Analysis of Laminated Composite Plates: A Comprehensive Bibliometric Review

Abstract

Laminated composite plates have become a crucial point of interest in the industry, with the need to ensure sustained and stable structures throughout the plates’ lifespan. This study conducted a bibliometric analysis using the Scopus database, gathering 8221 documents for further scrutiny based on the linked meta-data. Utilizing the VOS viewer software version 1.6.19, maps were generated from scientific publishing network data, illustrating connections between researchers’ nations and keywords. The investigation into co-occurring phrases associated with laminated composite plates employed author keywords. The results reveal a significant and close relationship among top authors, suggesting a strong research connection, with the United States and China leading the field. Top cited documents and keyword correlations are examined to gauge current research interests. These critical reviews serve as essential resources for scholars and practitioners in the field. Additionally, the review discusses the advancements in and practical applications of different theories for laminated composite plates, with a focus on a bibliometric study using the Scopus database. This paper categorizes models within the context of an equivalent single-layer laminate, analyzing variations in established theories and methodologies for modeling laminated composite plates to offer a nuanced understanding of approaches and assessments in this field.

1. Introduction

1.1. Laminated Composite Structures

A composite material is formed by combining various materials to achieve specific engineering properties such as bending, buckling, and stiffness. This production can result in superior properties compared to the individual components. A lamina is the essential unit of modern, composite-laminated structures, which is also referred to as a layer or a ply. Multiple layers of laminae form a laminate [1]. Composite materials provide a broad spectrum of advantages compared to traditional materials. Their favorable characteristics encompass superior stiffness-to-weight ratios and strength, extended fatigue life, lightweight construction, resilience against environmental deterioration, and the capability to produce structures that closely resemble their final form during the manufacturing processes [2]. One of the significant benefits of laminated composite plates is their capability to bear loads effectively in multiple directions. Among these plates, anisotropic ones stand out for their exceptional efficiency, primarily attributed to their remarkable strength-to-weight ratio, which makes them crucial in modern engineering applications [3,4]. To analyze the behavior of such plates, there should be mathematical representations that can describe their mechanics, which can be classified into different theories based on

Table 1. Plate theories classification [22].

Based on Elimination of Thickness Coordinate Z	Based upon Field Variable	Based upon Choice of Variable Description
Continuum	Stress
Asymptotic	Displacement	ESL
Axiomatic	Mixed	MLT

. During the last 7 decades, the equivalent single-layer (ESL) theories [1,3,4,5,6,7,8,9,10], which are the concern of this study, the layer-wise theories [11,12], and the zigzag theories (polynomial zigzag [13,14,15,16,17,18,19] and non-polynomial zigzag [20,21]) have drawn the attention of researchers.

Despite their performance, the anisotropy (presence of highly directional properties) of composite laminates leads to the complexity of designing and analyzing structures made of such materials, as compared to other types of structures, even under simple loadings. This is attributed to the occurrence of bending–extension coupling, which leads to a very complex failure process with no possibility, except in special cases, of finding analytical solutions [23].

In engineering applications, composite laminated structures are susceptible to failure when exposed to a variety of static loading conditions (out-of-plane and in-plane loads). One of the important failure modes is bending under transverse loading where bending–stretching effects have to be considered. Due to this effect, obtaining solutions in closed-form is not possible and, thus, many studies neglect the coupling effect through using symmetrically laminated angle-ply configurations [24].

In recent years, the engineering community has increasingly turned its attention to the development of variable stiffness composites as a promising solution to optimize structural performance and enhance system response [25]. Unlike traditional constant stiffness composites, which offer limited flexibility in material properties, variable stiffness composites enable engineers to tailor stiffness and strength characteristics to specific design requirements [26]. This ability to customize material properties opens up new avenues for innovation across various industries, from aerospace and automotive to marine and biomedical engineering.

Variable stiffness composites offer significant advantages over constant stiffness composites due to their ability to tailor stiffness and strength properties. One key advantage of variable stiffness composites is their ability to enhance structural efficiency. Traditional constant stiffness composites often exhibit over-engineered designs to accommodate worst-case loading scenarios, leading to unnecessary weight and material usage [27]. In contrast, variable stiffness composites enable the design of lightweight structures with localized reinforcement in areas experiencing higher loads, thereby reducing material consumption without compromising performance. Moreover, variable stiffness composites offer enhanced adaptability to dynamic loading conditions. By adjusting stiffness properties in real-time or in response to changing environmental factors, these materials can optimize structural performance and mitigate damage under varying operational conditions. This adaptability is particularly advantageous in applications such as aerospace, automotive manufacturing, marine engineering, and mechanics where structures are subjected to complex loading environments [28]. Additionally, variable stiffness composites facilitate improved damage tolerance and durability. By tailoring stiffness gradients, these materials can distribute loads more effectively, reducing stress concentrations and minimizing the risk of failure. This enhanced damage tolerance not only extends the service life of structures but also reduces maintenance requirements and associated costs [29]. Furthermore, variable stiffness composites enable the design of multifunctional structures with tailored mechanical, thermal, and electrical properties. By integrating functional additives or controlling fiber orientation, these materials can exhibit customized behavior to meet the diverse demands of modern engineering applications. This multifunctionality opens up new possibilities for innovation in fields such as smart materials [30].

1.2. Practical Application’s

Using laminated composite materials in structural design improves performance and dependability, including the specific modulus, stiffness-to-weight ratio, strength, impact resistance, total weight, and design flexibility [31,32]. Their physical structure and mechanical behavior are determined by fiber orientation, layer size, form, and the thickness of the lamina [33]. Additionally, adding piezoelectric layers to laminated composites transforms them into smart structures. Their unique properties make them indispensable in modern engineering applications including laminated composite plates, bridge decks, pressure vessels and tanks, pipes, aircraft structures, solar panels, wind turbines, helicopters, space station structures, marine constructions, gliders, fighters, and vehicles, to mention a few [34,35]. The most traditional structural elements in engineering applications that and made of composite laminated materials are plates and shells [36]. For further enhancement of the structural behavior of composite laminates, advanced composites are a promising structural material for use in numerous engineering fields. Recently, carbon nanotubes (CNTs) are gaining popularity among researchers for their electrical, thermal, mechanical, and chemical capabilities [37,38]. Also, functionally graded materials (FGMs) are inhomogeneous composites utilized in engineering applications where their characteristics vary smoothly in a specified direction [39,40]. Plates with different geometries such as circular, annular, and sector annular are universal configurations used in a wide range of industries and applications. They are widely utilized in aerospace, shipbuilding, petrochemical containers, aircraft manufacturing, civil construction, railway transportation, and other engineering industries [41]. In addition to their high load-carrying capacity, composite materials have the advantages of being durable and resistant to corrosion, particularly when utilized in marine and aerospace engineering [42]. Moreover, building-up structural elements using composite materials offers flexibility in design by allowing for different stacking sequences of fiber and matrix [43]. Consequently, all these applications and advantages have led to increased interest in optimizing laminate design techniques [44].

The primary motivation for undertaking this review is to conduct a comprehensive bibliometric study using Scopus, aiming to gain insights into the advancements made in the analysis of laminated composite plates. By examining various proposed and developed theories, this research seeks to not only elucidate the strengths and weaknesses of each model but also to contribute valuable knowledge to the broader research area by explaining past and recent methodologies. The utility of this review lies in its potential to provide a consolidated overview of the current state of research in laminated composite plate analysis.

2. Methodology

2.1. Data Collection and Search Approach

Bibliometric analysis is gaining prominence as a technique for determining the trend and pattern of studies [45]. The study patterns can be observed by classifying the publications by year, author, affiliation, or country. Additionally, the publication’s influence and performance can be quantified using matrices such as the number of citations, citations per year, h-index, and g-index [46]. The bibliometric search process was initiated by identifying pertinent keywords based on the search criteria, followed by the retrieval of results from various databases. Scopus was the primary search engine used due to its vast access to a large number of journals and its inclusion of resources for information analysis and bibliographic databases, which support the assessment and analysis of research performance. Additionally, it offers a range of analytical tools for conducting focused searches [47]. Covering documents published between 1956 and 2023, the search was conducted using the following query string; “Article title”, “Abstract”, Keywords “Analysis”, “Composite”, “Laminated”, and “Plates”. The analysis focused on journal articles within specific subject areas, including “Engineering”, “Material Science”, “Physics and Astronomy”, “Mathematics”, and “Civil Engineering”. After applying the specified filters, a total of 8221 articles were obtained and included in the bibliometric analysis and the data were extracted for all documents in this bibliometric analysis. The data were analyzed using two tools—Excel 365 and VOS viewer version 1.6.19 [48]—to determine the frequency of publication, visualize the bibliometric networks, and compute citation metrics. The diagram of the research methodology used to select and sort the related articles is shown in Figure 1.

2.2. Bibliometric Assessment

VOS viewer software version 1.6.19, a data mining tool that generates author, journal, and keyword maps from co-citation and co-occurrence data, was used in this stage of the analysis [48]. The software package includes a built-in viewer that allows users to conduct a comprehensive analysis of bibliometric maps.

These maps allow for a comparative analysis of different parameters (e.g., authors, journals, countries, etc.), enabling a comprehensive understanding of the research landscape. Also, VOS viewer version 1.6.19 offers multiple display options for maps, each emphasizing different aspects of the data. A typical representation of a bibliometric map consists of interconnected circles, where the size and distance between circles indicate the significance and inter-relationships of the associated terms [49]. In this study, VOS viewer version 1.6.19 was used to visualize, calculate, and analyze the impact of various journals, researchers, and countries involved in investigating the progress on laminated composite plates.

2.3. Qualitative Assessment

This evaluation process involved a comprehensive qualitative assessment of laminated composite plates. The qualitative assessment was informed by the findings of the science mapping, which provided insights into the structure and dynamics of this research field. The overall goal of this assessment was to contribute valuable knowledge to the broader research area, which has not been well-integrated, as well as other research topics in ESL theories.

3. Bibliometric Analysis of Results

3.1. Summary of the Obtained Information

A discernible trend was revealed from the bibliometric analysis of the published research contributions on laminated composite plates up to 2023 (Figure 2). In the current work, the authors considered the period from 1956 to 2024 for analysis. The reason behind this was that more than 75% of the 8221 papers in this field were published in past few decades. Research publications have grown rapidly and reached above 5000 publications per year from 1992 onwards. So, it is observed that the rise has increased substantially in last 4 decades only. The number of articles has consistently increased since 1992, experiencing a notable steep surge starting from 2010 onwards. The majority of research in this field was conducted between 1992 and 2023, accounting for 7443 research publications out of a total of 8221 published articles. Also, the graph clearly shows that the interest of researchers in this particular field is increasing continuously. during the decade 2000–2010, the number of publications was around 1600; in the last decade (2010–2020), this number reached to 2000. Early in 1969, the number of publications was 8; however, it was recorded to be 363 in 2023. This indicates that there is high growth in publications. Furthermore, the number of research documents published within this scope peaked at more than 400 articles in 2022 and 2023, demonstrating their considerable impact, as illustrated in Figure 2.

Figure 3 shows division on the basis of subject area, where Engineering has witnessed the most significant advancements in this particular domain, accounting for 44.6% of the progress. Martial Science contributed 27.1%, while Mathematics, Physics, and Astronomy, as a whole, maintained a dominant position at 17.2%. It can be observed that the major literature contribution came from Engineering, consisting of 7244 documents. Research in Material Science occupies the second rank with 4395 documents, followed by Physics and Mathematics with 1426 and 1378 documents, respectively.

All document types like articles, books, conference papers, and review articles were considered for analysis without any exclusions. The “analyze the search results’’ function available in Scopus was used to scrutinize the literature. The search showed 8221 documents, out of which research articles occupied almost 76.1% of the database. The distribution materials released on laminated composite plates are summarized in Figure 4, divided into 14 document categories (

Table 2. Distribution based on types of documents.

Document Type	Number of Documents
Article	6257
Book	14
Book Chapter	74
Conference Paper	1628
Conference Review	156
Data Paper	1
Editorial	2
Erratum	7
Letter	5
Note	3
Report	7
Retracted	2
Review	64
Short Survey	1

). More than half of all publications (76.1%) were classified as articles, followed by conference papers (19.8%). Until now, four types of documents (short survey, editorial, retracted, and report) have had fewer than ten published documents, with other publications accounting for 10 to 70% of the total documents.

3.2. Analysis of Journals

A bibliometric analysis of 8221 articles published across 111 different journals was conducted using the Scopus online database. The analysis, presented in Figure 5, mapped the sources of these journals, where a minimum criterion of at least two documents was set in VOS viewer version 1.6.19 to ensure a meaningful analysis, resulting in 111 journals that met this criterion. In the network representation of journal sources, the font and node sizes are directly proportionate to the number of publications originating from each journal. Therefore, journals with greater font and node sizes indicate a higher number of published articles included in the analysis. Additionally, the colors of the clusters and connecting lines reflect the degree of inter-relation among the various journals in terms of mutual citations.

3.3. Analysis of Keywords Co-Occurrence

Figure 6 illustrates a bibliometric map that visualizes the co-occurrence of laminated-composite-plate-related keywords used in the literature. To mitigate the impact of heavily cited sources within the network, counting methodologies known as “Author Keyword” and “Fractional Counting” were employed in conducting the analysis by VOS viewer version 1.6.19. The threshold for minimum occurrences was established at 5, leading to the identification of 1566 keywords from a total of 9954. The size of the circles in the map corresponds to the frequency of keyword appearances, while the thickness of the lines indicates the strength of association between the respective terms. Additionally, the proximity of the two terms on the bibliometric map reflects the intensity of their relationship. Notably, the keywords with the largest nodes on the map, namely “laminated composites”, “plates”, “finite element method “, and “shear deformation”, demonstrate the highest degree of co-occurrence.

3.4. Analysis of Co-Authorship Analysis

The co-authorship network was also investigated in the present bibliometric analysis using VOS viewer version 1.6.19. The analysis utilized a minimum criterion of two documents, resulting in 1329 authors meeting the criteria. Among them, only 160 authors were connected through the network, as depicted in Figure 7. Understanding the existing scientific collaboration networks in any field of study can provide researchers with easier access to funding, domain expertise, and valuable experiences. This can not only enhance productivity but also reduce isolation from the global scientific community. Ultimately, such collaborations can promote scholarly communication and lead to an increase in the volume of research output. Encouraging and supporting collaboration through research management and scientific policies has the potential to amplify research efforts, productivity, and impact [50].

A ranking of authors according to contributions in the concerned area is indicated in Figure 7. The top 10 authors are listed, with Singh and Reddy in the leading positions with 91 and 87 publications, respectively.

3.5. Contributing Countries Analysis

Using the author’s affiliations, VOS viewer version 1.6.19 was utilized to construct network maps. This observation underscores the significant contributions made by these countries in the field of analysis of laminated composite plates. The network map includes countries with at least eight documents, connecting 60 countries. The map revealed four distinct clusters of countries, as shown in Figure 8. Notably, the United States emerged as the country with the highest number of documents and achieved first place in overall link strength and linkages (1571), followed by China (1354) and India (1229). As seen by the contributions from these nations, the field of laminated composite plates has garnered significant interest from researchers.

4. Qualitative Assessment of Science Mapping Results

The science mapping exercise conducted in this study offers valuable insights that inform qualitative assessments of the literature pertaining to the analysis of laminated composite plates, with a specific focus on equivalent single-layer (ESL) theories. In recent years, the engineering sector has conducted significant analysis of composite laminated plates using ESL theories, which has drawn the attention of researchers. This assessment aims to gain insights into the advancements made in the analysis of laminated composite plates, by examining various past and recent proposed and developed theories.

4.1. Historical Progress of Laminated Composite Plate Theories

Several researchers have invested significant efforts in developing diverse approaches for analyzing laminated composite plates. The preceding discussion includes a concise overview of prior studies conducted on composite laminated plates. However, the literature search encompasses all relevant studies, offering a historical review and advancement analysis of the below-mentioned approaches, as illustrated in

Table 3. Comparative study of different approaches for laminated composite plates.

Theory	ESL	LWT	ZZ
Historical Period	Developed in the mid-20th century.	Developed in the late 20th century as an improvement over ESL.	Developed in the late 20th century as a mesh-free numerical method.
Model formulation	ESL is based on the equivalent single-layer theory, which simplifies the analysis of laminated composite plates by representing them as equivalent single-layer plates.	LWT accounts for the layer-wise variation in material properties and shear stresses.	It is an advancement of ESL and LWT models that superimposes an FSDT or HSDT with a zigzag function.
Suitable Usage	Suitable for analyzing laminated composite plates with simple geometries, uniform material properties, and moderate loading conditions. It provides accurate solutions for displacement and stress fields in thin to moderately thick plates.	Suitable for analyzing laminated composite plates with complex layer configurations, non-uniform material properties, and large deformations. It provides accurate solutions for both thin and thick laminates, including those subjected to high shear loads.	Suitable for analyzing laminated composite plates with arbitrary geometries, complex loading conditions, and large deformations. It offers the flexibility to model material and geometric nonlinearities, making it applicable to a wide range of practical engineering problems.
Shear Effect Consideration	Shear effects based on thickness to smaller dimension ratio which may lead to less accurate predictions of shear stresses and interlaminar effects, especially in thick laminates.	Includes shear effects, providing more accurate predictions of shear stresses and interlaminar effects compared to ESL.	Includes shear effects, allowing for accurate predictions of shear stresses and interlaminar effects in laminated composite plates.

.

Table 4. Historical review and advancement analysis for theoretical models based on ESL approach.

Theory	CLPT	FSDT	HSDT
Historical Period	Approximately 1850–1950	Approximately 1950–1980	Approximately 1980–present
Model name	Kirchhoff–Love plate theory	Ressiner–Mindlin plate theory	Extension of Ressiner–Mindlin plate theory.
Suitable Usage	Suitable for thin plates with thickness-to-width ratio less than 1/10 [51,52].	Suitable for thick plates with thickness-to-width ratio more than 1/10 [51,52].	This theory is utilized for analyzing thicker laminated anisotropic composite plates where FSDT is not accurate enough [51,52].
Shear Effect Consideration	Neglects shear effects	Includes shear effects	Includes shear effects

provides a concise overview of three prominent theories in the analysis of laminated composite plates, based on the ESL assumption. It should be noted that FSDT and HSDT are theoretical models describing the kinematics of displacements whereas ESL and LWT are approaches to applying these theories (FSDT and HSDT) to laminated composites. An FSDT can be an ESL and, at the same time, can also be a layer-wise model, if written for each layer in a laminate. Each theory has evolved over different historical periods and is tailored for specific applications based on plate thickness. CLPT, dating back to the 19th century, neglects shear effects and is suitable for thin plates. FSDT, developed in the mid-20th century, includes shear effects and is suitable for thicker plates. HSDT, an extension of FSDT, also considers shear effects and is applied to even thicker laminated composite plates where FSDT may lack accuracy.

The key distinction among the theories is clarified in Table 4 and Figure 9, which illustrates the orientation of the transverse plane after deformation. In the scenario of CPT, the transverse plane maintains its perpendicular and straight alignment to the midplane surface both before and after deformation, resulting in the elimination of shearing strains (${\gamma }_{\mathrm{y}\mathrm{z}} \mathrm{a}\mathrm{n}\mathrm{d} {\gamma }_{\mathrm{x}\mathrm{z}}$). In general, the theories used to analyze laminated plates may be divided into different major categories: (i) classical lamination theory (CLPT) [53], (ii) first-order shear deformation theory (FSDT) [54], (iii) higher-order shear deformation theory (HSDT) [55], (iv) layer-wise lamination theory (LWT) [56], (v) zigzag theories [57], (vi) 3D elasticity theories [58], and (vii) discrete-layer theories [59]. Each theoretical approach presented offers a unique perspective on comprehending the behavior of laminated composites. These theories cater to various levels of complexity and accuracy in their analyses, providing researchers with a range of options to study and understand the intricacies of laminated composites [60,61,62].

All the ESL theories are based on the concept of perfectly bonded layers, the assumption that the laminate is under plane stress, and assumption that each single lamina is orthotropic and linearly elastic. Due to their non-uniform behavior across the thickness and overall anisotropic nature, normal and transverse shear stresses are fundamentally present in composite plates [65]. The transverse and normal shear stress distributions in composite materials hold significant importance for several reasons. Firstly, when the in-plane stress exceeds the fiber materials’ yield strength, it can lead to the failure of interlaminar shear stress. This failure is caused by tangential slips occurring between the layers; the components are arranged in a specific relationship to each other. Secondly, a substantial increase in transverse normal stress can result in severe bonding failure. This occurs when two layers of the composite separate from each other, experiencing opposing forces that pull them apart [66]. The equations of displacements for CLPT, FSDT, and HSDT are provided in

Table 5. Equations of displacements for CLPT, FSDT, and HSDT.

CLPT	FSDT	HSDT
$u_1=u_0\left(x,y\right)-z\frac{\partial w}{\partial x}$	$u_1=u_0\left(x,y\right)+z{\varphi }_x (x,y)$	$u_1=u_0\left(x,y\right)-z\frac{\partial w\left(x,y\right)}{\partial x}+f\left(z\right)({\varphi }_x\left(x,y\right)+\frac{\partial w\left(x,y\right)}{\partial x})$
$u_2=v_0\left(x,y\right)-z\frac{\partial w}{\partial x}$	$u_2=v_0\left(x,y\right)+z{\varphi }_y(x,y)$	$u_2=v_0\left(x,y\right)-z\frac{\partial w\left(x,y\right)}{\partial y}+f\left(z\right)({\varphi }_y\left(x,y\right)+\frac{\partial w\left(x,y\right)}{\partial y})$
$u_3=w\left(x,y\right)$	$u_3=w\left(x,y\right)$	$u_3=w\left(x,y\right)$

where $u_0$, $v_0$, and $w$ denote the equivalent deformation of the midplane in the x, y, and z directions and $u_1$, $u_2$, and $u_3$ represent the coordinate point (x, y, and z) displacements in the plate element. Additionally, ${\varphi }_y$ and ${\varphi }_x$ are cross sectional rotations. $f\left(z\right)$ takes different expressions for different methods.

; these equations are crucial components in the analysis of laminated composite plates. These displacement equations, as outlined in Table 5, play a pivotal role in describing the deformations and behaviors of such plates under various loading conditions. CLPT provides a simplified but effective representation, assuming no shear deformation throughout the thickness of the plate. FSDT introduces shear deformation effects, offering a more realistic depiction of plate behavior. On the other hand, HSDT, with its higher-order consideration of shear deformations, further refines the accuracy of predictions. The comparison and understanding of these displacement equations contribute significantly to the selection of appropriate theories based on the specific requirements and precision needed for a given laminated composite plate analysis.

4.1.1. Classical Lamination Plate Theory (CLPT)

CLPT is a development of classical plate theory (CPT) and its adaptation for analyzing composite laminated plates can be attributed to Reissner and Stavsky in 1961 [67]. CLPT is the most fundamental approach to laminated structures, initially developed for analyzing homogeneous plates. It considers the laminate as a single layer in two dimensions, ignoring the three transverse normal stress and strain components, according to Kirchhoff [68], Love [69], and Rayleigh [70]. As a result, it is also referred to as the equivalent single=layer (ESL) theory, assuming that the layers in laminated composites behave as one equivalent single-layer. Although this theory has been applied extensively, it is most accurate for relatively thin plates, offering reasonably accurate predictions for such cases [5,60].

Volokh [71] presented an improved version of CLPT, which assumed that shear forces are statically equivalent to twisting and bending moments. This consideration contrasts with CPT, where the shear forces are unrelated to the Saint-Venant principle. In this proposed version of the theory, the treatment of shear forces serves as the foundation for describing the behavior of laminated plates. By incorporating this enhanced approach, the theory aims to obtain the exact deformation of laminated composite plates. As per Ressiner’s theory [54], neglecting shear stresses in the CLPT results in a decrease in or elimination of the three essential boundary conditions that must be met along the free edges. These boundary conditions encompass the bending moment, twisting, and normal force. Due to the exclusion of shear stresses in the theory, certain complexities are simplified. Wang et al. [72], Pagano [73], and Reddy [74] demonstrated the limitations of CLPT according to linear displacement when analyzing thick laminated plates throughout the entire composite laminate. This inadequacy results from the neglect of transverse shear deformation in CLPT. Khandan et al. [6] assured that CLPT remains a widely employed method for obtaining rapid and straightforward predictions, particularly concerning the behavior of thin-plated structures. The key simplification involves treating shells or 3D thick structural plates as shells positioned at their mid thickness or 2D plates. This simplification significantly reduces the number of variables and equations involved, leading to substantial savings in computational time and effort. Li et al. [75] conducted an analysis of a circular plate under uniformly distributed and concentrated transverse loads to calculate the rotation, stresses, and deflection at the center of the plate. The study utilized explicit expressions and the Bessel function for this purpose. The model was found to converge to the classical Mindlin assumption for the circular plate. Additionally, Bera et al. [76] extended these investigations to square and rectangular plates, expanding the scope of the study.

Based on the information provided, it can be inferred that CLPT is most suitable for thin-plate structures composed of balanced composite laminates and symmetric experiencing bending and pure tension. In such cases, CLPT can provide reasonably accurate results. However, its accuracy deteriorates slightly for unsymmetrically stacked, thin laminated plates and the deterioration of the solution becomes even larger as the plate thickness increases, due to the neglection of transverse shear stresses.

4.1.2. First-Order Shear Deformation Theory (FSDT)

FSDT is the simplest theory that considers the transverse normal and transverse shear deformations. Reddy [77] introduced a two-dimensional general shear deformation theory for laminated composite plates that encompassed CLPT and FSDT (Reissner–Mindlin). Reissner [54] considered the transverse shear deformability of the plate, taking into consideration the impact of the analysis on shear deformation. Consequently, the system of equations derived from this theory is such that three boundary conditions (simply supported, fixed, and free) need to be provided along the edge of the plate. Mindlin [78] incorporated a similar approach for including shear effects and rotatory inertia in the flexural motions of isotropic elastic plates.

Roufaeil et al. [79] proposed a FSDT formulation under the hypothesis that $w$ is constant along the plate’s thickness, while $v$ and $u$ varies linearly over each layer’s thickness. Additionally, the cross-sectional rotations ${\varphi }_y$ and ${\varphi }_x$ are considered constant across the plate. Chandrashekhara et al. [80] introduced a theory for shear-deformable laminated composite plates by utilizing Reissner’s principle to compute their in-plane response. Reddy et al. [55] discussed the analyzed shell and shear deformation plate theories proposed by Stavsky, which utilize straightforward, fundamental equations and exclude transverse shear strain. Ghugal et al. [81] conducted a revision of advanced shear deformation theories that aim to improve the accuracy of anisotropic and isotropic laminated plates, taking into account stress and displacement considerations. Reddy [82] provided a comprehensive analysis of the exact connections between solutions (such as buckling loads, natural frequencies, and deflection) obtained by employing CLPT and FSDT theories. Their focus was on sandwich plates and isotropic plates with various boundary conditions and shapes. Their study demonstrated that FSDT solutions could be readily derived from known CPT solutions for isotropic plates. Frederick [83] employed FSDT to investigate the bending behavior of thick, circular plates supported by elastic foundations. Wu et al. [84] assumed constant transverse shear stress, which is a simplification often used in plate theories. However, in order to satisfy the plate upper and lower surfaces boundary conditions, a shear correction factor (K) was developed to account for the transverse shear stiffness, thus affecting the validity of the outcomes utilizing FSDT. Maji et al. [66] proposed different shear correction factors for modifying the transverse shear stress in FSDT formulation. The selection of an appropriate shear correction factor is essential in achieving better agreement between the theoretical predictions and real behavior of the thick laminated plates. Pai [85] presented a novel approach to derive the shear correction factor via aligning the shear strain energy and exact shear stress results to be similar to that of FSDT. This formulation allows for obtaining shear correction factors without the need to solve a plate problem with particular boundary conditions and loadings. Furthermore, K for both symmetric angle-ply laminates and asymmetric are provided. Auricchio et al. [86] created novel mixed variational formulations for FSDT without using K. Using this innovative approach, out-of-plane shear stresses can be estimated directly without the use of any post-processing. By eliminating the reliance on shear correction factors, this formulation streamlines the analysis and ensures more accurate transverse shear effect predictions in laminated composite structures. Reddy and Nguyen et al. [87,88] derived the expression of transverse stresses and membrane stresses from the equilibrium equations. Qi et al. [89] presented a refined FSDT-based solution in both symmetric cross-ply laminated and homogeneous plates for the plane strain bending problem. This theory introduces effective transverse shear stiffness based on constant shear stress distribution throughout the thickness of the plate. The refined theory improves deflection predictions and includes variable transverse shear strain distribution. This assumption is supported and satisfied by the exact elasticity solution. Dobyns [90], Nelson et al. [91], and Medwadowski [92] investigated laminated orthotropic composite plate behavior based on refined FSDT. This study involved an investigation of various aspects, including transverse normal and transverse shear effects and quadratic displacement effects. Averill et al. [93] put forward an accurate solution for symmetrical laminated composite plates utilizing FSDT and considering the rotational inertia. Thai et al. [94] introduced a new approach based on FSDT, instead of the traditional original method, by reducing the number of unknown displacement functions from five to four; we assume that the angular displacement, theta (θ), replaces two rotations, ${\varphi }_x \mathrm{a}\mathrm{n}\mathrm{d} {\varphi }_y$, resulting in the displacement field represented by (u, v, θ, and w). This approach enables the derivation of analytical solutions for vibration, buckling, and bending for rectangular plate analysis using different boundary conditions, all without requiring the use of shear correction factors.

Consequently, as shown in

Table 6. Historical progress for shear correction factor.

Authors Name	Contribution
Reissner [54] and Mindlin [78]	Using K for laminate transverse shear in FSDT is an extension of Reissner and Mindlin’s theories for isotropic homogeneous plates.
Whitney [51,56] and Kaneko [95]	Discussed how to introduce K.
Stephen and Hutchinson [96]	Was the first to include a dependence on the cross-sectional aspect ratio.
Isaksson et al. [97]	Derived K for a corrugated board sandwich panel based on equilibrium stress.
Tanov et al. [98]	Utilized FEM to determine K.
Puchegger et al. [99]	Obtained K experimentally for validation.
Hadavinia et al. [100]	Derived the shear correction factor based using the energy equivalence method.

, K is chosen considering various aspects—the energy of shear deformation, material constants, the assembling pattern of laminate, geometry, configuration, loading, and end conditions [54,78].

Following the preceding analysis, it can be deduced that the utilization of FSDT, which neglects cross-sectional deformation effects, leads to a transverse shear stress that uniformly varies across the thickness of the laminate. Furthermore, the precision of the analysis is notably influenced by the shear correction factor employed. This inference prompts exploration into novel approaches that may eliminate the need for shear correction factors, potentially enhancing the accuracy of the analytical methods employed.

4.1.3. Higher-Order Shear Deformation Theory (HSDT)

HSDT is utilized for analyzing thicker laminated anisotropic composite plates where FSDT is not accurate enough for the predicting buckling loads, natural frequencies, stresses, and deflections of such plates. It considers the transverse shear stresses to follow a parabolic distribution and, notably, it does not necessitate any shear correction factors [101].

Reddy [102] proposed a HSDT for laminated composite plates that offers improved accuracy in predicting stresses and deflections, as compared to FSDT. Aydogdu [103] achieved a comparative analysis among various shear deformation theories (SDTs) for cross-ply rectangular plates with simply supported edges, concerning buckling, vibration, and bending behavior. Lan et al. [104] demonstrated that the newly proposed and straightforward HSDT, based on the principle of virtual work, offers improved reliability, accuracy, and computational efficiency compared to both FSDT and other HSDT theories. Matsunaga [105] provided two-dimensional theoretical equations, considering both transverse shear and bending deformations for evaluating displacement distributions and stress within a thick elastic plate subjected to fixed boundaries. Mantari et al. [106] introduced a novel HSDT in the case of composite laminated and sandwich plates. It assured that the plate’s boundary surface was free of tangential stress and ensured a precise transverse shear strain distribution throughout the thickness of the plate without using K. Dhuria et al. [107] provided an innovative HSDT for analyzing the mechanical behavior of antisymmetric and symmetric composite laminates in both angle-ply and cross-ply arrangements. This theory is suitable for buckling and static bending analysis. It incorporated a secant hyperbolic function within the displacement field, capturing non-linear displacement distributions. Shi et al. [108] introduced a new hyperbolic tangent SDT for the derivation of weak forms without K to analyze the static buckling, static behavior, and free vibration of laminated composite plates. These derived forms are then solved numerically using the iso-geometric analysis approach. Li et al. [109] created a comprehensive framework, unifying all HSDT theories. This framework is designed to model and analyze functionally graded and laminated plates. Shishehsaz et al. [110] employed layer-wise theory combined with third-order shear deformation theory, TSDT, and HSDT to examine the adhesive behavior within a uniformly loaded circular plate. Both analytical and FEM solutions are used to analyze and understand the adhesive behavior under these conditions. Kumar et al. [111] studied, for the first time, laminated composite plates with porosity using advanced TSDT. The present improved, third-order theory incorporating transverse shear stress continuity at each layer interface predicts better results than TSDT and FSDT.

While discussing HSDT, the distinction between polynomial and non-polynomial theories is discussed in

Table 7. Comparative analysis of polynomial and non-polynomial higher-order shear deformation theories for laminated composite plates.

Theory	Polynomial HSDT	Non-Polynomial HSDT
Computational merits	Involves simpler mathematical expressions.	Requires more complex formulations.
Accuracy	Offers less accuracy in capturing the complex deformation behavior of laminated composite plates.	Offers higher accuracy in capturing the complex deformation behavior of laminated composite plates.
Thickness functions	Utilizes simpler thickness functions, such as linear or quadratic variations, which may not adequately capture the true variation in thickness of laminated composite plates.	Allows for more flexible thickness functions, enabling a better representation of the actual thickness distribution in the plate.

.

Kant [112] analyzed the bending behavior of a rectangular plate with the help of a refined polynomial higher-order theory. The theory is based on a higher-order displacement model and the three-dimensional Hooke’s laws for plate materials, giving rise to a more realistic quadratic variation of the transverse shearing strains and linear variation of the transverse normal strain throughout the plate’s thickness. Additionally, he developed a theory that satisfies zero transverse shear stress conditions on the bounding planes of a generally laminated, fiber-reinforced composite plate subjected to transverse loads. The displacement model accounts for non-linear distribution of in-plane displacement components throughout the plate’s thickness and the theory requires no shear correction coefficients [113]. In addition, they presented analytical formulations and solutions to the static analysis of simply supported composite and sandwich plates using the same theory [114]. Moreover, they showed a general finite element formulation for the plate bending problem based on the same higher-order displacement model and the three-dimensional state of stress and strain [115]. Furthermore, Punera et al. [116] developed two-dimensional kinematic models to analyze CNT-reinforced sandwich cylindrical panels, with a focus on accurately estimating transverse interlaminar shear stress. Building on this work, they investigated the free vibration behavior of functionally graded, open, cylindrical shells using refined, higher-order displacement models [117]. Meanwhile, Chanda et al. [118] explored the impact of porosity on the free vibration and transient responses of functionally graded composite plates, utilizing a higher-order thickness stretching model. Additionally, they studied the electro-elasto-static behavior of porosity-gradient, smart, functionally graded plates, particularly focusing on piezoelectric fiber-reinforced composite materials [119].

In a series of studies using non-polynomial HSDT, the authors investigated various aspects of structural behavior and the performance of composite plates. Grover et al. [120] proposed new non-polynomial shear deformation theories to analyze laminated composite and sandwich plates, expanding the understanding of structural behavior. Soni et al. [121] extended this framework to functionally graded, carbon-nanotube-reinforced plates, exploring static analysis. Joshan et al. [122] introduced a coupled stress model within a non-polynomial framework to study the structural responses of laminated composite micro-plates, providing analytical solutions. Dhuria et al. [123] examined the influence of porosity distribution on the static and buckling responses of porous, functionally graded plates, considering material heterogeneity. Singh et al. [124] assessed the accuracy of new non-polynomial shear deformation theories for static analysis of laminated and braided composite plates, contributing to validation efforts. Chanda et al. [125] focused on stress analysis of smart composite plate structures, investigating practical applications. Singh et al. [126] analyzed the buckling behavior of functionally graded plates under various loading conditions, enhancing the understanding of stability. Chanda et al. [127] studied the flexural behavior of functionally graded plates with piezoelectric materials, exploring potential applications in energy harvesting. Lastly, Chanda et al. [128] proposed non-polynomial shear deformation theories for free vibration and transient analysis of plates with functionally graded materials supported on an elastic foundation.

Based on the above, it is concluded that 3D equilibrium equations need to be applied to achieve more satisfactory results. However, despite the progress made, researchers still need to investigate the integration of HSDT with other advanced theories or numerical methods to enhance its capabilities and applicability to a wider range of practical engineering problems.

4.2. Assessments of Various Methods to Model Laminated Composite Plates

4.2.1. Progress on Analytical and Numerical Models for Different Plates

Several researchers have put remarkable effort into analytically and numerically solving the boundary value problems of different laminated plates [129]. In this section, a brief summary of previously reported studies in the literature on the analytical and numerical analysis of laminated plates is presented. Prior to delving into the analytical and numerical solutions, various mathematical models exist that are based on shear deformation theory, as listed below:

The expansion technique (based on thickness) [130]:

• The mixed variational technique;
• The successive approximations method;
• An expansion method based on asymptotic analysis;
• Symbolic integrations technique;
• Initial functions technique.

The method of hypotheses (2D problem) [66]:

• The semi-inverse technique;
• The technique of trigonometric functions;
• The expansion based on Fourier series.

Analytical solutions are typically limited to cases involving simple geometries and boundary conditions in engineering and mathematics. This section is a review of some research related to analytical solutions of thin and thick laminated composite plates. Generally, analytical solutions are primarily applicable to situations involving specific boundary conditions, simplified theories, and particular cases of loadings.

Numerical techniques have been innovatively developed to efficiently find solutions for diverse structural components under various loadings and boundary conditions, as per Figure 10. These techniques offer the advantage of significantly reducing design time and associated costs for engineers and researchers. The fundamental idea behind FEM is to represent the solution region analytically or through approximation by dividing it into discrete elements (discretization). By assembling these elements in various configurations, it becomes possible to represent highly complicated shapes and structures [131].

4.2.2. Methods for Solution

Navier and Levy Solutions

These classical analytical solutions provide closed-form expressions for the displacement and stress fields in laminated composite plates. They offer simplicity and efficiency in solving specific boundary value problems with known material properties and boundary conditions. Whitney et al. [132] formulated the Navier solution for simply supported rectangular plates, addressing aspects of buckling and bending. Bshaskar et al. [133] investigated the equations of dynamic equilibrium using a combined approach involving the Laplace transform technique and Navier’s method based on FSDT. By applying this approach, the dynamic magnification factor can be calculated for both stresses and deflections. The study used various loading conditions and considered different geometric parameters. Mantari et al. [134] provided numerical results for static and free vibration assessments of cross-ply, simply supported plates using Navier’s solution. The generalized HSDT model has five unknown variables. Adim et al. [135] used refined HSDT to analyze the free vibration and static bending of simply supported laminated composite plates. He considered the parabolic evolution of transverse shear strains in the thickness direction. The theory comprises four unknown displacement components, which were enlarged using trigonometric functions and the Navier’s technique. Dan et la. [136] applied HSDT to study the free vibration response in simply supported beams with solid cross-sections. The Carrera Unified Formulation (CUF) was taken into account for displacement fields. The resultant differential equations were solved analytically using the Navier’s method. Reddy et al. [137] introduced Levy’s solution for symmetrically laminated rectangular plates, encompassing various boundary conditions—for example, free, clamped, and simply supported. These solutions were derived using FSDT. Xiang et al. [138] investigated rectangular plates’ natural frequencies by employing the state–space technique and Levy’s solution, in conjunction with FSDT. Atashipour et al. [139] used TSDT to analyze the bending of thick FG circular plates. Closed-form solutions for transverse plate deflection can be found by utilizing the boundary layer phenomena. Mohammadi et al. [140] presented buckling stresses in thin FG plates using CLPT and studied supported opposing edges and arbitrary boundary conditions along the remaining edges using Levy’s solution. CLPT posits that in-plane displacement components vary linearly with thickness, while material characteristics vary as a power function of the thickness coordinate. Naderi et al. [141] examined the buckling reaction of relatively thick FG sector plates using FSDT. The linked stability equations were decoupled and solved analytically using Levy’s solution for SS conditions on straight edges and free boundaries on circumferential edges.

Elasticity Solutions

Elasticity solutions utilize the principles of continuum mechanics to derive governing equations for laminated composite plates [142]. By solving these equations, engineers can obtain accurate predictions of displacement, stress, and strain distributions. Elasticity solutions offer high fidelity and can handle a wide range of loading and boundary conditions. The methodology begins with fundamental elasticity equations, incorporating lamination theory to model laminated composite plates [143]. This involves applying CLPT or FSDT and HSDT and developing constitutive models for material behavior. Finally, governing equations were assembled and solved using analytical or numerical techniques to predict displacement fields and stress distributions. However, these techniques also come with inherent limitations, particularly in the context of boundary conditions, simple geometry, and loading conditions [144]. In terms of boundary conditions, elasticity solutions for laminated composite plates may face challenges when dealing with complex boundary conditions. Although these techniques can effectively model certain boundary conditions, such as simply supported or clamped edges, they may struggle with more complex conditions involving free edge effects, discontinuities, or non-linear behavior. Similarly, elasticity solutions may encounter difficulties when applied to structures with non-uniform geometry or irregular shapes. However, these solutions can provide accurate predictions for simple geometries, such as rectangular or circular plates, but they may lack robustness when dealing with more complex shapes or geometries with discontinuities [145]. In such cases, approximations and simplifications inherent in the solution methodology may lead to deviations from reality, necessitating caution in interpretation. Regarding loading conditions, elasticity solutions are well-suited for analyzing structures under standard loading scenarios, such as uniform pressure or concentrated loads. However, they may struggle to accurately predict behavior under more complex loading conditions. In such cases, additional considerations and modifications to the solution methodology may be necessary to account for the effects of dynamic loading and time-dependent behavior. Hussainy et al. [146] and Srinivas et al. [147] derived solutions to the bending behavior of moderately thick, rectangular laminates plates under simply supported conditions based on 3D elasticity solutions. They explored the flexural response of two-sided composite plates with different edge conditions, while being supported on the remaining sides, considering the transverse shear effects. Pagano [148] used the same elasticity solution to develop a solution for rectangular laminates with pinned edges. The study included the resolution of specific cases, including a sandwich plate, which were then compared with corresponding solutions obtained using CLPT. Ray [149] derived 3D elasticity solutions for static analysis of rectangular, antisymmetric angle-ply plates. The resulting formulas for stresses and displacements are versatile and can be readily applied to calculate numerical results for plates with various fiber orientation angles. However, it is important to note that the exact solutions obtained for square antisymmetric angle-ply plates with a fiber orientation angle of 45° are not directly applicable to plates that have varying fiber orientation angles. Singh et al. [150] developed a 3D elasticity solution to analyze the mechanical behavior of composite plates with arbitrary support conditions, both in cross-ply and angle-ply, when subjected to patch loads.

Power, Trigonometric, and Fourier Series Approaches

These mathematical techniques decompose complex functions into simpler components, allowing engineers to represent displacement and stress fields in laminated composite plates using series expansions. They offer versatility and efficiency in approximating solutions, particularly for problems with periodic or symmetric loading conditions. Alipour [151] used the application of a power series method based on FSDT to analyze the stress and bending distribution in sandwich and angle-ply laminated composite plates subjected to non-uniform loads. This method offered analytical solutions over a broad spectrum of boundary conditions and diverse load distributions. The outcomes demonstrated its applicability to sandwich plates and moderately thick cross/angle-ply laminated composite, accounting for composite plate orientation, edge conditions, variable elastic foundation conditions, and non-uniform transversely distributed loads. Sayyad et al. [152] proposed a straightforward, double trigonometric series solution based on FSDT to analyze the buckling and bending of cross-ply laminated composite plates. The in-plane displacements were represented using sinusoidal functions in terms of the thickness coordinate, effectively incorporating shear deformation effects. The transverse displacement encompassed both shear and bending components without using K. The stresses, displacements, and critical buckling loads were obtained. Kabir [153] introduced an analytical solution using an innovative approach involving a boundary-continuous, double Fourier series solution based on FSDT, considering the shear flexibility of arbitrary rectangular laminates.

Rayleigh–Ritz and Galerkin Methods

These energy methods minimize the total potential energy of the system by approximating the displacement field using trial functions. Rayleigh–Ritz employed a discrete set of trial functions, while Galerkin used a weighted residual approach. Both methods offer flexibility and accuracy in approximating solutions for laminated composite plates. The Ritz method is an approximate energy-based technique utilized in mechanics problems. In this approach, the displacement function is estimated by composing it as a linear combination of trial functions. It is essential that this displacement function satisfies the geometric or fundamental boundary conditions [154]. It is a classical approach extensively employed to examine the dynamic, buckling, and static responses of structures. It is employed in the determination of moments, stresses, natural frequencies, and critical buckling in vibrating systems. Additionally, it can be applied to describe and solve various boundary value problems for shells, beams, and plates [155]. The Ritz method is among the well-known approaches that have been employed for solving many boundary value problems, which proves its capability and efficiency in predicting the bending behavior of composite laminated plates with boundary conditions including free edges [156,157,158,159,160,161,162,163]. Song et al. [164] employed the Rayleigh–Ritz approach to determine the natural frequency of laminated composite cylindrical shells under arbitrary boundary conditions. Nallim et al. [165] explored the application of orthogonal polynomials to the Ritz method, particularly concerning the examination of rectangular anisotropic plates to determine free vibration and bending deflection. Vaseghi et al. [166] used the hierarchical Rayleigh–Ritz approach to investigate the unilateral buckling of restricted rectangular plates. The study examined composite plate behavior under compressive in-plane stresses using a nonlinear elastic constraint on one side. Kharghani et al. [167] derived an efficient semi-analytical solution to assess composite plates’ load bearing capacity and flexural response. This solution employed the layer-wise HSDT theory in conjunction with the Ritz approximation technique. Belinha et al. [168] developed the Element-Free Galerkin approach to analyze anisotropic laminated plates using FSDT. Fazzolari et al. [169] introduced a trigonometric Ritz formulation for thermal and mechanical buckled anisotropic laminated plates, based on enhanced ESL, LW, and zigzag plate theories, to determine non-dimensional natural frequencies. Wang et al. [170] studied the free vibration of an FG porous cylindrical shell using higher-order sinusoidal theory. The Rayleigh–Ritz approach was used to determine the governing equations of plates under different boundary conditions.

Variational Methods

Variational methods formulate the problem as an optimization task, where the objective is to minimize a functional value representing the system’s energy. By varying trial functions and minimizing the functional value, engineers can obtain accurate solutions for displacement and stress distributions in laminated composite plates. The variational asymptotic approach, created using a mathematically rigorous method and group to analyze laminated structures, guarantees accurate results [171]. Hodges et al. [172] used the variational asymptotical method to split the solution of deformations in laminated plates into two distinct parts. The first part involves linear, one-dimensional analysis of the plate’s thickness, while the second part focuses on nonlinear, two-dimensional plate analysis to investigate the linear cylindrical bending of laminated plates with varying stacking sequences. Madenci et al. [173] introduced a variational approximate method with four nodes, each node having 11-DOF, using HSDT for laminated composite plates.

Differential Quadrature Methods (DQMs)

DQMs discretize the governing differential equations using a meshless approach, offering high accuracy and efficiency, especially for problems with irregular geometries. Chaudhuri et al. [174] introduced a semi-analytical solution to estimate interlaminar shear stress distribution in thick laminated plates. This method employs FEM with a quadratic displacement potential energy assumption for the analysis. Sherbourne et al. [175] examined the precision and convergence of differential quadrature method in relation to addressing various differential equations with changing coefficients related to the buckling of plates. Rectangular plates with different material properties, including anisotropic and composite laminates with orthotropic symmetric angle-plies subjected to linearly changing uniaxial compression, were considered. Jahromi et al. [176] introduced the generalized DQ approach to analyze the vibrations of relatively thick plates on a Pasternak basis, where the plate’s governing equations were derived using FSDT. Ferreira et al. [177] employed DQM to forecast both the free vibration and static deformation characteristics of cross-ply and variable-thickness laminated composite plates, spanning from thin to thick. The study encompassed a range of boundary conditions, including fully clamped or complete, simply supported boundaries, as well as mixed external conditions, involving combinations of free, simply supported, and clamped. Employing FSDT, Khalid et al. [178] introduced a novel method known as the two-dimensional inverse DQM for approximating solutions to the higher-order systems of differential equations related to laminated composite plates. Through numerical assessments, the buckling and bending of plates with varying boundary conditions and loadings can be determined. The proposed method’s accuracy was found to be comparable to FEM and Navier’s solution. Based on FSDT, Szekrényes [179] utilized DQM to analyze composite plates. The investigated plates were subjected to concentrated transverse forces at their midpoints. The study verified the utility of comparing the convergence of deflection obtained through the DQM with deflection values derived from spatial FEM and analytical methods. Civalek et al. [180] analyzed the vibration of laminated and FGM composite annular sector plates using the harmonic DQM, applying both FSDT and CLPT.

Engineering systems are typically governed by systems of high-order differential equations which require efficient numerical methods to provide reliable solutions, subject to imposed constraints. The conventional approach of directly approximating system variables can potentially incur considerable error due to the high sensitivity of high-order numerical differentiation to noise, thus necessitating improved techniques that can better satisfy the requirements of numerical accuracy, which are desirable in solutions of high-order systems. To this end, Ojo et al. [181] developed a novel inverse differential quadrature method (iDQM) for the approximation of engineering systems. A detailed formulation of iDQM, based on integration and DQM inversion, for the approximation of arbitrary low-order functions from higher derivatives is proposed separately. Also, they performed static analysis of laminated composite structures based on the theory of Unified Formulation (UF) and mixed methods, comprising a combination of the high-order finite element (FE) method and the new iDQM, to obtain adequate computational frameworks that can accurately estimate displacement and stress fields resulting from systems of high-order partial differential equations [182]. Additionally, Chanda et al. [183] proposed a new, high-order computational tool that combines the higher-order accuracy of the emerging inverse differential quadrature method (iDQM) and the simple kinematics of the Timoshenko beam theory for efficient and accurate prediction of the static behavior of both constant stiffness and variable stiffness curved beam structures. This novel application of iDQM to curved beam analysis is leveraged upon its excellent potential to mitigate differentiation-induced errors by using the so-called indirect approximation strategy.

iDQM approximation provides accurate solutions without losing computational efficiency and offers superior numerical stability over DQM solutions.

Carrera Unified Formulation (CUF)

The Carrera Unified Formulation (CUF) stands out as an exceptionally efficient tool in the realm of structural analysis for composite materials. Developed by Prof. Erasmo Carrera, CUF offers a systematic approach to generating higher-order theories, providing engineers with a powerful framework for analyzing complex composite structures. One of the key strengths of CUF is its ability to unify various classical and advanced theories into a single mathematical framework. By doing so, CUF streamlines the process of formulating higher-order theories, which are essential for accurately capturing the behavior of composite materials under different loading conditions. CUF’s efficiency stems from its unique hierarchical formulation, which allows engineers to systematically include additional terms in the analysis to account for higher-order effects. This hierarchical approach not only ensures accuracy in predicting structural behavior but also offers computational efficiency, making it suitable for analyzing large-scale composite structures [22,36,57,136,169,177,184,185]. Carrera et al. [186] developed a new, higher-order theory for micropolar plates and beams using the CUF approach. The higher-order theory is based on the variational principle of virtual power and the expansion of the 3D equations of the micropolar theory of elasticity into generalized series in terms of cross-section coordinates. The 1D theory is developed from general 3D equations of linear micropolar elasticity using the principle of virtual power. Also, in relation to the composite shells of revolution, they developed a variational principle of virtual power for the 3D linear anisotropic theory of elasticity and generalized series of the shell thickness coordinates [187]. Furthermore, they presented an accurate evaluation of complete three-dimensional (3D) stress fields in beam structures with compact and bridge-like sections. They employed a refined beam finite element (FE) formulation, which permits any-order expansions of the three displacement components over the section domain by means of the CUF [188]. Cinefra et al. [189] evaluated the robustness of plate elements based on the Mixed Interpolation of Tensorial Components (MITC) technique and the variable kinematics approach of (CUF), with respect to the problem of distorted meshes. They obtained refined plate elements by referring to high-order equivalent single-layer and layer-wise models expressed in the CUF for the analysis of multilayered anisotropic structures. The MITC technique was originally proposed for Reissner–Mindlin-type plates to develop shear-locking-free plate elements.

Recent Analytical Methods

Based on FSDT, Dorduncu et al. [190] presented a new approach using a nonlocal peridynamic differential operator to perform static analyses of simply supported laminated composite plates. In his formulation, the peridynamic differential operators were used instead of the traditional local spatial derivatives. Pathan et al. [191] presented a solution for the static analysis of functionally graded and smart laminated composite plates. The deformation kinetics within smart structures were represented using five newly introduced non-polynomial HSDT theories. Based on HSDT, Shukla et al. [192] investigated angle-ply laminated plates using a method that employed multiquadric radial basis functions in a mesh-free approach for flexure analysis. It enabled the calculation of deflection and stress results for both cross-ply and angle-ply laminated plates. Naik et al. [193] Top of Form introduced a unique analytical solution to examine the bending behavior of mechanically loaded, simply supported sandwich and laminated composite plates. The analysis employed HSDT, which utilized polynomial-type form functions up to the fifth order in the displacement field. Additionally, they presented the same theory formulated for the comprehensive analysis of the same plates undergoing bi-directional bending under transverse loads. In the displacement field, this theory involved nine unknowns, without using shear correction factor [194]. Li et al. [195,196] analyzed thin rectangular cantilever plates using the superposition method in symplectic space (SSM) to achieve a reasonable analytical solution for free vibration characteristics. The SSM technique provides a new analytical solution to the buckling response of thin rectangular plates. Chanda et al. [197] presented a non-polynomial, higher-order plate theory with zigzag kinematics, involving a trigonometric function and a local segmented zigzag function, for modeling the deformation of a smart piezoelectric laminated composite plate supported on an elastic foundation. This model has only five independent primary variables, like that of the FSDT, yet it considers the realistic parabolic behavior of the transverse shear stresses across the thickness of the laminated composites plates and also maintains the continuity conditions of transverse shear stresses at the interfaces of the laminated plates.

Finite Element Analysis

FEM is widely used for the numerical analysis of laminated composite plates due to its ability to handle complex geometries and boundary conditions. It offers high accuracy and flexibility in modeling various material behaviors and loading conditions. Roufaeil et al. [79] introduced a novel method for laminated plates by analyzing the finite strip solution and incorporating FSDT. The study predicted that displacements $v$ and $u$ vary linearly over the thickness of individual layers and maintain continuity at interfaces adjacent to each other. Additionally, throughout the thickness, constant displacement w was assumed. As a significant outcome of their work, they developed two and three-node finite strip elements. Pryor et al. [198] applied CLPT and formulated FEM using the principles of Mindlin and Reissner theories. In their approach, they assumed that the transverse displacement of the composite plate was constant across its thickness. Additionally, they assumed that the two normal direction displacements varied linearly within each lamina. Turvey [199] introduced both approximate and exact methods to examine the flexural behavior of simply supported, antisymmetric, rectangular, moderately thick laminated plates. He examined how transverse shear deformations impact the plates’ response to different lateral load distributions. Reddy et al. [200] examined how element type (linear or quadratic), mesh size, and the application of reduced integration impacted the accuracy of FEM with governing anisotropic, laminated, thick composite plates. Owen et al. [201] introduced a finite element local model that builds upon enhanced approximate theory for anisotropic thick laminated plates. They demonstrated the practicality and cost-effectiveness of this model by successfully predicting plates’ in-plane displacements, transverse shearing stresses, and static bending stresses. Rolfes et al. [202,203] developed a novel FEM technique to compute enhanced transverse shear stresses in composite laminated plates, applying FSDT. The core concept involved directly estimating transverse shear stresses based on transverse shear forces, without taking membrane forces into account. The method eliminated the need for shear correction factors. Additionally, they developed a method to compute the transverse normal stress in the direction of thickness in layered composite plate systems. Their examples of antisymmetric angle-ply and symmetric cross-ply laminates in numerical form demonstrated that the proposed approach closely approximated the exact 3D elasticity solution. This holds true for thin and relatively thick plates, with a slenderness ratio as low as five. Touratier [204] conducted an FEM to examine plates with large size-to-thickness ratios and to obtain precise shear constraints without using shear correction factors. The proposed theory was then assessed for accuracy by investigating various significant problems, including buckling and bending of three-layered (laminated and sandwich), simply supported, rectangular or square, symmetric cross-ply plates. Zhang et al. [205] obtained LDT18, which is a basic displacement-based, three-node, 18-DOF operator. This element was designed for linear FEM of both thick and thin laminated composite plates using FSDT. Timoshenko’s laminated composite beam functions were used to calculate the rotation and deflection along the element’s boundary, to prevent shear-locking problems. Sze et al. [206] presented a predictor–corrector approach for the FEM analysis of laminated composite plates. In this approach, Mindlin finite element models were used to compute predictive strain, deflection, and stress. A least square fit was used to estimate the corrector transverse shear stress, predictor in-plane stress, and condition of homogenous stress equilibrium. Subsequently, the corrector deflection was calculated, based on FSDT. Ge [207] developed an improved FEM, discrete, 15-DOF, triangular, laminated composite plate element based on FSDT. The proposed element incorporates the shear strain, which is suitable to analyze both thin and moderately thick anisotropic, laminated composite plates. Singh et al. [208] proposed a new rectangular material finite element based on FSDT, free from any locking issues, that had 5-DOF at each node. The shape functions, derived from the equilibrium equations of moment shear, incorporated the element coordinates, material properties, and thickness. Gim [209] created a 2D finite element based on FSDT to model delamination in laminated plates, allowing for computation of the plate element’s strain energy. Kosmatka [210] introduced a novel six-node triangular plate element for analyzing laminated composite structures. It was based on Hamilton’s concept and incorporated FSDT, without shear locking, and exhibited the correct rank. Through numerical validation, the new element demonstrated remarkable convergence performance compared to available triangular elements. Kabir [211] developed an FEM for fiber-reinforced, laminated, simply supported plates with cross-ply laminations under uniform transverse loads with various thicknesses, using an isoparametric, three-node triangular finite element. The element’s strain–displacement relations were based on FSDT, which accounted for transverse shear deformations. Also, in the context of FSDT, To et al. [212] introduced straightforward, three-node, 6-DOF per node, triangular shell finite elements designed for laminated composites. The outcomes demonstrated that the suggested simplified plate elements provided exceptionally precise results for length-to-thickness ratios higher than or equal to 20. Günay et al. [213] introduced a nine-node quadrilateral element, capable of shear deformation and heterosis. To create this element, the Kirchhoff constraints were adjusted using FSDT assumptions. Sadek [214] proposed three rectangular, eight-node elements and implemented them to analyze cross-ply square composite plates with different thickness ratios. These elements were designed to include transverse shear deformations by accounting for FSDT-HSDT terms in the thickness coordinate. Load, stiffness, and stress matrices were derived for these elements. Auricchio et al. [215] developed a four-node finite for analyzing laminated composite plates using FSDT, considering the element’s advantage of being locking-free. Additionally, this element also includes an algorithmic iteration for evaluating the shear correction factors. Alfano et al. [216] introduced novel nine- and four-node elements for analyzing laminated composite plates using FSDT. These elements are based on the model of tensorial components developed by Bathe et al. [217,218] and have the advantage of being locking-free. This method produces very good results for the normal and transverse stresses. Rao et al. [219] used Toledano–Murakami higher-order bending theory to construct a unique FEM formulation to examine thick laminated plates comprising layers of arbitrary orientations. The investigation involved changing the fiber orientation angle inside a rectangular laminate’s bottom and top layers to assess in-plane stresses and central deflection. Vinh et al. [220] presented an innovative approach for static bending analysis of functionally graded (FG) plates using a four-node quadrilateral plate element and HSDT. It is achieved by incorporating higher-order terms that have quadratic shape functions. Mathew [221] employed a least square finite element solution to obtain displacements and stresses through a state–space model; they applied this solution to anisotropic, linear elastic, multilayered rectangular plates subjected to arbitrary boundary conditions on one edge while being simply supported on the other edge.

Meshless Techniques

These methods avoid the need for mesh generation, offering advantages in handling complex geometries and reducing the computational effort associated with mesh generation and refinement. Meshless methods have been shown to be capable of handling a wide range of engineering issues, and they outperform traditional numerical methods, particularly when dealing with discontinuities and large deformation problems [222,223]. Wang et al. [224] introduced a meshless technique utilizing the kernel particle approach to investigate the buckling and flexural behavior of rectangular laminated composite plates. This method adopted FSDT and constructed displacement shape functions through reproducing kernel particles, thereby ensuring adherence to consistency conditions. By embracing this meshless methodology, an efficient solution approach to analyzing laminated composite plates was achieved. Singh et al. [225] employed a meshless approach to analyze laminated composite plates, and deflections were measured using HSDT. Various RBFs were used to discretize differential equations and approximate solutions. Sator et al. [226] analyzed the bending of thin and thick elastic FG plates utilizing CLPT, FSDT, and HSDT. They used a meshless approach with the moving least square approximation to govern field variables. Zarei et al. [227] developed a mesh-free method of radial point interpolation for conducting buckling and static analyses of thick laminated plates. This method takes into consideration the zigzag displacement field changes with plate thickness and ensures consistency of interlaminar transverse shearing stresses. Do et al. [228] investigated the bending response using HSDT and the buckling response using nth-order shear deformation theory [229] of multilayered composite FGM plates, while also using an enhanced, meshless, moving kriging approach.

Boundary Element Method

The boundary element method (BEM) focuses on discretizing only the boundaries of the domain, leading to reduced computational effort compared to FEM. It is particularly suitable for problems with infinite domains or where the boundary conditions dominate the solution. Hsu et al. [230] developed a boundary element method based on FSDT to address a diverse set of complicated problems linked to thick laminated plates, including deformations. This method is versatile and can handle coupling problems, which often occur in unsymmetric laminated composite plates. Garg et al. [231,232] examined the analysis of poorly bonded laminated composite plates under cylindrical bending. They also investigated thick laminated composite plates subjected to bi-axial bending using the scaled boundary finite element method (SBFEM), a relatively recent approach. Ramos et al. [233] used a pure BEM to analyze the static behavior of 2D and 3D structures made out of thin, unsymmetric, laminated composite plates. Zang et al. [234] presented a novel numerical approach based on isogeometric analysis; this involved utilizing the scaled boundary finite element method, focusing on improving the computing performance (such as accuracy, efficiency, and flexibility) of bending and free vibration analyses for laminated composite plates with rectilinear and curvilinear fibers constrained or free from elastic foundations.

Recent Numerical Methods

Ferreira et al. conducted several studies on the collocation method for the analysis of isotropic laminated composite plates and double-curved composite shells. He utilized Deslaurier–Dubuc interpolating wavelets to evaluate bending, static, and free vibration analyses [235]. FEM was employed to discretize the plates, assuming constant transverse shear deformation based on FSDT. The meshless approach based on wavelet collocation was used for the discretization of boundary conditions and equilibrium equations. Their research covered topics such as elastic buckling loads [236], bending response [237], and natural frequencies [238], employing the collocation technique with interpolating wavelets for structural analysis. The studies demonstrated the applicability of this method for various analyses in composite materials. Using FSDT, Kien et al. [239] developed a radial-basis-function-based finite element approach to construct shape functions for analyzing laminated composite plate buckling, bending, and vibration. Compared to the traditional FEM, it simplifies the construction of shape functions as the number of nodes in the element increases. Using HSDT, Shukla et al. [240] employed a radial-basis-function-based, mesh-free approach for analyzing the buckling of both antisymmetric and symmetric laminated plates. Sabherwal et al. [241] predicted the free vibration behavior of laminated sandwich plates using the wavelet finite element (WFE). Different kinds of mother wavelets, namely, the B-spline wavelet on the interval (BSWI), Gaussian, Haar, Daubechies 6 (db6), Biorthogonal 3.7 (bior3.7), Coiflet5 (coif5), Symlets (sym8), Morlet, Mexican hat (Mh), and Meyer mother wavelets, are employed in WFE for predicting vibration frequencies.

Artificial Intelligence Based on Artificial Neural Network Approaches

ANN techniques leverage machine learning algorithms to learn complex relationships between input and output data. By training ANNs on a dataset of known solutions or experimental data, engineers can develop predictive models capable of accurately predicting displacement and stress fields in laminated composite plates. ANN approaches offer versatility and the ability to handle nonlinearities and complex interactions within the material system. Statistically, within the engineering discipline, artificial neural networks appear to be one of the great successes of structural engineering computing [242]. Artificial neural networks are widely used in engineering due to their efficacy in handling strong non-linear relationships [243]. Especially valuable when conventional methods demand significant computational resources or time, these networks find applications in diverse areas, including building material studies, structural identification (e.g., analysis of laminated composite structures), geotechnical engineering (e.g., earthquake-induced liquefaction potential), civil engineering heat transfer problems, transportation engineering (e.g., traffic problem identification), construction technology and management (e.g., estimating building costs), and building services (e.g., analyzing water distribution networks) [244]. Rjoub et al. [245] used an ANN model to forecast the frequency of porous FGM plates with a side crack. Third-order plate theory is used to examine how porosity and cracking affect dynamic behavior. Saadatmorad et al. [246] used convolutional neural networks (CNNs) and two-dimensional wavelet transforms to detect damage in rectangular laminated plates. Martinez et al. [247] analyzed the reliability of a smart laminated plate with a piezoelectric, fiber-reinforced composite actuator layer using an artificial-neural-network-trained model. Atilla et al. [248] studied the stability and vibration of composite plates with circular cutouts.

The previously mentioned studies primarily focus on solutions tailored to specific stacking sequences or address particular boundary conditions, showcasing a limitation in applicability. Additionally, these investigations provide an extensive survey of diverse methodologies utilized for the analysis of laminated composite plates. The spectrum of approaches discussed spans classical analytical methods, finite element methods (FEMs), meshless techniques, boundary element methods (BEMs), and artificial-intelligence-based approaches, with a specific emphasis on artificial neural networks (ANNs). The discourse encompasses a comprehensive exploration of each method’s relevance to the study of laminated composite plates, encompassing aspects like bending, buckling, and free vibration. Furthermore, it offers a meticulous examination of various methodologies employed in the analysis of laminated composite and functionally graded (FG) plates, presenting valuable insights into the strengths and implications associated with each approach.

5. Conclusions

To summarize, this research has established the development and rapid expansion of a surprisingly comprehensive body of global literature on laminated composite plates, highlighting its potential as an effective and significant component. Using VOS viewer version 1.6.19, a powerful bibliometric analysis tool, bibliometric maps were generated to visualize the connections between various research areas, authors, and scientific publications using different types of mappings such as journals, authors, keywords, and countries. This analysis aided in identifying key research trends, influential authors, and emerging areas of interest, facilitating a deeper understanding of the field and guiding future research directions.

Based on the obtained analysis in this review, it can be concluded that, with every passing year, the rate of research in this area is consistently increasing. Due to the versatility of materials, authors are proposing different theories globally. Thus, it is a field with a tremendous scope for future research. The potential for new discoveries is high; development analysis of the literature indicates that the trend in research considering technology in laminated composite plates will significantly increase during the next decade. These findings imply that the literature in this field has vital significance; still, some areas warrant further investigation. For example, more research is needed to:

• Achieve global engineering methods that are more efficient and cost-effective, especially in the fields of composite materials and finite element modelling.
• Conduct experimental studies to validate theoretical findings.
• Apply advanced methodologies such as computational simulations or machine learning to analyze complex systems under various boundary conditions.
• Employ statistical techniques to quantify the significance of different factors and their interactions.
• Conduct pilot-scale and full-scale studies to evaluate the feasibility of laminated composite plate applications.
• Perform sensitivity analysis to assess the sensitivity of models or systems to changes in boundary conditions.