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Advances in Composite Laminate Theories Essay

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] Advances in Composite Laminate Theories Essay Paper

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). Advances in Composite Laminate Theories Essay Paper

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. Advances in Composite Laminate Theories Essay Paper

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. Advances in Composite Laminate Theories Essay Paper. 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]. Advances in Composite Laminate Theories Essay Paper

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. Advances in Composite Laminate Theories Essay Paper. 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]. Advances in Composite Laminate Theories Essay Paper

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. Advances in Composite Laminate Theories Essay Paper. 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. Advances in Composite Laminate Theories Essay Paper. 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]. Advances in Composite Laminate Theories Essay Paper