to increasing the propagator matrix which led to a complex solution. Nonetheless, Hosseini et al. [438] analyzed the FV and forced vibration of the FGM (Al/ Al₂O₃) plate by using the MF collocation methodology based on RBF to discretize the GEs. Besides, the FSDT was used to define the strains and displacements, with developing a modified leave one out cross validation (LOOCV) to obtain the shape parameters. The transformation matrix was employed to reduce the discretized equations, which were solved further by the RK method. The obtained results show that the NF increases by enhancing the GI. The outcomes were shown to be in good agreement with previously published data obtained by other MF methods and high-order approaches.

Fig. 52

The schematic of a multilayered FGM elastic plates with imperfect interfaces [437]

2.3.2 High Order Solution with Considering the SDE

By considering the LSP in NLET and the SGT a fully presentation of materials and structures in NS can be accomplished. In several studies available in literature the authors were focusing in developing a HO NLC SGT by implementing and combining various methodologies. For the purpose of considering the nonlocality in global domain combine with high order stresses, Lim et al. [439]. For instance, Zhang et al. [440] investigated the FV as well as stability (mechanical and thermal) characterizations of an annular plate consisting of FG-GPLRC by implementing the HSDT combined with GDQM. The enhancement of GPLs distribution patterns and WF, temperature rise patterns, axial load, temperature gradient, inner radius to thickness ratio, outer radius to inner radius ratio, and other responses on NF were analyzed. This article is considered an entail step for further investigation in composite structures. Dastjerdi et al. [441] employed the NCET utilizing the FSDT and the HSDT to define the decoupling of constitutive formulas for multi-layered NPs embedded in an elastic matrix. The classification of linear analysis by high order solution with/without considering the SDE shown in Table 4.

Table 4 Classification of linear analysis by high order solution with/without considering the SDE

2.4 Linear Analysis by Mixed Solution

2.4.1 Mixed Solution Without Considering the SDE

The significance of the semi-analytical solution is due to the uniqueness of having the ability to model a FEM with conserving the high accuracy and effectiveness of the analytical solution. The semi-analytical solution is based on generating the solution as a piecewise analytical formula followed by implementing polynomials to determine the coefficients by Xu et al. [446]. In this framework, an early initial FV analysis for rectangular plate made of orthotropic material was accomplished by Bert and Malik [447] where the DQM was implemented. In addition, a further investigation was presented by the same authors Bert and Mlik [448] also by implementing the DQM approach, but in this study the tapered rectangular plates were examined with considering isotropic and especially orthotropic material. Moreover, Jafari et al. [449] examined the FV of FG-GPLs by employing the HT model to evaluate the E of the structure. The NvS and Fourier expansions were implemented for SSSS BCs and essential variables of displacement fields, respectively. The obtained outcomes show that the plates’ frequencies increased due to the increase in the GPLs’ WF. Also, the maximum frequency values were obtained by implementing the FG-X patterns, while the lowest were obtained by implementing the FG-O patterns. Furthermore, Thai et al. [450] obtained significant reliable results by employing the weak form of Galerkin, which has been defined by an IGA, to examine the FV of isotropic FGM plates and for a sandwich plate with a core made of isotropic and a skin layer of FGM. Besides, Aydogdu et al. [451] presented a semi-analytical solution to analyze the FV of FG CPs where the fiber content exhibits bending based on the polar linear elastic principle, which leads to HO of NvS type PDEs. It has been seen that increasing the stiffness parameter of the fiber bending causes an increase in frequency values. However, whether the fibers were resistant or perfectly flexible in bending, the frequency values obtained by Ω₁₂ are usually lower than those of Ω₂₁, and the fiber distribution inhomogeneity affects all values of frequencies. Besides, the fiber distribution inhomogeneity effects on all values of frequencies. Moreover, Rostamijavanani et al. [452] investigated the FV and buckling behaviors of various LC structures such as curved panels, flat plates , and cylindrical shells by implementing a semi-analytical approach to FS method. Besides, the third-order Hermitian shape function and the first-order Lagrange were utilized to estimate the longitudinal and transverse directions. However, Qu et al. [453] presented a semi-analytical study that was considered as a foundation for hypothetical expansion, the clarification of certain physical phenomena produced by couple stress, structural optimization, and finally experimental measurement of microstructures. Moreover, Katariya et al. [454] employed the HSDT combined with FEA to investigate the FV as well as the bending of the SSPs. The obtained results for examining the NDFF of the SSPs with various BC are illustrated in Fig. 53. In addition, Yin et al. [455] analyzed the FV and bending performances of FGPPs with even and uneven distribution patterns by developing a semi-analytical methodology by introducing scaled boundary finite element method (SBFEM)-based 3D-TE while assuming the material properties to vary along the thickness of the plate. Besides, the SBFEM was driven by using Green’s theory combined with the principle of virtual work. The discretization of the plates’ middle plane was accomplished by using the two-dimensional (2D) high-order spectral element, and the radial solution was achieved by using the analytical matrix exponential function and solved by precise integration methodology. The obtained results show that, at n =0.1 the frequency values increase slowly and smoothly as the porosity parameter increases. Also, the maximum frequency values were obtained by subjecting the plate to SCSC BCs.

Fig. 53

The NDFF of SSPs with various BCs. & thickness ratio a at (0°/C/0°), b at (0°/90°/C/90°/0°) & c at (0°/90°/0°/90°/C/90°/0°/90°/0°) [454]

Furthermore, Baghaee et al. [456] explored the FV of rectangular LCPs (PVDF-Gr/Ep) that contain a single PE layer (PZT-4) and based on EF. The EOM and the BCs were defined by HP combined with the FSDT which was solved further by generating a base function based on the general electrical potential and the generalized displacements which were expanded by applying the Legendre polynomial series. However, to define the BCs in the energy formula a Lagrange multiplier was used. The developed semi-analytical approach was shown to be in good agreement with available approaches. Besides, Safarpour et al. [457] explored the FV and static performances of annular plates and other structures made of FG-GPLRC. The state-space approach and DQM were used to perform the semi-analytical solution. The obtained results show that the maximum frequency values were achieved by the GPL-X distribution while the minimum was achieved by the GPL-A distribution. Likewise, by considering the effect of BCs on frequency, it is seen that by subjecting the structure to CCCC BCs the maximum frequency values were obtained. However, Wang et al. [458] investigated the FV and stability analysis of NPs reinforced with CNT by multi-term extended Kantorovich method (MTEKM)-Galerkin methodology. Although, the ROM was used to define the material properties, and the Galerkin was used to discretize the governing PDEs to ODEs and solved further by the state-space technique. The obtained results shows that, the FF highly influenced by subjected the plate to CCCC BCs, FG-X dispersal, reducing the a/h, as well as increasing the V_F. Shufrin and Eisenberger [459] presented a novel semi-analytical method for rectangular NPs with cutoff by employing the variational extended Kantorovich approach. Meanwhile the solution accuracy was enhanced by developing MTEKM approach which also eliminate the singularities at the cutoff region without demanding assembly for predefined trial functions. Also, the developed approach allows the explanation of complex distinction of thickness and cutoff. By comparing and validating the result with FEM results the results obtain by the developed by MTEKM found to be more accurate. Furthermore, Shaban and Alipour [460] explored the FV analysis of FGM circular thick plates resting on PF subjected to elastically restrained edges. The FSDT was employed to drive the GEs assuming mechanical properties to vary along the plate thickness. The differential GEs were transformed to algebraic recurrence equations by using differential transform method (DTM). The obtained results reveal that increasing the plates' relative thickness from 0.05-0.5 caused a reduction in NDNF and by increasing the vibrational mode to the second as well as the third the relative thickness effect becomes more significant. However, the material dispersal parameter has a significant impact on the NDNF and it has a major role in obtaining efficient design

Moreover, Lü et al. [461] developed a new semi-analytical technique to examine the FV responses of an anisotropic laminated plates under cylindrical deflection. The transfer matrix methodology was used to solve the thickness domain, while the DQM was used to solve the in-plane domain. Significantly, this approach is capable to examine the arbitrarily thick laminates as well as arbitrary BCs. Besides, it overcome the drawback of the conventional state space technique. It is revealed that by increasing the ply angle, the frequency parameters reduced gradually. However, by changing the fiber orientations of anisotropic LCPs the mode shapes have the availability to switch. In addition, the differential quadrature element method (DQEM) was employed by Malekzadeh et al. [462] to analyze the FV responses of thick plates subjected to two opposite BCs based on FSDT. Where the plate was decomposed to a series of elements or sub-domains and the discontinuity in EF, material features, and geometry in one direction was handled. The main advantage of this method is it decrease the required storage time and CPU to generate the global equations with increasing the computational efficiency. It was observed that the FNF decreased as a consequence of reduction in stiffness that caused by decreasing the thickness ratio. Besides, an accurate solution for high vibrational modes was achieved. Alipour et al. [463] explored the FV analysis on variable thickness two-directional FGM circular plates subjected to an EF with various BCs. The GEs of the NDNF were defined by implementing the DTM approach in conjunction with the Mindlin’s plate theorem which solve further by utilizing the N-RI. The obtained results shows that, the first two NDNF increased due to increasing the radial exponent E. While, increasing the thickness ratio of FGM plates caused to reduce the NF. Liew et al. [464] investigated the FV behavior of plates (isotropic, homogeneous and thin) with rectangular shape and abrupt thickness deviation by employing the domain decomposition methodology. The global energy function was generated by KPT with considering coupled strain and kinetic energy of each subdomain element which further discretized by using Ritz approach. A significant agreements was accomplished by comparing the results of SSSS BCs with previous published data. Although, the mode shape obtained by CCCC and SSSS BCs are almost identical which may cause by the symmetrical nature of the BCs configurations. Additionally, Wang et al. [465] analyzed the FV behavior of the thick orthotropic annular sector plates subjected to general BCs. With circumferential arc as well as internal radial supports. The GEs defined by employing the RRM, MPT and stationary energy principle. Meanwhile an improved FS approach was used to replace the traditional FS. The obtained results revealed that at all various BCs, the frequency parameters reduced rapidly with increasing the sector angle. Besides, reducing the plates’ thickness cause to increase frequency parameters at all various BCs. Furthermore, a new model has been developed based on the state-vector equation theorem to analyze the FV of stiffened laminated plates with PE patches by Qing et al. [466]. The developed model eliminates any restriction on the plates’ thickness and stiffeners’ height. The obtained outcomes shows that by increasing the stiffeners height the NF of vibrational modes one and two increases. Meanwhile, the third mode NF increased the decreased and the opposite behavior was observed in mode four. Moreover, the same model has been further investigated by the same author, Qing et al. [467]. A developed mixed variational principle for PE materials has been accomplished, and the state-vector equation of PE plates was presumed directly followed by applying the semi-analytical solution. However, the plates and the patches were assumed to be two three-dimensional PE bodies, while a linear quadrilateral element was utilized to discretize both bodies. This solution can provide a continuous generalized displacement as well as stress at the interface between PE patches and plate. Moreover, Boreyri et al. [468] explored the FV responses of FGM plates with in-plane exponentially nonhomogeneous material. The mass density and E considered to vary along the plate thickness from between a metal-rich and a ceramic-rich zone. Besides, the NF was obtained by the Taylor series expansion and the solution of SSSS BCs by LS. It has been revealed that by increasing the nonhomogeneous parameters the NF increased, and the a/h directly affected the NF.

2.4.2 Mixed Solution with Considering the SDE

Owning to the quadratic scaling with respect the basis set size, the semi-numerical solution considers further efficient than other types of methods for analyzing the enormous basis sets as well as molecules stated by Liu and Kong [469]. On the other hand, with recent development in NMS structure the essential of examining the NLC parameters increased widely in literature, especially in analyzing NMS electro-mechanical systems such as MEMS, as stated by Twinkle and Pitchaimani [470]. For instance, Zamani [471] performed a semi-analytical solution combined with an iterative arithmetical algorithm to explore the FV analyses of FG-VE foam plates modeled by SDT combined with NDT. The P-LR used to define the porosity dispersal along the plates’ thickness. The HP in a complex domain utilized to define the GEs. Furthermore, Zhang et al. [472] employed the FSDT combined with the classical delamination approach to examine the vibrational characterization of an laminated composite double-plate (LCDP) structure with a uniformly distributed artificial spring, as shown in Fig. 54 (c). Besides, the displacement admissible function was formulated by utilizing the improved FS methodology. The analytical model was followed with numerical examples for validation. The obtained results show that the presented method has higher flexibility compared with the FEM. Also, it has significant efficiency and convergence. Furthermore, in the sensitive area of the numerical impact of the middle elastic layer’s stiffness in this area, the system’s vibrational characteristics are highly affected by variations in the appropriate parameters. However, Dastjerdi et al. [414] applied the NvS to investigate the analytical solution for FV of the SSSS sandwich plate after performing a numerical study to examine the influence of CNT aggregation states, distribution, and V_F as well as the geometrical dimensions of the plate. Also, it has been observed that the increasing of CNT V_F enhances the frequency parameters. Moreover, Nguyen et al. [473] developed an ES-MITC3 element found on the FSDT to investigate the FGP plates sited on a partially supported elastic foundation (PSEF), as shown in Fig. 54 (a). The FV analysis was studied analytically, followed by numerical solutions to validate the developed method, where HP was applied. The impact of various types of PSEF geometrical characterization and material properties on the FV behavior of FGP plates was considered. The FV was observed to be highly influenced by the material properties and the parameters of the foundation stiffness. In addition, various types of PSEF lead to different NF. Besides, Pham et al. [474] employed the TSDT to analyze the FV of NPs with a core made of an auxetic honeycomb, as shown in Fig. 54 (b). Furthermore, due to its high accuracy and reliability, the Eringen’s nonlocal theory (ENLT) has been applied to define the SDE. Besides, numerical examples were presented to validate the presented method. The findings of this study reveal significant primary data and design guidelines for improved sandwich NPs with superior characteristics, such as ultralight, high-strength, and auxetic honeycomb cores that have excellent energy absorption capability.

Fig. 54

a The FGP plates with PSEF [473], b The schematic of NP with auxetic honeycomb core [474] & c The schematic of LCDPS with general BCs [472]

Zuo et al. [475] developed a novel C0 LW wavelet FEM to investigate FV and static behaviors of CPs, and the ZZT has been utilized to introduce the zigzag effect in the multilayers. Besides, the HSDT combined with piecewise linear C0 was used to obtain the GEs. The transverse shear strain at the interface layer condition and the yield to stress-free BCs on the plates’ surface with the absence of shear correction factor. By considering and examining various numerical examples and validating them with previous data, it is seen that the proposed methodology has good efficacy, accuracy, and reliability compared with other available arithmetical solutions and 3D-TE. Additionally, it can be considered the basic approach to designing composite structures and optimization. Moreover, A semi-analytical simulation has been performed by implementing the stochastic simulation, founded on the Monte Carlo theory by Parviz et al. [476]. Besides, the FSDT has been utilized to determine the displacement field, and the intrusive polynomial chaos approach has been used to examine the probability space. A remarkable error in the predicted NFs has been observed that is caused by using the material features’ linear dependency on the temperature. However, Kermani et al. [477] employed a semi-analytical/numerical methodology (state space-based differential quadrature method) to investigate the FV of multi-directional FG annual and circular plates subjected to several BCs while considering the 3D-elasticity equations. It has been seen that increasing the plate thickness or the inner radius-to-outer radius ratios causes an increase in the NF. Additionally, Liu et al. [478] studied the bending and FV analyses of NC APs reinforced with FG GPLs with various distribution arrangements using the 3D-TE. Meanwhile, the multilayer FG-GPLRC APs subjected to various BCs were examined by employing the state-space DQM. Besides, the MP was evaluated through the HT micromechanics model and ROM. Additionally, the DQM has been implemented to solve the semi-analytical solution. From the results illustrated in Fig. 55, it can be noticed that the FF increased dramatically by increasing the a/h of the NFs until it reached the optimum values, and then it continued to increase slightly.

Fig. 55

The enhancement of dispersal & size on the FF of the FG-GPLRC multilayer APs [478]

Civalek and Avcar [479] employed the DSCM to detect the FV behavior of laminated nonrectangular plates made of FG-CNT, and the FSDT and the CPT were utilized. The computational analysis was followed with a numerical solution, which shows that the plates’ vibration is highly influenced by the number of layers of the plat structure, BCs, SAs, the V_F of the reinforcements, and it is distribution, a/h as well as the geometrical factors. Furthermore, Nguyen et al. [287] analyzed the FV, static linear elasticity, and stability of FGPPs reinforced with GPLs by utilizing the IGA and incorporating the FSDT and TSDT. Additionally, an arithmetical investigation has been performed, and the obtained results demonstrate that the inclusion of GPLs significantly enhanced the stiffness of the FGPPs. Besides, Basu et al. [480] presented an FV analysis of FG folded plates by implementing the FEM considering the FSDT as well as the rotary inertia. Additionally, numerical results showed that the NFs were significantly affected by the BCs and plate thickness. More significant variation in the NF was observed at higher crank angles and large thicknesses. Meanwhile, Tahouneh et al. [481] employed the 3D-TE to analyze the FV responses of sectorial plates with a sandwich structure which consists of FG-MWCNT/Ep core subjected to various BCs; the modified HT has been implemented to define its E. Besides, the EOM has been generated by adopting a semi-analytical method composed of 2D-DQM and a series solution. It has been revealed that the frequency parameter reduced due to raising the value of sector angle, as illustrated in Fig. 56, while increasing n at small values caused a sharp reduction in the first two NDNFs. Although it keeps increasing the n>15, it causes constant NDNF values at various BCs. However, Benhenni et al. [482] employed the HSDT associated with 3D-FEA to investigate the NF of hybrid carbon/glass cross-ply LCPs, and the HP has been utilized to define the EOM. Besides, the NvS has been used to determine the closed-form solutions. It has been seen that mesh refining and raising the number of elements in the FEA solution has no significant effect on the obtained results. Also, by comparing the obtained results with previous data, it was found that the maximum error was less than 10%. Further results show that the NF was reduced by reducing the weight percentage of the glass fiber. In general increasing the total value of V_F enhanced the NF. Moreover, Malekzadeh and shajaee [483] utilized the NLC two-variable refined plate theory to investigate the FV of NPs. Various BCs were considered Fig. 57 illustrates the deviation of ND NF versus the NLC parameter of CCCC with various a/h. From the obtained result, it is noticed that the rising NLC parameter caused a monotonic reduction in the frequency parameters. Besides, the lower modes were less affected by the NLC parameters than the high modes.

Fig. 56

The enhancement of sector angle on the first two NDNF parameter ah h/a = b/a = 0/2 & n = 1 [481]

Fig. 57

The deviation of ND NF parameters in contrast to the NLC parameter of CCCC BCs NP with various a/h [483]

The application of NLC TSDT in studying plates' bending and vibrational behaviors was studied by Aghababaei and Reddy [256]. A further comment article on the same topic has been published by the same authors, Golmakani and Rezatalab [484] to present the inaccuracy of the NSs of bending analysis. Furthermore, Challamel and Reddy [485] presented a further discussion article to improve the NvS by implementing the sinusoidal-based deflection. Likewise, an embedded Gr sheet with EF subjected to external thermal load has been considered by Jalaei and Civalek [257] to examine its dynamical instability by implementing the NSGT refined plate approach. Moreover, Babaei and Shahidi [486] investigated the FV analysis of quadrilateral SLGS NPs based on NLC continuum models by utilizing the Galerkin methodology in which the enhancement of SSE has been considered and followed with NvS. It has been observed that lower FR has been obtained at high NLC parameter values, and increasing the length of the ARs caused the FR to increase FR. An additional significant observation is that increasing the vibrational modes enhances the NLC effects. However, Allahyari and Asgari [487] employed the ENLT to perform an analytical analysis that investigated the FV performance of polymer NC plate reinforced with Gr nanosheet as presented in Fig. 58 and the HP has been utilized to drive taking into consideration the performance of nanostructure points affected by all further NLC points. Besides, the HT was implemented to determine the mechanical properties, while the GE has been solved through analytical solutions. It has been seen that at different NLC parameters, which were obtained analytically and numerically, the NF ratio increased by increasing the plates' length until a specific value, and then it remained constant. It was reduced due to increasing the pristine Gr sheets at an orientation angle equal to zero under SSSS BCs, various NLC parameters, and V_F. Furthermore, studying the impact of defective Gr sheets shows that when it increases, it causes a reduction in E. Additionally, the vacancy concentration does not significantly affect the NLC parameters at plates with SSSS BCs. Likewise, Yekani and Fallah [488] presented a Levey solution to explore various mechanical behaviors of MPs based on the SDE. By using the MCST. The GEs defined by using the HP and solved by employing the single FS expansion combined with state-space methodology. This study highlighted the fact of absence of an analytical solution for NF, stability, and bending of Mindlin MPs except NvS and HSDT within the MCST. The maximum DLFF was obtained by subjecting the MPs to SSCC BCs. Also, increasing the LSP causes to increase in the DLFF. Furthermore, Torabizadeh and Fereidoon [489] presented a mixed solution to explore the FV of LCPs based on SDE by employing various types of plate approaches. The GEs defined by using the CPT combined with FSDT. The obtained results show that increasing the modular ratio causes to increase in the FF. also, the FF reduced with increasing the a/h until it reached a specific value then it remained constant. However, Mohammadimehr et al. [490] examined the FV of VE double-bonded polymeric NPs reinforced with FG-SWCNTs with various dispersal patterns (FG-O, FG-V, FG-X and UD) and founded on VE foundation. The material properties were defined by using the extended mixture rule, while the GEs defined by using the HP combined with SSDT and solved by NvS. Besides, the SDE was considered through utilizing the MSGT. The output results shows that the lowest value of NDNF was accomplished by using the FG-O dispersal. As a significant outcomes the NC plate was steadily converted to beam model with decreasing in its rigidity, by increasing the a/b. Furthermore, Karimi and Shahidi [270] examined the FV, stability, and bending of orthotropic Gr NPs based on EF subjected to a uniform thermal load. The SSE and the surface energy effect were considered by employing the NLET Eringen and GM, and the GEs derived through HP. It was revealed that increasing the NLC parameter enhances the effect of surface stress on vibrational behavior. Meanwhile, the impact of surface energy on vibrational behavior increased when the BCs were less stiff and vice versa. In addition, by assuming orthotropic & isotropic material characterizations, raising the temperature changes caused to enhance the effect of surfaces’ degree on FV. Besides, Zhang et al. [491] analyzed the FV behavior of the NLC PE Kirchhoff plates founded on the VE foundation and subjected to arbitrary BCs. While the EOM as well as the BCs were defined by using the HP combined with NLET for PE material and solved by modifying a Galerkin strip distributed transfer function methodology. The obtained results show that, by increasing the NLC parameter the first two NLC FRs decrease remarkably at various BCs and electrical voltage. Additionally, within the framework of the FSDT, Wang et al. [492] investigated the stability and the FV performances of LCP reinforced with Gr with considering the SSE, isotropic and symmetric cross-ply laminated subjected to opposite sides of BCs. The governing PDEs were defined by using the MTEKM based on trigonometric trial functions to fulfill BCs. However, the Galerkin methodology was used to reduce the PDEs to ODEs which was further solved by spectral element method (SEM). The results obtained by examining the LCP reveal that the NDFF raises with the thickness ratios’ increments, at different BCs and nanoparticles dispersal. In contrast, the opposite behavior was observed in the case of increasing the a/h. Furthermore, Norouzzadeh and Ansari [493] explored the SDE of surface stress as well as the NLC enhancements on the FV responses of NPs in circular and rectangular structures made of FGM with distinct surface and bulk phases. Where the Eringen and G-M theories were employed to investigate the NLC and surface impacts, and the MT was used to obtain the material features. Besides, the differential GEs were defined by a novel matrix-vector form, while the solution was obtained by IGA in which the exact geometry was generated by NURBS. It was revealed that, at modes one, two as well as high modes increasing the NP thickness leads to eliminating the surface effects. While by increasing the ND NLC parameter and material index the NDNF in modes number 4 and 5 decreases. Nonetheless, Mahinzare et al. [494] examined the FV responses of the smart circular rotary FG PE material NP created based on FSDT and illustrated in Fig. 59. Where the electro-elastic feature assumed to vary along the NPs’ radius and thickness by using the power-law model. Besides, the MCST was used to consider the enhancement of SSE, and the HP was used to define the GE which was solved by DQM. It was observed that, at various voltages increasing the FG PE material power index leads to a decrease in the first and second NDNF in clamped and hinged BCs. However, raising the coefficients of the two directional FG PE material causes a reduction in the critical ND angular velocity of NPs after the buckling point. Furthermore, Salehi et al. [495] analyzed the FV and the forced vibration of a KV VE NP based on PF, with considering the SDE through implementing the MCST. The presented solution for FV analysis has been conducted by implementing the Galerkin semi-analytical approach considering various types of BCs. The analysis was established by considering MCST and CPT to compare the results of each approach. It was seen that, for high plate thickness the frequency values obtained by both approaches were similar, while for plates with small thickness, a considerable variation between the frequency values obtained by the two methods was observed. Furthermore, Ye et al. [496] presented a novel semi-analytical solution of static and FV behaviors of FGM (Al/Al₂O₃ and Al/Zr₂O₂-2 FG) NPs based on SBFEM. The principle of virtual work was used to drive the GEs based on 3D-ET. By comparing the obtained results with previously published data a good agreement was shown at all BCs and by comparing the results of the relationship between normalized NF and power index, various benchmarks can be obtained by using SBFEM. Additionally, Nie and Zhong [497] analyzed the FV and forced vibration of FGM circular plates subjected to various BCs based on 3D-ET assuming the mechanical characterizations to vary along the plates’ thickness. The solution has been accomplished by employing the DQM combined with the state space method. The obtained outcomes show that the minimum NDNF raised with raising thickness-to-width ratio (t/w) and the circumferential wave number. In contrast, increasing the material property graded index cause to reduce the NDNF. Besides, Tahouneh et al. [498] analyzed the FV of annular sector plates consisting of 2D FGM subjected to arbitrary BCs and simply supported radial edges, based on 3D-ET by implementing 2D-DQM with considering a new 2D power-law dispersal for ceramic V_F. Also, the MT approach was used to define the material characterization, although the effective material characterizations at a specific point have been defined in terms of the local V_F. The obtained results show, that the NF was significantly reduced by using the graded ceramic V_F in 2D over using the traditional FGM. Additionally, the multidirectional FGM has the potential to substitute the unidirectional FGM. Moreover, Tahouneh et al. [499] accomplished a semi-analytical solution by using the DQM to analyze the FV responses of annular plates made of continuous grading fiber reinforced resting on EF subjected to various BCs at the circular edges based on the 3D-ET. Significant results were obtained which shows that the NDNF parameters of FG fiber V_F were greater than those of discrete laminated as well as those of the two-layer which can be considered as a benchmark solution. Besides, the frequency parameters increased to some extent due to increasing the shear layer of the EF to higher values. The classification of linear analysis by mixed solution with/without considering the SDE is shown in Table 5.

Fig. 58

The geometry of a rectangular NC with pristine Gr sheet reinforcement [487]

Fig. 59

The circular rotary NP with tow-directional FG PE material [494]

Table 5 Classification of linear analysis by mixed solution with/without considering the SDE

3 Nonlinear Free Vibration Analysis

3.1 Nonlinear Analysis by Analytical Method

3.1.1 Analytical Solution Without Considering the SDE

Since the developed technology required extremely lightweight systems with significant accuracy and high speeds, which caused remarkable deformation in the systems, in consequence Hook’s law is considered as insufficient among many other factors. In this context, investigating the NL FV of the system or the structure becomes essential. Notably, Ribeiro and Petyt [505] performed an exact solution of the NL FV composite laminated plate made of Gr/Ep with CCCC BCs using the hierarchical finite element method (HFEM) and the Harmonic-balance method (HBM). The presented model also includes the VKT NL strain displacement and the mid-plane in-plane displacement and the effect of dissimilar fiber orientation was considered. Furthermore, Ebrahimi et al. [506] employed the KPT to analyze the NL FV of a thin annular FGP integrated with two uniformly distributed actuator PZT4 layers. The obtained results illustrate that rising the FGM n leads to a reduction in the NF. Besides, the results obtained by CCCC BCs were much more significant than those obtained by SSSS BCs in both FEM and analytical solutions. Moreover, Aboutalebi et al. [507] employed the CPT combined with VKT and HPs in a comprehensive study for the NL behaviors of sandwich plates consisting of a core made of electrorheological (ER) fluids, as shown in Fig. 60, and the GEs were defined by utilizing the displacement control strategy. The results show that the NL resonant frequency values primarily reduced and then rose by increasing the maximal vibrational amplitude. Besides, rising the core thickness caused a reduction in the NL NF while it increased with increasing the plates' loss factor. In addition, increasing the BCs' rigidity will enhance the effect of core thickness on the NL NF. Besides, Niu and Yao [508] investigated the NL/linear FV of composite tapered plates and cylindrical panels reinforced with GPLs subjected to various BCs. The material properties were obtained by utilizing the modified HM and ROM. While, the NF and the mode shapes were determined by utilizing the FSDT combined with the Chebyshev-Ritz methodology. Besides, the NL GEs were obtained by employing Lagrange’s formula. The obtained outcomes show that under various BCs except the CFFF BC, raising the taper ratio causes a reduction in the NDNF at modes 1–5. Besides, the chaotic vibrations were produced with a significant amplitude/frequency of excitation. Additionally, the FF highly influenced by the GPLs and various dispersal patterns shows significant improvement in structural stiffness and FF, the enhancement of GPLs V_F on FF is illustrated in Fig. 61. However, Parida and Mohanty [509] analyzed the NL vibration of FGM CPs based on PF in TE. The mathematical model was generated using the HSDT, and the GEs were defined by HP and solved by DIM. As a remarkable outcome, it is seen that having intermediate material characterizations does not guarantee results in intermediate NL NF values. Besides, the WF has a lower impact on the FF than the PF. However, increasing any foundation leads to an increase in linear and NL frequencies. Besides, the maximum amount of NL/linear frequencies were obtained by subjecting the CPs to CCCC BCs.

Fig. 60

The ER sandwich plate structure [507]

Fig. 61

The enhancement of the weight fraction of GPLs on relative FF for structures subjected to SSSS BCs [508]

The enhancement of porosity and CNT characterizations on the NL performance of CPs subjected to an EF was examined by Bidgoli et al. [510]. The geometrical-based formula and the kinematics were generated based on the FSDT. The VKT nonlinearity type was used to define the geometrical nonlinearity, the GEs of the system defined by using the HP and the Galerkin was used to discretize the NL GEs to an ODEs time-dependent. An analytical solution has been presented to obtain the NL frequencies. It is revealed, that the extreme NL frequency values were achieved by FG-VA, although the extreme NL/linear FR was achieved by FG-AV. Besides, changing in porosity has no significant influence on the NL/linear FR. Though, the NL/linear FR remarkably increased by increasing the maximum deflection to overall thickness ratio. Furthermore, by examining the NL hygrothermal impact on the FF and active vibration control of VE sandwich LCP consist of CFRP and magnetostrictive by Zenkour and Shahrany [46]. The KV model was employed to define the material properties of the viscoelastic plate layers, although the GE was defined by HP. However, it was revealed that as the damping value of the VE structural increased, the NF as well as the deflection decreases. Besides, thicker plate has superior vibration damping properties than the thin plates. Additionally, the time required for the plate to control its vibration is less sensitive to the a/h than thickness ratio. Nonetheless, the NL vibrational responses of CPs consist of two face sheet made of GNP and a honeycomb core has been examined by Rezaei Bidgoli and Arefi [511]. Notably, the model was generated by using the FSDT and the geometrical nonlinearity has been considered based on the VKT. Besides, the GE was define by HP and discretized by Galerkin and solved analytically. The obtained results shows that, increasing t/l cause to reduce the NL NF. Meanwhile, by considering the variation of t/l the maximum value of linear/NL NF ratio obtained at θ₀ = 75°. Additionally, by using the honeycomb structure in the core of sandwich CPs structure the maximum amount of NL NF can be achieved by using θ₀ = 35°. Furthermore, Chang and Jen [512] analyzed the NL FV responses of heated orthotropic plates with rectangular shapes subjected to different BCs. The VKT and Berger’s analysis were used separately to drive the GEs and the Galerkin methodology was used to obtain the Duffing-type form. The NL solution was accomplished by employing the approximate and complete elliptic cosine. The results obtained by VKT shows to be in good agreement with those of Berger’s. The NL/linear vibrational period decreased due to increasing the amplitude. However, Avey et al. [513] explored the NL FV responses of LCPs with CNT originating layers interacting with PF bases based on FSDT. Besides, the Galerkin approach was used to discretize NL ODEs which solved by semi-inverse methodology.

3.1.2 Analytical Solution with Considering the SDE

In NMS structures’ investigations, the SDE has been found hind to the presence of surface as well as interface stresses owing to Van Der Waals and NLC atomic interactions forces stated by Wang et al. [514]. Besides, the NLC SDE shows to result from geometrical properties such as curvatures’ variation Sahmani et al. [515]. Accordingly, Wang et al. [516] employed the HSDT for plates in order to define the EOM of NC plates reinforced with SWCNTs with EF and founded on TE. Besides, a perturbation technique has been developed to solve the EOM to define the NL frequencies. It was seen that increasing the V_F of CNT increased the NF and the NL/linear FR. In addition, reducing the foundation stiffness or increasing the temperature rise cause to increase the NL/linear FR and decreases the NF. Also, there was a remarkable enhancement in the NL vibrational characterizations caused by the FG reinforcement. Likewise, Shen et al. [517] implemented the multi-scale methodology to design and analyze the NL vibration of composite laminated plates of FG-GRCs founded on EF. The HSDT has been used to determine the EOM and the VKT has been utilized to determine the geometric NL, which is further solved by using the perturbation technique. Besides, the Gr efficiency parameter was introduced to consider the SSE as well as other effects on the material characterizations. The obtained results show that reducing the foundation stiffness caused a decrease in the NF, although the NL/linear FRs rose. The enhancement of foundation stiffness on the frequency-amplitude curves at various distribution patterns is shown in Fig. 62. Furthermore, Varzandian and Ziaei [518] performed an analytical solution of NL FV of isotropic thin NP utilizing NLET. The plate’s formula based on KPT combined with VKT-type nonlinearity, although the GEs were defined by NLET combined with VKT. The classification of nonlinear analysis by analytical solution with/without SDE shown in Table 6.

Fig. 62

The enhancement of foundation stiffness on the frequency–amplitude curves of (0∕90∕0∕90∕0)_S FG-GRCs laminated plates founded on EF [517]

Table 6 Classification of nonlinear analysis by analytical solution with/without SDE

3.1.3 .

3.2 Nonlinear Analysis by Numerical Method

3.2.1 Numerical Solution Without Considering the SDE

While, the numerical solutions present some error due to the approximation process, the validated results obtained in literature prove that it is applicable to use in wide range of applications especially in NL formulation, complex scheme and in multi dimensions domain. Besides, the obtained outcomes show to be in good agreement with high order and analytical results. Notably, Chandrashekhar and Ganuli [524] performed an NL vibrational investigation for LCPs with a sandwich structure and arbitrary material properties by utilizing the C0 assumption strain interpolated FE plate model founded on Reddy’s third-order assumption. Besides, the variation of TS was obtained by VKT NL, while the NL/linear variance was obtained through implementing the Monte Carlo associated with the Latin Hypercube sampling. However, it is revealed that at significantly high amplitude ratios, dual peaks were noticed in NL frequency values. Additionally, it increased with the probability density of non-Gaussian function owing to raising the vibrational amplitude values. Meanwhile, Mehar et al. [525] performed a numerical FG sandwich plate structure analysis while considering the NL G-LS within a uniform TE combined with HSDT. Meanwhile, the direct iterative method (DIM) has been employed for the FE solution. It has been observed from the results that the FG-CNT recorded higher NL frequencies than the UD-CNT distribution. In addition, increasing the a/h, and the CNTs’ V_F will increase the NF. Furthermore, in a novel study by Yue et al. [526], the NL geometrical oscillations for remarkable deflection of plates made of FG-CNT by considering a 3D plate model based on the 3D-TE associated with NL G-LS tensor were utilized. However, the analysis was accomplished through IGA, and four types of CNT patterns were explored. The ROM was used to extract various effectual mechanical properties. The presented numerical solution steps are illustrated in Fig. 63. Moreover, Wang et al. [527] employed the DQM to investigate the NL vibrational behavior of the FG-GPLRC plate under electrical loading, considering three various FG profiles. The effective medium theory (EMT) has been utilized to predict the required parameters of the structure analysis. However, the ROM has been utilized to obtain the material characterizations, and the HP has been employed to drive the GEs of the NL FV. The results show that the FR increased due to the increase in DC-voltage, and the maximum increase occurred by U distribution stress, as presented in Fig. 64 (a). The reason behind this increase was stretched in electrostatic. However, it shows that any changes in AC-frequency led to a sharp variation in the FR. Besides, Ramezani et al. [528] proposed a composite FE by utilizing the assumed natural strain method (ANSM) to explore the linear/NL mechanical behaviors of plates of FG-GPLRC subjected to various BCs. The GEs were defined by implementing a modified FSDT, and the HT approach was used to extend the effective E. Besides, the FV and stability response were obtained by implementing the iterative arc-length methodology. The results of this study were compared and validated with various methods, such as CPT and MF. Accordingly, the CCCC BCs required higher NF values than SSSS in all various methods for square plates with central cutouts. However, Gholami et al. [529] employed the HT and ROM to define the material characterizations of rectangular plates made of FG-GPLRC, followed by implementing the HPs and variational differential quadrature techniques (VDQT) to obtain the weak form of discretized NL formulas of motion. Besides, the Galerkin, PAL, time-periodic discretization methodology combined with the modified N-RI to define the NL harmonically excited vibrations of the structure. The obtained results show that vibrational amplitude is reduced by raising the GPLs’, although the WF, causes greater NF values.

Fig. 63

The numerical solution strategy [526]

Fig. 64

The relationship between a DC voltage & FR, & b AC frequency & FR [527]

Furthermore, Ragb et al. [530] investigated the FV of irregular CPs subjected to WPF by implementing convolution and indirect MF methodologies, and the displacement field was obtained by employing FSDT. Besides, after reducing the GEs the DIM was implemented to solve the NL Eigen-value problem. The efficiency and accuracy were achieved by finding the CPU and comparing the exact solutions through RRM and classical quadrature (CQ). It is noticed from Fig. 65 that increasing the shear modulus, WPF, and thickness ratio leads to an increase in the NF, and increasing the foundation parameters increases the vibrational wave along discontinuity.

Fig. 65

At (θ = 45, α = β = 0.5, linear foundation γ = 0.2 & K₁ = K₂ = 100, K₃ = 50); a–b the deviation of NF for plates subjected to SSSS BCs with shear and E gradation ratio, & c–d the deviation of NF for plates subjected to SSSS CCCC BCs with thickness of skew plate [530]

Janane Allah et al. [531] employed the TSDT to explore the NL vibrational performance of plates made of FGM and with various material properties. The HP has been utilized to drive the resulting EOM, which is further solved by using the implicit algorithm. It has been observed that the maximum FF was recorded when the SSFF BCs were subjected and considered, although the values recorded by using CCCC BCs greater FF values were achieved compared with those of SSSS BCs. Besides, the NF values obtained by P–FGM and S-FGM were almost identical in all modes' shapes, and the V_F highly influenced the NF. Besides, Wang et al. [532] employed the stochastic MF associated with Reproducing Kernel to investigate the NL FV of CFRC plate with rectangular geometry. Also, the CPT has been utilized to establish the NL mathematical modeling of geometry, and the governing EOM has been determined through the principle of virtual displacement. The obtained results show that the BCs have a remarkable impact on the NL FV behavior, where the absence of any effect caused by the plate length has been noticed. Besides, under various BCs, it is seen that the DLFF increases due to the increase in amplitude, which explains the hardening spring-type NL in three various BC sets. Although the rate of increasing the NL FF in SSSS BCs was much more significant than those of CCCC and CSCS BCs with the rise in amplitude. However, Sit and Chaitali [533] performed a third-order NL model numerical and experimental investigations that analyzed LC structures' NF and FV behavior subjected to thermal enhancement conditions. The NF response of glass fiber reinforcement composite (GFRC) composite laminated plates with a variable number of layers by the enhancement of temperature variation has been studied experimentally, and the mathematical modeling has been achieved by utilizing the Green–Lagrange NL FE model based on TSDT and the obtained results by both methods were compared and validated. The results show that employing the fine mesh has achieved an excellent convergence for NDNF of CPs subjected to thermal load. Also, the NF values were reduced dramatically by increasing the temperature in both numerical and experimental methods, and the structure's stiffness decreased with temperature increment. Similarly, the same authors published a further study which as well focused on studying the influence of temperature in producing NL effect on the FV performance of FRP of bridge deck Sit and Chaitali [534]. However, Harrras et al. [535] employed HP to generate a theoretical model to analyze the geometrical NL FV response of a plate of CFRP fully clamped BCs to investigate the nonlinearity effect of the NL resonance frequency and the NL fundamental mode shapes. The first nine mode shapes were defined numerically and experimentally. Furthermore, the NL stability and FV of multilayer FG-GPLRPC, which experienced a PPB, were investigated by Gholami et al. [536], the modified HT and the ROM were used to drive the material characterization. The VKT-type nonlinearity, VDQ, and Lagrange formula for the enormous bend of FG-GPLRPC, although the Parabolic SDT of the plate has been implemented to obtain the discretized couple NL EOMs. The obtained outcomes reveal that raising the weight fraction of GPL shows to enhanced the CBL and raising the subjected compressive in-plane force in the buckled regime caused to reduce the plates’ NF as illustrates in Fig. 66. Besides, further increasing tended to have zero NF at the systems’ buckling load which owing to the reduction in the plates’ total stiffness as well as subsequently dropping the stability. The highest stability has also been recorded at FG-X and the lowest by FG-O dispersals. The plates’ strength increased when subjecting higher amounts of GPLs near the upper and lower surfaces, which consequently caused them to have greater NF values at FG-X patterns. Moreover, Zhang et al. [537] investigated the NL vibrational behavior of cantilever CP with a honeycomb sandwich structure by implementing the HP associated with Reddy’s TSDT to drive the GEs. Besides, the RRM has been used to determine some modes of NF, and the Galerkin methodology has been utilized to transfer the NL PDEs to NL ODEs. Periodic and multi-periodic motions, as well as chaotic motions, were observed in the CP structure when it is subjected to in-plane and transversal excitations. Additionally, Gholami et al. [538] implemented a unified HSDT to examine the NL dynamic behavior for moderately thick and thick polymer NC plates with rectangular geometry reinforced with GPLs. The HT and ROM were used to define the material characterizations, and the HPs combined with VKT were used to drive the unified mathematical formulation. The NL vibrational solution was obtained through GDQM and the discretization of GEs by Galerkin. The obtained results show that the NL/linear FR obtained by MPT is much lower than those obtained through HSDT, which may be because the HSDT does not require a shear correction factor. In addition, greater linear and NL frequencies were obtained by increasing the GPLs’ WF. However, it reduced the NL FR and the hardening performance.

Fig. 66

The FV behavior of PPB FG-GPLRPC plates (a) & (b) at different GPL weight fraction values at a/h = b/h = 10, & (c) at different GPL dispersal patterns at a/h = b/h = 10, w_GPL = 0.4% [536]

Furthermore, Xu et al. [539] examined the NL FV of MEE CPs via the VKT NL strain–displacement approach under HSDT. Besides, the HPs have been utilized to define the GE, while the Galerkin combined with HBM to solve the NL dimensionless motion equation. The obtained results show that at constant vibrational amplitude, raising the span-thickness ratio leads to a decrease in the NL FV FR. Also, the NL FV FR increased due to the increase in the a/h. In addition, the NL FV behavior is significantly affected by the negative magnetic/electric potential. Besides, Adhikari et al. [540] investigated the geometric NL deflection FV performance of LCP made of FG-CNTRC by implementing the G-LS field based on VKT. Besides, the Lagrange motion formula based on Reddy’s HSDT has been utilized to determine the GEs, and the NTI scheme has been utilized to encapsulate the NL vibrational responses after defining the NL eigenvalues. Besides, the direct iteration approach has been used to analyze the NL Eigenvalues for FV performance as well as the dynamical responses. Moreover, Adhikari and Dash [541] employed the HNPSDT to investigate the FV of LCPs subjected to various BCs, and the FEM was utilized to drive and discretize the GEs. The first mode shapes of at CCCC and SSSS BCs cross ply [0°/90°/0°/90°/0°] are illustrated in Fig. 67. It has been observed that increasing the amplitude ratio until a certain point leads to an increase in the FR followed by a reduction due to the sudden variation in the NL stiffness of the plate. Besides, at low vibrational amplitude, increasing the modulus ratio increases the degree of nonlinearity. However, increasing the span-thickness ratio caused a decrease in the NL behavior. Likewise, Swain et al. [542] employed the HO polynomial SDT and FEM to investigate the NL FV of LCPs. By examining the SSSS BCs with a/h = 10, the results show an increase in the degree of nonlinearity caused by rising amplitude ratios and the number of layers. However, there was a reduction in the softening type of nonlinearity due to increasing the number of layers, while the hardening type of nonlinearity increased with increasing amplitude ratios. Besides, by studying the enhancement of various BCs on the FR, it has been seen that the greatest FR has been achieved by implementing the CCCC BCs while the minimum by CCFF and the softening type nonlinearity obtained by CFCF BCs at amplitude ratio equal 0.4. Besides, Fig. 68 shows that by increasing the fiber angle to 90°, the FR has reduced, and the variation of FR is greater at a smaller fiber angle.

Fig. 67

The first mode shapes of (a &b) CCCC BCs, (c & d) SSSS BCs cross ply [0/90/0/90/0] with a/h = 1000 & W_max /h = 1.1 for a while for b W_max /h = 1.2 for c W_max /h = 1.6 & d = W_max /h = 1.7 [541]

Fig. 68

a The FR at various amplitude ratio values & lamination scheme at SSSS BCs & a/h = 10, & b the enhancement of fiber angle on the FR in 2 layers plate [542]

The FV and NL behaviors of a novel NC double variable edge plates made of FG-GRC and immersed in a liquid subjected to explosive loads, were comprehensively designed and analyzed by Ha et al.[543]. The HT micromechanical model has been utilized to define the material characteristics, while the GEs have been defined by implementing the CPT combined with Galerkin. Various reinforcement distribution patterns were considered. It has been seen that increasing the fluid density leads to a reduction in the NF, and increasing the immersed depth will cause the same effect as well. The weakest distribution pattern type was the FG-O distribution, though the FG-X distribution guarantees the highest durability.

3.2.2 Numerical Solution with Considering the SDE

The accessibility of NL and NLC models capable of exactly apprehending and forecasting the behavior of the structure is of supreme significance for many structural analysis applications as recorded by Patnaik et al. [544]. In this context, Fan et al. [545] employed the NURBS based on IGA associated with MCST of elasticity to examine the NL vibrational behavior of FGPPs with diverse porosity distributions. It has been seen from the results of U-FGPPs, that the plates exposed to CCCC BCs recorded superior NDNL frequency than those subjected to SSSS BCs. Besides, the NL frequency was observed to reduce by increasing the MP GI. However, for the excellent plate deflection values w/h > 0.8, the effect of MP GI becomes vice versa. It has been revealed that the MCS-based FR becomes minimum at a specific l/t value, which is further influenced by raising the porosity index of the FGPPs material properties. Besides, Zhang et al. [546] employed the Kp-Ritz to analyze SLGS, while the NLET was implemented to consider the SSE. The obtained results show that at SSS, CSCS, and CCCC BCs at all side lengths, by raising the amplitude, the NL frequency increases. In addition, overall, the greatest amount of NDF was recorded at SSS. In contrast, the least amount of NDF was recorded at CCC. Furthermore, as a significant result, it is revealed that the linear response of Gr was found to be less sensitive to the supplementary mass than those of NL response. Furthermore, Nguyen et al. [547] performed numerical modeling by employing the IGA NURBS analysis to examine the NL dynamical (FV and forced vibration) and static behaviors of MSPs made of FG-GNFs. The MCST and VKT were utilized through the analysis, and the EOMs solved by implementing the N-RI and NTI. As seen in Fig. 69 the best reinforcement performance has been achieved using the FG-X arrangement with the lower central deflection amplitude. In comparison, the FG-O arrangements achieved the most significant central deflections’ amplitude. However, this study is considered the first to focus on enhancing Rayleigh damping on the FG-CNT CPs’ oscillations subjected to three different dynamic loads with various steps.

Fig. 69

The enhancement of Graphene nanofillers (GNFs) dispersal on the NL dynamic performance of SSSS BCs FG MSPs (l/h = 50, a/h = 20 & W*_GNF = 1.0%) [547]

Mirzaei and Kiani [259] investigated the NL FV of plates made of FG-CNT subjected to CCCC and SSSS BCs in which the ROM was employed to define the equivalent material characterizations where the SDE was considered and VKT type of geometrical NL has been included in the formula. Besides, the Ritz methodology has been used to obtain the GEs. Also, the Galerkin methodology to eliminate the time dependency and the direct displacement control approach (DDCA) has been utilized to solve the NL eigenvalue problem. A mode redistribution has been seen in both SSSS and CCCC BCs, and the CNTs’ V_F and distribution patterns greatly affected the NL frequencies. Additionally, the amplitude frequencies obtained by SSSS are lower than those obtained by CCCC BCs. However, Zhu et al. [548] explored the vibrational performance of an NP with a composite sandwich structure, which consists of a core made of PMMA and two face sheets of PMMA matrix with FG-CNTs contemplating the NPs as well as the NLC effect agglomerations. Eshelby-MT approach has been implemented to define the material properties, while the NSGT combined with Reddy’s HSDT and VKT NL deformation theory were utilized to drive the EOM. Besides, the Galerkin and the N-RI were utilized to perform the solution of the SSSS plates’ vibration and impulsive problems. It has been obtained that greater FV frequency can be achieved as a consequence of improving the NPs’ stiffness, which is caused by increasing the content of CNT, less CNT agglomerations, smaller functional gradient, weaker size effect, greater a/h as well as greater a/h as well as reducing the core thickness. Additionally, the structural stiffness can be enhanced significantly by reducing the size effect and the agglomeration of the material characterizations. Moreover, Cho [549] employed the VKT in the framework of a 2D NEM combined with (1,1,0)* hierarchical model, corresponding to the FSDT, to examine the NL FV of FG-CNTs CPs. Besides, as a reflection of the SDE material characterizations'; the CNT efficiency factors were considered. The results show that the NL/linear FF ratio is significantly affected by the CNT's gradient patterns, and the maximum nonlinearity has been noticed by applying the FG-O distribution patterns, while the lowest in FG-X is entirely contrary to the plates' stiffness. Besides, increasing the plate thickness or changing that plate geometry to square will cause an increase in the NL/linear FF ratio, and Fig. 70 illustrates the NF in three mode shapes. In addition, the FV nonlinearity obtained by the non-CNTRC isotropic plate was much lower than that obtained by the CNTRC plate.

Fig. 70

The lowest mode shapes (V^*_CNT = 0.11), a (1,1) mode, b mode (1,4) & c mode (2,1) [549]

Furthermore, Ashraf et al. [550] examined the NL FV of annular sector NPs made of GPLs by implementing the modified HT to define the mechanical properties. Besides, considering the geometrical nonlinearity properties, the EOM was determined using the HP combined with FSDT. Meanwhile, the numerical solution was obtained using the 2D-GDQ, N-RI, and HBM. It has been observed that the elastic medium controls the enhancements of GPLs weight fraction in all distribution patterns and the sector angle under subjecting the plates to various BCs on the NL vibrational responses. Also, increasing the inner-outer radius ratio causes an increase in the FR. Likewise, Adhikari et al. [551] employed the MT mode combined with FEM representative volume element (RVE) methodology to investigate the NL vibrational performance of a plate made of FG-CNTRC with several RVEs dispersal as shown in Fig. 71 (a), and founded on a new shear strain function through implementing the IGA process. The extended-MT considered various types of CNT fiber orientation in the matrix. Lately, the obtained solution has been promulgated to the FG-CNTRC model to define the NL response of the geometry. Besides, the Green-LS and the NL NTI methodology were implemented to determine and solve the NL GEs. The relationship between the molecular mechanics bond representation and the continuum mechanics bond representation is illustrated in Fig. 71 (c). Figure 71 (b) shows that there was no discernible difference in the NL time response of the FG-V plate for various types of RVEs, and the most excellent values of NF were recorded by using the FG-X distribution while the lowest by using the FG-O distribution. The results obtained by FEM RVE were very similar to those obtained by extended-MT at a/h of 15 and 7% V_F of CNT. However, the low stiffness value is the main reason for achieving a great degree of nonlinearity by implementing the FG-O. Moreover, Lewandowski et al. [552] employed the refined ZZT to explore the FV of laminated plates that consist of VE layers by considering volumetric and deviatoric strains based on the separate fractional Zeners material model.

Fig. 71

a Several RVEs dispersal of CNT fiber, b The NL time response of SSSS-1 & CCCC with FG-V distribution under UDL sinusoidal time dependent load at nL = 15, Λ = 10 & b/h = 100, & c the relationship between the molecular mechanics bond representation and the continuum mechanics bond representation [551]

Chu et al. [553] employed the MSGT to investigate the NL large-amplitude oscillations of an FGPPS composite with rectangular geometry. Besides, the novelty of this study was that it considered the first study that focused on analyzing each microstructural tensor enhancement individually on the NL free oscillation of the FGPPs with and without the central cutout through implementing the IGA. Consequently, the central cutout has been displayed to verify C⁻¹ continuity founded upon the refined higher-order plate formulation. It has been observed that the symmetric rotation gradient tensor enhances the FF by 1.2%, while the dilatation gradient tensor achieved higher enhancement by around 3.27%. Besides, a significant enhancement in the FF of about 9.43% has been obtained by contemplating the deviatoric stretch gradient tensor. Additionally, as long as the NL frequencies related to positive oscillation amplitudes were lower than those related to negative ones, it caused an unsymmetrical frequency response of the anisotropic character of the FGPPs. However, Kiani [554] employed the NURBS formulation based on the IGA to examine the large amplitude FV of LCPs reinforced with Gr, and the HT approach was utilized to obtain the material properties. Besides, the GEs were obtained using the TSDT combined with the VKT type of kinematic assumptions. It has been seen that the NL/linear FR of LCPs at different w/h obtained by SSSS BCs was higher than those obtained by CCCC BCs. Additionally, the NF reduces with increasing the temperature at various BCs. However, it leads to an increase in the NL/linear FR, and these results were accepted as long as the material characterizations were assumed to be dependent on temperature. Else, Mehrabadi and Farahani [555] used the finite strain tensor to examine the NL FV of skew plates consisting of FG-CNTs, and the ROM associated with MDS results was utilized to approximate the material features. However, the CPT based on the Green-LS tensor has been used to drive the governing EOM. Furthermore, the DQM, Galerkin methodology, and the Frechet derivative were implemented to solve the NL equations. Nevertheless, Torabi and Ansari [556] investigated the NL FV performances of CPs consisting of CNT subjected to thermal loads, and the material properties were assumed to vary continuously along the direction of thickness. The FSDT has been utilized to determine the GEs and solved by arithmetical variational methodology, and various numerical approaches were implemented to complete the analysis. The results show that the linear NDNF was reduced, and the hardening effects increased due to an increase in the initial thermal load. Besides, the NL effects and the NL/linear FR became enormously significant by increasing the thickness-to-outer radius and the inner-to-outer radius ratios. Additionally, the axisymmetric solutions found overestimated NF values and inaccurate results. Furthermore, Yue et al. [557] investigated the NL dynamical buckling characterization of NMPs with randomly distributed CNTs as illustrated in Fig. 72 in PPB regimes and subjected to various BCs through implementing NCs combined with quais-3D plate theory, besides the Galerkin combined with PAL to accomplish the NCs based NL FR and frequency-load deformation curves.

Fig. 72

The schematic of a quais-3D heterogeneous NMP with randomly distributed CNT [557]

A model of NP has been developed by Gholami et al. [558] to examine the NL FV, size-dependent as well as the large amplitude of a multi-ferroic composite NPs with rectangular geometry and founded on EF (WP/WF) through utilizing the HSDT, NLET, as well as unified NLC with VKT strain tensor. Besides, the HP has been used to define the GEs and BCs, and the large amplitude vibrational problem has been solved by numerical solution through implementing the shifted Chebyshev–Gauss–Lobatto grid and Galerkin. The obtained results shows that the structures’ stiffness reduced due to raising the a/h, which, in consequence, reduce the NF. Besides, utilizing the linear mathematical method for forecasting the FV behavior causes unreliable results. Moreover, the maximum nonlinearity of spring hardening was obtained by subjecting the structure to CCCC BCs, while the minimum was by subjecting the structure to SSSS BCs.

Moreover, in a novel study Barati and Shahverdi [559] investigated the NL FV of NLC 4-variable FG NPs subjected to an NL hardening EF. Besides, the porosity in the NPs was considered by utilizing the ROM, and the GEs were defined by utilizing the VKT and solved by Galerkin combined with homotopy perturbation method (HPM). It is revealed from the obtained outcomes that the CCCC BCs can increase the stiffness of the NP, which, in consequence, leads to higher NF compared with SSSS BCs. On the other hand, having more voids or higher porosity caused a reduction in the NL NF, which increased with the increase in the inhomogeneity index. Besides, Naghinejad and Ovesy[317] studied the NLC VE NPs’ FV performance with rectangular cutouts, although the surface effects and various BCs were examined. The NLC FEM was developed by using the principle of total potential energy, and the CPT drove the formulas. Besides, the variational principle has been utilized to define the Eigenvalue problem. The results revealed that increasing the NLC parameter reduced the real and complex Eigen-values, although the maximum complex Eigen-values were recorded at CCCC BCs. Moreover, Gholami et al. [560] explored the FV and the stability of METE, considering the size dependency NP model by employing the TSDT combined with NLET. By using the HP, the GEs have been defined and solved by implementing the GDQM approach, and for reducing the discretization, the Galerkin was applied. It is seen that the magneto-electromechanical load has a more significant influence on the NF and stability compared with thermal load conditions. Nonetheless, Shakouri et al. [561] performed an FV analysis using the Galerkin of NPs modeled based on NLC CPT. The influence of Poisson’s ratio, NLC parameter, and a/h on vibrational responses was examined. It is seen that by applying FFFF BCs to the NLC NP model, a remarkable effect caused by Poisson’s ratio can be obtained. Besides, Li et al. [562] presented a closed-form and explicit solution by employing the separation-of-variable methodology for FV analysis of the NLC Mindlin plates model. Besides, for formulation of the NLC plate model, Rayleigh’s principle was used. It is revealed that under all various types of BCs, increasing the NLC parameters caused a reduction in the FF ratios, while it increased by increasing the side length of the plate until a certain level, then it became constant. However, the FR shows an increase due to increasing the t/l. Furthermore, Dangi and Lal [563] explored the NL thermal impact on the FV of FG Mindlin NPs of Bilinearly changing thickness by using the ENLT. The GEs obtained by HP, although the frequency equations for various types of BCs defined by using the GDQM and solved by MATLAB. It is revealed that rising the NLC factor causes to reduce the variation in the frequency parameter at various temperature difference and various plates’ length as well as various nonuinform factors. Likewise, Daneshmehr et al. [564] examined the enhancement of SDE on the FV analysis of CPs based on NLET and HSDT. The GEs was defined by the minimum potential energy and solved GDQM. It is revealed that by increasing the NLC parameter, the SDE increases in all BCs besides, the FR obtained by CCCC BCs shows to have the highest SDE behavior. Furthermore, Mohammadi et al. [565] explored the FV and the dynamic pull-in instability of circular MPs under electrostatic and hydrostatic forces. Besides, the SGET combined with KPT were used to account the SDE. Meanwhile, the HP was employed to define the BCs and the GE which solved further by using the GDQ. It has seen that at specific values of ND LSP and normalized NF, the MSGT predicts greater pull-in voltage of those of MSGT. Besides, the rate of normalized NF remains constant by increasing LSP. The classification of nonlinear analysis by numerical solution with/without considering the SDE shown in Table 7.

Table 7 Classification of nonlinear analysis by numerical solution with/without considering the SDE

3.3 Nonlinear Analysis by High Order Theory

3.3.1 High Order Solution Without Considering the SDE

To accomplish significant accuracy in predicting frequency responses, deformation as well as stress in a multilayered CPs. There is a necessity to implement high order solution. Above all types of high order methods, the predictor corrector have remarkable potential for prediction, Noor et al. [580]. Besides, it achieved accurate stresses and displacements’ dispersals along the CPs’ thickness. Although the higher-order approaches required C¹ continuity in FE applications. In this perspective, Janane-Allah et al. [581] presented a novel methodology based on the TSDT, which is identified as the high-order implicit algorithm to investigate the NL FV as well as the forced vibration of FGPPs shown in Fig. 73. Besides, the HP has been utilized to determine the GEs and the approximations of the resulted equations were computed through implementing the FEM. Else, various types of porosity were considered and examined, and it was noticed that increasing the porosity fraction at a V_F ≤ 1 causes a decrease in the FF gradually, and the plates’ stiffness depends on porosity distribution, V_F exponent, as well as l/t. Additionally, it has been proven that in the case of investigating the NL vibrational behavior, the porosity effect must be considered, and the enhancement of uneven porosity in maximum deflect is much lower than that on even porosity distribution. Furthermore, Mahesh et al. [582] investigated the NL FV of multifunctional sandwich plates consisting of an auxetic core BaTiO₃-CoFe₂O₄ and two face sheets of MEE subjected to various BCs through developing and implementing a higher-order FEM. HP and Reddy's HSDT were utilized to define the global EOM. The obtained results show that increasing the thickness of MEE sheets can achieve greater coupled frequency. However, the uncoupled/coupled frequencies are significantly affected by different parameters such as the rib length ratio, micro topological arrangement, the inclination angle of the auxetic, and the SA. Also, reducing the length ratio of the rip and the inclination angle improved the NF remarkably, as shown in Fig. 74 However, the auxetic inclination angle depends on the rib's length ratio.

Fig. 73

a The schematic of FGPPs with the longitudinal & transverse axes, b the material deviation along the plate thickness, & c the flowchart of high-order implicit algorithm the for defining NL dynamic system [581]

Fig. 74

The enhancement of auxetic inclination combined with electromagnetic circuits & the length ratio of the rib on the FF performance of the structure at a/h = 100 & a/b = 1;2–1-2 a when the plate subjected to CCCC BCs & b when the plate subjected to SSSS BCs [582]

Moreover, Thinh and Tung [583] explored the FV and dynamical responses of FGM CPs subjected to tangential BCs. The EOM defined by VKT was combined with CPT and solved analytically by Galerkin for nonlinearity relation, and the 4th-RK was used to trace the deflection-time trails. The results reveal that the dynamic responses are significantly influenced by the tangential BCs at high temperatures. Besides, the enhancement of porosity type on the NF depends on n, where increasing n causes an increase in the NF. However, it is noticed that at high n values, the NF is reduced by increasing the porosity V_F in an even dispersal type. Nonetheless, Deepak et al. [584] analyze the FV responses of LCPs made of Gr/Ep with an outer face sheet made of PZT-5A by performing a solution by employing the predictor–corrector to solve the GEs that discretized by Galerkin to time depends on ODEs. Besides, the Monte Carlo model was used to measure the variability of parameters on NL NF. The primary purpose of this study was to explore the enhancement of thermal load and the randomly applied actuation potential on the NL NF by subjecting the plat to various BCs. It is revealed that at positive control potential difference and by increasing the temperature, the NF reduces, and NL/linear FR rises. In contrast, it is reversed when implementing negative control potential difference. Beside, Akhvan and Ribeiro [585] explored the NL FV of LCPs with geometrical imperfection, containing curvilinear fibers and subjected to various BCs. The TSDT has been utilized to define the displacement function and the EOM defined by p-version FEM. The modified RK-Fehlberg combined with Cash-Karp methodology was used to control the error with adaptive step size. It is observed that the NL FR increases due to the increase in the geometrical imperfection. Likewise, increasing the NL FR in perfect plates causes to have hardening effects, and this effect becomes softer by generating and increasing the imperfection. Also, the softening behavior can be observed only at a slight deflection, while the hardening behavior is dominant for large deflection. However, Lee et al. [586] performed NL FV analysis for symmetric/unsymmetric CPs. A reduction process has been performed by Duffing-type modal by utilizing the FEM reduction methodology. The RK and HBM were used to solve the Duffing-type, and the RK was used to define the accurate FR at the minor and high modes of the CPs at several deflection values. The results of the linear system were found to form a parabola function, while the NL system is a sinusoidal function. Furthermore, by examining the positive and negative amplitude frequency versus maximum deflection, both amplitudes were shown to have the same tendency. Furthermore, in a novel investigation, Dat et al. [587] presented a huge vibrational amplitude NL analysis for imperfect 3D penta-Gr plates subjected to a TE. The density function approach determines the thermal expansion coefficient and the elastic constants. However, Reddy’s HSDT were used to drive the EOM and compatibility. The Galerkin methodology has been employed to obtain the linear frequency and NL/linear FR in closed-form expression. Besides, the dynamic behaviors were obtained by 4th-order RK, and the optimum NF values were obtained using the Bees algorithm. It has been revealed from the results that at significantly high amplitude values and the NL NF rises as a consequence of increasing the b/h. Moreover, the primary imperfection amplitude has an undesirable impact on the dynamic behaviors. Additionally, the PF has a more significant influence than the WF. Moreover, Cong and Duc [588] studied the NL dynamic and vibration responses of CPs made of multilayer of FG–GPLs utilizing the FSDT. The E of the GPL has been obtained by using the modified HT micromechanics approach. All various types of foundation shows to influence the NF. However, the most significant influence on NF recorded by using the Visco-PF. However, Zhu et al. [589] investigated the relation between NL FV and forced vibration of VE plate structure based on the Reddy HSDT combined with VKT. The VE model was adopted by employing the KV. Besides, the HP was utilized to define the GEs which solved by combining the RK with HBM. The results reveals that the NL hardening behavior occurs at various material characterizations, various subjected load and different geometrical features. Furthermore, at large ND amplitude the a/h has a slight impact on the NLNF. In addition, Yan et al. [590] presented a fast semi-analytical solution to examine the NL vibrational responses of variable stiffness plates. The developed formula is based on the mixed variational theory, while the spatial dependence is handled by using the Ritz-like. Besides, the direct integration methodology, HBM, perturbation techniques, and the method of averaging were developed and compared for the temporal dependence. The obtained results reveal that the structural performance was characterized by reducing raised values of the corresponding fiber angle T₀. Besides, by considering the effect of fiber steering the tensile force was found to effectively enhance the NL frequency. However, Shih and Blotter [591] investigated the NL FV analysis of arbitrarily laminated, cross-ply laminated, angle-ply laminated, and orthotropic thin plates in rectangular shape founded on EF and subjected to various BCs. The GEs defined by VKT consider linear WF, PF, and NL WF which are discretized by Galerkin and solved by RK. It was revealed that EF has a higher impact on plates subjected to SSSS BCs than those of CCCC BCs, and the FR achieved by SSSS BCs significantly higher than those of CCCC BCs. Moreover, Dinh Dat et al. [592] analyzed the NL FV of an auxetic laminated plates with two face sheet made of electro-elastic under blast, electrical, magnetic and thermal load condition and founded on PF as shown in Fig. 75. The basic equations defined by Reddy’s HSDT and the solution was accomplished by using the 4th-RK and Galerkin. It has been revealed that by increasing the EF coefficients the NF has increased, although increasing the electrical potential, temperature increment and reducing the magnetic potential caused to reduce the NF and increased the FR. Besides, the NF and FR reduced by increasing the inclined angle of the unit cell in the auxetic core layer.

Fig. 75

The schematic auxetic laminated plates with two face sheet made of electro-elastic [592]

3.3.2 High Order Solution with Considering the SDE

The NLET adopts the hypotheses that stresses at a point in an elastic system depends on the strain fields at all the points in the entire region stated by Li et al. [562]. Besides, the NLC parameter must be considered and applied firstly. Notably, Gupta et al. [593] developed and employed the hyperbolic HSDT combined with NDT to explore the NL flexural and vibrational performance of a shear deformable FGP. The obtained results show that the presented method shows an outstanding conjunction with the mesh refinement process for frequency parameters, and a good agreement with the previously published data has been noticed. Besides, various types of imperfection have various effects on FR, and the enhancement of amplitude on NL FR was weak on some types such as L1, G1, and G2, as shown in Fig. 76. Moreover, Cong et al. [594] explored the NLC effects as well as the Kerr foundation on NL mechanical performance of CPs consisting of FG-GPL in NMS. The CPT combined with NLET was employed to define the basic formulas, and the VKT strain displacement was employed to define the geometrical and motion compatibility formulas. Dynamic and static nonlinearity were solved by implementing the Bubnov-Galerkin and the stress function methodologies. Furthermore, the 4th-order RK was employed to solve the GEs for the dynamical model, as seen in Fig. 77, where KU, KS, and KL defined the upper, shear, and lower spring constants, respectively. It is seen from Fig. 77 (a) that KS has the most significant influence on the FR-amplitude correlation. From Fig. 77 (b), the most significant effect of NLC on the FR-amplitude correlation was found to be at μ = 5 × 10^–20, which means that increasing the NLC parameter enhances the FR-amplitude. However, Karimipour et al. [595] analyzed the dynamic responses of electrostatic MPs based on the SDE by employing the MCST. The HP combined with KPT were utilized to define the EOM. Besides, the mode shapes and NF values were obtained by employing the mixed-EKM and DTM. The 4th-order RK was used to solve the dynamic EOM in the displacement field domain. From various parameters selected to examine, it revealed that the FNF reduced until it reached zero due to a rise in the DC-voltage. Increasing the a/h causes to increase the NF values. The NF decreases and increases due to subjecting the CPs to compressive and tensile loads respectively. Furthermore, the LSP enhanced the CP stiffness as well as the NF. Furthermore, Thanh et al. [596] explored the NL dynamic behavior as well as FV analysis of an imperfect FG-CNTRC subjected to an EF, based on Reddy’s TSDT. The material properties and temperature dependency obtained by ROM. Besides, the NF and the NL dynamical behavior were defined by employing the 4th-order RK, Airy stress function as well as the Galerkin approach. It is revealed that the geometrical factors have a notable impact on the NL vibrational behavior of the CPs. Also, by reducing the ratios of geometrical factors the variation amplitude of the imperfect structures’ has reduced. Moreover, Shishesaz et al. [597] analyzed the NL FV of circular MPs based on NLET and exerted from HP by implementing the HPM. The model formulation was based on the NL model of VKT strain in polar coordinates combined with CPT. Besides, the Galerkin was employed to discretize the GEs. Furthermore, the 4th-order RK was used to solve the NL equations. The results obtained by HPM and 4th-order FK were in significant arrangements. The t/r highly affected the NL FR based on Eringen’s, and the NLP parameters highly influenced the ND FR. By comparing the results of CPT and non-CPT, it is seen that in the case of employing the CPT, the mode shapes have no significant impact on the NL FR. However, the reverse is in the case of NLET. Furthermore, Sahmani et al. [598] accomplished analytical solution of FGPPs NMP made of FG-GPLRC subjected to axial load with considering SDE for NL/linear vibrational performance within PPB regimes. The SGT associated with geometric nonlinearity were employed in the refined exponential SDT, and the HT and the closed-cell Gaussian random field were used to obtain the mechanical properties of the FG-GPLRC. Besides, an improved perturbation approach combine with Galerkin methodology were employed in an analytical expressions in an explicit form for SDE linear frequency-load as well as deflection NL frequency behavior. It was observed that at a specific value plates’ deflection the FR increased due to rising the porosity coefficient value, although this observation was dissimilar in various porosity distribution patterns. In addition, by considering the NLC the enhancement of GPL V_F decreases which lead to decrease the gap between the linear frequency-load curves.

Fig. 76

The relationship between NL frequency & amplitude ratio for various geometric imperfection [593]

Fig. 77

The enhancement of (a) Kerr function and (b) NLC parameter; on the FR-amplitude relation of NMS FG-GPL [594]

3.4 Nonlinear Analysis by Mixed Solution

3.4.1 Mixed Solution Without Considering the SDE

The mixed solution have the ability to utilize the 1D-DQM, Galerkin as well as iteration process to determine the vibrational responses for CPs subjected to various BCs stated by Kitipornchaiet al. [599]. In this framework, Gao et al. [600] employed the DQM to examine the NL FV behavior of an FGPPs NC with WPF analytical. The E was obtained by implementing the HT micromechanics model. Also, the CPT with considering the VKT strain displacement relationship and HPs were utilized to define the GE. The various examined porosity dispersals are shown in Fig. 78. The obtained results show that the porosity dispersal has a more significant influence on the dynamic performance of the plates than the GPL dispersion pattern. Additionally, the NF can be enhanced by adding a fraction of GPLs into metal foam, which leads to remarkable effects on the structures’ stiffness. Moreover, Nguyen et al. [601] employed the Bézier extraction of NURBS as well as C⁰-HSDT to investigate the NL performance of innovative porous plates made of FG-GPLs. As shown in Fig. 79, the FG-GPLs porous core has been settled in between two PE layer, the upper considered as an actuator layer and the lower as a sensor layer. Furthermore, the GEs were solved using the N-RI methodology combined with Newmark’s integration. Besides, the two PE layers were connected by a closed loop by adding an amplifier and controller components. The obtained results show that adding a velocity feedback controller gains cause to reduce the central deflection magnitude and the periods of motion of the NL dynamical responses. However, Khoa et al. [602] performed a unique mixed solution investigation for the NL FV and the dynamic performances of FG-CNTRC LC polymer NPs. Reddy’s TSDT combined with VKT geometrical nonlinearity was utilized to obtain the NL equation and solve it analytically, followed by applying the Galerkin methodology associated with RK. The obtained results show that reducing the a/h increased the plate’s thickness, which caused an increase in the plate’s strength, which, in consequence, will enhance the amplitude to fluctuate significantly. Besides, FG-X, which was modeled as the outer layers, can obtain greater load capacity, and it has been proved that FG-X is the best choice in developing FG-CNTRC for mechanical applications. Additionally, subjecting the structure to the EF reduced the structural amplitude, and increasing the weight fraction of CNTs caused the amplitude reduction.

Fig. 78

Various porosity dispersal in the FG-GPLs reinforced porous NC plates [600]

Fig. 79

The schematic of porous plate consist of FG-GPLs & two face sheets of PE layers [601]

Furthermore, Lal et al. [603] employed the HSDT and VKT type NL strain displacement relation to develop basic formulas that investigated the NL FV behavior of LCPs based on EF, which has arbitrary system properties. Besides, to solve the random NL generalized eigenvalue problem, the first-order Taylor series combined with the perturbation technique was implemented. The obtained results show that as the amplitude ratio increased, the NL frequency increased. Also, higher frequency values were obtained by using the PF. In addition, plates with square geometry have lower NF than rectangular geometrical shapes. Besides, Lore et al. [604] employed the non-polynomial HSDT with seven degree of freedom (DOF) to investigate the NL FV of LCPs and various geometries of shell panels. Also, an NL FE model has been established and implemented to define the discretized NL formulas, and the nonlinearity of the geometry has been considered in terms of VKT through the Green–Lagrange. The obtained results show that the degree of nonlinearity is reduced at higher vibrational modes, although it rises due to a high amplitude ratio. Besides, the first and the greater NL frequencies obtained by the cross-ply (0°/90°/0°/90°/0°) were lower than those of the angle-ply (45°/-45°/45°/-45°/45°) up to a specific value of amplitude ratios, while the opposite is true after this value. Nonetheless, based on Lewandowsk et al. [552] and Litewka et al. [605] analyzed the NL Eigen-value problem using the HM combined with quasi-Newton methodology. The analyzed model has a remarkably greater damping ratio than traditional plates with sandwich structures with VE core or metallic outer faces, and the redefined ZZT has proven to be highly efficient for laminated plates without the necessity for HSDT or NL SDT. However, Dastjerdi et al. [606] presented a semi-analytical and numerical analysis by implementing the SAPM as well as DQM to develop and study the dynamic and static analyses of NPs and MPs made of FGM and subjected to various BCs by utilizing an exact 3D elasticity in TE the exact solution included the implementation of FSDT and the analysis has been performed for NL and linear response. It was seen that increasing the plate thickness for all various types of BCs will cause an increase in first and second-frequency resonance, while the 3D approach achieved higher NF values compared with the NF values obtained by using the FSDT. Besides, at CCCC BCs, greater values of ND second frequency were achieved. Hence, referring to the original articles for more information regarding thick plates is recommended. Furthermore, Shi and Dong [607] employed the tangent HBSDT combined with IGA to investigate the NL bending and vibrational analysis of composite laminated plates. It was found that increasing the vibrational amplitude and plates’ thickness caused an increase in the nonlinearity FR degree. The results obtained by TSDT were slightly higher than those obtained by HBSDT, possibly due to the NPSDTs. Similarly, it has been remarkably noticed that when applying the SSSS BCs at specific amplitude values, the FR decreased suddenly, followed by a rise, which proves that the plats exhibited a hardening behavior in those exact values. These phenomena may be explained due to the variation in the plates’ stiffness. Figure 80 illustrates the relationship between NL FR and amplitude ratio at different BCs and span-to-thickness ratios.

Fig. 80

The diversity in NL FR at various amplitude ratio at various BCs & various span thickness ratios [607]

Ćetković and Vuksanović [608] performed a local–global study to explore the FV, stability, and deformation behaviors of a sandwich and LCPs by implementing the LW displacement model. The EOM defined by employing the HPs, and MATLAB has been utilized for the analytical and FE solutions parts. The obtained results illustrate that the plates' stiffness and frequencies were much higher in thin plates. However, Rafiee et al. [609] employed the FSDT and VKT to examine the NL dynamic stability of a PSP FG SWCNT-CNTRC thin to thick plate. Additionally, the Galerkin method has been implemented to obtain the NL equation with cubic and quadratic NL terms and solved analytically, followed by employing the HBM to define the NL dynamic stability. The impact of the applying constant voltage on the thin plates is significantly distinguished compared with thick plates. In contrast, the opposite is correct in the case of increasing the temperature. Various types of CNT distributions were considered in this study, Fig. 81 presents the impact of these various distributions on the transverse mode with unstable regions. Furthermore, it has been observed that the stability areas’ size grows as the volume percentage of CNTs rises. Besides, as the V_F of CNTs increases, the vibrational amplitude will decrease. Furthermore, Kurpa and Shmatko [610] explored the NL FV and buckling performance of FGM sandwich plates based on a modified Timoshenko-type theorem of the first order. Besides, the proposed methodology is based on the R-function theorem as well as the Ritz variational approach which considered the inhomogeneity of the plate. The obtained results show that the thickness ratio has a quite effect on NL NDNF and the results of complex shapes were insufficient. The in-plan uniformly compressed plates with three-layer FGM with high power law above 5 achieved sign-frequencies and critical loading than those obtained by plate type 1–1 and type 1–2.

Fig. 81

The impact of various CNT distribution on the transvers mode with unstable regions [609]

3.4.2 Mixed Solution with Considering the SDE

The SDE cannot be explained indirectly using classical elasticity theories since there is a lack of material LSP. Consequently, higher order continuum theories such as micro-polar elasticity, CST, MCST, SGT, MSGT, and NLET, which include further material LSP and the classical material constants, have been proposed stated by Talebitooti[611]. Accordingly, Jain et al. [612] investigated the NL FV and forced vibration of an FGM MPs under a transverse patch load condition. The CP model was generated using the HSDT combined with the MSGT, although the FGM was modeled by employing the P-LF. Besides, the GEs were defined using HP and solved using the Galerkin methodology. Furthermore, the increment HBM has been implemented to obtain the NL vibrational behavior. Furthermore, the VDQ, GDQ, and Newmark-beta integration were employed to analyze the NL vibrational performance of circular plates made of FGP by Ansari et al. [613]. Besides, Voigt’s ROM has been utilized to estimate the hygro-thermomechanical characteristics, and the GEs were defined by FSDT combined with VKT geometrical NL relations and HP. The results show that the porosity causes a reduction in the vibrational frequency. In addition, the vibrational amplitude may increase due to variations in moisture concentration. The classification of nonlinear analysis by mixed solution with/without considering the SDE shown in Table 8.

Table 8 Classification of Nonlinear analysis by mixed solution with/without considering the SDE

4 Discussion

Due to the various significant applications, of the plate structure it is considered one of the most important structures, especially in small nano/micro-scale sizes. The CPs/LCPs have an advanced property. Specifically, its mechanical and electrical characterizations and physical properties lead to higher application and demands. These unique characterizations of CPs make them suitable for MEMS, actuators, energy harvesting, transportation, and biomedical applications. However, the CPs are categorized as either carbon or metal-based and may include various types of NPs or NFs. The composite can also be porous, such as the porous Gr, to leverage the benefits of porous, which, overall, will influence the CP structures.

The linear/NL FV analysis is extensively required in mechanical and civil structures. To understand the dynamical responses of the structure, the influence of geometrical, NLC, material, mechanical, structural, reinforcements, and environmental parameters under various BCs and constraints on the NF/FF must be performed. However, it is worth mentioning that examining the enhancement of material and geometrical parameters on the FV responses in each study is essential. However, the FV analysis can be performed by implementing various solutions, which are analytical, numerical, high-order, and mixed solutions. The solution is usually applied after defining the GEs of the system with different theories and approaches by considering material and mechanical characterizations, BCs environmental effect, geometrical parameters, NP dispersal patterns, tensors, structure parameters, and in some cases, the NLC parameters or size dependency. Above all mentioned solutions, the analytical solutions show outstanding results as they provide benchmark results and achieve high accuracy for various CP structures under a variety of methodologies as well as FE equations. However, the main drawback is that it requires numerous complex mathematical equations that are challenging and frustrating to solve. Furthermore, it has been seen that some authors studied each parameter dependently. In contrast, others considered various parameters in their studies to explore the enhancement of combining two or more parameters on the FV behavior, such as BCs with weight fraction.

It is revealed that increasing the NLC parameter reduces the structure rigidity, which, in consequence, will reduce the NF. The SSE was observed to increase due to increasing the NLC parameter, which consequently reduced the structures’ stiffness, which means higher flexibility. Also, neglecting the surface effects caused a reduction in the damped NF. For thick NLC NPs models, the NFs primarily increased, followed by constant values. Meanwhile, for thin NLC NP models, the NFs have a climbing tendency associated with imperfection amplitude.

Besides, the non-locality efficiently predicts the composite waves’ dispersive nature. However, increasing the a/h cause to have the same impact on NF. The maximum porosity dispersal and the n are shown to control the NPs’ material characterizations. In most articles studied, the sandwich CPs’ material characterizations were assumed to diverge constantly along the thickness direction according to the V_F of the constituents. The enhancement of the porosity parameter becomes crucial in increasing the material index, which leads to a reduction in the frequency while increasing the deflection. Nonetheless, it is revealed in circular NPs with smaller thickness the impact of surface effects becomes more significant. Which in consequence will affect the vibrational responses. Notably, lower frequency values were obtained by increasing the surface density.

Additionally, the plates’ stiffness decreased owing to rising porosity, which, in consequence, increased the deflection and reduced the NF. However, minor stiffness causes more significant deflections. The FGM NPs recorded higher frequency values than FGP due to their lower deflection abilities. Likewise, the remarkable reinforcing enhancement was obtained by employing the FG-X distribution arrangements since it has the lowest value of central deflection, though the NF is the greatest. That is caused by the fact that, after the second order of NF, the system becomes tough and does not easily deform. Moreover, changing the mechanical characterizations in various directions in the material causes weaker materials than homogeneous ones. These changes lead to valuable material characterizations such as significantly light with high strength and heat resistance.

Investigating various plate models revealed that when the ND LSP increased, the variance between these plates reduced. Likewise, the MPs made of FGM recorded more significant NF values with greater a/h values. Besides, its vibrational behavior is highly affected by the material characterization in the case of greater ND LSPs. Also, the NL responses region for a specific structure increased owing to applying softer edge/BCs. Additionally, the NDNF of FGM MPs correspondingly evaluated for various material characterizations such as a/h and ND LSP.

For defining the material properties, the following methodologies and approaches were considered and employed: the MT approach, which is considered for effective reinforced structures’ characterizations, has been applied in many studies. Meanwhile, the modified HT model was utilized for defining the NCs E. Most studies have used the ROM, especially for obtaining the mass density and the Poisson ratio. The porosity V_F is usually included through the modified ROM, which, in consequence, leads to considering the FGPs’ material properties as continuously varying along the direction of the plate thickness based on the V_F. Besides, the enhancement of porosity decreased significantly with huge imperfection amplitude.

On the other hand, the Chebyshev-Gauss-Lobatto was employed to define the grid points of the mesh in numerous analyses. Besides, the DQM has been implemented to solve the DEs precisely using less CPU and fewer grid points. Nonetheless, the DQM is considered one of the best methods that can be used in various solutions, numerical, semi-analytical, and analytical. Likewise, the Galerkin approach has similar capability but with some limitations compared to DQM. However, The PDEs were revealed to be solved significantly by using the Cauchy problem, which also defined BCs and particular parameters. However, from a local/NLC perspective, it is revealed that reducing the LSP can reduce its impact on the NF in the local models. It is observed that the enhancement of the TS deformation on the NF is considerably undervalued by using the CPT at small plate thickness, which ends up with the necessity of combining the SGE into the classical HSDT to emphasize the size dependency plates’ model. In the scenario of all material LSP or at least two of them being set as zeros, the advanced NLC CPT can degenerate into the MCST-HSDT and the classical HSDT. As a remarkable observation, the FSDT, TSDT, HSDT, as well as CLPT are considered unpractical and inaccurate to use for equitably thick plates; for this reason, there is a necessity to apply the 3D-ET. Also, for FGM abstemiously thick plates, it is recommended to employ the 3D-ET. Furthermore, based on the NLC theory, the FSDT cannot be considered perfectly appropriate for analyzing the NS structures. However, in some cases, the FSDT might be suitable in NP dynamic analysis FV/forced vibration when combined with other theories or by using specific elements. In addition, the HSDT with four unknown observed to be more accurate in FGM plates than LCPs and sandwich plates Furthermore, Shear-locking can be prevented in NPs when they become thinner by implementing the shear correction factor. Previous studies have revealed that there are many techniques to eliminate the essential shear correction factor. Furthermore, in analyzing laminated plate the necessity of SDT and HSDT can be avoided in case of using ZZT. However, applying IGA solution instead of analytical solution for analyzing complex structure shows to have significant output.

In most investigated studies, the supreme implementing methods for defining the GEs HP, potential energy and driving the stability formula are essential for the NCP exposed to in-plan compressive loads. Besides the NvS is used widely specially in plates subjected to SSSS BCs Meanwhile, in the framework of applying the high-order theories, it is observed that authors show more interest in using the RK approach compared to Adam’s and the predictor-corrector. However, as an enormous fragment of reviewed studies for CPs and LCPs with various structural geometries, most authors were interested in performing purely numerical or analytical solutions, and more studies focused on investigating the linear FV compared with those of NL FV. In another word the NL FV analysis is still limited.

However, it is revealed that the porosity has more effect of the NF than FGM nanoparticles. Besides, the NF of LCPs with angle-ply were higher than those obtained by cross-ply until a certain amplitude level the opposite behavior may occur. Enhancing the SDE cause to reduce the structure stiffness. The GI seems to cause a reduction and an increase in NF depends on the mode shapes. Nonetheless, the degree of nonlinearity reduced at high vibrational modes, although it increase with increasing the amplitude. Else, in most analysis the first and second mode shapes have similar behavior while at the third mode shape the behavior start to vary. Also, high modes were less sensitive to the difference in foundation stiffness. In, addition the FF at low modes were significantly affected by the V_F Moreover, a hardening behavior was noticed in some case studies, which occurs owing to the variation in plate thickness and sudden reduction in the FR followed with increases. In contrast, a softening behavior was seen in NPs with positive surface effect which occurs as a consequence of NLC parameters. The NFs in all various analysis that obtained by subjecting the structures to CCCC BCs were more significant than any other type of BCs. Likewise, the NFs of the structure were directly affected by the implementing method during the mathematical modeling, for instance by applying the CPT and MCST much lower NF values were obtained than those of MSGT. However, this observation can be affected by other factors such as type of structure and material properties. Besides, increasing the nonlocality cause to reduce the NF. However, in case of subjecting the structure into a fluid environment the friction between the fluid and the structure surface cause a reduction in the NF. Furthermore, applying positive and negative external voltage have different impact on the NF, where the negative voltage can achieved more significant NF than positive voltage. Additionally, moisture and temperature have various effects on the linear and NL FV responses which may influence and affected by other parameters at the same time. Besides, increasing the imperfection interface parameters cause a reduction in the NF. However, adding a smart layer or increasing the thickness of a smart layer can reduce and harmful effect that may have an impact on the dynamical responses. Nonetheless, the residual stress and E have direct impact on the NF values. In some cases the distribution patterns of the reinforcement’s nanoparticles may have more significant effect on the dynamical responses than the VF of the reinforcement’s nanoparticles.

5 Conclusion

This study conducts and illustrates a comprehensive review of linear/NL FV analysis of various CPs and LCPs structures on nano/micro-scale with and without considering the SDE. Several types of CPs are investigated, such as SC-based, PE-based, VE-based, FG-based, LCPs-based, Gr-based, MEE and MEMS etc. The primary purpose was to investigate the researcher's interest through the last decade in analyzing the FV of developed CP structure based on the recent improvements and highly engineering application of the CPs by various solutions, which are analytical, numerical, high order, and mixed. Moreover, it highlights the enhancement of numerous parameters on the FV performance. Additionally, as a consequence, it provides main principles and guidance for further future studies in this framework. To accomplish this work, several databases were used, such as Scopus, Google Scholar, ScienceDirect, etc.

The linear and NL definitions and applied theories, several composite nanostructures and nano/micro-plates, and FV analysis for various types of structure on a SSE with addressing some examples from optimization and machine learning algorithms, as well as the basic plates' methodologies were briefly discussed in the introduction section with some flowchart. The classification is mainly categorized under two subtitles, linear FV, and NL FV, followed by a further classification based on the employed solution type without considering the NLC parameters. A tabular classification was presented at the end of each section.

The applied and developed theories utilized in literature to define various material and mechanical properties are presented. The approaches that are usually implemented for defining numerous parameters such as BCs, tensor, environmental, NLC, EF, weight faction, and distribution patterns of the nanocomposite/nanofiller and geometrical factors for driving the GEs and EOM are highlighted. In addition, the methodologies that were employed for solving the GEs, Eigen-value problem, and discretization are addressed. A separate discussion for each study is presented, highlighting at least one of the most significant features that impact the FV behavior.

Based on this extensive study, numerous FV analyses for small-size structures were generally available in the literature, and many researchers show interest in investigating the stability of the structure besides the FV analysis. However, by comparing the number of studies on linear FV analysis with NL FV analysis of CPs, it is revealed that researchers show more interest in examining the linear FV performance. Likewise, the studies on NL FV can be considered limited compared with those of linear analysis. The researchers are more interested in investigating CPs with rectangular and square geometries than rotating, trapezoidal, circular/ annular, and elliptical CPs. The dominant applied solution types are analytical and numerical. Besides, the analytical solutions indicate exceptional outcomes as they provide benchmark results and advanced accuracy for various CP structures.

In contrast, the number of studies considering the high-order solution is limited in linear or NL FV analysis. The axisymmetric solution may be accurate, and the obtained NF values have some overestimation values. The dominant approaches for defining the GEs are HP, the principle of energy, and work theories. MT was utilized for VE parameters to define the material properties by ROM or the modified ROM, while the HT model was utilized for defining the NCs Young's modulus. Besides, the G-M used to identify all various types of surface effects in the formulations. However, the SSDT was found to satisfy the shear strain and stress. Likewise, the shear correction factor and several methodologies can prevent shear locking. Researchers show interest in considering the inclusion of a homogenous infinite matrix by employing Eshelby's tensor.

Furthermore, researchers show significant interest in exploring the impact of SDE on dynamical responses, where the modified NLC overcomes the drawback of CPT. Meanwhile, the SDEs have an essential role in the mechanics of the structure at NS, and it is challenging to perform exact experimental dimensions at NS. The high CPU related to the MDS, and the continuum modeling of NS has involved substantial attention. Amongst several modified continuum-based theories, the NLET and the NLESGT were implemented to evaluate the mechanical characterizations of small-size structures. Moreover, adequate recorded data regarding implementing the Cosserat approach in modeling small-scale CP structures is absent. While the SDTs were remarkably employed. Various implementations of different MF methodologies through analyzing the FV with or without considering the SDE were observed. Moreover, studies that analyze double nano CPs have a unique behavior in the vibration phase, where the double CPs combine and behave as single MPs without internal EF enhancement.

The symmetric rotating gradient tensor has a lower impact on the FF than the dilatation gradient tensor. The porosity type and V_F have significant effects on the FV behavior. The highest impact of distribution patterns was observed by FG-X and BCs in CCCC. Furthermore, increasing or decreasing the structure stiffness by geometrical parameters or other types of parameters directly impacts the frequencies. All available studies in the literature were validated by comparing them with previously published data or by detecting the CPU. Using theoretical models in 2D shows to provide accurate as well as good results as the 3D models, which should be considered instead of 3D models for computational costs. Besides, most of the validation process for 2D-SDT is accomplished by comparing it with 3D-ET in the absence of comparison with HOT.

As a gap of research, it is revealed that very few studies focused on performing machine learning and optimization algorithms, which means that there is a compulsory requirement for implementing artificial inelegance in this framework.

This context provides the potential for further research studies such as a comprehensive review of NL/linear vibrational response of various smart composite small-size plate structures. Besides, it can be considered as an initial point to accomplish further modified methods.