Abstract
This work is considered as the first comprehensive review, that covers all types of meshfree method including the traditional or developed meshless techniques that have been implemented for the purpose of investigating static analysis (bending, stability) besides dynamic analysis (free vibration, force vibration and other types of dynamic behaviors) of linear and nonlinear mechanical system. The secondary methods utilized together with the meshless methods are also highlighted such as; Hamilton’s principle, first-order shear deformation theory, high-order shear deformation theory, Monte Carlo, local/nonlocal theories and others. Also, some computational mechanics approaches are briefly addressed. The basic fundamental equations of meshfree methods and the error analysis are presented. Various types of schematics and structure size are discussed. Else, the implementation of composite material in solid mechanics are concisely highlighted. As a key finding, in each unique schematic in specific scale, various implemented parameters like boundary conditions, thickness to length ratio (t/l), as well as the aspect ratio have different impacts on the mechanical performance in both static and dynamic analysis. Additionally, as each meshfree method is considered unique by itself and has its own developed mathematical model, each method has different application and numerical problems to solve. Galerkin, reproducing kernel particle method, moving least square are the most common meshfree. Based on the literature, many studies mainly show interest in investigating the piezoelectric and diverse distribution of carbon nanotubes, and some in fictional graded material in different structures. This review is recommended for researchers interested in solid mechanics analysis at various scales using meshfree techniques.
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1 Introduction
1.1 Meshless/Meshfree
Meshless/Meshfree methods are defined as a group of numerical algorithms [1]. While they differ in how shape and test functions are created, they all possess the capacity to ease the finite element methods (FEMs) strict connectivity constraints. Belytschko [2] defined meshfree as a series of numerical techniques that are intended to overcome the re-meshing issue in many FEM simulations. In another words, it depends on the interaction of each single node with its neighbour nodes. Qiao et al. [3] presented another definition of meshfree methods as numerical methods that have absence of fixed connectivity between the discretization nodes. These methods are preferable to use in simulating impact failure, fragmentation and penetration as well as in continuum fluids and solids. Over the last two or three decades, a number of these strategies have emerged and reached a highly developed stage. Meshfree approaches avoid employing elements to generate the approximation, in contrast to the FEM [4]. Since the superiority of the approximation is considered to be influenced by the geometric placement of nodes as well as the domain influence of each node, the placement of the nodes is rather arbitrary. Compared to the FEM, the connectivity between the nodes that is determined by how these zones of influence overlap can be designed more freely. Figure 1 illustrates the difference between FEM analysis processing and the meshfree method analysis processing.
A comparison between the formulation process of FEM and meshfree
The basic meshfree methods and some types of developed meshfree method are illustrates in Fig. 2. Generally speaking, the most common meshfree approaches that have wide application are; Galerkin method [5, 6] which include element free Galerkin (EFG) as well, the particle difference method (PDM) which is categorized as a robust form meshfree technique and discretizes the governing equations (GEs) [7], Hp-meshless cloud method [8, 9], the moving least square method (MLS), which has been combined with the EFG technique [10], the reproducing kernel particle method (RKPM) [11, 12] an example of RKPM shape function of the first derivative by a tri-linear polynomial basis for microstructure dynamic shear band is shown in Fig. 3 [13], the smoothed particle hydrodynamics method (SPH) [14, 15] which have been employed to analyze the laser fusion additive manufacturing process [16]. The same method has been used recently to analyze the grinding processes [17]. There are some advanced developed meshless approaches as shown in Fig. 4, such as; the meshfree local petrov–Galerkin method (MLPG) [18, 19] it has been employed to analyze the shear loading [20] and for elastic moduli analysis of the woven fabric composite [21], the partition of unity approach meshless method [22,23,24], the partition of unity finite element method (PUFEM), and the diffuse approximation method (DAM) which can be utilized to generate smooth estimation of functions known at certain sets of locations and, to estimate its derivatives accurately [25], the smoothed peridynamics (SPD) which was utilized for extremely large deformation and caking problems [26], Peridynamic (PD) based elastodynamic (Peri-ealstodynamics) which is a nonlocal (NLC) meshless method and a scale independent generalized method mainly applied to visualize the ultrasonic wave and the acoustics in a plate and in nanodevice [27].
Classification of meshfree methods: elementary traditional types of meshfree methods and some developed meshfree methods
A 3D of RKPM/meshfree shape function and its first derivatives generated by the polynomial basis (tri-linear) [13]
Classification of Developed meshfree methods
An example of a schematic of sandwich structure nanocomposites plate reinforced with carbon nanotubes (CNTs) proposed to thermoelastic dynamic analysis by the MLS meshfree method is illustrates in Fig. 5 [28]. However, it is worth to mention that the MLS meshfree methodology have the potential to transfer the NLC variables model to a local model such as the densities of the geometrically necessary dislocations (GNDs) [29].
The schematic of sandwich structure nanocomposites plate reinforced with CNTs and their clusters [28]
Out of all methods of meshfree theory, the SPH as well as the material point method (MPM) are considered as the most important methods. This is because they overcome the large mesh deformation also the requirement for remeshing that are frequent towards finite element (FE) models in the Lagrangian framework, and the challenges associated with tracing history of the material in the Eulerian formulation.
Owning to the higher order smoothness of the meshless form functions and the avoidance of mesh-dependence problems, it has been demonstrated that the employment of Galerkin based approaches can reduce the need for sensitivity filtering [30, 31]. The uniqueness of meshfree methods is that there foundation is based on particle hydrodynamics and gas dynamics, and instead of relying on mesh or lines, they use nodes particles or points [32]. The key benefit of using the meshfree methods is that the issue domain does not need to be meshed. The selection of shape functions is the primary and most crucial problem in meshless approaches [19]. Due to the fact that most of the application of meshfree methods are in rational functions and the meshfree methods are based on Gauss quadrature, which considered as insufficient to accomplish an exact integration. Recently many studies worked on vibrational adaptively to develop the support domain size for meshless methods’ shape functions. The importance of finding developed algorithms to enhance the efficiency of meshless methods is because the old techniques of meshes and other like-stabilized nodal integration may complicate the process to a bath way far off the theory of meshless.
The meshfree approach is most importantly applied in simulating plastic materials where a large deformation may occurs [33]. Additionally, 3D high complexity objects, cracks simulation, linear and nonlinear (NL) material behaviors (singularities/discontinuities), bending simulation, etc. The variety of methods in meshfree approaches have extensive applications. Ren [34] developed an explicit meshfree Galerkin formula to detect the ductile fractures throughout the finite inelastic deformation. This advanced implementation has these features: 1. Capable with thermal–mechanical coupling and stability in an adiabatic split algorithm, 2. Effective dynamics meshfree algorithm even in high speed impact, 3. Automated crack algorithm which provide the crack growth lifespan, and 4. Adopting modified forward Euler tangent algorithm for NL thermal–mechanical inelastic constitutive relation. The crack growth analysis by meshfree have been used quite often in research study as [6, 35,36,37]. Bui et al. [38] have developed meshfree methods to perform a transient dynamic analysis fracture variables of a functional graded crack. The novel method for the partition of unity extended meshless Galerkin method founded on moving kriging interpolation (MKI) was developed. Diverse kinds of correlation functions, including quartic polynomial, Gaussian, as well as truncated quartic polynomial, were used to improve the MKI shape functions. The main enhancement is the removal of the correlation parameter effects that were frequently included in the MKI shape function. Lu et al. and Guan et al. [39, 40] employed the Galerkin technique for the purpose of simulating the plastic flow during the equal channel angular pressing technique (ECAP), in viscoplastic/rigid-plastic. The MPM [41] and the SPH [42] shows significant and reliability results by studying the ECAP. Furthermore, when comparing the straightforward hp-cloud technique to other meshless techniques, the fundamental boundary conditions (BCs) may be applied directly using the simple hp-cloud method since it has the Kronecker delta property [8]. Recently, the attention of researchers has been drawn to the radial point interpolation method (RPIM) and it is another meshless methodology similar to the EFG method, and have been utilized to solve various issues in the applied mechanics field [43]. For instance, Kazime et al. [44] have employed the RPIM method to accomplished an analysis for viscoplastic problem based on the Cartesian transformation method (CTM). Hu et al. [45] have performed the metal forming analysis by using the RPIM parallel point. A comprehensive review about mesh free methods which have been used in studying the fracture mechanics and cracks has been published recently [46]. Nguyen et al. [47] utilized the Galerkin radial point interpolation (GRPI) technique to investigate the crack propagation in the elastic solids. Additionally, Ha and Bobaru [48] examine the dynamic crack in brittle materials by using the meshfree PD theory. Guiamatsia et al. [49] implemented the EFG method for the first time to analyze the delamination (interlaminar) and the interlaminar matrix microcracking in the composite laminates. Yaghoobi and Chorzepa [50] have performed a micropolar predynamic investigation of fracture analysis of a fiber reinforced concrete structure. Selim et al. [51] have employed Element-free IMLS-Ritz technique with the high-order shear deformation theory (HSDT), to perform a vibrational controller of functionally graded graphene nanoplates (FG-GNPs) with piezoelectric layers as presented in Fig. 6.
The control process with 2 positions; a case 1 and b case 2 of piezoelectric sensor and actuator layers [51]
Recently, Behdinan et al. [52] established the MLS method to facilitate the governing thermal equation to examine the effects of graphene (Gr) and CNT on steady-state as well as transient heat transfer performances of the functionally graded (FG) nanocomposite axisymmetric cylinders under different BCs set. Furthermore, Abdollah and Boroomand [53] employed the Galerkin method to solve the problems in NLC integral models, and for inspecting the efficiency of the methods. Also, they presented some examples with smooth and non-smooth. However, Pan et al. [54] presented a review article about employing the meshfree methods in solution to image stresses of dislocations. Chen et al. [55] have published recently a review article about the developments in meshfree methods, its classes and their applications generally over the last two decade. Li and Liu [56] have published a surveyed meshless particles methods. Also, the authors noted its role in the applied mechanics field. Also the review has covered three main methodologies and the nanoscale mechanism. A recent review paper which provides a methodological investigation into potential and the current meshless approaches for computing the partial differential equations (PDEs) that control solid mechanics difficulties has been published by Liu et al. [57]. It discusses the general phases of a node-based meshfree technique with displacements as the primary variables. Node-based interpolation methods, cell-based interpolation methods, function smoothing techniques, and MLS approximation methods are part of function approximation techniques. The special strain construction, direct differentiation, as well as the gradient smoothing, are all components of the gradient approximation. Strong-form (SF), weak form (WF), weak-strong-form (WSF), local weak form (LWF), as well as weakened weak form are all formulation approaches, a brief summary of SF, WF, WSF and LWF with a single example for each is illustrated in Fig. 7
A summary of comparison between WF, LWF, WSF and SF with a single example for each
A review focused on NL free and force vibration of shell structure and works done by meshless methods were discussed on that review [58]. Shapiro et al. [59] presented a comprehensive discussion about CAD/CAE, highlighting the application and the classification of meshfree methods in CAD. Wu and Liu [60] presented an overview which highlights the meshfree method that were specifically applied to perform analyses for sandwich structure and laminated composites of the FG elastic/piezoelectric materials (FGEMs/FGPMs) shells as well as plates including various types of BCs. Liew et al. [61], published a review about meshfree methods that have been employed in laminated and functionally graded shells and plates specifically. Grag et al. [62] accomplished a comprehensive review which cover all theories and approaches such as; meshless, local NLC elasticity theories and etc. that have been implemented for the purpose of investigating various types of composites nanoscales structures. However, the recent development in the computational approaches that have been performed for investigating the dynamic behavior of complex structure for real time hybrid simulation (RTHS) have been studied by Betancur et al. [63]. In addition, the review highlighted that there is still a need for more studies which focused on the foremost challenges RTHS control theories such as; complex BCs, NL and uncertainties, computational efficiency as well as the multi hazard engineering application. Gupta et al. [64] accomplished a comprehensive review which focused on the recent progresses in the isogeometric analysis (IGA), especially the NU rational B-splines (NURBS). Additionally a brief comparison between IGA, FEM, meshfree as well as the coupled meshfree with FEM methods has been addressed. Moreover, Gu et al. [65] performed a concurrent analysis technique for investigating the various deformation systems which needs the integration of material properties from nanoscopic to macroscopic dimensional. The radial basic function (RBF) interpolation has been employed for the purpose of determine the continuum subdomain. Where the molecular dynamic (MD) analysis was carried out for the atomic subdomain.
Meanwhile, this presented review is considered as the first of its type that covers the static analysis (bending and buckling) and dynamic analysis (free/forced vibration as well as other dynamic analysis) for all different types of structures and geometries in various scales (nano, micro, and macrosalces), and summarized literature works. However, this review article is a valuable resource for researchers interested in the field of solid mechanics analysis at various scales using meshfree techniques, where the secondary methods utilized together with the meshless methods, numerical and analytical analysis are addressed. Besides, different type of material were mentioned and the enhancement of composite distribution and the impact of various parameters on deformation, static and vibrational performance are discussed. The analyses which are considered in the scope of this comprehensive review are illustrated in Fig. 8.
The applications of meshfree methods in solid mechanics analyses and in the scope of this review
1.2 Literature Methodology
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:
As the first search cord "meshfree” OR "meshless" OR "mesh-free" OR "mesh free” has been used then AND ((("solid mechanics" OR "static analysis" OR "static" OR "dynamic analysis" OR "dynamic")) AND ("beam" OR "plate" OR "cylindrical" OR "shell" OR "sandwich" OR "cylinder")) AND ("nano" OR "micro" OR "macro") as a search in the search cord. 573 documents emerged. Not all of the 573 documents are exactly into the scope of these review; so many some keywords were excluded. Owing to the fact that Scopus does not exclude the other alloys and metals from these search cord. The overall, documents were found by Scopus after exclude some other key words is 135, but again some documents were not relevant.
Later on, additional articles were collected from other databases such as; Google Scholar, ScienceDirect and ScienceWeb. To generate the theoretical background of the research; where the supportive articles help in emergent the contexts of the review generally.
Figure 9, shows Scopus statistics of meshfree/meshless applications in solid mechanics’ analysis for different structures in nano, micro, and macro size.
Scopus statistics of meshfree/meshless applications in solid mechanics analysis for different structures in nano, micro, and macro size (2001–2023)
The graphs show that the interests in mesh free starts at the early beginning of twenty-first century. With an upward tendency, a significant increase is shown since 2022 until today. The greatest interests were found to be in engineering areas with 40.9%, followed by 15.5% of publication in the area of mathematics. The lowest percentage of documentation was in environmental science with only 0.8%. However, the highest number of publications was recorded by China with 37 documents. Followed by 27 documents have been published in Iran. Canada has relatively fewer number of publications than USA were it recorded 12 documents and 16 documents in USA. While almost half of this number of documents have been published UK, Georgia, and India. The records show fewer numbers in Germany, and Portugal. Australia, has published the same number of documents as in KSA.
1.3 Developing and Improving the Meshfree Regular Methods
According to Seleson [66] research, the meshfree approach is a desirable discretization technique since it is simple to implement numerically, and has a low computing cost compared to other discretization techniques. In addition, this method meanwhile has some problems with accuracy and convergence. A lack of an ideal selection of quadrature points and an approximate estimate of the areas of the intersecting regions (commonly known as partial areas) among neighbor cells and a point's neighborhood are the main causes of these problems.
Among the most popular and crucial meshless techniques is the EFG approach. However, it has the drawback of poor computing efficiency when compared to the FEM. Sun et al. [67] have presented an improved element-free Galerkin method (IEFGM) in order to resolve two dimensional elastic complications that is based on the dimension splitting MLS (DS-MLS) approach as well as combining the Galerkin vibrational WF with integral coordinate transformation. The DS-MLS technique improves the speed and precision of shape function calculations by reducing the dimension as well as complication of matrix operations while resolving shape functions. Besides, the meshless approach developed from the DS-MLS approach has the ability to increase computational effectiveness. The IEFGM suggested in that work can also obtain higher accuracy and uses slighter central processing unit (CPU) time than the EFG approach for some elastic issues, according to the numerical example results of the elasticity problems. Roque et al. [68] expanded the trigonometric theory for beams to include composite and sandwich plates. In addition, it demonstrated outstanding performance in free vibration (FV) analysis. For the purpose of improving some of the limitations of the global RBF approach, various authors individually indicated a local RBF method under several names such as local multiquadric-based differential quadrature (LMQDQ) technique [69], and RBF in a finite difference mode [70]. The LMQDQ has been used by, Ding et al. [71] in order to resolve 3-D incompressible Navier–Stokes (N–S) formulas in the primitive-variable format. For the first time the radial basis function based on differential quadrature (RBF-DQ) was employed to examine the FV behavior of laminated composite arbitrarily shaped plates [72]. Furthermore, Shu et al. [73] simulated the inviscid compressible flows through shock wave by using RBF-DQ. Wright and Fornberg [74] developed a method to reduce the number of stencil nodes with remaining high accuracy, by employing RBF with finite difference (RBF-FD). In this study, different models were used to test and validate the method. For the first time, the multiquadric radial basis function (MQRBF) approach was utilized to interpolate the motion equations as well as BCs to perform FV analysis of plates. Oruç [75] has developed a developed methodology based on the RBF-FD, for the purpose of solving the coupled damped Schrödinger systems in (1, 2, and 3) dimensions. Tornabene et al. [76] produced the moving least squares differential quadrature (MLSDQ) created on the RBF for solving various structural geometries made of doubly curved shells designed by composite materials as illustrates in Fig. 10 as well as Fig. 11. The first ten mode of vibration for each shape were obtained. The study concludes that with varying curvature radii while considering laminated composite materials and anisotropic lamination structures, once the structures having varied radii of curvature were incorporated, it is proved that the methodology is quite accurate compared with the semi-analytical solutions for plates, although there are some constraints must be taken. Additionally, the approach permits the application of any BCs, including both natural and essential conditions.
The geometry of three shell structures is defined as: grid, local coordinate orientation system, and position vectors r(α1,α2) [76]
The doubly-curved surface geometry of the Ding-dong Revolution [76]
In a further studies the same method has been applied for the purpose of investigating the FV [77], besides static examination [78] of the functionally graded plates (FGPs) simply supported (SS) modeled by third order shear deformation theory (TSDT), the obtained outcomes were found to agree with previous analytical and numerical solutions. The Mori–Tanaka technique has been implemented in both studies for the purpose of defining the material properties. To investigates a moderate laminated thick composite sandwich plates SS, Ferreira [79] present a formulation of the MQRBF. The author examined both stresses and displacement. High accurate results were observed and the model was compared with FEM and exact formulations. Ferreira et al. [80], developed a new meshfree method which is a finite point founded on the MQRBF, and has been employed with the third-order theory of Reddy for the purpose of analyzing and discretizing different types of plates (sandwich plate, composites laminated plates and isotropic plate). Nevertheless, the BCs interpolation has been schematically drawn out. The outcomes prove that the presented method efficient and accurate more than previous other method proposed before. Ferreira [81] has used the MQRBF again for the same structure as well as same types of plates, but in this study the FSDT and TSDT were applied. Besides, the layerwise theory was accompanied by independent rotations in each individual layer and schematically formulates the BCs interpolation. The shear correction factor is not necessary for this presented method also it combine the constitutive equation by the transverse shear stress directly calculated. The combination of layerwise with the MQRBF discretization consider as simpler yet remarkable effective numerical approach for thin and thick different types of plates (sandwich laminates and composites). For the purpose of examining Mindlin plates and Timoskenko beams for FV behavior, Ferreira and Fasshauer [82] presented a novel numerical methodology with high accuracy, by combining the pseudospectral (PS) method with collocation RBF meshfree method. Furthermore, the RKPM has been utilized to compare the natural frequency (NF) results for each type of plates with previous obtained results by Rayleigh–Ritz and by shear-deformatble plate’s theory established on FSDT meshfree method. The deformations of laminated composite, sandwich plates as well as shear flexible isotropic were investigated by a novel numerical scheme [83]. For producing a high accurate results, the collocation by RBF was viewed as a PS method and the layerwise theory of sandwich or laminated plate was applied. The optimum value of shape parameters variable was considered. Additionally, for studying the free behavior the harmonic solution was considered and the NF results were very close to the previous obtained results by using the Rayeigh-Ritz solution. The corresponding 3D views first eight modes of the square Mindlin/Reissner plate with clamped–clamped (CC) BCs is illustrates in Fig. 12.
The Modes (5–8) of vibration of a CCCC square Mindlin/Reissner plate with ℎ/a = 0.01,v = 0.3, 3D view [83]
However, Ho et al. [84] employed the integrated radial basis function (IRBF) and the conic programming for kinematic yield design computed homogenization of ductile and periodic composite materials. Xiang, et al. [85] employed the meshless radial point collocation (MRPC) technique founded on the thin plate spline RBF as well as the nth-order shear deformation theory for evaluating laminated composite plates FV. Besides, to determine the NF of the plates while considering different BCs, material properties, side-to-thickness (a/h) ratios, the zero transverse shear stress BCs on the top and bottom surfaces of the plate were satisfied by the nth-order shear deformation theory. The nth-order theory (n = 3) was considered as a superior instance of Reddy's third-order theory. Nie et al. [86] have introduced parallel node placement technique by bubble simulation (PNPBS) applying the basic domain decomposition (DD) by the METIS software, for an random number of processors. The flowchart of the processes is presented in Fig. 13.
The PNPBS methodology flowchart; a The complete process b dynamic simulation for each specific processor [86]
According to [6, 87, 88], the majority of meshfree methods rely on the MLS shape function [32,33,34], that neglects the Kronecker-delta property. Although absence of the Kronecker delta property tends to make treating BCs more difficult, this difficulty is usually resolved through using additional methods such as Lagrange multipliers and the Penalty method. Liew et al. [89] provided the boundary element-free method (BEFM), which is a direct meshless method of boundary integral equation (BIE) for 2D elastodynamic difficulties. Likewise, it incorporates the improved moving least squares (IMLS) approximation with the BIE system for 2D elastodynamics in the Laplace-transformed domain. Sansour and Skatulla [90], recorded that employing the MLS approximations in a NL Cosserat continuum-based formulation has impacts for the necessary BCs enforcement as well as the rotation field updating. Furthermore, Wang and Liu [91] performed the proposed RPIM to satisfy Kronecker-delta, which leads to find the necessary BCs easily.
Rosca and Leitao [92] stated that the governing integral equation in meshfree methods which is based on WF is derived out of the WF of elasticity local domains or over global. Besides, they offer an improvement of meshless method by using the Monte Carlo integration methodology, and considered as more efficient, simpler as well as straightforward in comparison to other local or global integration technique. Khosravifard and Hematiyan [93] stated that the most common draw back for meshfree methods are low accuracy, additional time consuming, complexity, and high CPU capacity. To overcome these problems, they modified a developed version of the CTM to treat the territory integrals in meshfree approaches. The CTM method has been used earlier to evaluated the territory integrals in the boundary element technique by [94, 95]. At the end of twentieth century, researcher’s interest in developing the SPH started to increase in different mechanical engineering fields [96,97,98]. There were various purposes of developing the SPH method such as; seeking to solve tensile instability [99, 100], zero energy mode [101], the absence of interpolation consistency [102], and enforcing the BCs in different types of analysis [103]. However, the main drawback of SPH, is that it cannot be used to signify the rigid body motion properly as recorded by Liu et al. [12]. As a consequence of this problem, the main concept of improving the SPH is generating a corrective kernel function and remaining the initial kernel as well. To investigate the fluid-elastic structure of the incompressible fluid, Khayyer et al. [104] established an incompressible smoothed particle hydrodynamics (ISPH)-SPH meshfree technique. Liu and Liu [105], presents a review of SPH meshfree method. In addition, it highlights the recent development of the SPH, its advantages and the recent applications. Furthermore, the basic concept of KPM approximations and smoothing kernel function that have been used to develop the SPH were reviewed. Figure 14 illustrates the schematic of two different scenarios of SPH kernel approximations in exact and not exact second order accuracy. In Fig. 14a, W which is smoothing function is not shortened by the boundary, the smoothing function integration is unity, while the integration of the smoothing function's first moment term and the surface integral are zero, while in Fig. 14b, W is condensed by the boundary, the smoothing function integration is not unity, and the surface integral and the integration of the smoothing function's first moment term are not equal to zero. Rahmat et al. [106], developed a novel three dimensional numerical model based on discrete multi physics (DMP), the MLS meshfree method, and a mass spring model (MSM).
The schematic depicting the intersection of the support domain and the problem domain, in which a the SPH kernel approximations are of second order precision. b The SPH kernel estimates do not have perfect second order precision [105]
Silling and Askari [107] have employed the PD theory to develop meshfree methods in cracks modeling. The key benefit of that method was to redefines the underlying mathematical representation of solid mechanics to ensure the identical equations hold each on or off of a discontinuity involving a crack. Besides, even in case of having a huge quantity of cracks the treating process can still be accomplished with their mutual interaction, in another words it is preventable to handle the other side of cracks separately. The conclusion of that study shows that the PD method have theoretical as well as practical advantages over other meshfree methods in fracture mechanics. As a theoretical advantages, it has the ability to determine various sets of continuum equations. Also, its integral equations have the potential to persist directly applicable when a discontinuity seems, which can’t be guaranty by the classical theory PDEs. Furthermore, the PD approach cross over the popular issue (tensile instability) which usually shows in SPH methodology. However, the PD model is found on the interaction between pairs of nodes, which in consequence eliminate the requirement of great number of neighbor nodes to define spatial derivatives and it is counts as a practical advantages. Another significant practical fact is that the cracks emerge spontaneously as a natural consequence of the equation of motion as well as constitutive models. Amiri et al. [108] provides a model of phase-field for fracture in Kirchoff-Love thin shells that utilizes meshfree local maximum entropy (LME) approach. This technique was created to assess brittle materials and structures with complicated geometry and topology. The study includes constructing a fourth order phase-field model to be utilized with smooth approximants like IGA or LME, with a goal of boosting accuracy and efficiency while enhancing the level of convergence. Zhang et al. [109], work is considered as the first study that incorporate HSDT with an meshfree approaches for the purpose of examining a FV response of CNT reinforced FGPs. Seleson and Littlewood [110] presented a Conjunction work in meshfree PD simulations. In 3D, they estimated numerical partial volumes utilizing a recursive subdivision and sampling technique. For the purpose of avoiding the numerical estimation of the singular integrals in the boundary element technique and to eliminate the singularity of the fundamental solutions, Fu et al. [111] developed singular boundary method (SBM) which is defined as meshfree technique by claiming the conception of origin intensity factors. The developed schematics are shown in Fig. 15 as well as Fig. 16. Additionally, due to the fact that the SBM avoids the singularity in the fundamental solution, Lin et al. [112] employed the SBM to investigate the linear elastic half plane with cavities of buried under incident P and SV waves for the purpose of verifying the seismic response.
The top-shaped axisymmetric schematic structure [111]
The periodic with one line scatterers schematic structure [111]
An investigation of the application of an undisturbed influence functions with a finite support for enhancing the accuracy as well as convergence of the mathematical solutions in PD was also conducted. Liu et al. [113] presented a review which focused on boundary element method (BEM) and its application in improving various types of methods such as the symmetric Galerkin formulations and the boundary meshfree methods. The MLPG meshfree method has been implemented with the MD method to perform a multiscale simulation. Additionally, both multiple time steps and multiple length scales were performed in the mathematical cases. The obtained results demonstrated that the suggested method is optimum in reducing the computational cost and decreasing the reflection of phonons [114]. Tang et al. [115] applied the MLPG to analyze the materials with strain-gradient effects. Since there is no FEM having the ability to solve the strain gradient theory (SGT) formulation, the nodal rotational degree of freedom (DOF) nodal were not considered. The results show that superior results to mixed FE procedures or primal of MLPG were obtained in fourth-order elliptic problems. However, the redefined SGT has been employed by, Limkatanyu et al. [116] to investigate the beam bulk material nanoscale effect in a study which focused on examining the bending, stability and FV of a nanobeam structure substrate moderate model. Besides, the free energy of the surface has been obtained by utilizing the Gurtin–Murdoch surface elasticity theory. Yvonnet et al. [117] demonstrated a new meshfree natural element method (NEM) expansion. A visibility requirement was incorporated in this methodology, and designated the constrained natural element method (C-NEM), for the sake of selecting natural neighbors for computing the shape functions. The particular shape functions were computed using an adjusted, limited Voronoi diagram. A few of the obstacles associated with this method in non-convex domains were averted. In addition, the study of problems requiring fractures or discontinuities became simpler. Unlike other meshfree approaches, the introduction of essential BCs is relatively easy since the NEM satisfies the Kronecker delta features. Bessa et al. [118] developed a connection between the meshless state-based PD approach and further meshless approaches, specifically MLS reproducing RKPM. It has been determined that discretizing state-based PD immediately yields to estimate the derivatives produced by RKPM. Nevertheless, state based PD has achieved similar results at a considerably less computational cost. In response to the results; an update to the method was provided, in which the constraints of the utilization of BCs as well as the application of non-uniform (NU) grids were modified by employing RKPM. The PD NLC meshfree method was employed for comparing the empirical findings of carbon fiber reinforced polymeric-halloysite nanotubes (CFRP/HNT) laminates under in-plane shear and flexural loads, where the PD analysis prove that the halloysite nanotubes (HNT) obstruct the cracks growth [119].
To investigate the complex phenomena spall fracture that is caused by the shock wave, Ren et al. [120] have developed and employed the RKPM method with multiscale micro-mechanics theory that has been suggested by Wright and Ramesh [121]. Furthermore, Mishra et al. [122] developed a closed-form mathematical expression called multibody dissipative particle dynamics (MDPD) for simulating a complex wall analytically. MDPD is an adjusted version of the particle-based meshfree methodology dissipative particle dynamics (DPD). Based on unlimited advantages of meshfree method, it has various application in biomedical and pharmaceutical fields. For example, Imai et al. [123] recorded that BEM is the highest rigorous process for Stokes flow, and it is exclusively beneficial for infinite flow complications. The semi-implicit method (MPS) and DPD were extensively utilized as well for cellular flow difficulties. Also, the meshfree methods have applications in food industry [124]. Liu et al. [125] presented a review of DPD, and it included the formulations, methodology, implementation procedures, some related numerical aspects and applications in different fields. Figure 17 illustrates the different time and length scale corresponding computational methods, and it is observed that for each single length scale, different method must be adequate.
The differentiation in time and length scale corresponding computational methods [125]
Where the boundary moving least square (BMLS) satisfy the BCs easier, it has a benefits over other traditional meshfree methods in solving higher dimensional problems. It has been applied to solve two dimension elasticity problems [126]. Chen et al. [127] have combined the smoothed discrete particle hydrodynamics (SDPH) and FVM as shown in Fig. 18, in order to achieve higher effective performance in solving a multiphase flow difficulties of gas-particle. The Ritz-Galerkin discretization has been employed with Kirchhoff–love hypothesis, equilibrium configurations and the kinematics of the shell for generating the mathematical modeling of the thin shell [128]. Some of the unique numerical examples that have been examined are presented in Fig. 19. Moreover, Thai et al. [129] have employed the Galerkin-WF meshfree technique for bending and stability analyses as well as vibrational analysis models for functional graded material (FGM) sandwich isotropic plates to generate the discrete formulas of the system. This study focused on applying the generalized shear deformation theory for FGM for static and dynamic examination, based on the IGA approaches.
SDPH and FVM combination algorithm [127]
The Pinched hemisphere b bunny c connected pipes [128]
Recently, Faghidian et al. [130] reproduced an effective meshfree numerical method with a sequence of solutions for kinetics as well as kinematics field parameters. However, the mixture stress gradient theory of elasticity and the stress gradient theory through a stationary variation basis were performed for FG nanobar in tension. Figure 20 shows 3D schematic of the variance of an axial nondimensional (ND) displacement for a FG nanobar \(\overline{u}\) exposing linear axial loading against ND abscissa \(\overline{x}\).
The FG nanobar is exposed to axial linear loading with fixed–fixed BCs \(\overline{u}\) Vs. \(\overline{x}\) and ζ for fixed α = 0.2, 0.6, 1.0 [130]
A study by Chen et al. [131] focused on investigating the application of RKPM for NL structures with large deformation analysis considering elasto-plastic and hyperelastic materials, the computational results revealed that RKPM manages severe material distortion more efficiently compared with FEM owing to smoother shape functions then, as a consequence, gives results with greater accuracy subjected to huge deformation. Contrary to traditional FEM, nodal spacing NU in RKPM does not result in an irregular mesh shape, which greatly reduces the correctness of solution. Whenever NL RKPM is applied to virtually incompressible hyperelasticity and perfect plasticity circumstances, no volumetric locking is seen. The basics of MLS method and its developed formation and interpolations were presented by Breitkopf et al. [132]. Żur and Faghidian [133] developed a novel numerical meshless theory for unified gradient elasticity theory of torsion in nanobar. The presented meshless technique was founded on the principle of Reissner stationary vibrational. Likewise, unconventional series solutions for kinetic and kinematic field variables were discussed. An outstanding agreement between the meshfree technique and the exact solution was detected. Figure 21 illustrates a torsional performance of the unified gradient elastic bar as influenced by NLC besides gradient characteristic characteristics at the nanoscale. In a further study, for the rigorous investigation of the dynamic properties of elastic nanobeams, the mixture unified gradient theory of elasticity has been utilized by Faghidian et al. [134]. A coherent vibrational framework was created, and the dynamic equilibrium boundary-value problem was recognized with the appropriate shape of the extra non-standard BCs. Besides, the elastic nanobeams’ FV and wave dispersion response were investigated analytically. Additionally, the demonstrated augmented elasticity theory has shown to be effective in achieving the softening and stiffening responses of nanobeams. A novel arithmetic benchmark for dynamic modelling of elastic nanobeams has been discovered. The developed MUGEM offers a realistic solution to addressing nano-scale structure dynamics in pioneering MEMS/ Nanoelectromechanical systems (NEMS).
The torsional rotation of the regularly loaded fixed end nanobar in 3D [133]
Sladek et al. [135] implemented the micro-dilatation theory to develop a novel meshfree method for porous elastic material which is known as the MLPG. Additionally, the MLS was employed to define the spatial variation of displacement and micro-dilatation. In a recent study, Kwak et al. [136] presented a novel meshfree method for the annular plates and cylindrical shells of ply drop-off laminated conical in which the FV analysis has been investigated. The conceptual framework of FV examination was founded on FSDT, then the field functions were estimated using an innovative meshless Tchebychev-RPIM approach. In addition, the shape function was founded on Gaussian radial functions and Tchebychev polynomials. The GEs as well as BCs for ply drop-off laminated composite shell substructures were developed. Implementing an artificial spring approach generalizes the boundary and continuous conditions, and the type of BCs determined by the spring stiffness values. A schematic of the ply drop-off laminated conical shell is illustrates in Fig. 22. And Fig. 23 illustrates the geometry of the combined region.
Schematic of the ply drop-off laminated conical shell [136]
The geometry of combined region [136]
1.4 Mathematical Modeling
In this section the basic formula of traditional meshfree method are presented. The following equation of meshfree estimation for a scalar function u in terms of the material (Lagrangian) coordinates has the basic approximations [1]:
where: the shape function ΦI: Ω → ℝ, the nodal values at particle I located at position xI uI, and the set of nodes I for which ΦI (x) ≠ 0 is ℓ
The general formula of SPH is defined as following equation [137]:
Although the continuous form of SPH is second order complete, it can be simply exposed that the discrete SPH form is given by [1]:
where the RKPM is developed from the continuous SPH approximation, but the C (x,y) correction function is presented into the approximation so as to rise the order of completeness of the approximation [12].
where K (x,y) = C(x,y) w (x − y) with C(x,y) is defined such that the approximation is n th order consistent p.
Numerical integration must be used to assess this continuous statement. In the RKPM, a discrete variation of the RKPM is obtained through this stage.
For MLS approximation; uh: Ω → ℝ of the function u: Ω → ℝ is posed as a polynomial of order m but with non-constant (NC) coefficients. The local approximation around a point \(\overline{x }\) ∈ Ω, evaluated at a point x ∈ Ω is given by
where p (x) is a complete polynomial of order m, and a (x) contains NC coefficients which depends on x.
The formula of PUFEM is define as [24]:
where coefficients vjI are nodal unknowns and \({\varnothing }_{I}^{0}\left(x\right)\) is the shape function based on the Lagrange polynomials.
The approximation in the hp-clouds method can be enhanced by adding the different partitions of unity and written as follow at any point of x ∈ Ω [138, 139]
While the \({\varnothing }_{I}^{m}\left(x\right)\) and the \({\varnothing }_{I}^{k}\left(x\right)\) both are meshless shape functions of the orders m and k respectively.
A set of test functions Ψ are chosen and the system of equations is determined by sitting εh
Orthogonal to set of functions for Weighted residual (WR) methods
Usually, the collocation method formula is not use as it is; for better illustration the formula of elastic foundation (EF) with the governing equation of a string is in assist. And by choosing a specific parameter of a = 0.01, c = 1 and f = -1 for boundary points × 1 and × 2 the familiar equation can be written as follow:
The exact solution of Galerkin method is given by:
The general formula of the Discrete equation for elastostatics is given by:
The integration is evaluated only for the direct nodal integral which defined as integration point as well:
where adding additional stress points to the nodes is enhancing avoiding the instabilities due to rank deficiency, so the stress point integration is written as follow [2]:
For the support-based integration for the background mesh or cell structure over performing Gaussian quadrature
where ξ is local coordinates and det \({J}^{\xi } (\xi )\) is the determinant of the Jacobian.
For modification of the intrinsic basis such as EFGM which can be defined based on crack kinematics, the Westergaard solution is introduced into the basis P [140]:
Approaches founded on an extrinsic MLS enrichment to model cracks in meshfree techniques is to introduce the analytical
For methods based on an extrinsic PU enhanced XFEM
Typically, the solutions for PDEs with discontinuous coefficients have discontinuous derivatives along the discontinuity, and the discontinuous derivatives is given by;
1.5 Error Estimation
Concerning finding the best values for the variables involved in the approximation of the error, some studies have worked on improving meshfree methods for the purpose of studying the error function. For example, Gavete et al. [141] employed the EFG approach in combination with several selection strategies of the MLS approximations, for different structures such as plate, rod, and beam. Lu and Chen [142] have provided a great summary of adaptive Galerkin meshfree techniques. Chung Belytschko [143] presented an error estimator established on residuals that compares the variation between an improved stress field and raw EFG field, similar to conventional ZZ error estimator in FEM. Durat and Oden [144], have employed the a hp-adaptive approach to accomplish an a posteriori error estimator, using a dual meshfree technique, global stringent constraints on the energy are found in [145]. But, Krongauzr and Belytschko [146] have established a posteriori approximation error for meshless algorithms for refining the corrected derivative adaptively. Bordas and Duflot [147,148,149] have worked on some studies which focused on error measuring for the extended FEMs which are considered for further developing in error estimation in meshfree/meshless methods with intrinsic. Moreover, h-adaptively is more easily incorporated into meshfree methods than mesh-based approaches because of their absence of involved meshing. Hence, a meshless framework makes it conceptually simpler to construct p-and r-adaptively. While, a local error estimator, or at the very least an indication, is required to drive the adaptively. Some other studies that have worked on error estimation were published such as, Combe [150] who worked on adapting approach for the EFGM, the RKPM adaptation by [151], as well as the error approximation of EFG which been considered by Gavete et al. [141]. Zhang et al. [152] implemented RPIM method combined with suitable implicit time integration to perform transient wave propagation investigation by the dispersion analysis for different structure and geometries. The dispersion error of the numerical findings was extensively examined through a dispersion analysis. Furthermore, it has been determined that the RPIM with sufficient support domains of quadrature points is capable of producing nearly no spatial dispersion errors that are significantly less than those resulting from the conventional FE methods having the same node scatterings. Therefore, monotonically enhancing the computation precision of the arithmetical outcomes by employing reducing time steps was guaranteed. Figure 24 illustrates the graphs of the overall dispersion errors from several numerical techniques. It shows that several unique CFL numbers as well as wave propagation angles were taken into consideration owing to clearly as well as comprehensively evaluate the dispersion properties.
The overall dispersion error obtained along the wave propagation angle θ = 45.∘ Vs. the normalized numerical wave number kh for several methods: a FEM-T3; b FEM-Q4; c RPIM with αs = h; ( RPIM with αs = 2h; e RPIM with αs = 3h [152]
Rabcuk and Belytschko [153] employed the posterior error estimation for the adaptive refinement for 2D and 3D structures. Bayona et al. [154] described how to anticipate an error of a solution via RBF-FD approach with a one (constant) shape parameter value. This was founded on analytical formulas developed for the local estimated inaccuracy of RBF-FD equations. Because error indication is reliably, and the possibility of estimating the optimal form parameters accurately, which reduces the solution error to the minimal value, has been demonstrated. Furthermore, Duan et al. [155] investigated the error of displacement and established an analysis of different structure after developing a meshfree method through the consistency of the derivatives of nodal shape functions in meshfree Galerkin methods, and a novel framework was proposed. Additionally, the presented three-point integration scheme employed background triangle elements, of which the smoothed (corrected) nodal derivatives at quadrature sites were computed via a discrete divergence consistency (DDC) for quadratic approximations, which consist of the MLS approach with the quadratic basis. It also reveals that the corrected nodal derivatives satisfy the differentiation of the approximation consistency (DAC) for quadratic approximations, hence the proposed methodology is known as the quadratically consistent three-point (QC3) integration scheme since it fulfills both the quadratic DAC and the quadratic DDC. Hou et al. [156] provide error analysis based on the vibration analysis results of a strain gradient plate model to examine the error ratio among analytic solution, MLS approximation, and MKI for the model with varied small-scale parameters. Fu et al. [157] have presented a developed BPM by using the Laplace transformation theory, to perform a numerical modeling of the time fraction diffusion equations. Although the recursive composite multiple reciprocity technique (RC-MRM) was employed in purpose of converting the inhomogeneous problem to higher-order homogenous. The error variation of LTBPM is illustrates in Fig. 25. This example demonstrates that the suggested LTBPM performs efficiently for 3D fractional diffusion equations. Correspondingly, all final errors have a high probability of reducing rapidly and subsequently when M boundary increases it will increase as well. Although the turning points of error curves were controlled by the accuracy of NILT input data. Besides, it implies that the final inaccuracy is primarily owing to the Stehfest numerical inverse Laplace transform instead of the BPM computation. Iliopoulos and Michopoulos [158] have performed an absolute error distributions calculations for random grid method (MRG) and traditional meshless method. Figure 26 shows the respective absolute error of the MRG.
LTBPM error variations with respect to Mon different time instants a T = 0.01, c T = 1, e T = 100; and LTBPM solution atz = 1 cross section on different time instants b T = 0.01, d T = 1, f T = 100 [157]
The respective absolute error of the MRG, the strain and absolute error contours for nodal noise level of 1 × 10–2 pixels and loading angle of ß = 45°. And the domain is assumed to span 1600 × 1600 pixels [158]
As recorded by Wu et al. [159] in a proposed study of modeling and analyzing different composite solid structure (pressure vessel, cantilever beam and unite cell) by implementing an immersed meshfree method based on Galerkin method, it was proved that the optimal error estimate in the energy norm is satisfied by the non-conforming Galerkin method in problems such as elasticity interface. Additionally, the author proved that this method can be used in estimating the error in micro and macroscales. Dehghan and Najafi [160] implemented the RBF high order mesh free method with other high order methods for numerical solution of a non-classical one-dimensional two-phase Stefan problem. A comparison between the methods results and error analysis were performed. In a short communication article presented by, Roque et al. [161] the examination of the error for the obtained fundamental frequency (FF) after employing the MQRBF combined with HSDT to investigates the FV behavior of the FGPs was accomplished. In overall, the relative error is minimal and decreases as the number of points/side increases. In certain circumstances, the error is contrary to this trend, which was previously discovered and discussed in many articles. Furthermore, when compared to exact results previously published, the error is less than 1% for each case in the evolution of ND frequency with the number of nodes for a/h ratios of a SS plate. The ND frequency has been compared to the available exact results. The number of nodes in a regular grid was modified and investigate several exponents p. The plate is rather thick, having a/h = 5 and the errors = 0.4%. Ferreira [162] implemented the root mean square (RMS) error, after employing the MQRBF as well as the inverse MQRBF approaches to examine the static behavior of laminated composite beams (isotropic, symmetric and orthotropic) under uniform bending load. The CC and the SS BCs were considered. Furthermore, the isotropic three point bending beam was considered to evaluate the method. An excellent performance was observed by validating the results.
1.6 The Implementations of FGM and Other Composite Materials in Solid Mechanics Structures
For the high importance and advantages of the FG composite material, the researcher shows high interests into it in various engineering complex structure from MEMS to bridge and the various classification of FGM are illustrated in Fig. 27. As an examples, Civalek [163] employed the DSC to discretized the BCs and the GEs for the purpose of studying the statically bending behavior of a composite laminated thick symmetric cross-ply. Besides, for presenting the singular convolution algorithm both the Lagranange delta sequence (LDS) kernel as well as Regularized Shannon’s delta (RSD) kernel. Nian et al. [164] proposed an innovative nature inspired functionally graded lattice filled protection structure (FGLPS) for the purpose on enhancing the structural energy absorption characterizations subjected to ship impact load. The novel structure proved to be highly efficient in reducing the prolong impact time and the peck impact force as well as play a prominent protecting role. Furthermore, Mellal et al. [165] employed the HSDT and generate the equation of motion by using the Hamilton’s principle (HP), for the purpose of examining the buckling and the FV behaviors of perfect/imperfect FG beams with an EF. Also, the Navier’s method has been utilized for solving the system analytically.
The various classification of FGM
Hadji et al. [166] developed a novel NLC hyperbolic shear deformation beam theory (HSDBT) for the purpose of examining the FV response of a porous FG nanobeam. Also, the equation of motion has been defined by using the HP. Mercan et al. [167] performed a FV study for a FG cylindrical shells that founded on love shell theory by utilizing the DSC method which guaranties an accurate NF results. In a novel study focusing on a laminated conical shell problem, Civalek [168] study the FV behavior of a laminated conical shells by employing the DSC methodology and the governing equation has been formulated by utilizing the Love’s first approximation classical thin shell theory. Furthermore, the same author has employed the same methodologies as well as for the purpose of determining the FV analysis but this time for a composite orthotropic laminated conical as well as a single isotropic conical shells [169]. Likewise, Vinh et al. [170] investigates the bending and FV behavior of a FG sandwich various plates structure. The analysis has been performed by developing a new superior first order mixed plate element (IMQ4). The parabolic distribution shear stresses has been assumed and it enhance the transverse shear stresses due to the fact that it satisfy the free condition of the transverse shear stresses at lower and upper plates’ surfaces. Besides, the FSDT has been combined with the mixed FE formulation for the purpose of generating the fundamental equations of the novel element. A further study has been established by, Hadji et al.[171] which focused on examining the FV of a beam structure made of CNTRC subjected to different BCs. Besides, for generating a new refined beam method, the HP combined with trigonometric shear deformation theory (TrSDT) were employed for the purpose of developing the equation of motion and the outcome equation has been defined analytically. Additionally, the presented developed model has the ability to address the shear deformation effect as well as dealing with three unknowns just like the FSDT for Timoshenko beam with eliminating the shear correction factor. Also, the authors addressed that the CNTRC beam can be analyzed analytically under various BCs. Moreover recently the HP has been employed combined with NLC elasticity to study the FV of the FG nanoplates by, Hadji et al. [172] followed by an analytical solution for the obtained equation. Rizov [173] accomplished a delamination analysis under NL creep behavior of beam made of FGM by employing the Ramberg–Osgood stress strain relation. Ersoy et al. [174] employed the DSC as well as the harmonic differential quadratic method (HDQ) approaches for analyzing the FV of curved structural components and annular plates as shown in Fig. 28. Besides, the governing equation of motion has been generated based on HSDT. FGM, laminated and isotropic material properties were considered. The obtained results shows that the shells’ accuracy and convergence significantly affected by number of grids in the circumferential directions. Also, for capturing the demanded accuracy with high efficiency for shells; the author recommended using 11 grids in the circumferential direction. Additionally a good agreement has been observed between the obtained results and previous published data.
The annular plates and curved shells schematics with various materials [174]
Avcar et al. [175] studied the enhancement of the deviation of exponent of volume fraction (VF) of CNTs on the NF of CNTRC in beam structure under various BCs. The equations of motion were found based on HP and TSDT followed by analytically solution. Furthermore, a recent comprehensive review has been published by, Alibar et al. [176] which aimed to investigate the vibrational behavior of various nanocomposites structures affected by porosity as well as CNT distribution and VF. Besides, the FGM reinforced CNTs were addressed. Moreover, Nuhu and Safaei [177] accomplished an inclusive review about analyzing the vibrational behavior of nano/microscale assemblies made of composite materials in various engineering systems such as nano actuators, biosensors, as well as electromechanical by implementing the NLC/nonclassical continuum theories of elasticity. Chakraborty et al. [178] accomplished a review paper which aimed to cover the enhancement of CNT on panel’s and plates’ vibrational behavior and stability. Figure 29 illustrates a pie chart of the covered literature which shows a comparison between FG-CRTRC, RD-CNTRC and 3-phase composites. Kim et al. [179] used various types of fillers such as; CNT/silver nanowire (AgNW) and others in a ternary systems for a stretchable Polydimethylsiloxane (PDMS) composite electrodes, for the purpose of examining the electromechanical behaviors. It has been observed that high mechanical reliability can be obtained with significant rupture strain under tensile deformation. Another remarkable outcome of tensile was that when the AgNW networks disconnected on the outer layer of the composite, it result in a fifty percent rise in resistance under extreme tensile strain. Additionally, a gradual rise in the resistance combined with remaining the electromechanical stability has been observed under large elongation values.
A representation of the research interests in CNTRC concerning stability dynamic response and vibration characterizations in percentage [178]
Jankowski et al. [180], employed the NSGT combined with Reddy HSDT for the purpose of studying the dynamic response of a smart nanobeam founded on NEMS and made of a sandwich functionally graded porous (FGP) with piezoelectric layers in various length, affected by nonlocal parameters. The results outcomes shows a diversion between FV resulted from classical and NLC approaches. Kurpa et al. [181] employed the R-function method combined with Rayleigh–Ritz approach for the purpose of investigating the FV of a sandwich plate structure made of FGM as a face layer and a core made of ceramic with various porosity distribution. Pajand and Masoodi [182] accomplished an analytical analysis for the purpose of investigation the NF as well as stability load of a curved beam columns made of FGM and considering semi-rigid connectors. In a novel study, Pajand et al. [183] investigated the static analysis of a FG non-prismatic sandwich curved beam structures. While the Ritz methodology and the total potential energy were employed to define the governing equations, although the Redyy and Timoshenko beam theories were utilized to develop the formulation. The obtained results shows that the Shear impacts were observed to be significantly greater in FG curved sandwich beams with high curve as well as low slenderness ratios. Moreover, the cross sectional type, taper ratio, and the gradient index were also found to have a substantial impact on static performances. Fantuzzi et al. [184] performed a FV analysis for various arbitrary designed FG-CNT reinforced plates, the mapping technique has been accomplished via NURBS and the FSDT has been employed to determine the mechanical performance of the structure. Besides, the governing equation strong form has been solved by implementing the generalized differential quadrature method (GDQM). Arefi et al. [185] employed the sinusoidal shear deformation theory (SSDT) as well as the NLC elasticity theory for the purpose of analyzing the FV performance of a nanoplate made of FG polymer composite and reinforced with GNPs and based on Pasternak foundation. I Arefi et al. [186] work, the same methods were employed and the same Pasternak foundation has been utilized for the purpose of investigating the bending behavior of a multilayered curved nanocomposite beams made of FG polymer composite and reinforced with GNPs and a shear layer has been embedded on the top of foundation. Also, several patterns dispersal of nanoplatelets were considered. Tornabene et al. [187] employed the FSDT and the GDQM for FV analysis of a laminated nanocomposite shells (conical and cylindrical) and plates in which each layer is generated as a three-phase composite that include CNT/fiber/polymer and the CNTs has been added to the polymer matrix. Tornabene et al. [188] investigated the enhancement of CNT agglomeration on the FV of the layered composite doubly curved panels and shells. Different CNT exponential dispersals as well as numerous laminations schemes along the structure thickness. The theoretical and numerical models has been performed by utilizing the Carrera’s unified formulations (CUF) and HSDT and the GDQM respectively. The authors published a further study by implementing the same methods but the aim of this study was to investigate the enhancement of CNT agglomeration on the linear static behavior of the same mentioned structures [189]. Sobhani and Safaei [190] employed the harmonic differential quadrature method (HDQ) to examine the FV of cylindrical shells structure made of the reduced Gr oxide nanocomposite (rGON) where the Halpin–Tsai (HT) homogenization technique has been utilized (which is a developed method that is used to determine the composites reinforced by nano/micro-particles) [191]. Besides, the HP, the general shell theory combined with the FSDT were employed to determine the main formulas. Additionally, the following distributions of the FG types of the reduced Gr oxide; UD, FGA, FGV, FGO and the FGX. The enhancement of various distribution on the NF compared with the pure polymer has been considered.
1.7 Computational Methods in Mechanics
The meshfree techniques can also be classified into numerical and analytical methods and some of these methods were classified according to, Liu [57] such as strain constructed PIM (SC-PIM), smoothed PIM (S-PIM), etc. which consist as L/NL elliptic PDEs. Also, as a NL hyperbolic PDEs SPH, MWS and LRPIM. In addition some other computational meshfree were mentioned by, Xiao et al. [192] such as; discontinuous deformation analysis (DDA), material point method (MPM), as well as the discrete element method (DEM). Although there are many other analytical methods which are in used to analyze solid mechanics and fluid mechanics such as; FEM, finite difference method (FDM), (BEM), Finite volume method (FVM), smoothed FEM (SFEM), and eXtended FEM (XFEM) [192]. Besides, the most communal difficulties that faces the progress in computational mechanics software’s are summarized in Fig. 30. Also, Fig. 31, illustrates the various types of programming languages implemented in analyzing computational mechanics with it is s advantages as well as it is shortcoming [192]. It is worth to mention that the SFEM methodologies consist as a group of methods developed by combining the standard FEM with some meshfree approaches which leads to have higher accuracy and greater convergence rates than regular FEM [193]. Safaei et al. [194] utilized the unit cell methodology (UCM) to define the entire impact of various parameter on the FV of a honeycomb sandwich structure(HCSS). In addition the representative volume element (RVE) approach has been employed. The main objective of that study was to examine the stress state, modal as well as the FV analyses of the HCSS under various BCs. Besides examining the impact of various material and structure properties on the NF of (HCSS). The results were validated with analytical and arithmetical previous published data. In a further study proposed also by, Safaei et al. [195] in which the authors analysis the HCSS as a core of a beam under static as well as harmonic loads and compare its behavior with the normal hexagonal shape sandwich as a beam core by implementing the FEA. The obtained results shows that using carbon fiber instead of metal as a skin layers for the sandwich beam have more advantages such as; higher stiffness and lighter weight. Besides, the sandwich beam strength can be enhanced by the ply angle alignment as well as layers quantity. In the other hand, the NF increased due to the rise in core thickness as well as rising the skin layers, in another words increasing the height of the core will influence the NF and the shear stress. The most popular meshfree methods and meshbased methods which are utilized in the computational mechanics are illustrated in Fig. 32.
The most communal difficulties that faces the progress in computational mechanics software’s [192]
The various types of programming languages implemented in analyzing computational mechanics with it is s advantages as well as it is shortcoming [192]
The meshfree methods and meshbased methods utilized in the computational mechanics
Sheikh and Behdinan [196] presented a comprehensive review for the implementation of significant discrete scale as well as multiscale computational techniques for analyzing composite materials. Various types of composites materials were discussed such as; metal/polymer based composites, CFRP, nanocomposites and etc. also various characterizations were presented such as; fracture and failure analysis, dynamic and FV. Ramteke and Panda [197], published an inclusive review which concentrates on linear/NL analysis of various porous graded (PG) structures’, it is computational modelling as well as the experimental challenges. The review has covered vibration, linear/NL static and the transient deflection, although the stress parameters were considered. Another comprehensive review has been published by, Gierden et al. [198] where the authors were interested in the FE-Fast Fourier transformation (FE-FFT) methods based on the 2-scale methodology, which have been utilized for computational mimic of macroscopic material performance and the microstructure evolution. Geers et al. [199] presented a review which focused on investigating the developed multi-scale computational homogenized methods. The review covers various issues such as first order homogenization in solving mechanical systems, thin structure beams as well as shells as shown in Fig. 33, thermal problems and others. Cueto et al. [200] published an inclusive review about the natural neighbor Galerkin methodologies (NEM) which cover the explanation of Laplace and Sibson in fluid and in solid mechanics. In addtion, the authors, highlighted some drawback of meshfree method such as; handling of secondary variables, imposing the essential BCs. Lorong et al. [201] evaluated and highlighted the most recent innovations in computational mechanics' involvement in the computer modelling of machining and blanking. Besides, the contribution of computational mechanics which has been developed at a very global scale entitled macroscopic scale, while it required an illustration of the deformations of the part. Furthermore, the FD can be considered as a good alternative for solving homogenization issues in heterogeneous multi scale methodology [202]. Rafiei et al. [203] presented analytical simulations for a double skin composite wall shear strength and the resulted outputs were validated. In an earlier study, Rafiei et al. [204] developed FE model for the purpose of mimic the performance of a unique composite shear wall system subjected to in-plane loading. Artioli et al. [205] employed the high-order virtual element approach for the purpose of solving the homogenization problem in NL composites with long fiber. Nevertheless, the same method has been implemented by, Marino et al. [206] to investigate the homogenization of polycrystalline materials.
Schematic of a shell in 2-scale homogenization problem by thickness representative volume element (RVE) [199]
However, the self-consistent clustering analysis (SCA) has been utilized by [207, 208] to investigate the NL mechanical and damage behaviors of a braided woven composites. Civalek and Akgöz [209] investigated the FV of a microscale annular sector as well as sector shaped graphene with an EF (Winkler-Pasternak) by implementing the NLC elasticity method. The numerical solution was found by employing the DSC. Also, the thin plate theory has been employed combined with 8 nodes curvilinear element in order of converting the governing equation of motion to a computational region. Besides, various BCs set and NLC parameters were considered. In an earlier work, Civalek [210] employed the DSC for the purpose of examining the FV as well as buckling behavior of a various structure of composite plates. Although the computational domain has been mapped through employing the geometric mapping transformation with a 4 nodes element. The validation results prove that the DSC is an accurate simple algorithm for analyzing the mentioned system. The same author utilized the same method in a further study [211] to investigate the FV of a straight-sided quadrilateral plates where the geometrical transformation has been employed for the purpose of converting the BCs and the GEs to a square computational domain. By validating the results a remarkable accuracy and convergence were obtained compared to exact and numerical solutions. The HDQ has been employed in an earlier study by, Civalek and Ülke [212] for the purpose of examining the linear bending behavior of a circular plates both analytical and numerical solutions were presented. Also, the error-grid numbers relationship has been calculated by HDQ, FE, DQ and FD. Civalek [213],employed the HDQ in a novel study for the purpose of investigating the NL of a thin isotropic plates with a rectangular geometry based on Winkler-Pasternek EF subjected to different types of dynamic loads. The obtained results prove that the Pasterank parameter (G), has less effect on the plate displacement that the Winkler parameter K. besides, a reduction in the plate deflection has been observed due to the increasing in the G. The modified strain gradient elasticity theory (MSGT) has been implemented by Akgöz and Civalek [214] for the purpose of studying the buckling behavior of a protein microtubules, and the Matlab code has been generated based on the MSGT. Various mode numbers, lengths, and material parameters were chosen to examine the enhancement of scale effect and material parameters on the buckling performance. Jalaei et al. [215] employed the Navier’s and inverse laplace transform methodologies for the purpose of solving the GEs which obtained by Hamilton’s variation in a study focused on investigating the transient response of porosity-dependent viscoelastic FG nanobeam subjected to a dynamic load in a magnetic fields as shown in Fig. 34. Besides, the Kelvin-Voigt visco constitutive model was implemented to examine the model’s internal damping effect. Additionally the effect of thickness stretching has been studied through the implementation of nonlocal strain gradient theory (NSGT).
The schematic of A the nanobeam made of visco-imperfect FG, B The porous FG in nanobeam under dynamic loading excitation and magnetic field [215]
For the purpose of detecting the macroscopic failure through acoustic tensor in macroscale as illustrated in Fig. 35, Spahn et al. [216] employed the FEM combined with FFT- for micro formulation. The macroscopic destruction behavior has been predicted using NL material properties generated by microscopic physical phenomena on a smaller length scale. Additionally, for computational simulation of periodic microstructures recently the FFT-based method has been demonstrated to be a quick and accurate tool [198] .
The onset macroscopic failure detection through acoustic tensor [216]
Recently, Numanoğlu et al. [217] investigated the FV, thermomechanical and size dependent of a nanobeam. HP has been implemented based on the Timoshenko beam theory to obtain the size dependent dynamic formulas, then combined with the stress formula of the NLC elasticity approach. While the eigenvalue problem has been solved by FE and separation of variables total methodology. Also, the local NLC stiffness as well as mass matrices were generated through the WR approach of finite element analysis (FEA). Suchde et al. [218] accomplished a review about point cloud generation (PCG) and it is implementations in meshfree methodologies. Various approaches and software employed for PCG in meshless, although the initial territory discretization processes for the time dependent issues has been addressed. Additionally, a car model has been presented and analyzed followed by generating a mesh with a node density similar to Gmsh FE mesh generator. Figure 36, illustrates the generated mesh with PCG that created by vertices of mesh. Safaei et al. [219] employed the FEA to investigate the CBL and bending of a sandwich beam made of carbon foam as a core and the outer face sheets made of carbon fiber under thermal as well as axial compression loads. Also, the heat distribution along the sandwich layers has been examined. The author of that study recommended using the aerogel instead of fiberglass as an insulation in electronics and aerospace industries based on their unique obtained results. Balokhonov et al. [220] suggested an approach for digital modelling of a 3D structure of materials with reinforced particles of complex irregular shapes which has been identified experimentally, which implied that the scale invariance of natural mechanical disintegration. The particles were distributed at random across the matrix and coating computationally domains to construct two-phase structures of metal-matrix composites (MMCs) as well as coatings of varying spatial sizes. Nevertheless, where resilient coatings and composites material are becoming gradually more important in various industrial applications which are considered as main application of tungsten carbide (WC) nanoparticles, Asmael and Memarzadeh [221] presented a comprehensive review that focused on the latest accomplishments as well as challenges facing the electrochemical machining of WC. Fallahi et al. [222], employed a high order 1D computationally effective model founded based on the framework of CUF for the purpose of studing the enhancement of fibers orintation path on the mechanical performance (stability and FV) and to analyze the variable angles in tow panels statically. However, many researchers shows interesting in studying the mechanical properties of various composite materials by implementing the MD approaches [223,224,225,226,227,228,229,230,231]. Pajand et al. [232] studied the NL performance of FG sandwich shells as well as plates by implementing the equivalent single layers method (ESL) with rule of mixture methodology. The investigation has been accomplished by FEA and numerically, where in FEA for structural discretization has been obtained through employing a triangular shell element, and for justifying the locking problems the mixed interpolation of the Tensorial components methodology (MITC). It has been demonstrated that an extension of the unstable part of the equilibrium path as well as a drop in buckling load cause by decreasing the core E. besides, the load capacity in FG-sandwich shell and beam of two FG faces sheets with a homogenous core is lower that of with FG core and homogenous face sheets. Also, the buckling load has been observed to decrease by raising the power index (n), owing to the conversion of material properties as of ceramic toward metal. Milić et al. [233] proposed a NL geometric IGA framework for a shell structure created of a composite laminates counting piezoelectric layers considered by the electro-mechanical coupling. While the Mindline-Reissner theory which include the transvers shear impact has been implemented for the kinematics of the shell. For structural discretization the NURBS, ACShell9 and the Shell91 (Ansys) finite elements were utilized. Milić et al. [234] employed the same approach again to investigate the same structure but in this article only NURBS has been employed to develop the formulation of the IGA shell and the Greville’s points has been utilized for defining the normal vectors of the surface. Also, in this article the Mindline-Reissner theory was combined with FSDT for the purpose of maintain the level of arithmetical determination in an adequate realm.
The PCG around the car utilizing the Gmsh, A mesh wedge left side and the associated segment of the point cloud made up of mesh nodes right side, and the point cloud merely shows the points behind the displayed mesh slice [218]
1.8 Boundary Conditions in Solid Mechanics
There are various classifications of BCs sets that vary based on the structure aimed to investigate such as; SS, FF, CC, SF, CF, and CS these BCs can be used in beams structure. While in case of analyzing plates the BCs vary to SSSS, CCCC, CFCF, SFSF, SCSC and FFFF. In this review the majority of mentioned studies from literature considered studying the enhancement of various BCs on buckling, bending and vibrational behaviors. Also in the studies which were interest in piezoelectric material the enhancement of BCs on electrical load has been considered. Where implementing essential BCs is an important challenge in meshless approaches. Since meshless interpolation is unable to evaluate the Kronecker delta property, and therefore enforcement of prescribed values is more difficult than using the FEM [235]. However, the authors purposed a comprehensive overview designated techniques for enforcing essential BCs in Galerkin methods. Moreover, some studies such [236], finds that in various BCs sets the estimated deflection by the designed model was always slighter that the classical approaches. To eliminate the shear locking, the CS model supposed to be applied for the thin plates [237]. The obtained results in [238] shows that there is a relationship between the enhancement of BCs with the and type of distribution of reinforcements in various structures. More significant results of the effect of BCs is presented in Tables 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10. Liu and Karamanlidis [239], stated that when the chosen shape functions fulfil the Kronecker-delta characteristic therefore the essential BCs might be directly satisfied. Lei et al. [240] stated that the Kronecker delta property is not accessible through the shape function; so the authors implemented the transformation method [131], for the purpose of enforcing the essential BCs. In that study the following sets of BCs were investigated with UD, FG-O, FG-X; SSSS, SCSC, SFSF, SSSF, CCCC and CCCF. The full transformation methodology has been presented by [241], in order to implement the essential BCs. Also, the Lagrange multipliers has been utilized to implement the essential BCs [242]. The implementation of crucial BCs in meshfree computation requires extensive CPU time owing to the loss of Kronecker delta characteristics within meshfree shape functions, Chen and Wang [243] developed a novel BCs treatments in meshless computation of contact difficulties. However, the maximum buckling load effected values were recorded in CCCC. In a novel investigation of high order plate theory, Neves et al. [244] employed the principle of displacements under CUF for the puprpose of defineding the BCs as well as the explicit governing equations. Wang et al. [245] employed the penalty methodology to apply the essential BCs. While the equation of motion and the BCs were obtained by employing the HP by [246]. Karami and Malekzadeh [247] focused on employing the DQM to investigate the FV of a quadrilateral thin plate various BCs were examined. It has been noticed from the results that adding a clamped BC cause an increase in the NF of all nodes in rhombic plates. Also, in trapezoidal plate the CCCC recorded higher NF values that SSSS. Furthermore, the challenge of enforcement of BCs for systems with governing differential equations as well as at least two BCs at the boundary point is one of the main difficulties in the classical utilization of DQM. Several strategies were suggested for handling this challenge. Bert et al. [248], proposed two established approximate techniques for the purpose of solving dynamic problems in structures the first methodology is a developer of complementary energy approach, and the second methodology was the δ technique which eventually emerged as the DQM. Jang et al. [249], established a static analysis (bending and stability) of different structures subjected to different BCs by utilizing the DQM. The results were compared with previous results which was obtained numerically by Galerkin. Moreover, the CCCC as well as the SSSS for the first time were subjected to thin skew plates by, Wang et al. [250] by using the DQM as well as the oblique references axes were employed.
In a recent study, Noroozi et al. [251] the obtained results shows that by applying the CCCC higher NF values were recorded than SSSS. Liew et al. [252] stated that, generally, plates having clamped ends possess a larger load carrying capacity, unless when the BCs contain a free edge (e.g., SSFC). Furthermore, the dimensionless (DL) axial compression obtained by utilizing the clamped BCs was almost doubled the DL axial compression obtained by utilizing the simply supported [253]. Artan and Batra [254] employed the initial values methodology for the purpose of examining the FV of a strain gradient beam by utilizing three different sets of BCs through considering the higher orded BCs. Zhao et al. [255] examined the vibrational behavior of thick rectangular plates made of FGP (Al/Al2O3) as shown in Fig. 37 analytically with arbitrary BCs. The obtained results shows that the plates with CCCC BCs recorded higher ND frequency parameters (FPs) that plates with SSSS BCs in all examined modes.
A thick rectangular plates made of FGP with various porosity models [255]
Zhao et al. [256] found that the square palates’ made of aluminum/alumina and subjected to CCCC in all modes recorded significantly higher values of critical buckling temperature than the same plates’ subjected to SSSS. Chen et al. [257] stated that the FGM plates under various types of edges when it is subjected to CCCC BCs it recorded greater buckling load factors than those subjected to SSSS BCs can be obtained. Ahmadi [258], observed that the NFs obtained by CC multiple-nanobeam system (MNBS) BCs were similar to NFs obtained by free-free (FF) MNBS BCs. Else, the author observed that increasing the Winkler stiffness k the NF of the clamped chain of triple-nanobeam system (TNBS). Pradhan et al.[259] observed that the FG cylindrical shells subjected to CC and FF BCs recorder greater values of NF than those subjected to a SS BCs. Additionally, Ansari and Darvizeh [260] stated that among all BCs alternatives, the clamped FGM shell exhibits the largest NF. Barati et al. [261] implemented an advanced four variables plate theory in a novel study for the purpose of investigating the FG piezoelectric plates’ with porosity vibrational as well as the electro-thermal–mechanical performance. The results of that study shows that the DL frequency in different thermal atmospheres in (uniform, linear and nonlinear) temperature rises obtained by applying the CCCC BCs is much greater than the values obtained by applying the SSSS BCs. Besides, changing only one condition in each BCs sets to clamp can also enhance the DL frequencies. On the other hand, [262] found that the greatest fundamental frequencies (FFs)for all values of lamination angle φ were obtained by applying the SSSS BCs. Also, it has been reported that the electromechanical coupling values obtained by plates subjected to SSSS BCs is higher than the values obtained by plates subjected to CCCC. This results were found based to the fact that the CCCC allows to have more constraints which leads to have smaller range of displacement as well as rotations for nodes [51].
Natsuki and Natsuki [263] explore the enhancement of BCs and resonant frequency on the elastic modulus (E) of a nanowires made of CNT based on EF and subjected to a clamped BCs as illustrated in Fig. 38. The authors stated that the clamped length highly influenced the vibrational frequency reliance with an E, especially if E > 1.0TPa. In addition, owing to the calculation errors between E as well as vibrational frequency; the impact of the clamped BCs on predicting the E cannot be neglected. Wang et al. [264] employed the Adomian decomposition methodology combined with the modal analysis to proposed an iterative model for the purpose of investigating the forced vibration behavior of a NU beam with NL BCs (hardening NL spring boundary). It has been observed that, the degree of nonlinearity reflected in the vibration response differs with varied base excitation amplitudes with the fixed NL supported spring parameters. Besides, the beam VA and its period remained within a particular boundary stiffness range varied with the extent of boundary nonlinearity under the same base motion excitation. While the excitation frequency as well as the amplitude were defined, the vibration status can be modified by modifying the NL spring property. However, that study solution can be considered as an initial step for further studies of investigating the vibrational performance of NU beams with various NL BCs subjected to diverse excitations.
The nanowires made of CNT based on EF subjected to a clamped BCs [263]
2 Static Analysis
2.1 Bending Analysis
2.1.1 Bending Analysis in Nanoscale
Significant reliable bending results of impact of the embedding nanofillers and stress responses as well as to static electromechanical deflection of the SMP were obtained. The examined structure of FG nanocomposite porous plates under static electromechanical load is shown in Fig. 39 [265]. Safaei et al. [266] have applied the EMT’s approach with HSDT combined with a MLS meshfree method and the meshless for of strain to investigate mechanical loading (static bending) and thermal gradient of the PNSPs with the FG CNT porous core of clusters/polymer nanocomposite layers.
Structures of the proposed FG nanocomposite porous plate bonded with 2 piezoceramic layers under static electromechanical inputs [265]
By applying three different uniform case load 0, 500, 100 kPa to study the central deflection performance for the PNSP with square geometry and SS, subjected to a thermal gradient load on. The result shows that intensified deflection is caused by the direction of the moment at Tb > Tt. In addition, by applying a thermal gradient loads the mechanical deflections observed to be moderated specifically when Tb < Tt as shown in Fig. 40. Besides, Fig. 41 illustrates the unique sandwich plate structure reinforced CNT.
The relation between central normalized deflections versus the bottom surface temperature in square SS PNSP under different uniform load case [266]
Sandwich plate structure with a FG porous core with 2 nanocomposite as an outer layers reinforced with CNTs and their clusters[266]
Dastjerdi and Aghadavoudi [267] investigated the deflection and the stress distribution in sandwich plates presented in Fig. 42 by the first-order shear deformation theory (FSDT) founded on meshfree method and the MLS meshfree approach has been employed for approximating the displacement field. Remarkable results were recorded at CNT VF = 5% and six vacancy defects in the CNT configuration caused a rise in the sandwich plate deflection up to 41.67%.
The sandwich plate schematic with 2 defected CNT/polymer face sheets resting on two-parameter EF [267]
For analyzing the bending performance of the FG nanoplates with diverse geometries, the nonlocal strain gradient meshfree method (NSGM) based on NSGT and HSDT were emloyied. Additionally, moving kringing (MK) meshfree method was employed to resolved the deflection [268]. It has been observed from the obtained results that the displacements decreased when length to thickness ratio (l/t) increased. For the increased in n, strain gradient parameter and the NLC parameter the displacement increase and decrease. Chinh et al. [269] employed the point interpolation (PI) meshfree method which is a combined of RBF with the polynomial basic function (PBF). For the purpose on analyzing the static behavior for a sandwich beam structure with a porous core that has vary mechanical properties through the depth direction, and with a FG face sheets as illustrates in Fig. 43. The obtained results shows that the 1–8–1 beam with a higher soft-core ratio softens and has more defections than the 1–2–1 beam. Besides, as the power low index (k) goes up, the beams deflection decreases. The reason for this is due to the fact that the effective elastic modulus of FGMs improves as k rises. Additionally, by analyzing the effects of the porosity coefficient (e0) on the DL transverse displacement at the mid-span of thick beams (L/h = 5) and thin beams (L/h = 20) for three types of BCs SS, clamped-hinged (CH), and CC. It has been observed that the displacement is greatest for SS beam and least for the CC beam.
Schematic of porous sandwich beam with its coordinate and geometrical parameters [269]
2.2 Bending Analysis in Microscale
The mechanical load has been applied to the CNTR-FG cylindrical panels and caused a large deflection; for studying this deformation through geometrically NL analysis, Zhang et al. [279] applied the Kp-Ritz approach. CNTs were considered as classified in the thickness direction of the cylindrical panel, as well as effective material characteristics were evaluated using an Eshelby–Mori–Tanaka-based micromechanical model. The formulation relies on the FSDT. The RKPM approximates the two-dimensional displacement field. To enhance computational effectiveness and to prevent shear and membrane locking, the stabilized conforming nodal integration scheme has been implemented to evaluate bending stiffness of the system. Additionally, the direct nodal integration (DNI) methodology was used to determine the membrane and shear terms. It is observed from Fig. 44 that when CNT is increased by volume, the ND central deflections decrease. This phenomenon was observed also at different types of CNT panels.
The Variation in the ND central deflection with load @ fully CC BC for CNTR-FG cylindrical panels exposed uniformly distributed radial pressure @ h = 0.004 [279]
Thai et al. [280] accomplished a bending analysis of FG isotropic microplates exposed to sinusoidal distributed transverse load and a homogeneous square microplate SS exposed to sinusoidal dispersed load. In both cases results were validated and compared to insure the chosen method reliability. The results show that the meshfree method guaranty a remarkable reliability of the fundamental sizes for support domain. Besides, rising the material l/t led to reduction in the normalized displacement value. In addition, this study recommends a selection of α = 2.6 in order to yield accurate results and, less bound values in contrast with analytical solution. Nevertheless, by examining and validating the results of the isotropic microplates it has been observed that for all cases of l/h ≠ 0, the analytical TSDT model were essentially smaller than the ND displacements from the IGA- refined plate theory (RPT) model, though the displacements of these models have nearly identical results. Utilizing the same RPT model, the presented outcomes founded on moving Kriging integration meshfree technique were to some degree closer to the reference value as validated and compared. Roque et al. [281], accomplished a bending analysis for micro isotropic plates SS by using a meshfree method with a modified couple stress theory (CST), founded on FSDT of plates. The improved CST includes only one length scale parameter, which simplifies the theory owing to the fact that the single scale value is easier to establish experimentally. The FSDT bending equations were developed using a meshless approach that utilized collocation with RBF. The numerical methodology was demonstrated to be simple to apply and to produce accurate results that were in outstanding agreement with the analytical solutions. Sahmani and Safaei [282] investigated the NL bending performance of the micro/nano-beam subjected to distributed load, made of a bi-directional functionally graded material (BDFGM). The pseudo-arclength continuation method (P-ALCM) combined with GDQM applied to define the NLC strain gradient load–deflection responses of NL bending. The results observed that the bending stiffness decrease by enhancing the lateral as well as the axial material charactristics gradient indices; due to this to execute the required maximum deflection value it is necessary to applied lateral load. Figure 45. Illustrates the NLC strain gradient uniform lateral load- maximum deflection responses related to NL bending.
The NLC strain gradient uniform lateral load- maximum deflection behavior related to NL bending for CC and SS BCs [282]
2.3 Bending Analysis in Macroscale
Li et al. [285] analysis of the shear bending in large deformation stated that, in any numerical simulating of strain localization under large deformation, the meshfree techniques can be used whether the structure is 2D or 3D. Gong et al. [30] reported an investigation of varied loading circumstances and compliance reduction under stress limitations using the EFG approach with Gaussian integration. The design variables are the densities at the particles, and the meshless shape functions were used to interpolate the densities just at computational Gauss points. In order to enhance the density field's smoothness and exclude the checkerboard problem, Li et al. [286] employed the SPH to optimize linear structures (cantilever plate, CC beam, and Messerschmitt–BÖlkow–Blohm (MBB)) under various load situations. The findings demonstrated that, the superposition of the topologies achieved by exposing a single load condition and the optimized topology for several load instances are not equal. Moreover, SPH solution provides a more logical optimal topology than the EFG approach by using the FEM solution as a reference. Figure 46 illustrates the various topologies.
a The cantilever plate topologies derived from various discretization’s, b Optimal topology of the beam with C–C BCs c F1 = 2, F2 = 1 for multiple loading case and c Optimal topology of MBB beam for multiple loading cases (c) [286]
Krysl and Belytscko [242] have studied the EFG on different shapes of Kirchhoff thin plates. Findings from this analysis demonstrates that Mindlin-Reissner theory or techniques like discrete Kirchhoff theory are not necessary. The usage of a quadratic polynomial basis satisfies the consistency requirements. A baseline mesh for numerical integration is provided using a subdivision technique resembling finite elements. Lagrange multipliers impose the necessary BCs. It is demonstrated that random grid geometries, CC and SS BCs, as well as orderly and disorderly grids, may all be used to attain excellent accuracy. The optimal support, according to numerical research is around 3.9 node spacing’s. Also high-order quadrature is essential [287]. The thin plate bending issue has been resolved using the MLPG method. The method adopts a local symmetric WL and the MLS approximation to interpolate the solution parameters. On regularly formed domains (generally, spheres in three-dimensional applications) in addition to their boundaries, all integrals can be efficiently evaluated. The penalty technique was performed for enforcing the important BCs. For the purpose of demonstrating how the MLPG technique is implemented and performed, several numerical examples were looked into. The outcomes of numerical examples demonstrated that given CC and SS edge conditions, excellent accuracy can be attained for arbitrary nodal distributions. As the MLS approximation-based original solution from the current technique is of C2 type, no additional post processing is necessary to calculate the strain and stress [288]. According to Liu [289], high efficiency in analyzing static deflection of plate structure with complex shapes have been achieved by applying the EFG with MLS. The EFG method combined with HSDT has been utilized to investigate the static deflection of thin and thick laminated composite plates. Results shows that TSDT gave superior accurate outcomes than the classical or FSDT and there is no necessity for shear correction factor. In addition, for shear correction factor K = 56, the solutions obtained from FSDT were quite close to those of TSDT and ELS [290]. Dastjerdi et al. [265] have developed meshfree method combined with Reddy’s TSDT of plates, for solving the electromechanical governing coupled equation of smart multidisciplinary plate (SMP) subjected to static loads.
Peng et al. [291] accomplished the bending examination of a folded laminated plates through applying the FSDT meshfree technique. Liew et al. [292] has performed a bending analysis by using the EFG method to investigate the SMA beam pseudoelastic’s behavior. The EFG formula has been developed by introducing the MLS shape function parallel with continuum tangent stiffness tensors to an equilibrium equation that is considerable in WF. The load versus deflection resulted by bending test is illustrated in Fig. 47. A nearly identical load–deflection response in the three different nodal arrangements is observed, which leads to expose the major advantage of the meshless techniques over FEM that is, a highly smoothed and continuous discretization field caused by the weight function. As a consequence of this unique and required discretization, a reduction in demanding modal preparation and more accurate solution in small node sets or in extremely irregular nodal arrangement will be revealed.
The load versus deflection results from bending test [292]
Dastjerdi et al. [293] have applied the MLS with FSDT to scrutinize the static performances of the nanoclay-reinforced plates with piezoelectric layer as shown in Fig. 48. The static behavior is improved by applying the exfoliated morphology with nanoclay. In addition, it is observed that the plate’s deflections have reduced by increasing the input voltage owing to the mechanical loading, although a further increase may cause a reverse deflection. Likewise increasing the nanocomposites layer thickness will lead to decrease in the deflection.
The nanocomposite plate schematic structure [293]
The EFG method was employed to analyze arithmetical modeling of the size effects with the gradient elasticity formulation. The bending analysis for cantilever beam CC from one end were performed, and the load was subjected to the other free end. The results of this analysis shows that by decreasing the displacement, the stiffness ratio decreases as well but the strain ratio will increase [294]. Ferreira et al. [295] employed the RBF collocation with two higher order Zig-Zag theories for analyzing laminated plates made of three layers as shown in Fig. 49. Furthermore, the Taylor expansion up to fourth-order has been employed for the equivalent single layer (ESL) method. Figure 49 illustrates the procedure of assembling ESL approach on a k layer. After obtaining the results and validated with HSDT theories, is shows clearly that for normal and transverse shear stresses, transverse displacements and all values of a/h ratios; the FSDT is not efficient for utilizing the thick laminates. Besides, applying the SINUS-ZZ theory is more significant than z3-ZZ theory for thicker plates.
The arrangement of the ZZ assumption for the three-layered laminate and the ESL approach assemble process [295]
Ferreira et al. [296] implemented the SSDT besides the RBF collocation, aiming to analyze the laminated spherical shells geometry illustrates in Fig. 50. Besides, the CUF was employed on behalf of obtaining BCs and the equation of motion. The obtained results presented by the different set of nodal were compared with results obtained previously by FSDT, HSDT and the results shows to be almost identical.
The geometry and notations for the doubly curved multilayered shell [296]
Ferreira et al. [297], in a novel study employed the trigonometric shear deformation theory used for displaying a discretized symmetric composite sandwich plates via the MQRBF. The aim of this method was to employ the trigonometric functions over the entire thickness direction, permitting for zero transverse shear stresses at the bottom also the top of plate surface. A numerical analysis was done for three different types of plates subjected to different loading conditions; square sandwich plates made of three layers subjected to uniform load, square cross-ply laminated plate made of four layers [0°/90°/90°/0°] exposed to a sinusoidal load, as well as a thick square isotropic plate subjected to uniform load. All obtained outcomes of in-plane displacement, shear and normal stresses, were compared with previous work published by the same author, exact results, FEM results and shows a good agreement. Besides, the results prove that using these methods guarantee excellent solution for the composite plates. Neves et al.[298] implemented the quasi-three dimensional SSDT with developed employment of CUF and the base of collocation RBF for addressing the thickness stretching matter on static analysis response of FGM plates. A novel algebraic model was proposed, the bending analysis was performed on isotropic FGPs and square plate with sandwich structure filled with FGM core, and 91 mathematical layer were considered, a sinusoidal load was applied for both types of plates. The obtained results shows good agreement with previous available results found by other meshfree methods and FEM. Furthermore, the author stated that it is necessary to consider the σzz in the formulation. Ferreira et al. [299], employed the MQRBF parallel with the TSDT (for beam and plate), to analyze the static bending of beam structure and plate structure with different geometrical characterization. The BCs interpolation for beam and plate for SS and CC supported were obtained based on the MQ interpolation. The beams (symmetric laminated composite, orthotropic and isotropic) all were subjected to a uniform load and both CC and SS BCs. Where the plate made of four layers with the same thickness oriented at [0°/90°/90°/0°], was exposed to sinusoidal load (perpendicular pressure), and the BCs were set as SS. By validating the results it is observed that this method has high accuracy for composite laminates. Additionally, accurate prediction of transverse displacement, normal stresses and transverse stresses were accomplished. In a novel study published by, Neves et al. [300] the collocation RBF according to a SSDT for plates was employed, for examining the bending analysis of the FGM square plate. The development of displacement besides stresses throughout the thickness direction for different values of the exponent k was demonstrated via a 21 × 21 grid. A considerable modification in the transverse displacement and in the normal stress were observed by considering εzz ≠ 0. Ferreira et al. [301] studied the static bending deformation of a square sandwich plate made of three layers SS subjected to uniform load, and a square cross-ply laminated made of three layer [0°/90°/0°] SS exposed to sinusoidal load. The analysis was established by employing the meshfree MQRBF combined with the layerwise deformation theory. The Kansa’s unsymmetric collocation method has been implemented to predict the BCs by RBF. By comparing and validating the results it is proved that this method is highly significant and it is not required to use the shear correction factor. Ferreira et al. [302] implemented the RBF meshfree technique founded on the layerwise deformation theory for the purpose of investigating the bending performance of laminated composite spherical shell that has a layer orientations as [0°/90°/0°] and [0°/90°/90°/0°]. The spherical shell was SS, under sinusoidal vertical pressure. The results were compared with layerwise analytical, HSDTs and FEM and found to be in good agreement. Ferreira et al. [303] employed the collocation RBF method and the HSDT, for the purpose of investigating the static deformation of the SS FGPs with square geometry. The cross-validation technique has been applied for choosing the optimum values of the shape parameters in the RBF. Moreover, for assuming the effectual properties of FGM the Mori–Tanaka homogenization approach has been considered. In a recent study.
2.4 Static Analysis Buckling
2.4.1 Buckling Analysis in Nanoscale
The MLS has been applied in parallel with the modified Halpin–Tsai approach and Reddy’s TSDT, for examining the mechanical buckling for multifunctional smart sandwich plates (MFSSP) that contains a Gr nanocomposite and porous layers with two active piezoelectric layers as displayed in Fig. 51. The outcomes shows that the MFSSP’s mechanical buckling response improves significantly as the Gr VF increases. Moreover, embedding extra pores in the core layer enhance the MFSSP’s thermal stability behavior, without significantly lowering their mechanical stability responses [308].
The MFSSP’s Schematic including 2 piezoelectric layers, 2 Gr/polymer nanocomposite layers and 1 porous core resting on a two-coefficient (ks and kw) basis and exposed to mechanical loads (Fx and Fy) and electrical input (V0) [308]
To study the buckling behavior of single walled carbon nanocones (SWCNCs), Yan et al. [309] has applied a meshfree computational framework originated from the MKI. Figure 52 demonstrates the relationship between the average energy per atom relative to the compression ratios in various kinds of CNCs with identical geometrical parameters. A significant energy jump emerges at each energy curve, which corresponds to CNC buckling. A fascinating fact is that the essential axial compression displacements of all types of CNCs were recorded as 0.22 nm. Besides, consequently the corresponding axial compression ratio rises as a function of (1/cosθi). Where in CNC the θi considered as the apex angle of ith types.
The average strain energy/atom Vs. compressive ratio for CNCs with identical side edge length of 7.48 nm. [309]
The electrostatic and buckling analysis of a thin beam have been examined by using different set of uniformly distributed load and BCs (pined-pined, fixed–fixed, fixed-free, and fixed-pined) using the local point interpolation method (LPIM) method. The results show that at fixed–fixed, the critical the highest value of buckling load λ was at four different nodes [310]. A novel active sandwich plate have been developed and analyzed by Dastjerdi et al. [311] where a porous CNTRC core was implemented in a sandwich structure within two piezoelectric layers. FG patterns of CNT dispersal and porosity dispersal were investigated for the core layer. The suggested active multidisciplinary sandwich plate (AMSP) thermal as well as mechanical buckling resistances were highlighted. The governing Eigen value equations of the buckling evaluation for AMSP were found using Reddy's TSDT. Using a meshfree solution devised, the critical buckling resistance values of AMSP were derived from the GEs. In a comprehensive study framework, the impacts of porosity, CNT, and geometric dimensions on the buckling resistance of AMSPs were examined. Additionally, a remarkable increase in buckling resistance have been achieved by using the CNTs as a reinforcement in the core layer. Figure 53 presents the schematic of the sandwich structure of the piezoelectric layer under the biaxial buckling load.
The schematic of the AMSP subjected to a biaxial buckling loads [311]
The MLS approach has been employed to generate an interpolation function in order to carry out numerical forming of SWCNTs which guaranty a spontaneously satisfaction of the high order continuity, and that it is using the displacement as the only nodal degree of freedom. Besides their bending buckling response were numerically simulated [312]. The (15, 0) CNT of 12.78 nm contour’s subjected to axial compressive strain within buckling is shown in Fig. 54. Prior to buckling, there are two minor compression zones near the tube's ends. Figure 54a. As an outcome, two snap buckles formed in the two compression zones Fig. 54b. The maximal axial compressive strain was indicated as a significant inducer of kinking besides can thus be utilized as a criterion of bending buckling. Furthermore by comparing the buckling bending angles that found by MLS and by the atomistic simulation both methods gave identical values and only at very high bending angles a slight difference in the average energy per atoms has been observed between both methods. Moreover the obtained results prove that this method can be utilized to determine the mechanical deformation of SWCNTs accurately. Also, a true displayed of bending buckling can be defined.
The (15, 0) CNT of 12.78 nm contour’s subjected to axial compressive strain within buckling: a configuration just before buckling; b the final buckling pattern [312]
Liew et al. [313] have used the Galerkin approach founded on the FSDT to study stability analysis for unstiffened/stiffened of orthotropic plates (trapezoidal corrugated plate) thin and thick plates by applying elastic buckling. A uniaxial in-plane compression in y-direction was considered. The results were validated and the error was calculated. Safaei et al. [314] investigated the thermal besides mechanical buckling of light-weight polymeric nano-composite sandwich plates structure as shown if Fig. 55. The MLS meshfree method was employed to facilitate the governed buckling equations. While implementing the FSDT and TSDT for determining the sandwich plates total energy functions. It has been observed that the thermal and mechanical buckling response were enhanced by adding the CNTs into the outer layers. Besides, the critical mechanical loads were reduced by porosity, although it enhanced the thermal buckling.
Porous sandwich plate reinforced with randomly-oriented CNTs and clusters in between two outer nanocomposite layers [314]
Rafiee et al. [315] investigated the square plates (piezoelectric to CNTRC) to define the effect of applied voltage (Vs0), and the thickness ration (hp/h) on buckling temperature (Tcr = ΔTcr + T0) subjected to a uniform temperature rise (Vc* = 0.12). Furthermore, Kiani [316] performed an axial buckling analysis of magnetic affected stocky nanowires carrying electric current by using Galerkin-RKPM meshfree method. To examine the compressive critical buckling load (CBL) of doubled nanowires in magnetic field, Kiani [317] applied the RKPM meshless method and the assumed mode method (AMM) and significant results were obtained. Additionally, the four coupled equations of the CBL have been solved through RKPM. Shahmohammadi et al. [318], employed the Galerkin method for solving the extended NL differential equilibrium system equation and the compatibility equation of the doubly curved imperfect shallow shells comprised of FG-CNTRC as shown in Fig. 56, and subjected to lateral and in plane loads. Ipek et al. [319] investigated the stability behavior of a nanoplate made of FG-CNT based on Winkler EF as shown in Fig. 57, subjected to a compressive load in an elastic as well as thermal atmospheres through employing FSDT for plate. Followed by implementing the Galerkin for solving the critical biaxial compressive loads and various CNT distribution were considered and investigated.
Schematic of a FG-CNTRC doubly-curved shell [318]
The schematic of a nanoplate made of FG-CNT based on Winkler EF [319]
2.4.2 Buckling Analysis in Microscale
Thai et al. [280] presented a buckling examination of FG isotropic as well as sandwich microplates founded on refined FSDT and MCST. The Galerkin-WF generates a discrete system equation; these equations were solved by MKI meshfree method. The same standard of selecting the BCs in FEM were used due to the fact that the MKI method satisfies the function of Kroncker delta. The first six mode shapes of isotropic FG circular microplates fully CC under buckling analysis by n = 10, h/R = 0.1 and l/h = 1 are presented in Fig. 58.
The results of the first six mode shapes of isotropic FG circular microplates fully CC under buckling analysis by n = 10, h/R = 0.1 and l/h = 1 [280]
The ND CBL parameter observed to dramatically decrease; due to the increasing in thickness-to-radius ratio. On the contrary, as n levels increase, the ND CBL parameter rises as well. Furthermore, once material l/t increases, the ND CBL parameter increase. Additionally, the CBL variate from the current size-dependent model (l/h = 1) and the traditional model is essential, although the variation reduce as the thickness of the microplate increases.
For stability analysis for the random checkerboard composite cylindrical microshells, Yang et al. [253] have developed the MKI for size dependent shell model shown in Fig. 59, which includes rotational gradient tensor. The snap-through phenomena observed by tracking the post-buckling paths. The importance of the stiffening property corresponding to the rotation gradient tensor expands with increasing the VF of GNPs. Likewise, for a given value of GNPs VF, decreasing the l/w proportion of GNPs reinforcements increases the significance of couple stress size dependability in the CBL as well as critical shortening of an axially compressed cylindrical microshell. The same schematic have been employed in a further study [320], to investigate the NL stability analysis for a microshell with random reinforcements exposed to a collective load (axial and lateral). The Kriging meshfree based numerical strategy has been implemented for enforcing the essential BCs the nodes. Besides, satisfying the Kronecker delta by employing the correct MK shape function. The obtained results show that, considering adding the lateral compressive load for the collective loading condition with axial compression dominanceit was obvious that the region of snap-through expertise developed largely as well as the corresponding minimum load shifts to induced more deflection, yet a smaller shortening. Moreover, for specific VF values as well as specific particular area of graphene nanofillers (GNFs), merging of axial or lateral compressive load will cause a reduction in the critical stability values loads of collective compressed of the microshells roles of microscale-dependent gradient tensors. Figure 60, illustrates the shortening behavior of unsystematically reinforced nanocomposite shell subjected to a domination lateral load condition.
The schematic of a cylindrical microshell with GNPs reinforcement distributed in checkerboard random pattern [253]
The shortening behavior of randomly reinforced nanocomposite shell subjected to domination lateral load condition with nanofillers with several aspect ratios (l0 = l1 = l2 = 40μm,VGPL = 1.5%) [320]
Alshenawy et al. [321] applied the MK meshfree approach combined with the 3D strain gradient continuum mechanics to investigate the stability response of the thermos-electro-mechanical loaded FG piezoelectric with microshell cylindrical structure as shown in Fig. 61. The MK method has the ability to showcase the transition buckling mode containing various gradient tensors with microsize-dependent. Three different micro-strain gradient tensors were investigated. The obtained outcomes of the second NL stability under compression show that the stiffening character was significantly greater that the first one as shown in Fig. 62. However, the electric actuation role can be neglected by shifting to the postbuckling territory. Lower values for stiffening character of the 3D mechanical/electrical bifurcation load gradient tensor compared with the values obtained for the first postbuckling load.
The schematic of piezoelectric microshell thermo-electromechanical loaded FG and Chebyshev, the regular uniform and irregular the 3D nodal arrangements patterns for the MKM-based shell models [321]
The 3D mechanical/electrical load–deflection graphs of FG piezoelectric microshells with and without the micro-strain gradient tensors @ (V0 = − 500 v, ΔT = 0°C, η = 0.5) [321]
Sahmani and Safaei [282] investigated the size-dependent postbuckling performance of the micro/nano-beam subjected to axial compressive load, made of BDFGM. The P-ALCM combined with GDQM were employed to define the NLC strain gradient postbuckling configuration. The results determined that the nonlocality decreases CBL despite enhancing the impact of strain gradient size. The axial compressive load–deflection for several NLC parameter values and CC and SS BCs is presented in Fig. 63. Furthermore, at deeper into the postbuckling regime, the NLC size impact rises, resulting in a larger gap between curves. Nevertheless, a reverse observation completed on strain gradient size reliance. These expectations were identical for both CC and SS BCs. In a recent research published by, Yuan et al. [322] and focused on analyzing the NL buckling (NLB) of the hydrostatic pressurized FG composite with conical microshells structure. The GDQM of discretization patter combined with Galerkin technique were applied to define the size-dependent NL problem and different types of homogenization scheme were investigated. The outcome shows that the impact of the material property gradient index over the NL critical buckling pressure is observed to decrease for a FG composite conical microshell with a larger semi-vertex angle. Furthermore, it has been discovered that the coupled stress size reliance is more essential in NLB. Figure 64 illustrates the difference of the improved couple stress-based critical buckling pressure of FG composite conical microshells with ratio of R1/h resulted from several minor scale parameters. In both types of BCs, it has been observed that the NLB behavior with low ratio of R1/h highly effected by the couple stress size reliance. The NL lateral buckling of a cylindrical microshell has been numerically investigated based on the TSDT of a deformable shell model and modified SGT in continuum mechanics [323]. Additionally, the arithmetical solution designed for the microstructural-dependent NL has been conducted utilizing MK meshless approach, which is capable of accurately adapting the critical BCs.
The axial compressive load–deflection for different NLC parameter values; CC and SS BCs [282]
The difference of the improved couple stress-based critical buckling pressure of FG composite conical microshells with ratio of R1/h resulted from several minor scale parameters [322]
The buckling analysis of an electro-thermomechanical NL stability made by FG piezoelectric with cylindrical panels shape, subjected to compression load with SS and CC BCs has been performed [324]. The existence of the buckling mode transition in the microstructure was predicted by employing the MK meshfree method. Hence, the implementation of MK method in this article was considered as one of its types. The panel models with MK meshfree method base and various arrangement of nodes established on the Chebyshey, ordinary uniform and unordinary patterns are shown in Fig. 65. It is found that when the microsize-dependent gradient tensors were considered, the first and second bifurcation loads, in addition to the first and second minimal postbuckling loads, increase concurrently. Furthermore, due to the size dependency, the lowest NLB loads related to the first and second buckling modes were shifted toward a less panel deflection besides a greater panel shortening. Furthermore, taking into account the microscale size dependencies, the micropanel end shortening associated to the first and second bifurcation loads rises. Furthermore, the first and second loads, along with the lowest postbuckling loads, decrease as the property gradient index increases, resulting in a higher VF of PZT-5H phase when compared to PZT-4. The property gradient index value, in contrast, has a minor effect on both first and second critical shortenings in addition to shell deflections at the first and second minimal NLB points.
The panel models with MK meshfree method based and various arrangement of nodal established on the Chebyshey, regular uniform and irregular patterns [324]
Liu et al. [325] employed the MK method for the purpose of solving the size dependency problem. The study’s objective was to inspect the postbuckling as well as the NLB of a cylindrical microsized shells comprised of checkboard arbitrarily reinforced nanocomposites under combined load (lateral and axial compression). The SS and the CC BCs were taken into account in this study. The eccentric load-shortening behavior of arbitrarily reinforced microshells subjected to lateral dominated load encountered with nanofillers with diverse aspect ratios (l = 40 μm, V GPL = 1.5%) is illustrated in Fig. 66. It is shown that the stability of coupled compressed composite microshells rises by increasing the aspect ratio of nanoscale reinforcements. This outcome was achieved since the plane-type of shape allied with the GNFs creates the significant surface area specification in enhanced reinforcing of the resin matrix. An example of investigating the postbuckling in a flowchart has been presented by, Liew et al. [252] and shown in Fig. 67.
The eccentric load-shortening behavior of arbitrarily reinforced microsized shells subjected to lateral dominated load having nanofillers with diverse aspect ratios (l = 40μm, VGPL = 1.5%) [325]
The postbuckling analysis flowchart in a plate [252]
2.4.3 Buckling Analysis in Macroscale
The stability behavior of the discreetly thick plates having different geometries as presented in Fig. 68 have been investigated by using the hp-cloud method combined with Mindlin’s theory. The analysis has been accomplished by applying pure shear load, uniaxial and biaxial in-plane compressive. Findings from this analysis reveal that the local buckling is found to be coefficients with point supports and intermediate. Moreover, the hexagonal and quadrilateral plates have 25 scattered nodes, triangular plates have 15 or 25 scattered nodes, circular plates have 13 scattered nodes and the semicircular plates have 9 scattered nodes [8].
Different geometry shapes of thick plates [8]
The RBF method has been used by, Liew et al. [328] to investigate thick plates (trapezoidal and skew) subjected to a NU load in buckling analysis. The discrete system equations were solved to find the initial stresses pre-buckling. The RBF proved to be efficient in buckling analysis in these circumstances. Liu [329] employed the RPIM to analyze shear deformable laminated plates subjected to numerous BCs for static buckling. The displacement and CBLs were determined using the TSDT. The results shows that the critical buckling stresses of the three unique plate models grow as the modulus ratio (E1/E2) increases. Additionally, it has been noted that as E1/E2 rises, and the error concerned between CBLs expected with classic laminated plate theory (CLPT) as well as TSDT expand. The RKPM method has been employed to investigate the large deformation problems NL elastic materials under different case load (plain strain compression and tension and axisymmetric compression) [131]. The buckling analysis for square shape with a hole, subjected to uniaxial tension was examined. A reduction in the maximum strain value in the numerical model and an increase in the reaction force will cause an increase in the internal length parameter value. As a consequence, the stiffness ratio will reduce but the strain ratio will rise [294]. Ferreira et al. [295] employed the RBF collocation with two higher order Zig-Zag models to mimic laminated plates and four layer [0°/90°/90°0°] as well as three layer [0°/90°/0°] square cross-ply laminates SS and CC supported. Both uni-axial and bi-axial buckling load were applied. Figure 69 illustrates the first four buckling modes for bi-axial buckling load SS laminated plate of three layer [0°/90°/0°] for SINUS-ZZ formulation. The obtained results shows that for all various grids the results obtained with applying CC BCs were higher than those obtained with SS.
The first four buckling modes for bi-axial buckling load of three-layer [0°/90°/0°] SS laminated plate for SINUS-ZZ formulation [295]
Ferreira et al. [330] examined the stability of elastic plates (isotropic plates and cross-ply laminated plates) under uni-axially and bi-axially buckling load hence; the eigenvalues were considered as the buckling loads. The BCs were set as SS. The analysis was established by using the collocation RBF and TSDT of Reddy. It has been observed that results obtained by uniaxial load were much greater than those obtained by bia-axially load Fig. 70 presents the mode shapes of (45/ − 45/ − 45/45) laminates for uniaxial loading, with several a/h ratios, and N = 15. Excellent agreement was obtained by comparing the results with previous methods.
The mode shapes of (45/ − 45/ − 45/45) laminates for uniaxial loading, using various a/h ratios, and N = 15 [330]
Kumar et al. [331] employed the MQRBF for discretizing the governing differential equations (GDEs), after proposing a novel inverse hyperbolic HSDT (IHHSDT) so as to analyze the stability of porous FGM plate on the base as shown in Fig. 71. While, three different types of porosity dispersal were taken into consideration. By investigating the enhancement of number of nodes, and span to thickness ratio (a/h) of the FGM-4 plate on the buckling load parameter subjected to biaxial loading were a = b = 1 and n = 0.5. It has been observed that increasing a/h leads to decrease the buckling load parameters wither in 1, 2, and 3 dimension. As an overall observation the maximum value of buckling load parameters in all different values of a/h were recorded at node distribution 7 × 7.
The schematic of FGM plate with a rectangular geometry and porosity inside the coordinate system [331]
3 Dynamic Analysis
3.1 Free Vibration
3.1.1 Free Vibration in Nanoscale
A FV investigation for several CNT reinforced composites (CNTRC) plates in thermal atmosphere was conducted using the EFkp-Ritz technique. The analysis was run to evaluate NL vibration, and dynamic buckling. The outcomes illustrates that the fundamental ND NF's greatest and smallest values have been determined for CNTRC plates with CCCC and SFSF BCs. In addition, the maximum and minimum values of ND fundamental NF were observed for CNTRC plates with CCCC and SFSF BCs correspondingly, for various forms of CNTRC plates with varying BCs [338]. Furthermore, the dynamic vibrational analysis of nanocomposites plates reinforced using wavy CNT by employing the MLS method have been performed [339]. The MLS method has been used to examine the resonance as well as the FV of a finite length FG nanocomposites cylindrical reinforced with a wavy CNT [340]. The analysis shows that the vibrational behavior is highly influenced by the waviness of the CNTRC in cylindrical structure. As observed from Fig. 72, the lowest values of speed in radial vibration and the greatest values of amplitudes were shown in the first mode of resonance, considering the fact that the frequency of loading is improved during the second and third modes of resonance. Furthermore, cylinders with wavy CNTs exhibit greater amplitude and lower speed compared to cylinders with straight nanotubes. In addition, when CNT volume increase, the vibration amplitude will reduce, while it speeds for straight and wavy CNTRC cylinders in the all resonance modes.
Three first modes of radial vibration of the X-CNTRC cylinders with ri = 0.1, ro = 0.2, L = 0.4 m, AR = 1000 for a–c w = 0 and d–f 0.425 at mid radius [340]
Thai et al. [270] employed the MKI meshfree for the purpose of investigating the FV of nanoplate with and without hole. The obtained results show that, in the plate with complicated hole the frequencies rise with an increase in the l/t as well as the value of V ∗ CNT. Additionally, the FGX and FGO patterns gave the major besides the minor values of ND natural frequencies, correspondingly. Also, the highest frequency values were recorded. Ansari et al. [341] employed the Galerkin for examining the NL FV for the frictional viscoelastic nanobeams. To determine the FV performance of a single layered graphene sheet (SLGS), Zhang et al.[342] applied the Kp-Ritz technique combined with the NLC continuum model as an element-free computational framework (EFCF). In addition, the Eringen NLC constitutive equation joint with the classical plate theory were combined by the NLC continuum model, and it revealed to have the capability of accounting for the small scale effect. Furthermore, the Kp-Ritz method proved to be efficient for solving SGLGS vibration of NLC continuum model. Figure 73 illustrate the first eight vibration mode shapes of CCCC BCs for SLGS.
The first eight vibration mode shapes of SLGS @ CCCC BCs [342]
Ghayoumizadeh et al. [343] has applied the local integral equations (LIEs) founded on the MLPG technique to investigate the displacement field as well as finite length FG nanocomposite reinforced by CNT’s elastic wave propagation under a transient dynamic analysis. The arrangements of the aligned CNTs was deduce to differ as three types of FG dispersion along with the uniform distribution (UD) throughout the radial direction of axisymmetric reinforced cylindrical composites. A unit step function was employed to examine the function in the LWF that led to local integral equations (LIEs). The examined territory was separated into smaller sub-domains with a circular shape. Ejabati and Fallah [344] implemented the meshfree finite volume method (MFVM) so as to investigate the enhancement of air drag on dynamic analysis for the moving nanoparticles. The GEs were integrated over control volumes (CVs) boundaries and approximated by Gauss quadrature methods. A comprehensive analysis was carried out to validate the possible applicability of the MFVM to solve problems in the nanoscale state when compared to exact solution methodologies. Besides, the MLS method has been utilized to estimate the unknown field variables. Dastjerdi et al. [345] have investigated the aspect ratio and the impact of CNT waviness on the vibrational performance by applying the MLS meshfree method and the FSDT for a sandwich plate made of a pure polymeric core and the outer layer made of two CNT/polymer. Also, the Winkler spring has been used on the foundation as shown in Figs. 74 and 75.
The sandwich plate structure with 2 FG-CNTRC face sheets with the elastic basis [345]
The MLS method was employed with TSDT for the purpose of resolve the couples governing eigenvalue equations of the smart multifunctional sandwich plates proposed in Fig. 74. The results of this study shows that the implanting of pores inside the core will improve the NF of the plate owing to the significant decrease in the weight of the structure, as shown in Fig. 76 [346].
The arrangement of the layers of the smart multifunctional sandwich plate structure [346]
The relation between the essential frequency of SMSP with porosity volume for a porosity dispersions and b environment temperatures [346]
An essential study for vibration analysis of structure with nanoscale was established by Kiani [347] were he employed the RKPM meshfree technique to examine the FV of nanorod as shown in Fig. 77 with two different set of BCs (fixed–fixed and fixed-free). The AMM model was execute to check the precision of the RKPM methodology. The attained natural frequencies by the models were similar. Furthermore, the fluctuation of FFs relative to the support stiffness is supplementary visible at lower support stiffness levels. The impact of the support stiffness at one end on the essential frequency decreases as the stiffness of the support at the other end increases. It specifies that if the right side end at the nanorod have no traction, the left support stiffness on the basic f of the nanostructure extends the maximum value.
The schematic of the elastically nanorod with fixed–fixed BCs [347]
A linear FV analysis of FG-CNT of plates structure with elliptical geometry has been performed by using the MLS meshfree method with FSDT and HP [348]. The outcomes shows that the vibration characteristics can be remarkably improved by the thickness direction of the FG-CNT. Additionally, the ND NF is decreased by rising the aspect ratio. High stability in the plates can be guaranteed by increasing the CNT VF, it also guarantees higher values of ND fundamental frequencies. Figure 78 illustrated the first nine mode of vibration.
The first 9 mode shapes of FG-CNT elliptical plate SS; where a/b = 2, b/h = 100 and V.*cn = 0.17 and FGX [348]
Jankowski et al. [349] examined the FV of FGM sandwich nanobeam structure as presented in Fig. 79, which is SS by employing the Differential Quadrature Method (DQM) meshfree method. For the purpose of employing the NSGT with the basic Reddy TSDT for beam modeling assumption for developing a vibration base piezo electrically actuated smart nanoactuator model. The results of distributing the porosity in combination with external electric potential shows that increasing the L/H parameter will increase DL natural frequencies. Reducing l/t ratio will increase the DL natural frequencies.
The schematic of porous FGM sandwich nanobeam subjected to axial in-plane forces and external electric potential [349]
Mohammadimehr et al. [350] investigated the viscoelastic double bonded polymeric (DBP) nanocomposities plate reinforced by FG-CNT’s with a base design by viscoelastic foundation on MSGT to define the FV as shown in Fig. 80, by the MLS meshfree method with Navier’s to define the NF of the nanocomposites. Furthermore, different dispersal of SWCNTs were examined for instance; FG-O, FG-V, FG-X and UD. The obtained results show that, due to the fact that the distribution of FG-X SWCNTs were discrete steadily in all the territories of the polymeric nanocomposite plate as presented in Fig. 81. It recorded the highest non-dimensional natural frequency (NDNF), and the lowest value of the NDNF in the FG-O. Additionally, increasing the aspect ratio of a/b will reduce the NDNF. Another remarkable outcome was that the nanocomposite was gradually converted to beam model with decrease in its rigidity, by increasing in a/b.
The schematic of the BDP viscoelastic (nanocomposite plate structure) implanted in the visco-Pasternak foundation [350]
Different distributions of SWCNTs in the nanocomposite plate structure; a UD b FG-V c FG-X d FG-O [350]
The RKPM has been applied to define the natural flexural frequency of nanotubes, to obtain the mechanical performance of the double walled carbon nanotubes (DWCNTs) for sensing the nanoparticles at various BCs [351]. For all BCs sets that have been considered, the deviation rate of first NDNF of the nanosensor in expression of the entire number of RKPM's particles seems to reduce. As the quantity of RKPM particles rises, the predicted results derived from different NLC technique combine to the specified values for each set BCs rise. Furthermore, a free flexural vibration study of the DWCNTs with random BCs by RKPM meshfree method has been performed [352]. It has been observed that in small scale parameters the natural frequencies decrease, especially at high mode of vibration. Also, at all various kinds of applied BCs, the natural frequencies express an improvement through the surrounding matrix lateral stiffness. Additionally, by adding the clamped-free (CF) BCs, the highest enhancement of the surrounding matrix rotational stiffness on the first natural vibration variations were obtained. For analyzing the FV performance of the FG nanoplates with diverse geometries, the NLC strain gradient meshfree method founded on the NSGT and HSDT was developed. Furthermore, the MK meshfree approach was used to define the natural frequencies [268]. The results show that the natural frequencies decreased when l/t increase. For the n, as strain gradient parameter and the NLC parameter increases, the natural frequencies remains the same with these changes. Alshenawy et al. [353] developed a meshfree collocation technique to discretize the NL GEs using an amalgamated of polynomial and multiquadric basis functions (MQBF), in order to evade any potential singularity in a bridge form of an energy harvester nanobeam alongside SS and CC BCs as shown in Fig. 82. The obtained results shows that the implementation of nanoscale energy harvesters improves the magnitude of surface stress impacts on acquired voltage from 24.77 to 25.59% for SS BCs and from 26.86 to 27.38% for CC BCs. Furthermore, a greater cluster volume proportion leads to rise the required period of time. As a result, the time for the SS energy harvester at nanostructures increases from 34.50 to 35.21 μs founded on the classical model, as well as from 32.98 to 33.61 s based on the surface elastic-based model. The total required period for the CC energy harvester at nanostructures increases from 32.88 to 33.46 s using the classical model, and from 31.54 to 32.08 s using the surface elastic-based model.
The schematic of nanobeam-energy harvester subjected to time-dependent mechanical load [353]
Sobhani et al. [354] investigated the FV of combined shells (hemispherical-cylindrical-conical) made of a hybrid porous nanocomposites (nanoscale 3D Gr foams, epoxy matrix as well as macroscale carbon fiber) as shown in Fig. 83. The GDQM has been employed in order to define the governing differential equations, and HP has been used to determine the governing equation of motion. The obtained results shows that in all patterns, rising the porosity factor will leads to reduce the NF. Besides, for all she thickness’ values the hemispherical part meridional angle increments’ enhanced the NF of the combined structure under CC and FC BCs. Sobhani et al. [355] accomplished a vibrational analysis of joined conical-conical-cylindrical shell (JCCCS) by presenting 30 different geometries all made of Gr nano-platelet (GNP) as a nanocomposite (GNPN) as illustred in Fig. 84. The purpose of using the GNPN is to influence the vibrational behavior of the JCCCS. The study has been accomplished by implementing a variety of methods where the Donnell’s simplification and FSDT were employed for accessing the core relationships related with the JCCCS, and HP to define the differential motion equations (DMEs). To verify the mechanical marks; the HT combined with rule of mixture hypotheses were implemented. While the GDQM has been utilized for discretizing the DMEs. The outcomes of that study shows that, in all cases the highest NF values were recorded in the structures which has PX patterns dispersal of GNP, while the PO patterns dispersal has the minimum values of NF. Also, at greater modes the influence of GNP dispersal on the NFs is more significant. The researchers who are interested in ocean engineering generally are recommend to check the article for more details. The same methodologies GDQM, Donnell’s simplification, FSDT and HP were employed by, Sobhani et al. [356] also for FV analysis but in this article the authors were interested on studying the FG-GNP as well as FG-CNT sandwich composite JCCCS. The enhancement of various BCs sets, several geometrical parameters and material conditions on NFs were considered. Sobhani [357], performed a recent study for the purpose of investigating the vibrational performance of a marine structure in which two various full-spheroidal and semi-spheroidal shells were connected to a conical shell.
The hybrid matrix carbon fiber nano-composite combined (hemispherical-cylindrical-conical) schematic shell structure [354]
The concept of the GNPN-JCCCS schematic with its geometrical features [355]
Sobhani and Safaei [358], employed the GDQM for discretizing the equation of motion of a nanocomposite conical shells structure made of Gr oxide powder (GOP) based on Winkler as well as Winkler-Pasternak EF as shown in Fig. 85. Also, the HT and the rule of mixture techniques were utilized for discovering the nanocomposites’ operative values. While for defining the essential connections of the structure the FSDT combined with Donnell’s shell theory were used. Additionally, the equation of motion has been defined by HP. It has been observed that increasing the parameters of Winkler stiffness caused an increase in all presented models responses. Also, in case of using the Winkler-Pasternak the shell models’ responses raised owing to the fact that Pasternak layer support a hardening condition. However the hardening condition play a major role in the effect of Pasternak on the system response. Additionally it is observed that even the minimum values of NFs dropped down by changing the BCs to FF. In another recent study for the same authors, Sobhani and Safaei [359] the same methods were employed for investigating the NF of a porous nanocomposite by embedding CNT and coupled hemispherical-cylindrical marine shell-system.
The schematic of GOP nanocomposites shell based on Winkler-Pasternak EF [358]
3.1.2 Free Vibration in Microscale
A FV analysis for (homogeneous square, isotropic FG square, sandwich FG square) microplate have been done by Thai et al. [280]. The observation shows that in homogeneous square microplate, the maximum value of the first NDNF were recorded at l/h = 1.0 as well as a/h = 100 and it reached 13.6470. While for isotropic FG square, a close enough value = 13.5617 were recorded at the same l/h ratio but with different a/h = 20 and n = 0. However, for sandwich FG square the maximum value of NDNF was much more smaller compared to the other two geometrical structure were it reached 3.9325 at a/h = 10, l/h 1.0 and n = 0.5. Figure 86 illustrates the first six modes shape of dynamic performance of this geometry at the same a/h and l/h values but n = 1.
The sandwich FG square microplate first six modes shape of dynamic response of this geometry at the same a/h and l/h values but n = 1 [280]
To investigates the FV behavior of microtubules founded on an atomistic-continuum model Xiang and, Liew [371] applied a meshfree numerical scheme method with a higher-order gradient continuum constitutive. Figure 87, illustrates the microtubules structure. The effect of various length and BCs on the microtubules NF were examined, and showed that by increasing the vibration mode the predicting natural frequencies were increasing. Sahmani and Safaei [372], employed the Galerkin meshfree technique collective with the basic of GDQM and the pseudo-arc-length continuation approach, for generating a numerical solution. For the purpose of investigating the NL FV of a micro/nano-beams made of bi-directional functionally graded materials (2D-FGM) as illustrated in Fig. 88. The SS and CC BCs were considered. Figure 89 illustrates the NDNF deflection response of the 2D-FGM micro/nano-beams effected by the NLC size effect. The obtained results shows that the NL frequency to linear frequency (WNL/WL) ratio increased at an extreme specific value of the deflection by the NLC size effect. Although (WNL/WL) ratio decrease by the small scale effect of the strain gradient. This configuration considered as more noticeable for greater amount of the maximum deflection. Yuan et al. [373] published an extensive study for a NL FV analysis of a conical microshells made of FGM in-plane with composite material in micro/nanoscale as shown in Fig. 90. The Galerikin method combined with the GDQM were used to obtain a mathematical model for obtaining the NL modified couple stress-based on the frequency of the structure.
a The microtubules structure and its subunits and b the homogenized model of protofilaments fraction [371]
The schematic of a 2D-FGM micro/nano-beam [372]
The DL NL frequency-deflection response of the 2D-FGM micro/nano-beams effected by the NLC size effect [372]
The schematic of a FG composite conical microshell and the associated geometrical parameters [373]
3.1.3 Free Vibration in Macroscale
According to Liew et al. [334], the RKPM with FSDT of the Mindlin/Reissner type have been exposed to be an exact and effective tool for finding the arithmetical solutions for FV of plates with diverse geometrical shapes and BCs. The lower order vibration modal shapes of a CC circular Mindlin/Reissner plate are presented in Fig. 91.
FV modal shapes of a CC circular Mindlin/Reissner plate. a first Mode; b second Mode; c fourth Mode; d sixth Mode; e seventh Mode; f ninth Mode; g eleventh Mode; h thirteenth Mode; i fifteenth Mode; j seventeenth Mode [334]
The goal of Zheng et al. [379] was identify the FV continuum structures with the EFG and SIMP and to maximize the essential eigenvalue while taking the nodal density into consideration as a design variable. Gong et al. [380] have used the modal topology optimization method to maximize the first order NF by applying the EFG with SIMP technique. Wu and Chiu [381] established RMVT-based MC and EFG methods for quasi-3D FV analysis of multilayered composite and FGM plates. The essential field parameters were interpolated utilizing DRK interpolants, which was generated using uniformly or NU distributed nodes, in these approaches. The exhibited examples show that the present MC and EFG solutions converge faster and have a high correlation with the earlier published approximate and precise 3D solutions of FGM sandwich plates. Where the highest order of basic functions (n) is suggested to be n = 3, the number of uniformly distributed nodes (Np) is suggested to be Np = 21, as well as the radius of the influence zone (a) is suggested to be (a) is assumed to be 3.1 times the distance among the adjacent nodes. The impact of aspect ratio, shear deformations, layer thickness ratios, and material characteristics gradient indices on frequency characteristics and their associated through-thickness dispersion of various modal field variables are investigated using a parametric approach. Additionally, the illustrative examples demonstrate that as the material characteristic gradient index grows, the variations of the through-thickness dispersion of different variables between FGM plates (kp ≠ 0) and homogeneous plates (kp = 0) becomes progressively important. Likewise, the free-traction conditions on the sandwich plate's lateral surfaces, in addition to the displacement and transverse stress continuity conditions at the interfaces among neighboring layers, were precisely achieved. Figure 92 demonstrates the varying through-thickness dispersals of several modal field parameters with the material characteristics index, kp, where kp = 0, 1, and 10. It is demonstrated that the dispersals of in-plane and out-of-plane variables for the homogeneous plate, where kp = 0, are higher-order polynomial fluctuations across the thickness coordinate, whereas there are higher-order polynomial variations for the FGM sandwich plate, where kp = 1 and 10. Also, The free-traction features of the sandwich plate's lateral surfaces, in addition to the displacement as well as transverse stress continuity constraints at the interfaces between nearby layers, were perfectly met.
The various modal field variables of the FGM sandwich plates' through-thickness distributions [381]
Ferreira et al. [382] have used the MQRBF to examine the symmetric laminated composite plates' FVs. The set of motion equations for the FSDT define an eigen-problem that can be solved using a variety of methods. The output results are in excellent agreement when compared to previously published results of FEM or meshfree techniques. The results for CC skew plates are always greater than those for supported plates because; of the collocation technique around the plate corners. This approach is a straightforward yet effective substitute for conventional meshless or FEM for analyzing the laminated composite plates for FV.
The meshless local collocation technique founded on a thin plate spline RBF has been employed by Xiang and Kang [383], to evaluate the FV behavior of laminated composite plates. The FSDT GEs and BCs have been discretized utilizing the local thin plate spline collocation approach. The MRPC technique has been used for developing a mathematical model as well as examining the FV of isotropic plates (square, circular). However, the results illustrate that at n = 9, the highest amount of ND frequencies were gained at mode 8 for square and circular with h/b = 0.1, and equal to 10.9057 and 125.45 respectively [384]. These method provides an opportunity to investigate with the optimum order number to estimate the structural issue less than inquiry and absence of any shear correction factor. Moreover, the FV analysis of symmetric composite plates by meshless MQRBF and trigonometric layerwise deformation theory discretized have performed. Besides, exceptionally solutions for the FV of composite and sandwich plates was provided [68]. The frequency analysis for five different thin plate’s geometry were processed by using EFG with MLS meshfree methods [289]. A high efficiency analysis with significant convergence were observed. Additionally, the maximum value of NF of lateral FV in all different shapes of plates were recorded at the 10th mode. Dai et al. [290], used the HSDT with EFG to investigate the NF and static deflection of thin and thick laminated composite plates. A penalty technique execute fundamental BCs by numerous arithmetical models for laminates with diverse a/h, material coefficients, BCs or ply angles. The outcomes shows that the first three frequencies corresponding to the rigid displacements were = 0.0, for the thick laminates CPT constantly over predicting natural frequencies [290].
It has been discovered that the superposition-Galerkin approach can be employed to accurately analyze the FV of a rectangular plate that is totally free. A general analysis of all probable modes of the totally free plate is conceivable by coupling the given sets of building blocks with related sets driven along the edges ŋ = 0 and ξ = 0. To obtain the extension in capabilities as most of the components of the larger eigenvalue matrix linked to the described problem, some additional research is required. One might be concerned about the requirement that pairs of building blocks can be used instead of individual ones in the new method for thin plates. On the condition of the Mindlin plate, a set of three blocks was used in place of the earlier procedure's individual blocks [385]. For the FV analysis of composite laminates with complex plate shapes, the EFG approach has been developed. Several numerical examples were provided and compared to previous findings to demonstrate the efficacy of this method in determining the natural frequencies of laminates. A good convergence for the EFG can be observed. For plates with regular shapes, there have been some significant agreements with other numerical results. Besides, it was investigated to show how the fiber orientation affected the frequencies for laminates with various forms and BCs. The outcomes shows that the EFG technique is reliable for the purpose of evaluating the FVs of laminated composite plates with complex shape [386].
Jayakumar et al. [387] have investigated the NL FV analysis of plate structure (SS piezo-laminated) with material properties and arbitrary actuation electric potential difference by using the Galerkin meshfree method. Furthermore the same methodology was utilized for determining the linear/NL FVs for plates made of composite and fiber metal laminated in a close form solution [388]. Besides, Morozov et al. [389] applied the same technique to analyze the vibrations for fully CC sandwich plate. The arithmetical solution, analytical approach, as well as FEA were used to investigate the impact of panel size, elastic characteristics, and face and core densities on the basic frequency. The research reveals that the critical solution presented in that work can calculate the first NF with appropriate precision. A novel plate structure made of sandwich panels with geometry of square honeycomb cores have been analyzed in a purpose of studying FV, bending and buckling. Also, the HP parallel with Galerkin method were performed [390]. The meshless MLPG has been applied with the HSDT as well as normal deformable plate theory to examine the force and FV of a thick plates structure with rectangular shapes by Qian et al. [391].
The same structure has been used by Safaei et al. [392], to investigate the FV response by MLS under steady state thermal gradient loads. Figure 93, presents the impact of cluster volume and CNTs fraction exponent to the sandwich plate frequency at thermal gradient loads of Tt = 400 K and Tb = 300 K. As a consequence of rising the volume exponent of CNTs to n = 1, an instantaneous increase in (FPs) was noticed, followed by dramatically decrease. Furthermore, the schematic is shown in Fig. 94.
The central point timeline vibrations of sandwich plates with n = 1, μ = η = 1 (fully-scattered), ℎ/a = 0.1, hf/h = 0.1 and ks = kw = 0 for a frmax = 0,02 and b frmax = 0.05 [392]
The schematic of sandwich plate on 2-parameter EF [392]
Gu and Liu [310] examined the FV of a thin beam by applying the LPIM and Euler–Bernoulli (EB) beam as an analytical solution. FV analysis was performed with diverse BCs (pined-pined, fixed–fixed, fixed-free, and fixed-pined) at 21 uniformly distributed nodes as presented in Fig. 95. In addition, the time step and error % were considered in the force vibration analysis.
FV modes of thin beams with different BCs: a pinned–pinned; b fixed–pinned; c fixed–free; d fixed–fixed [310]
The local radial point interpolation (LRPIM) has been utilized by Liu and Gu [393], to study the boundary value problems for FV analysis purpose in 2D solids beam. The results show that these method is capable for use in both cases of regular and irregular nodal distributions. According to Liu et al. [394], it is recommended to use α = 2 in huge local support domains. Have used the point PI method which is a part of LRPIM meshfree approach to study the mode shape frequency and the static load of a 2D piezoelectric structure (disk and beam). However, a set of scattered nodes have been utilized to represent the problem boundaries and domain. Besides, the shape functions were derived founded on the PI approach. Both cases of ordinary and unordinary nodal distributions show good results. A FV analysis for a highly skewed plates have been investigated by Wu et al. [395], by applying the least-squares finite difference method (LSFD). Zhu and Liew [396] performed the FV analysis of moderately thick graded plates, by applying the local Kriging meshfree method. Zhang et al. [397] investigated the FV of a triangular CNT reinforced composite plates under plane stresses by employing the FSDT element-free technique. The various BCs and angles α for the mode shapes of the composite plate’s reinforcement with FG-X-CNT were calculated and are shown in Fig. 96. Dastjerdi et al. [398] accomplished dynamic analysis (FV and stress wave), by subjecting the SWCNTs to impact case load using MLS. The NF characterization and stress wave propagation behavior caused by the improvement of type of dispersal and VFs of CNT and cylinder thickness. Figure 97 demonstrated that the velocity of stress wave propagation is nearly identical for diverse forms of CNT dispersion. The stress amplitudes in the FG-V and FG-X kinds are larger compared to the other types. It has been shown that the UD and FG-X types act almost identically, yet the amplitude of axial and shear stresses rises in the FG type.
The vibrational mode shapes of FG-X CNT-reinforced composite triangular plate with t/b = 0.01, α = 30° and k/kcr = − 0.5 @ boundary condition CCC [397]
The Time history of a Radial stress, b hoop stress, c axial stress, d shear stress and e radial vibration at inner radius of first model of CNTRC cylinders for different kind of CNT distribution [398]
Liu [329] used the RPIM based on radial basis functions to analyze shear deformable laminated plates subjected to various BCs for FV. The natural frequencies was determined using TSDT. Based on a set of distributed nodes in the problem territory, the technique approximates field variables. The enhancement of the aspect ratio, (E1/E2), a/h, and the BCs, on NF were studied. The results indicate the FFs rises as the a/h rises. Additionally, it is observed that as the a/h of laminates increases, the disparity between the natural frequencies identified by CLPT, FSDT, and TSDT drops. The stabilization and vibrational persistent integration have been demonstrated to be crucial for achieving optimal alignment for nodal integration, which is an intriguing outcome recorded by Hillman and Chen [332]. Also, the eigenvalue evaluation demonstrated that the addition of vibrational consistent test functions did not sacrifice stability, and CPU time measurements demonstrated that the addition of vibrational consistent integration did not significantly reduce efficiency. The effectiveness of the suggested strategy in offering stable solutions for significant deformation problems was also demonstrated in a number of examples. For the purpose of developing accelerated, convergent and stable nodal integration for RKPM of plate and beam structures by RKPM. Figure 98 shows the eigenvalues of stiffness matrices. Figure 98a illustrates that the first lowest 30 modes in both cases fully integrated FEM agrees fairly well. However, when the initial smallest 200 modes were considered, VC-NSNI exhibits significantly greater agreement as in Fig. 98b.
Eigenvalues of stiffness matrices: a 30 lowest modes and b 200 lowest modes [332]
A FV simulation for the nano clay-reinforced plates with piezoelectric layer has been performed by employing the MLS meshfree method. A substantial rise in amplitude of vibration has been noticed by reducing the nanocomposites core thickness. Furthermore, the vibrational amplitude decreases by increasing the thickness of piezoelectric layer. However, there is no evidence that changing the thickness structure will affect vibrational frequency [293]. Figure 99 illustrates the behavior of transvers vibration besides time history of the generated voltage effected by layer. Givli et al. [399] considered the FV for a sandwich panels (delaminated unidirectional with transvers flexible core). The numerical analysis was based on HP and HSDT. The results of NF were validated with previous data found by Galerkin meshfree method and found to be almost identical. Wang et al. [400] applied the MLS and Kp-Ritz meshfree methods for multiscale NL vibrational analysis of FG-base PMMA/CNT composites plate as presented in Fig. 100. The macroscale simulation reveals that the placement of CNTs have a substantial impact on the NL vibration behavior of FG-based composite plates. The established multiscale technique offers an effective framework for exploring the macroscopic performance of composite structures while taking mechanical response from microscopic constituents into consideration.
Time history of central point a deflections b generated voltages of smart sandwich plates subjected to periodic mechanical load with ωt = 500 Hz [293]
An Outline of the SWCN- reinforced PMMA composite at various length scales; a the atomistic models of the PMMA polymer chain and the SWCNT segment; b the molecular model of the PMMA/SWCNT nanocomposite; c the equivalent nanocomposite; and d the macroscale PMMA/CNT composite plate [400]
Ferreira et al. [295] employed the RBF collocation with two higher order Zig-Zag theories for analyzing the FV NF of the [0°/90°/90°0°] square cross-ply laminates SS. The obtained results shows that at all different grid values the z3-ZZ formulation is more efficient than SINUS-ZZ formulation for v23 = 0.25 and more close to previous studies by exact and other meshfree methods. Neves et al.[298], implemented the quasi-three dimensional SSDT with developed implementation of CUF and the base of collocation RBF in order to solve the thickness stretching difficulty on static analysis of FGM plates. The FV analysis was performed on isotropic FGPs SS. For the purpose of achieving a comparable material properties, the Mori–Tanaka scheme was employed to compare the results. The obtained results shows good agreement with previous available results found by other meshfree methods and analytical solution. At εzz ≠ 0 results noticed to match the exact solution. This model considered as a comprehensive model to be considered in both static and vibration analysis of thin and thick FGM plates for BCs and equation of motion equations. Ferreira et al. [296] implemented the SSDT and the RBF collocation, for analyzing the FV behavior of the laminated spherical and cylindrical shells. The ND natural frequencies were found for various cross-ply spherical shells and compared with previous analytical solution results with excellent agreement. Tornabene et al. [401] used the RBF meshfree approach on a general higher-order equivalent single layer (GHESL) formulation to explore the FV behavior of a doubly-curved laminated composite in panels and shells. The various geometrical structures were defined by using the differential geometry (DG) for singly-curved, degenerate shells, and doubly-curved. RBF and the placement of pointes on the provided computational domain influence the shape parameters. Nonetheless, changing the placements of the points will modify the shape parameters. Furthermore, after the shape parameters have been set for a specific grid distribution, they can be rounded off and utilized for any type of construction. This is an important aspect because when a fixed parameter is utilized, the RBF methodology transforms into a "parameter free" numerical method. Besides, the maximum natural frequencies were recorded at the 10th mode in all methods which have been applied and it is strongly agreed with compared data. The schematic description of the doubly-curved shell and the co-ordinate system is presented in Figs. 101 and 102 illustrates the various shell and panel analyzed structures and the relative RBF discretization for computer applications.
The schematic description of the doubly-curved shell and the co-ordinate system [401]
The various shell and panel analyzed structures and the relative RBF discretization for computer applications [401]
Ferreira et al. [402] investigated the FV performance of the symmetric laminated composite plates (rectangular and square), by an advanced numerical scheme by combining the PS methods with the collocation RBF, where the enhancement of various t/l, thickness to side ratio (h/a) on NDNF were concerned. An earlier convergence observed at CC BCs at high t/b ratio regardless of a/b ratios. The outcomes indicated that increasing the a/h, will reduce the (FPs). These statistics additionally demonstrate that increasing the thickness ratio leads to a reduction in (FPs). The reason of this phenomena is that; the shear deformation as well as rotating inertia. These impacts are greater in higher modes. More significant frequencies were obtained with CCCC borders similarly, to what was expected because stronger constraints result in higher frequencies. Ferreira et al. [301] considered the natural frequencies of a composites laminated plate [0°/90°/90°/0°] and isotropic plate. Both plates were square shape and SS. The analysis was established by employing the meshfree MQRBF combined with the layerwise deformation theory. The obtained results prove that the presented method is significant in predicting the NF when compared with the 3D method and with superior formulations. Additionally, the layerwise formulations may handle the deformation behavior of some laminates, specifically sandwich laminates with very diverse core and skin materials more significantly. Ferreira et al. [302] implemented the RBF meshfree technique founded on the layerwise deformation theory to investigate the FV behavior of laminated cross-ply cylindrical and spherical shells orientated as [0°/90°/0°] and [0°/90°/90°/0°]. The NF results were compared with HSDT and FSDT and noticed to have outstanding agreement with analytical solution. Kumar et al. [331], the MQRBF meshfree method has been employed for discretization in an analysis of FV behavior of porous FGM plate on the basis. By investigating the enhancement of number of nodes, and a/h of the FGM-2 plate with SS BCs on the NF parameter were a = b = n = 1. It has been observed that increasing a/h to result in decreasing the NF parameters wither in 1, 2, and 3 dimension. As an overall observation the maximum value of NF parameters in all different values of a/h were recorded at node distribution 9 × 9. Besides, at a/h = 50 and a/h = 100 almost identical NF parameters were observed. Recently, Kim et al. [403], employed the meshfree shape function (MSF) which is categorized under the MLS method, in order to approximate the nodal displacement. The objective of this work was to inspect the FV behavior of a composite laminated double-plated system (CLDPS) as shown in Fig. 103 and it is considered as the first study of its type.
The schematic of the CLDPS; a BCs of a single CLP, b geometry of single CLP, c CLDPS bonded by a number of elastic springs [403]
3.2 Force Vibration
3.2.1 Force Vibration in Nanoscale
Safaei et al. [412], investigate the force vibrational response by MLS under steady state thermal gradient loads. It is observed from Fig. 104, that the thermal gradient loads enhance the initial values of the vibrational (static deflection) along with the vibrational amplitudes of the sandwich plates. A significant improvement in the structure stiffness, increasing number of CNTs into the face sheets led to having slighter amount of vibrational amplitude. Moreover, the schematic is shown in Fig. 105. Gu [310], investigated the force vibration of a thin beam by applying the LPIM and EB beam as an analytical solution, where the LPIM is usually used for analyzing the thin beam with pinned–pinned BCs forced vibration. The results after measuring the error shows great consistency with the systematic results, where significant LPIM stability was observed as well.
The central point timeline vibrations of sandwich plates with n = 1, μ = η = 1 (fully-scattered), ℎ/a = 0.1, hf/h = 0.1 and ks = kw = 0 for a frmax = 0,02 and b frmax = 0.05 [412]
The schematic of sandwich plate on 2 parameter EF with 2 nanocomposite face sheets reinforced with CNTs [412]
In order to increase the stiffness of CNTRC plates, it has been determined that reinforcements applied to the top and bottom of the structure are more durable in comparison to those placed in the mid-plane. Figure 106 display how the four sides of various types of CNTRC plates are fully CC while altering the plate aspect ratio (ß = a/b) from 1.0 to 2.0. The CNT VF V*CNT is considered, and the plate width-to-thickness ratio (b/t) is specified to be b/h = 10. It is observed that when ß of different types of FG-CNT and for UN-CNTRC plates rise, the ND FFs reduced. Also, variations in the plate aspect ratio enhance the distribution types of CNTs in the plate [338]. Sahmani and Safaei [413] employed GDQM combined with the Galerkin method to determine the surface elastic based NL frequencies of the conical nanoshell made by FG composites. The meshfree methods were employed after constructed the NC shell model by integrating the Gurtin–Murdoch elasticity theory into a HSDT of conical shell. It was established that decreasing shell thickness, thereby raising the surface to volume ratio, raises the gap in the middle of the surface elastic-based and classical FR, indicating that the increase in the gap becomes more important as the amplitude of the NL vibrations rises. Furthermore, raising the material characteristic gradient indices increased the importance of surface stress impact on NL FV in the structure itself [414].
The NDNF variation of various types of CNTRC plate’s four edges fully CC versus plate aspect ratio [338]
3.2.2 Force Vibration in Microscale
Sahmani and Safaei [419] implemented and solved the GDQM and Galerkin meshfree methods to obtain the amplitude respond related to NL primary resonance of the 2D-FGM micro/nano-beams, as well as the NL NLC strain gradient FR. Hence, for validation issue the strain gradient size dependency was ignored. Both SS and CC were considered as a BCs. Figure 107 depicts the effect of strain gradient size on the DL FR associated to the NL basic resonance of the 2D-FGM micro/nano-beams. It shows that the associated excitation frequency and the oscillation amplitude peak both increases by the locality. Nevertheless, the strain gradient size effect has a reverse impact and cause to reduce the associated excitation frequency and the oscillation amplitude peak. These remarks owing to the drop in the stiffness of the structure produced by the NL size influence, which results in greater geometrical NL. According to Yuan et al. [420] the GDQM and Galerkin meshfree methods can be combined and used to predict the NLC NST-based dynamic features of the FGM composite shortened conical microshell integrated by a magnetostrictive facesheet. Wang et al. [421] employed the meshfree Local Kriging method to investigate the behaviors of microelectromechanical system (MEMS) devices, exposed to a NL loading. The Kriging interpolation technique is used to attain the form function, which possesses partition of unity and delta function features. The GEs for MEMS devices were derived in their local Petrov–Galerkin-WF. The Local Kriging method was designed to evaluate the performance of micro-switches and micro-tweezers, which are commonly employed in MEMS devices. Also, Wang et al. [422], shows interest in studying the MEMS, but in this study the analysis was accomplished by implementing the meshfree point weighted least-squares method (PWLS). Besides, the trial function has been generated based on the field nodes. However, the PWLS guaranty very high stability of the MEMS system. The Galerkin method has been employed by Rafiee et al. [315] for the purpose of linear/NL dynamic stability examination for plate structure exposing in-plane load harmonic applied voltage and thermal electrical loading. As a findings, the VF in CNTs increase the vibrational amplitude will reduce. Besides, the size of stability areas grows as the volume percentage of CNTs increases. It has been observed that the SWCNT distribution patterns have a considerable impact on the size of stability areas.
The impact of the strain gradient size that influence the DL FR related to the NL primary resonance of the 2D-FGM micro/nano-beams [419]
3.2.3 Force Vibration in Macroscale
Muthurajan et al. [423] studied the NL vibration analysis of primarily stressed thin laminated rectangular plates on EF by the EFG. Results of the CC square and SS were provided. Additionally, by applying the lateral and compressive in plane forces increasing the amplitude; will enhance the NL frequency. Chen et al. [424] have used the Galerkin technique to investigate the NL vibration of initially stressed plates. The outcomes demonstrate that the NL vibration FR depend on; initial stress, Poisson ratio, vibration amplitude, and plate theory. According to the author’s findings, there are some unexpected discrepancies across the numerous plate theories, indicating that transverse normal strain, NU shear stress, as well as starting stress have a significant impact on how plates vibrate when subjected to NL vibration. Chen et al. [425] have performed a NL vibrational analysis for plates (orthotropic) through initial stresses that’s considered as an amalgamation of extensional stresses and pure bending stresses. The EF (Winkler and Pasternak) were examined with two parameters. It has been discovered that the NL foundation stiffness or compressive stress increase the frequency ratio value, while the EF stiffness or tensile stress decreases. The frequency ratio increases as the E1/E2 or biaxial stress ratio increases. A plate with initial compressive stress resting on a low stiffness basis and high E1/E2 exhibits NL vibration behavior. Under high foundation stiffness, increasing the bending stress ratio has a much smaller effect on the frequency ratio. Furthermore, the plate frequency ratio rises together with the vibration amplitude once the foundation stiffness seems to be low. A large foundation stiffness can minimize the plate's NL vibration behaviors. The EFG method have been implemented for discretizing the equation of motion after obtaining the NL elastic region characterization in solid by performing the preisach-may-eroyz space (PM space), to investigate the NL wave propagation of a plate. It has been uniformly distributed and harmonic force with magnitude 3 kHz were observed. The elastic modulus shows a decrease by increasing the harmonic load. While in plane stress case, the harmonic amplitude was observed to increase by increasing the fundamental amplitude acceleration [426]. Liu et al. [427] investigated the dynamic behavior of a corrugated cylindrical shells exposing NL low-velocity impact as shown in Fig. 108. An innovative computational technique for NL impact force in the local contact area has been utilized. In order to start, a novel comparable shell model for corrugated cylindrical shells was developed. The GEs were then constructed employing the Love thin shell theory and HP, subsequently converted into ordinary differential equations using the Galerkin method and for discretization, following a numerical solutions which was produced using the Duhamel integration and small-time increment technique. The findings show that this novel method agrees with results obtained previously. There was some variation due to the varied shell/plate theories and methodologies for calculating contact force. Besides, the convergence findings were received in all time histories when the wave numbers rise to m = 50 and n = 50, respectively. Hence, the shortened coefficients was used. Various corrugation types have a minor effect on contact time and extreme impact force. However, it is observed that the corrugation may overall improve shell stiffness, and the corrugated shell has a slighter transverse central displacement compared with the noncorrugated shell. Additionally, the radial displacement of the round corrugated cylindrical shell is least significant. This means that in all circumstances, the round corrugated cylindrical shell has a superior energy absorption effect and the greatest potential for enhancing impact resistance as shown in Fig. 109.
The Schematic of a corrugated cylindrical shells exposed to NL low-velocity impact [427]
The impact of corrugation type on impact response of the corrugated cylindrical shell [427]
Bui et al. [410] applied the RPIM meshfree technique to investigates the frequency and the transvers response of the sandwich beam with inhomogeneous FG core subjected to dynamic load. The obtained results show that, increasing the core stiffness will enhance the NF to increase. As long as a sufficiently small time steps were applied such as Δt ≤ 2 × 10−4, stable results were guaranty additionally, when great time-steps are considered through time the displacement amplitude will decay. Moreover, in case of reducing number of nodes the obtained accuracy of the computed frequency is reduced.
3.3 Other Types of Dynamics Analysis
The vibration of a non-homogeneous (NH) visco-elastic rectangular plate with a linearly changeable thickness exposing a thermal gradient has been studied. Galerkin's technique has been utilized to solve the governing differential equation of motion. Time period, deflection, and logarithmic decrement at various points for a NH visco-elastic rectangular plate CC on two parallel edges and SS on the remaining two edges were calculated for various values of thermal gradients, NH constant, taper constant, and aspect ratio [428]. In a magnetic field, the vibrational response of bilayer graphene sheets (BLGSs) was investigated by Zhang et al. [429]. The kp-Ritz technique was used for the purpose of obtaining the PDE. The enhancement of aspect ratio, BCs, side length, magnetic parameters, and NLC parameters on vibrational behavior (frequency) were also investigated. Figure 110 depicts a schematic of BLGSs exposed to an in-plane magnetic field. The obtained results reveal that the magnetic field has a considerable effect on the second interlayer vibrational mode frequency, and as a result, at high magnetic fields, the first interlayer vibration mode frequency will eventually become asymptotic to the second interlayer vibration mode frequency. Besides, the NLC parameters, aspect ratio, BCs and the side length were found to have no influence on the second interlayer vibration mode frequency.
The schematic of the BLGSs subjected to in-plane magnetic field [429]
The same method has been used by the same author in a further study to investigate the Impact of powder size on the crystallization response through laser solid forming Zr55Cu30Al10Ni5 bulk amorphous alloy [430]. Lei et al. [431] applied the kp-Ritz to perform elastodynamic analysis for CNT reinforced FGPs. The influence of CNT b/t, VF, aspect ratio, type of load type of distribution of CNT, and BCs on the dynamic responses of CNTR-FG-plates were investigated. Figure 111 presents the VF of CNTs as VCNT = 0.17 for the center vertical displacement time-profile curves of several types of FG-CNTR plates exposed to a transverse abrupt dynamic load. The results reveal that the CNTR plate's stiffness is excellent at high CNT VFs, and rising the CNT VF for UD and the other three FG-CNTR plates reduced the central vertical displacement.
The Central vertical displacement time–history for FG-O CNTR-FG plate with different CNT VFs [431]
Furthermore, the Galerkin technique was employed by Li et al. [432], to examine the the dynamic shear propagation behavior in plates under symmetric impact loads. The results show that by increasing the impact velocity, the characterization of the dynamic shear band will variate. In addition, the front region of shear band tip express to have high strain rate region. Medyanik et al. [433], presented an analysis of initiation and the propagation of the dynamic adiabatic shear bands (ASBs) for 4340 steel and OFHC copper. To examine and to confirm the suggested criterion by investigativing the propagation, formation and the post bifurcation of the ASBs, large scale meshfree simulations were conducted. Kiani et al. [434], have employed the RKPM method parallel with HP to find the SWCNT-based sensor discrete form of the equation of motion. The study has focused on analyzing the potential application of the SWCNT as a nano-sensors with various BCs. The schematic of equivalent continuum structure (ECS) is presented in Fig. 112.
The Schematic illustration of ECS with attached nanoparticles [434]
Recently Almasi et al. [435], implement the ISPH to examine the behaviors of various multi-phase flow system. Dastjerdi and Behdinan [436], employed the MLS for solving dynamic equation of the bridge type of the piezoelectric energy harvesters (PEH) with a multifunctional nanocomposite substrate. The schematic and cross sectional of the PEH bridge type are illustrated in Fig. 113. The results show that increasing the CNT volume will slightly have some effects on the dynamic responses, where the oscillation frequency will enhance and the amplitude will decrease. Liu et al. [437] employed the meshfree particle method (MP) to investigate the high velocity impact of micron particles at different speeds with different angles. Wu et al. [438] employed the strain-morphed NLC meshless method combined with interactive particle enrichment algorithm for the purpose of analyzing the dynamic ductile fracture. To analyze the behavior of an active constrained layered damping (ACLD) treatments on a smart laminated composite beams Sahoo and Ray [439], employed the EFG meshfree method to drive the meshless model of the system. Error analysis was carried out and the NF was determined.A further study has been performed by, Civalek [440] were the DQM combined with discrete singular convolution (DSC) have been employed to resolve the time wise integration in a NL dynamic as well as static responses of a shallow spherical sells with a Winkler-Pasternak EF as shown in Fig. 114. The obtained results shows that both static and dynamic response highly affected by the foundations’ shear parameters. Besides, the shell curvature as well as the damping coefficient of the shell have a remarkable impact on the deflection results. Generally, increasing the foundation parameters will cause a reduction in the static as well as dynamic responses. That study prove that the combination of DSC-DQM counts as an accurate and effective approach for solving the NL static as well as dynamic analyses of the shell assembly.
The schematic and cross sectional (rectangular nanocomposite porous plate) of the PEH bridge type, with 2 SS edges and located between 2 piezoceramic faces [436]
The schematic of shallow spherical shell with EF and it is coordinate system [440]
4 Discussion
Meshfree or meshless methods are a group of method mainly based on Lagrange equation and have been developed to overcome the drawbacks of FEM and other traditional methods. The main concept of meshfree methods is that the functional evaluation is based on the nearby samples or nodes and it does not require net-mesh or remeshing where the nodes or points can be arranged uniformly or NU. These group of method can be utilized to solve and to handle the discretization of the linear and NL, partial differential equation (PDE), ordinary differential equation (ODE), local and NLC in many applications rather than dynamic and static analysis. The plate’s structure in different geometry have been favorable for studies and investigations in static and dynamic analysis. However, in theoretical terms, a meshfree method can be generated by combining function approximation, gradient approximation, and formulation techniques, resulting in a matrix that consist of a huge number of alternative methods.
Enforcement of essential BCs has been considered as a critical issue to investigate in many studies. Furthermore, meshfree modeling only requires a collection of unorganized points containing the domain of interest. Because of their simple topological data structures, meshfree particle techniques provide straightforward adaptable refinement, parallelization, and flexible interpolation in a deformed field. This class of methods has a high potential for meeting the requirements of modern software, such as error estimators, multiresolution analysis, HP adaptively, edge detection, and sampling approximations. Although great progress has been made in the last decade, there are still numerous responsibilities and challenges ahead, and optimizing the CPU is still considered a difficulty.
Among these challenges are the cost-effective meshfree Galerkin technique, scaled execution of important BCs, exact nodal integration strategies, and stabilization schemes that accommodate discretized WF and collocated strong form formulations. Although the essential for sensitivity filtering is implemented by employing the Galerkin based methods. Meshfree particle approaches are expected to play a large role in next generation computational mesomechanics, or computational micro-mechanics, which is an integral part of nanotechnology and supercomputing technology. Nonetheless, most meshfree techniques have a significant computational cost, which makes large-scale investigations prohibitively expensive. Although the hybrid strategy combines the beneficial features of mesh based and meshfree systems, its high level of complexity and computational cost remain unresolved challenges. It is critical to examine the use of high performance computing (HPC) in the development of a multi-physics, multi-scale model. The MPM and SPH is able to overcome the huge mesh deformation. Due to this advantage both methods are considered as the most important methods. But there are still some restrictions on SPH method which inhibit its usage in indicating the rigid body motion accurately. To overcome this problem, researchers have developed a corrective kernel function, while keeping the original kernel. For the purpose of approximating the field variable, the EFG meshfree method uses the MLS mesfree method, while in discretizing an algebraic equation system, it follows the Galerkin-WF based method. Satisfying the BCs became much easier by developing the BMLS. As a result, it is capable to solve higher dimensional problems. Furthermore, the MK meshfree approach is capable to enforce the essential BCs straightly at the assigned node with the absence of any penalty method. The RBF has been developed in many studies based on various mathematical modeling.
The stabilizing conforming nodal integration approach is commonly used in static and dynamic analysis to predict bending stiffness, while the DNI strategy is used to avoid shear locking. The first, third and fourth-orders shear deformation plate and shell theories were employed and combined with the meshfree methods in most studies. Furthermore, based on some literatures, there are many techniques to develop meshfree method by mathematical modeling and numerical method based on the aim to be accomplished in the study. Nevertheless, in static and dynamic analysis, researchers were interested in defining the influence of BCs, VF, and t/l ratio mainly on buckling strength, bending, and NF. Besides, some other parameters were considered due to some structural nature that needed investigation such as hole size and geometry in plates with holes. Based on some literature, it is important to highlight the error analysis to guaranty high performance and accuracy of the developed meshfree method no matter how the analysis was carried out. It has been observed that most of the studies were interested in investigating CNT, FGPs ceramic to metal composite, FGM, and polymeric composites. Another observation is that there are a quite large number of studies showing interest in investigating static analysis, especially dynamic analysis of the piezoelectric. However, these unique group of methods are not limited specifically on solid mechanics field (static and dynamic) analysis. Besides, it has many other various applications, especially in scientific fields and mechanical engineering in particular such as thermal analysis which can be performed by Galerkin, fluid and particles analysis which can be investigated by SPH, for developing new material characterizations and properties the RBF method is employed, band structure and grain structure as well Quasi-brittle which can be examined by Galerkin, and Material defected crack failure microstructure changes and in civil engineering (concrete slab) which can be analyzed by kernel. Moreover, problems such as elasto-viscoplastic deformation, food engineering, biomedical, and temperature and heat transfer analysis also can be utilized by meshfree methods.
5 Conclusion
This is the first comprehensive review of meshfree methods, including traditional and developed meshless methods, that have been used to study static (bending, buckling/stability) and dynamic (FV, force vibration, and other types of dynamic analysis) for linear and NL analysis. For all different types of structures and geometries at various scales (nano, micro, and macrosalces). In addition to the meshless approach, secondary methods and principles such as HP, FSDT, HSDT, Monte Carlo, local and NLC theories, and others are highlighted. Likewise, some computational mechanics methods are discussed. The essential equations of meshfree methods are presented, as well as the error analysis. The basic mathematical equations of meshfree methods are illustrated. There is discussion of several sorts of schematics and structural size are addressed. Furthermore, this inclusive review is a significant resource for academics and researchers interested in the field of solid mechanics analysis at various scales utilizing meshfree techniques, as it addresses the secondary approaches used in conjunction with meshless methods, numerical and analytical analysis. Furthermore, several material types were described, as well as the enhancement of composite distribution and the impact of several variables on deformation, static and vibrational performance. The application of BCs and its effect as well as the implementation of FGM and other composite material for system which are examined by meshfree approaches are clarified.
Meshfree methods demonstrates excessive possibility to come across the demands of modern software, error estimators and various types of analysis. Meshfree technique, considered in its individual form represent the impending generation of computational numerical developed methods. Meshfree modeling requires simply a set of unstructured points that encompass the territory of interest. Meshfree particle approaches enable easy adaptive refinement, parallelism, and flexible interpolation in a deformable territory due to their simple topological data structures. Numerous complications that cannot currently be addressed by FEM or finite difference methods have been demonstrated to be tractable by meshfree approaches. Despite the fact that tremendous progress has been made over the last decade, there are still many responsibilities and challenges ahead. Among the issues to be addressed are the cost-effectiveness of the meshfree-Galerkin technique, scalable application of essential BCs, accurate nodal integration strategies, and stabilization procedures for both discretized WF and SF. Meshfree particle methods are expected to play a key role in the future generation of computational meso-mechanics. Also known as computational micro-mechanics that is an integrated aspect of nanotechnology and computer systems technology. The high benefits and the unique characterizations of the RBF method makes it possible to be combined with a variety of different methods, therefore making its application limitless, especially in FG composite material structure for NL analysis. It is highly important to estimate the error in each study for the purpose of validating the developed mathematical method as well as for comparing the accuracy of the founded data with the data that have been published earlier.
Most of the research works carried out investigations in nanoscales structure in both static and dynamic analysis. Regarding dynamic analysis, most focused on FV analysis rather than forced vibration analysis. However, for static analysis it has been observed that in thermomechanical and electromechanical simulations, the authors were considering the buckling analysis and not bending analysis. The gap in literature is that a few studies focused on investigating force vibration. Also, absence of applying meshfree methods for analyzing pico-scale system. Besides, there is limitation in recorded data about applying the meshfree methods on examining the impact. Else, a few studies investigated the Gr copared with FGM and CNT. The limitation of this study that it discussed the fracture mechanics briefly so as a future work a comprehensive review about implementation of meshfree methodologies for studying cracks and failure caused by porosity may be accomplished.
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Al Mahmoud, Z., Safaei, B., Sahmani, S. et al. Implementation of Different Types of Meshfree Technique in Computational Solid Mechanics: A Comprehensive Review Across Nano, Micro, and Macro Scales. Arch Computat Methods Eng 31, 725–838 (2024). https://doi.org/10.1007/s11831-023-09999-6
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DOI: https://doi.org/10.1007/s11831-023-09999-6