This paper reviews the Composite Laminate Theories that have already been proposed and developed in the recent years. These theories mainly focus on the macro mechanical analysis of the composite laminates which provides the elastic relations of the lamina. Stress-induced failure can occur in multiple ways in composite materials. Hence to understand and predict transverse shear and normal stress accurately, various composite laminate theories have been developed. The advantages and disadvantages of each model are discussed in detail. In this study, the Composite Laminate Theories are divided into two parts: (1) Single Layer Theory, where the entire plate is considered as one layer and (2) Layer Wise Theory, where each layer is treated separately for the analysis. It starts with displacement-based theories from very basic models such as Classical laminate theory to more complex higher-order shear deformation theory. [6]
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INTRODUCTION
The requirement of composite materials has grown rapidly. These materials are ideal for applications that require low density and high strength. Composite materials provide great amount of flexibility in design through the variation of the fiber orientation or stacking sequence of fiber and matrix materials. The mechanical behavior of laminates strongly depends on the thickness of lamina and the orientation of fibers. Hence, the lamina must be designed to satisfy the specific requirements of each particular application and to obtain maximum advantage from the directional properties of its constituent materials. The normal stresses and through-thickness distributions of transverse shear for composite materials are very important because in laminate composite plates, stress-induced failures occur through three mechanisms. For instance, when the in-plane stress gets too large, then the fiber breakage occurs. However, normally before the in-plane stresses exceed the fiber breakage point, inter laminar shear stress failure occurs when one layer slips tangentially relative to another. Alternatively, transverse normal stress may increase enough to cause failure by which two layers pull apart from each other. Therefore, it is imperative to understand and calculate transverse shear and normal stress through the thickness of the plate accurately. In general, two different approaches have been used to study laminated composite structures, which are: (1) single layer theories and (2) discrete layer theories. In the single layer theory approach, layers in laminated composites are assumed to be one equivalent single layer (ESL) whereas in the discrete theory approach, each layer is considered separately in the analysis. Also, plate deformation theories can be categorized into two types: (1) displacement and (2) stress -based theories. A brief description of displacement-based theories is given below: displacement-based theories can be divided into two categories: classical laminate theory (CLT) and shear deformation plate theories. Normally, composite laminate plate theories are described in the CLT, the first-order shear deformation theory (FSDT), the global higher-order theory, and the global-local higher shear deformation theory (SDT).
DESCRIPTION:
In the studies carried out in last few decades, many different theories were presented to overcome various issues and explain the behaviors of composite materials more accurately. In this paper, these theories are reviewed, categorized, and their advantages, weaknesses and limitations are discussed in detail.
LAMINATED COMPOSITE PLATES
Classical Laminate Theory (CLT)
The simplest ESL laminate plate theory is the CLT, which is based on displacement based theories. In the nineteenth century Kirchhoff initiated the two-dimensional classical theory of plates and later on it was continued by Love and Timoshenko. The principal assumption in CLT is that normal lines to the mid-plane before deformation remain straight and normal to the plane after deformation. The other assumptions made in this theory are (1) the in-plane strains are small when compared to unity (2) the plates are perfectly bonded (3) the displacement are small compared to the thickness. Although these assumptions lead to simple constitutive equations, it is also the main limitation of the theory. These assumptions of neglecting the shear stresses lead to a reduction or removal of the three natural boundary conditions that should be satisfied along the free edges. These natural boundary conditions are the bending moment, normal force and twisting couple. Despite its limitations, CLT is still a common approach used to get quick and simple predictions especially for the behavior of thin plated laminated structures. The main simplification in this model is that 3D structural plates ( with thickness ) or shells are treated as 2D plate or shells located through mid-thickness which results in a significant decrement of the total number of equations and variable, consequently saving a lot of computational time and effort. Since they are present in closed-form solutions, they provide better practical interpretation and their governing equations are easier to solve [6]. This approach remains popular because it has become the foundation for further composite plate analysis theories and methods. This method works relatively well for structures that are made out-of a balanced and symmetric laminate, experiencing either pure tension or only pure bending. The error which is introduced by neglecting the effect of transverse shear stresses becomes trivial on or near the edges and corners of thick-sectioned laminate configurations. It is observed that the induced error increases for thick plates made of composite layers. This is mainly due to the fact that the ratio of longitudinal to transverse shear elastic moduli is relatively large compared to isotropic materials [2]. It neglects transverse shear strains, under predicts deflections and overestimates natural frequencies and buckling loads [3]. Composite plates are, subjected to transverse shear and normal stresses due to their discontinuous through-thickness behavior and their global anisotropic nature [3]. In order to achieve better predictions of the response characteristics, such as bending, buckling stresses, torsion, etc., a number of other theories have been developed which are presented in following sections [6].
Figure1. Deformation Hypothesis [Taken from class notes. Advanced Plate Theory.1]
Displacement and strain field for CLT are given below:
[Taken from class notes. [1]]
First-order shear deformation theories (FSDT)
Reissner and Mindlin developed the conventional theories for analyzing thicker laminated composite plate which also considered the transfer shear effects. These theories are popularly known as the shear deformation plate theories. Many other theories, which are extension of SDT, have also been proposed to analyze the thicker laminated composite. These theories are primarily built on the assumption that the displacement w is constant through the thickness while the displacements u and v vary linearly through the thickness of each layer. In general, these theories are known as FSDT. The primary outcome of this theory is that the transverse straight lines will be straight both before and after the deformation but they will not be normal to the mid-plane after deformation. As this theory postulates constant transverse shear stress, it needs a shear correction factor to satisfy the plate boundary conditions on both the lower and upper surface. The shear correction factor is introduced to adjust the transverse shear stiffness values and thereby, the accuracy of results of the FSDT will depend notably on the shear correction factor. Further research has been undertaken to overcome the limitations of FSDT without involving higher-order theories to avoid increasing the complexity of the equations and computations [2, 7]. Authors like Bhaskar and Varadan [23] used the combination of Navier’s approach and a Laplace transform technique to solve the equations of equilibrium. Onsy et al. [4] presented a finite strip solution for laminated plates. They used the FSDT and assumed that the displacements u and v vary linearly through the thickness of each layer and are continuous at the interfaces between adjacent layers. They also postulated that the displacement w does not vary through the thickness. These assumptions provide a more realistic situation (when compared with CLPT) where in the shear strains are not continuous across the interfaces between adjacent lamina. The other limitations are (1) assumption of constant shear stress is not correct as stresses must be zero at free surfaces. (2) FDST produces accurate results only for very thin plates. In order to calculate transverse shear more accurately, to satisfy all boundary conditions and to analyze the behavior of more complicated thick composite structures under different loading condition and to overcome the limitations the use of higher-order shear deformation theories are imperative[1].
Figure2. Reissner – Mindline Plate [picture taken from MAE 557 class notes. 1]
Higher Order Shear Deformation Theory:
The limitations of the CLT and the FSDT have persuaded the researchers to develop a number of global HOSDT. The higher-order models are based on an assumption of nonlinear stress variation through the thickness [1]. These theories are developed for thick plates but are predominantly 2D in nature. These theories are capable of representing the section warping in the deformed configuration. At the layer interfaces, some of these models do not satisfy the continuity conditions of transverse shear stresses. Although the discrete layer theories do not have this concern, they are computationally slow when solving these problems because of the fact that the order of their governing equations purely depends on the number of layers [24]. Whitney attempted to examine the problem with inter laminar normal stress [25]. Several authors were involved in developing this theory , for instance the calculation of inter laminar normal stress was studied by Pagano [26], a boundary layer theory by Tang [6], the perturbation method by Hsu and Herakovich , and an approximate elasticity solutions by Pipes and Pagano. In most of these models, the laminate is assumed to be reasonably long. The stress singularities were considered in a model presented by Wang and Choi. In order to determine the stress singularities at the laminate free edges, Wang and Choi used the Lekhnitskii’s [27] stress potential and the theory of anisotropic elasticity. The Eigen function technique developed by them uses a collocation system at every ply interface to satisfy continuity. The major limitation of this theory is that it can be applied to only relatively thin laminates [17]. In order to explain plate deformation for composite laminate plates with thickness, Ambartsumian proposed a higher-order transverse shear stress function. Various different functions were proposed by Reddy [2], Touratier , Karama and Soldatos. The results of some of these methods were compared by Aydogdu [23]. For example, a 2D higher-order theory is developed by Matsunaga to investigate buckling in isotropic plates for in-plane loads where the effects of transverse shear and normal deformations are predicted in his study. Higher-order theories, which consider the complete effects of transverse shear, normal deformations and rotary inertia, have been studied for the vibration and stability problems of specific laminates. In general, researchers who have wanted to simulate plates have used the third-order shear deformation theories (TSDTs) which was first published by Schmidt and later developed by Jemielita. This theory is also known as parabolic shear deformation plate theory (PSDPT). Researchers like Phan and Reddy [30] applied this theory for the free vibration, the bending and the buckling of composite plates [23]. The same unknown displacements as those used in FSDT were used. This theory also satisfies transverse shear-free conditions at the outer surfaces. The results obtained show that for the thick laminates the in-plane stresses are predicted much well than those identified using FSDT, but still these results have errors when compared with 3D models. This theory is not based on the layer-wise type, therefore, unlike most of other ESL theories, it does not satisfy the continuity conditions of transverse shear stresses between layers [9]. Vuksanovic proposed another parabolic distribution of shear strains through the laminated plate thickness which has a cubic variation of in-plane displacement. The results confirm that this model can predict the global laminate response better than previous used parabolic methods. The primary limitation is that it is challenging to accurately compute the inter laminar stress distributions [9]. In the third-order shear deformation theories assumes (1) the in-plane displacements are a cubic expression of the thickness coordinate (2) the out-of-plane displacement is a quadratic expression. Carrera presented a third-order shear deformation theory which based on the model which was presented by Vlasov for equation of bending plates. By imposing homogeneous stress conditions with correspondence to the plate top-surface the reduced third-order shear deformation model with only three displacement variables was obtained. This was further modified in the same research for the non-homogeneous stress conditions[6].
Figure2. Displacement field and transverse shear stress field for the various composite laminate theories. [* Figure taken from class notes. Advanced_plate_theory.pdf]
Trigonometric shear deformation plate theory (TSDPT)
Shear deformation plate theories which use trigonometric functions are called TSDPT. In this theory the ability to predict accurate solutions has been enhanced by combing trigonometric terms with the algebraic through the- thickness terms assumed for the displacements. Touratier [48] chose transverse strain distribution as a sine function [23]. Stein developed a 2D theory wherein the displacements are stated by trigonometric series. Stein and Jegley studied the effects of transverse shear stress on the cylindrical bending of the laminated composite plates [31]. The results obtained from these theories show that this theory calculates the stresses more accurately than other theories. Kassapogolou and Lagace used the principle of minimum complimentary energy to introduce a straightforward method to analyze symmetric laminate plates which are subjected to tension/compression [15, 16]. Afterwards Kassapogolou [28] generalized and modified this approach for general unsymmetrical laminate loads i.e. in-plane and out of- plane common moment and shear loads. The shortcomings of this model are that it does not solve the weaknesses of inequality in Poisson’s ratios. Becker [29] made use of cosine and sine functions for warping deformation of v and w displacement, respectively and developed a closed-form higher-order laminated plate theory. Mortan and Webber presented an analytical method which took into consideration the thermal effects in their model and by using the same approach as Kassapoglou and Becker. Lu and Liu [22] proposed an inter laminar shear stress continuity theory in which the inter laminar shear stress is directly obtained from the constitutive equations. This theory was postulated in order to develop an accurate theory for inter laminar stress analysis by considering both the transverse shear effects and continuity requirements. The drawback in this model is that the deformation in the thickness is neglected and therefore it cannot calculate the inter laminar normal stress directly from the constitutive equations. Later, Lu and Liu [21] developed the interlayer shear slip theory based on a multilayer approach model by investigating the effect of interfacial bonding on the behavior of composite laminates. Finally, Lee and Liu also derived the closed-form solutions for the general analysis of inter laminar stresses for thin and thick composite laminates under sinusoidal distributed loading. Both inter laminar shear stress and inter laminar normal stress at the composite interface were satisfied in this model and also the inter laminar stresses could be calculated directly from the constitutive equations. Touratier [20] proposed a theory based on using certain sinusoidal functions for shear stress. After comparing the results obtained numerically for the bending of sandwich plates it was shown that this theory is more accurate than both FSDT and HOSDT [17].
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Hyperbolic shear deformation theory (HSDT):
Soldatos proposed the hyperbolic shear deformation theory [6]. Timarci and Soldatos combined the various shear deformation theories to formulate this HSDT. The major advantage of this unified theory is the ability to vary the transverse strain distribution [19]. Authors like Ramalingeswaran and Ganesan [18] have used parabolic and hyperbolic function to uniform external pressure and a simply supported cylindrical shell for cross ply laminated composite by considering an internal sinusoidal pressure [17]. Karama et al. proposed an exponential function for the transverse strain for his study of the bending of composite [6].
Layerwise Theory (LT): (Zigzag effect)
In order to present accurate results many new theories like the layerwise theory and individual layer theory have been formulated. Some of the eminent researches involved in developing these theories are Wu, Chen, Plagianakos[13], Saravanos, Fares and Elmarghany[14]. The basic technique employed in these theories is that assuming certain displacement and stress models in each layer of the composite laminates and in order to reduce the unknown variables, equilibrium and compatibility equations are defined at the interface. These theories are often computationally time consuming and very expensive to obtain accurate results due to the fact that they use many different unknowns for multilayered plates. To predict both gross response and the stress distributions accurately a number of layer wise plate models which can represent the zigzag behavior of the in-plane displacement through the thickness have been developed in the recent past. However, in the Layerwise Theories the major drawback is that the number of unknown increases significantly with the number of layers and consequently the computational weight becomes considerably heavier and higher. To overcome this problem, various solutions were suggested. Cho and Parmerter presented a model where in the number of the unknowns is independent of the number of layers. They achieved it by superimposing a cubic varying displacement field on a zigzag linearly varying displacement [6]. This method was very efficient as it satisfied the transverse shear stress continuity at the layer interfaces and shear-free surface conditions. The theories that have been developed to justify through-the-thickness piece-wise behavior of stresses and displacement are often subjected to zigzag theories (ZZ). The zigzag effect can be termed as the different tangential elastic compliances of the plies which cause the displacement components to show a quick change of their slopes in the thickness direction at each layer interface. To summarize, the in-plane stresses can be discontinuous at each layer interface, while the transverse stresses, for equilibrium, must be continuous. In ZZ theory, the compatibility of the displacements and the inter laminar equilibrium of the transverse stresses in the thickness direction are assured by defining a new stiffness matrix [12]. Lekhnitskii was one of the pioneers who tried to define a ZZ theory. The main drawback for this model is the limitation of the approach to only multi-layered composite where each layer is isotropic. Ren later improved this model by using an extension of the theory developed by Reissner to multi-layered plates. This approach used a Lagrange function with five parameters, which represents the DOF of the structure.
These are the set of displacement field equations used in zigzag theory. This equation are taken from the class notes, advanced plate theory [1]
Mixed Plate Theory:
In order to overcome the limitations of each composite laminate theory researchers have started to unify the different laminate theories. Unified equations have been proposed for mixed layer wise and mixed ESL theories. The main aim is to formulate these unified theories in the most general way for users to be able to choose from the approaches like ESL, Layer wise zigzag, etc. and at the same time choose the order of the expansion of displacements and transverse stresses. This class of model has been contemplated over the last few decades. The so-called mixed variation approach based on the variation principles developed by Hellinger was proposed and then improved by Reissner. In this theory the number of variables that must be computed should be at least 2n, where n is the total number of layers. By using the weak form of Hooke’s Law the number of variables can be significantly reduced, which shows the variables in terms of the three displacements only. Shimpi et al. [11] derived two novel formulations with only two variables, which work perfectly for moderately thick isotropic plates. The major limitation is that it requires accurately calculated shear correction factors for transverse shear stresses in multilayered composite plates.
Conclusion
In this literature review, various composite laminate plate theories have been categorized. The advantages and limitations of each model have been discussed in detail. This paper mainly focused on how efficiently and accurately the various models can predict the transverse shear effects. It is explained that CLT and FSDT are unable to efficiently predict transverse shear stresses of both moderately thick and thick laminated composite plates. Higher order shear deformation theories were developed in order to obtain accurate transverse shear stresses. Also, the zigzag theories satisfying inter laminar continuity of transverse shear stresses at interfaces is unable to accurately compute transverse shear stresses directly from constitutive equations. 3D equilibrium equations have to be adopted to accurately obtain transverse shear stresses which also require heavy computational processing because of the large number of variables which depend on the number of layers, they become impractical for engineering applications. Some researchers have recently tried to use the transverse shear and warping effect in highly anisotropic composite to passively control the composite structure. Smart passive adaptive structures are a new technological approach for introducing smart and predictable composite materials with wide ranging applications. They can be used to exploit the effects of shear and elastic coupling and link stretching to bending to twisting of the structure. This requires a higher understanding and in-depth knowledge of inter laminar shear. Therefore, predicting transverse shear effect accurately and in practical way for various engineering applications is imperative.
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