Stability of Thin Rectangular Laminated Composite Plates

Contents

CHAPTER ONE. INTRODUCTION
1.1 General Introduction
1.2 Research Objectives
1.3 Book Overview
1.3.1 Developments in the Theories of Laminated Plates
1.3.2 Numerical Techniques
1.3.3 The Past Work of Buckling Analysis

2 CHAPTER TWO. FIBER REINFORCED LAMINA
2.1 Introduction
2.2 Structure of Composites
2.3 Mechanical Properties of a Fiber Reinforced Lamina
2.3.1 Analytical Modeling of Composite Laminates

3 CHAPTER THREE. MATHEMATICAL FORMULATIONS AND NUMERICAL MODELING
3.1 Introduction
3.2 Fundamental Equations of Elasticity
3.3 The Numerical Method

4 CHAPTER FOUR. VERIFICATION OF THE COMPUTER PROGRAM
4.1 Convergence Study
4.2 Validation of the Finite Element (FE) Program
4.2.1 Comparisons with Theoretical Results
4.2.2 Comparisons with the Results of ANSYS Package
4.2.3 Comparisons with Experimental Results

5 CHAPTER FIVE. NUMERICAL RESULTS AND DISCUSSIONS
5.1 Effect of Lamination Scheme
5.2 Effect of Aspect Ratio
5.3 Effect of Material Anisotropy
5.4 Effect of Fiber Orientations of Layers
5.5 Effect of Reversing Lamination Scheme
5.6 Effect of Boundary Conditions

6 CHAPTER SIX. CONCLUDING REMARKS

Dedication

In the name of Allah, the merciful, the compassionate All praise is due to Allah and blessings and peace is upon his messenger and servant, Mohammed, and upon his family and companions and whoever follows his guidance until the day of resurrection.

To the memory of my mother Khadra Dirar Taha, my father Mohammed Elmardi Suleiman, and my dear aunt Zaafaran Dirar Taha and my second mother Niemat Ibrahim Suleiman who they taught me the greatest value of hard work and encouraged me in all my endeavors.

To my first wife Nawal Abbas Abdelmajied and my beautiful three daughters Roa, Rawan and Aya and granddaughter Lana whose love, patience and silence are my shelter whenever it gets hard.

To my second wife Limya Abdullah Ali whose love and supplication to Allah were and will always be the momentum that boosts me through the thorny road of research.

To Professor Mahmoud Yassin Osman for reviewing and modifying the manuscript before printing process.

To the memory of Professors Elfadil Adam and Sabir Mohammed Salih.

This book is dedicated mainly to undergraduate and postgraduate students, especially mechanical and civil engineering students plus mathematicians and mathematics students where most of the work is of mathematical nature.

To Mr. Osama Mahmoud Mohammed Ali of Daniya Center for Printing and Publishing Services whose patience in editing and re – editing the manuscript of this book was the momentum that pushed me in completing successfully the present work.

To my friend Professor Elhassan Mohammed Elhassan Ishag, Faculty of Medicine, University of Gezira, Medani, Sudan.

To my friend Mohammed Ahmed Sambo, Faculty of Engineering and Technology, Nile Valley University, Atbara, Sudan.

To my homeland, Sudan, hoping to contribute in its development and superiority.

Finally, may Allah accept this humble work and I hope that it will be beneficial to its readers.

Acknowledgements

I am grateful and deeply indebted to Associate Professor Dr. Tagelsir Hassan for close supervision, constructive criticism, and provision of useful parts of his papers and/ or other relevant materials during his stay in Russia, and also the valuable recommendations during the various stages of building up the present study, without which this work would not have been accomplished.

I am also grateful to Professor Dr. Mahmoud Yassin Osman who supervised me during the first period of this study.

I am also indebted to many people. Published texts in mechanics of materials, numerical techniques have been contributed to the author's thinking. Members of mechanical engineering department at Tehran University of Science and Technology, Nile Valley University, Red Sea University, Sudan University of Science and Technology and Blue Nile University have served to sharpen and refine the treatment of my topics. The author is extremely grateful to them for constructive criticisms and suggestions.

Special appreciation is due to the British Council's library for its response in ordering the requested reviews and papers.

Also, thanks are extended to the Faculty of Engineering and Technology, Atbara for enabling me to utilize its facilities in accessing the internet and printing out some papers, reviews and conference minutes concerning the present study.

Special gratitude is due to Professor Mohammed Ibrahim Shukri for the valuable gift "How to write a research", which assisted a lot in writing sequentially the present study. Thanks, are also due to Associate Professor Izz Eldin Ahmed Abdallah for helping with the internet.

Thanks, are also due to Faculty of Engineering and Technology, Nile Valley University administration for funding this research in spite of its financial hardships.

Abstract

Finite element method (FEM) is presented for the analysis of thin rectangular laminated composite plates under the biaxial action of in – plane compressive loading. The analysis uses the classical laminated plate theory (CLPT) which does not account for shear deformations. In this theory it is assumed that the laminate is in a state of plane stress, the individual lamina is linearly elastic, and there is perfect bonding between layers. The classical laminated plate theory (CLPT), which is an extension of the classical plate theory (CPT) assumes that normal to the mid – surface before deformation remains straight and normal to the mid – surface after deformation. Therefore, this theory is only adequate for buckling analysis of thin laminates. A FORTRAN program has been developed. The convergence and accuracy of the FEM solutions for biaxial buckling of thin laminated rectangular plates was verified by comparison with various theoretical and experimental solutions. New numerical results are generated for in – plane compressive biaxial buckling which serve to quantify the effects of lamination scheme, aspect ratio, and material anisotropy, fiber orientation of layers, reversed lamination scheme and boundary conditions.

It was found that symmetric laminates are stiffer than the anti – symmetric one due to coupling between bending and stretching which decreases the buckling loads of symmetric laminates. The buckling load increases with increasing aspect ratio, and decreases with increase in modulus ratio. The buckling load will remain the same even when the lamination order is reversed. The buckling load increases with the mode number but at different rates depending on the type of end support. It is also observed that as the mode number increases, the plate needs additional support.

ملخص الكتاب

في هذا الكتاب تمَّ استخدام أسلوب العنصر المحدَّد (FEM) لتحليل الألواح الشرائحية الرقيقة مستطيلة المقطع المسلَّط عليها حمل إنضغاط في محورين. يستخدم التحليل نظرية الألواح الشرائحية الكلاسيكية (CLPT) التي تتجاهل تأثيرات تشوه القص. في هذه النظرية يتم إفتراض أنَّ اللوح يكون في حالة إجهاد مستوٍ، وتكون الطبقة المفردة مرنة خطياً، ويكون هنالك رابط جيدَّ بين الطبقات. نظرية الألواح الشرائحية الكلاسيكية التي هي إمتداد لنظرية الألواح الكلاسيكية (CPT) تفترض أن الخط المستقيم الوهمي المتعامد مع منتصف سطح اللوح قبل التشوه يظل مستقيماً ومتعامداً مع منتصف السطح بعد التشوه، بالتالي فإن هذه النظرية تتجاهل تشوه القص، عليه فهي تكون صالحة فقط لتحليل الألواح الرقيقة. لهذا الغرض تمَّ تأليف برنامج فورتران (FORTRAN) وتمَّ تأسيس تقارب ودقة برنامج أسلوب العنصر المحدد (FEM) بمقارنته بالعديد من الحلول النظرية والمختبرية. تمَّ الحصول على نتائج عددية جديدة للإنبعاج ثنائي المحور وذلك للتحقق من تأثيرات إتجاه الألياف، نسبة النطاق ، تباين الخواص للمادة ، عكس إتجاه الألياف والشروط الطرفية للوح.

وُجد في هذه الدراسة أنَّ الألواح المتماثلة تكون أكثر صلابة من الألواح الغير متماثلة وذلك نتيجة لتأثير الإزدواج بين الإنحناء والإستطالة الذي يخفِّض أحمال الإنبعاج للألواح المتماثلة. يزيد حمل الإنبعاج بزيادة نسبة النطاق ويقل بزيادة نسبة المعاير . يظل حمل الإنبعاج ثابتاً حتى في حالة عكس إتجاه الألياف. يزيد الإنبعاج بزيادة عدد الأنماط بمعدلات مختلفة إعتماداً على نوع الإسناد الطرفي. يُلاحظ أيضاً أنه كلما زاد عدد الأنماط فإن اللوح سيحتاج لإسنادات إضافية.

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CHAPTER ONE. INTRODUCTION

1.1 General Introduction

The objective of this book is to present a complete and up to date treatment of uniform cross section rectangular laminated plates on buckling. Finite element (FEM) method is used for solving governing equations of thin laminated composite plates and their solution using classical laminated plate theory (CLPT). Plates are common structural elements of most engineering structures, including aerospace, automotive, and civil engineering structures, and their study from theoretical and experimental analyses points of view are fundamental to the understanding of the behavior of such structures.

The motivation that led to the carrying out of the present study has come from many years of studying classical laminated plate theory (CLPT) and its analysis by the finite element (FEM) method, and also from the fact that there does not exist a publication that contains a detailed coverage of classical laminated plate theory and finite element method in one volume. The present study is an attempt to fulfill the need for a complete treatment of classical laminated theory of plates and its solution by a numerical solution.

The material presented is intended to serve as a basis for a critical study of the fundamentals of elasticity and several branches of solid mechanics including advanced mechanics of materials, theories of plates, composite materials and numerical methods. It includes certain properties of laminated composite plates, and at the end of this chapter the most important objectives of the present book are cited, this subject may be used either as a required reading or as a reference subject. Developments in the theories of laminated plates, several numerical methods and the past work of buckling analysis are also presented. Mathematical formulations and numerical modeling of rectangular laminated plates under biaxial buckling loads are also introduced. The present finite element (FEM) results are validated with similar results generated by FEM and/ or other numerical and approximate analytical solutions in chapter four. Additional verification with ANSYS package and experimental results has been done in this chapter. In chapter five, the effects of lamination scheme, aspect ratio, material anisotropy, fiber orientations of layers, reversed lamination scheme and boundary conditions are investigated. In chapter six, the most important results have been summarized.

The theoretical background of this study is suitable as a textbook for an advanced course on theories of plates and finite element techniques in mechanical and civil engineering curricula. It can be used also as a reference by engineers and scientists working in industry and academic institutions.

1.2 Research Objectives

The present work involves a comprehensive study of the following objectives, which have been achieved over a period of five years:

1. A survey of various plate theories and techniques used to predict the response of laminated plates under buckling loads.
2. The development of a theoretical model capable of predicting buckling loads in a thin laminated plate as a new and unprecedented approach.
3. The development and application of the finite element technique for the analysis of rectangular laminated plates subjected to a buckling load.
4. Investigation of the accuracy of the theoretical model through a wide range of theoretical and experimental comparisons.
5. Further investigations on the influence of coupling between bending and extension and/or twisting on the response of laminated plates could be carried out.
6. Generation of new results based on classical laminated plate theory (CLPT).

1.3 Book Overview

1.3.1 Developments in the Theories of Laminated Plates

From the point of view of solid mechanics, the deformation of a plate subjected to transverse and / or in plane loading consists of two components: flexural deformation due to rotation of cross – sections, and shear deformation due to sliding of section or layers. The resulting deformation depends on two parameters: the thickness to length ratio and the ratio of elastic to shear moduli. When the thickness to length ratio is small, the plate is considered thin, and it deforms mainly by flexure or bending; whereas when the thickness to length and the modular ratios are both large, the plate deforms mainly through shear. Due to the high ratio of in – plane modulus to transverse shear modulus, the shear deformation effects are more pronounced in the composite laminates subjected to transverse and / or in – plane loads than in the isotropic plates under similar loading conditions.

The three – dimensional theories of laminates, in which each layer is treated as homogeneous anisotropic medium, (see Reddy 6) are intractable. Usually, the anisotropy in laminated composite structures causes complicated responses under different loading conditions by creating complex couplings between extensions and bending, and shears deformation modes. Expect for certain cases, it is inconvenient to fully solve a problem in three dimensions due to the complexity, size of computation, and the production of unnecessary data specially for composite structures.

Many theories which account for the transverse shear and normal stresses are available in the literature (see, for example Mindlin 7). These are too numerous to review here. Only some classical papers and those which constitute a background for the present book will be considered. These theories are classified according to Phan and Reddy 8 into two major classes on the basis of the assumed fields as: (1) stress based theories, and (2) displacement based theories. The stress – based theories are derived from stress fields which are assumed to vary linearly over the thickness of the plate:

(Where is the stress couples, h is the plate thickness, and is the distance of the lamina from the plate mid – plane).

The displacement – based theories are derived from an assumed displacement field as:

Abbildung in dieser Leseprobe nicht enthalten

Where: , and are the displacements of the middle plane of the plate. The governing equations are derived using principle of minimum total potential energy. The theory used in the present work comes under the class of displacement – based theories. Extensions of these theories which include the linear terms in in , and only the constant term in , to account for higher – order variations and to laminated plates, can be found in the work of Yang, Norris and Stavsky 9 , Whitney and Pagano 10 and Phan and Reddy 8.

Based on different assumptions for displacement fields, different theories for plate analysis have been devised. These theories can be divided into three major categories, the individual layer theories (IL), the equivalent single layer (ESL) theories, and the three-dimensional elasticity solution procedures. These categories are further divided into sub – theories by the introduction of different assumptions. For example, the second category includes the classical laminated plate theory (CLPT), the first order and higher order shear deformation theories (FSDT and HSDT) as stated in Refs. {11 – 14}.

In the individual layer laminate theories, each layer is considered as a separate plate. Since the displacement fields and equilibrium equations are written for each layer, adjacent layers must be matched at each interface by selecting appropriate interfacial conditions for displacements and stresses. In the ESL laminate theories, the stress or the displacement field is expressed as a linear combination of unknown functions and the coordinate along the thickness. If the in – plane displacements are expanded in terms of the thickness co – ordinate up to the nth power, the theory is named nth order shear deformation theory. The simplest ESL laminate theory is the classical laminated plate theory (CLPT). This theory is applicable to homogeneous thin plates (i.e. the length to thickness ratio a / h > 20). The classical laminated plate theory (CLPT), which is an extension of the classical plate theory (CPT) applied to laminated plates was the first theory formulated for the analysis of laminated plates by Reissner and Stavsky 15 in 1961 , in which the Kirchhoff and Love assumption that normal to the mid – surface before deformation remain straight and normal to the mid – surface after deformation is used (see Figure(1.1)) , but it is not adequate for the flexural analysis of moderately thick laminates. However, it gives reasonably accurate results for many engineering problems i.e. thin composite plates, as stated by Srinivas and Rao 16, Reissner and Stavsky 15. This theory ignores the transverse shear stress components and models a laminate as an equivalent single layer. The classical laminated plate theory (CLPT) under – predicts deflections as proved by Turvey and Osman 17, 18, 19 and Reddy 6 due to the neglect of transverse shear strain. The errors in deflection are even higher for plates made of advanced filamentary composite materials like graphite – epoxy and boron – epoxy whose elastic modulus to shear modulus ratios are very large (i.e. of the order of 25 to 40, instead of 2.6 for typical isotropic materials). However, these composites are susceptible to thickness effects because their effective transverse shear moduli are significantly smaller than the effective elastic modulus along the fiber direction. This effect has been confirmed by Pagano 20 who obtained analytical solutions of laminated plates in bending based on the three – dimensional theory of elasticity. He proved that classical laminated plate theory (CLPT) becomes of less accuracy as the side to thickness ratio decreases. In particular, the deflection of a plate predicted by CLPT is considerably smaller than the analytical value for side to thickness ratio less than 10. These high ratios of elastic modulus to shear modulus render classical laminate theory as inadequate for the analysis of composite plates. In the first order shear deformation theory (FSDT), the transverse planes, which are originally normal and straight to the mid – plane of the plate, are assumed to remain straight but not necessarily normal after deformation, and consequently shear correction factors are employed in this theory to adjust the transverse shear stress, which is constant through thickness (see Figure (1.1)). Recently Reddy 6 and Phan and Reddy 8 presented refined plate theories that used the idea of expanding displacements in the powers of thickness coordinate. The main novelty of these works is to expand the in – plane displacements as cubic functions of the thickness coordinate, treat the transverse deflection as a function of the and coordinates, and eliminate the functions , , and from equation (2.2) by requiring that the transverse shear stress be zero on the bounding planes of the plate. Numerous studies involving the application of the first – order theory to bending, vibration and buckling analyses can be found in the works of Reddy 20, and Reddy and Chao 21.

In order to include the curvature of the normal after deformation, a number of theories known as higher – order shear deformation theories (HSDT) have been devised in which the displacements are assumed quadratic or cubic through the thickness of the plate. In this aspect, a variationally

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Figure (1.1) Assumed Deformation of the Transverse Normal in Various Displacement Base Plate Theories

Consistent higher – order theory which not only accounts for the shear deformation but also satisfies the zero transverse shear stress conditions on the top and bottom faces of the plate and does not require correction factors was suggested by Reddy 6. Reddy's modifications consist of a more systematic derivation of displacement field and variationally consistent derivation of the equilibrium equations. The refined laminate plate theory predicts a parabolic distribution of the transverse shear stresses through the thickness, and requires no shear correction coefficients.

In the non – linear analysis of plates considering higher – order shear deformation theory (HSDT), shear deformation has received considerably less attention compared with linear analysis. This is due to the geometric non – linearity which arises from finite deformations of an elastic body and which causes more complications in the analysis of composite plates. Therefore, fiber – reinforced material properties and lamination geometry have to be taken into account. In the case of anti – symmetric and unsymmetrical laminates, the existence of coupling between stretching and bending complicates the problem further. Non – linear solutions of laminated plates using higher – order theories have been obtained through several techniques, i. e. perturbation method as in Ref. 22, finite element method as in Ref. 23, the increment of lateral displacement method as in Ref. 24, and the small parameter method as in Ref. 25.

In the present work, the analysis uses the classical laminated plate theory (CLPT) which does not account for transverse shear deformations. In this theory it is assumed that the laminate is in a state of plane stress, the individual lamina is linearly elastic, and there is perfect bonding between layers. The classical laminated plate theory assumes that normal to the mid – surface before deformation remains straight and normal to the mid – surface after deformation. Therefore, this theory is adequate for buckling analysis of thin laminates. A FORTRAN program has been compiled, the convergence and accuracy of the FEM solutions for biaxial buckling of thin laminated rectangular plates are established by comparison with various theoretical and experimental solutions and new numerical results are generated.

1.3.2 Numerical Techniques

Several numerical methods could be used in this study, but the main ones are finite difference method (FDM), dynamic relaxation coupled with finite difference method (DR), and finite element method (FEM).

In the finite difference method, the solution domain is divided into a grid of discrete points or nodes. The partial differential equation is then written for each node and its derivatives are replaced by finite divided differences. Although such point – wise approximation is conceptually easy to understand, it becomes difficult to apply for system with irregular geometry, unusual boundary conditions, and heterogeneous composition.

The DR method was first proposed in 1960th; see Rushton 26, Cassel and Hobbs 27, and Day 28. In this method, the equations of equilibrium are converted to dynamic equations by adding damping and inertia terms. These are then expressed in finite difference form and the solution is obtained through iterations. The optimum damping coefficient and the time increment used to stabilize the solution depend on the stiffness matrix of the structure, the applied load, the boundary conditions and the size of mesh used.

In the present work, a numerical method known as finite element method (FEM) is used. It is a numerical procedure for obtaining solutions to many of the problems encountered in engineering analysis. It has two primary subdivisions. The first utilizes discrete elements to obtain the joint displacements and member forces of a structural framework. The second uses the continuum elements to obtain approximate solutions to heat transfer, fluid mechanics, and solid mechanics problem. The formulation using the discrete element is referred to as matrix analysis of structures and yields results identical with the classical analysis of structural frameworks. The second approach is the true finite element method. It yields approximate values of the desired parameters at specific points called nodes. A general finite element computers program, however, is capable of solving both types of problems and the name" finite element method" is often used to denote both the discrete element and the continuum element formulations.

The finite element method combines several mathematical concepts to produce a system of linear and non – linear equations. The number of equations is usually very large, anywhere from 20 to 20,000 or more and requires the computational power of the digital computer.

It is impossible to document the exact origin of the finite element method because the basic concepts have evolved over a period of 150 or more years. The method as we know it today is an outgrowth of several papers published in the 1950th that extended the matrix analysis of structures to continuum bodies. The space exploration of the 1960th provided money for basic research, which placed the method of a firm mathematical foundation and stimulated the development of multi – purpose computer programs that implemented the method. The design of airplanes, unmanned drones, missiles, space capsules, and the like, provided application areas.

The finite element method (FEM) is a powerful numerical method, which is used as a computational technique for the solution of differential equations that arise in various fields of engineering and applied sciences. The finite element method is based on the concept that one can replace any continuum by an assemblage of simply shaped elements, called finite elements with well-defined force, displacement, and material relationships. While one may not be able to derive a closed – form solution for the continuum, one can derive approximate solutions for the element assemblage that replaces it. The approximate solutions or approximation functions are often constructed using ideas from interpolation theory, and hence they are also called interpolation functions. For more details refer to Refs. {29 – 31}.

In a comparison between the finite element method (FEM) and dynamic relaxation method (DR), Aalami 32 found that the computer time required for the finite element method is eight times greater than for DR analysis, whereas the storage capacity for FEM is ten times or more than that for DR analysis. This fact is supported by Putcher and Reddy 23, and Turvey and Osman {17 – 19} who noted that some of the finite element formulations require large storage capacity and computer time. Hence due to the large computations involved in the present study, the finite element method (FEM) is considered more efficient than the DR method. In another comparison, Aalami 32 found that the difference in accuracy between one version of FEM and DR may reach a value of more than 15 % in favor of FEM. Therefore, the FEM can be considered of acceptable accuracy. The apparent limitation of the DR method is that it can only be applied to limited geometries, whereas the FEM can be applied to different intricate geometries and shapes.

1.3.3 The Past Work of Buckling Analysis

Composite materials are widely used in a broad spectrum of modern engineering application fields ranging from traditional fields such as automobiles, robotics, day to day appliances, building industry etc. This is due to their excellent high strength to weight ratio, modulus to weight ratio, and the controllability of the structural properties with the variation of fiber orientation, stacking scheme and the number of laminates. Among the various aspects of the structural performance of structures made of composite materials is the mechanical behavior of rectangular laminated plates which has drawn much attention. In particular, consideration of the buckling phenomena in such plates is essential for the efficient and reliable design and for the safe use of the structural element. Due to the anisotropic and coupled material behavior, the analysis of composite laminated plates is generally more complicated than the analysis of homogeneous isotropic ones.

The members and structures composed of laminated composite material are usually very thin, and hence more prone to buckling. Buckling phenomenon is critically dangerous to structural components because the buckling of composite plates usually occurs at a lower applied stress and generates large deformations. This led to a focus on the study of buckling behavior in composite materials. General introductions to the buckling of elastic structures and of laminated plates can be found in e.g. Refs. {33 – 46}. However, these available curves and data are restricted to idealized loading, namely, uniaxial or biaxial uniform compression.

Due to the importance of buckling considerations, there is an overwhelming number of investigations available in which corresponding stability problems are considered by a wide variety of analysis methods which may be of a closed – form analytical nature or may be sorted into the class of semi – analytical or purely numerical analysis method.

Closed – form exact solutions for the buckling problem of rectangular composite plates are available only for limited combinations of boundary conditions and laminated schemes. These include cross – ply symmetric and angle – ply anti – symmetric rectangular laminates with at least two opposite edges simply supported, and similar plates with two opposite edges clamped but free to deflect (i.e. guided clamp) or with one edge simply supported and the opposite edge with a guided clamp. Most of the exact solutions discussed in the monographs of Whitney 47 who developed an exact solution for critical buckling of solid rectangular orthotropic plates with all edges simply supported, and of Reddy {48 – 51} and Leissa and Kang 52, and that of Refs. 39 and 53. Bao et al. 54 developed an exact solution for two edges simply supported and two edges clamped, and Robinson 55 who developed an exact solution for the critical buckling stress of an orthotropic sandwich plate with all edges simply supported.

For all other configurations, for which only approximated results are available, several semi – analytical and numerical techniques have been developed. The Rayleigh – Ritz method 53 and 56, the finite strip method (FSM) 36 and 57, the element free Galerkin method (EFG) 58, the differential quadrature technique 59, the moving least square differential quadrature method 60 and the most extensively used finite element method (FEM) 61 are the most common ones.

The Kantorovich method (KM) {62 – 64}, which is a different and in most cases advantageous semi – analytical method, combines a variation approach of closed – form solutions and an iterative procedure. The method assumes a solution in the form of a sum of products of functions in one direction and functions in the other direction. Then, by assuming the function in one direction, the variation problem of the plate reduces to a set of ordinary differential equations. In the case of buckling analysis, the variation problem reduces to an ordinary differential eigenvalue and Eigen function problem. The solution of the resulting problem is an approximate one, and its accuracy depends on the assumed functions in the first direction. The extended Kantorovich method (EKM), which was proposed by Kerr 65, is the starting point for an iterative procedure, where the solution obtained in one direction is used as the assumed functions in the second direction. After repeating this process several times, convergence is obtained. The single term extended Kantorovich method was employed for a buckling analysis of rectangular plates by several researches. Eienberger and Alexandrov 66 used the method for the buckling analysis of isotropic plates with variable thickness. Shufrin and Eienberger 67 and 68 extended the solution to thick plates with constant and variable thickness using the first and higher order shear deformation theories. Ungbhakorn and Singhatanadgid 69 extended the solution to buckling of symmetrically cross – ply laminated rectangular plates. The multi – term formulation of the extended Kantorovich approach to the simplest samples of rectangular isotropic plates was presented by Yuan and Jin 70. This study showed that the additional terms in the expansion can be used in order to improve the solution.

March and Smith 71 found an approximate solution for all edges clamped. Also, Chang et al. 72 developed approximate solution to the buckling of rectangular orthotropic sandwich plate with two edges simply supported and two edges clamped or all edges clamped using the March – Erickson method and an energy technique. Jiang et al. 73 developed solutions for local buckling of rectangular orthotropic hat – stiffened plates with edges parallel to the stiffeners were simply supported or clamped and edges parallel to the stiffeners were free, and Smith 74 presented solutions bounding the local buckling of hat stiffened plates by considering the section between stiffeners as simply supported or clamped plates.

Many authors have used finite element method to predict accurate in – plane stress distribution which is then used to solve the buckling problem. Zienkiewicz 75 and Cook 76 have clearly presented an approach for finding the buckling strength of plates by first solving the linear elastic problem for a reference load and then the eigenvalue problem for the smallest eigenvalue which then multiplied by the reference load gives the critical buckling load of the structure. An excellent review of the development of plate finite elements during the past 35 years was presented by Yang et al. 77.

Many buckling analyses of composite plates available in the literature are usually realized parallel with the vibration analyses, and are based on two – dimensional plate theories which may be classified as classical and shear deformable ones. Classical plate theories (CPT) do not take into account the shear deformation effects and over predict the critical buckling loads for thicker composite plates, and even for thin ones with a higher anisotropy. Most of the shear deformable plate theories are usually based on a displacement field assumption with five unknown displacement components. As three of these components corresponded to the ones in CPT, the additional ones are multiplied by a certain function of thickness coordinate and added to the displacements field of CPT in order to take into account the shear deformation effects. Taking these functions as linear and cubic forms leads to the so – called uniform or Mindlin shear deformable plate theory (USDPT) 78, and parabolic shear deformable plate theories (PSDPT) 79 respectively. Different forms were also employed such as hyperbolic shear deformable plate theory (HSDPT) 80, and trigonometric or sine functions shear deformable plate theory (TSDPT) 81 by researchers. Since these types of shear deformation theories do not satisfy the continuity conditions among many layers of the composite structures, the zig – zag type of the plate theories introduced by Di Sciuva 82, and Cho and Parmeter 83 in order to consider interlaminar stress continuities. Recently, Karama et al. 84 proposed a new exponential function {i.e. exponential shear deformable plate theory (ESDPT)} in the displacement field of the composite laminated structures for the representation of the shear stress distribution along the thickness of the composite structures and compared their result for static and dynamic problem of the composite beams with the sine model.

Within the classical lamination theory, Jones 85 presented a closed – form solution for the buckling problem of cross – ply laminated plates with simply supported boundary conditions. In the case of multi – layered plates subjected to various boundary conditions which are different from simply supported boundary conditions at all of their four edges, the governing equations of the buckling of the composite plates do not admit an exact solution, except for some special arrangements of laminated plates. Thus, for the solution of these types of problems, different analytical and / or numerical methods are employed by various researchers. Baharlou and Leissa 56 used the Ritz method with simple polynomials as displacement functions, within the classical theory, for the problem of buckling of cross and angle – ply laminated plates with arbitrary boundary conditions and different in – plane loads. Narita and Leissa 86 also applied the Ritz method with the displacement components assumed as the double series of trigonometric functions for the buckling problem of generally symmetric laminated composite rectangular plates with simply supported boundary conditions at all their edges. They investigated the critical buckling loads for five different types of loading conditions which are uniaxial compression (UA – C), biaxial compression (BA – C), biaxial compression – tension (BA – CT), and positive and negative shear loadings.

The higher – order shear deformation theories can yield more accurate inter – laminate stress distributions. The introduction of cubic variation of displacement also avoids the need for shear correction displacement. To achieve a reliable analysis and safe design, the proposals and developments of models using higher order shear deformation theories have been considered. Lo et al. 87 and [ 88] reviewed the pioneering work on the field and formulated a theory which accounts for the effects of transverse shear deformation, transverse strain and non – linear distribution of the in – plane displacements with respect to the thickness coordinate. Third – order theories have been proposed by Reddy {89 – 92}, Librescu 93, Schmidt 94, Murty 95, Levinson 96, Seide 97, Murthy 98, Bhimaraddi 99, Mallikarjuna and Kant 100, Kant and Pandya 101, and Phan and Reddy 8, among others. Pioneering work and overviews in the field covering closed – form solutions and finite element models can be found in Reddy 90, 102, 103, Mallikarjuna and Kant 100, Noor and Burton 104, Bert 105, Kant and Kommineni 106, and Reddy and Robbins 107 among others.

For the buckling analysis of the cross – ply laminated plates subjected to simply supported boundary conditions at their opposite two edges and different boundary conditions at the remaining ones Khdeir 108 and Reddy and Khdeir 51 used a parabolic shear deformation theory and applied the state – space technique. Hadian and Nayfeh 109, on the basis of the same theory and for the same type of problem, needed to modify the technique due to ill – conditioning problems encountered especially for thin and moderately thick plates. The buckling analyses of completely simply supported cross – ply laminated plates were presented by Fares and Zenkour 110, who added a non – homogeneity coefficient in the material stiffnesses within various plate theories, and by Matsunaga 111 who employed a global higher order plate theory. Gilat el al. 112 also investigated the same type of problem on the basic of a global – local plate theory where the displacement field is composed of global and local contributions, such that the requirement of the continuity conditions and delamination effects can be incorporated into formulation.

Many investigations have been reported for static and stability analysis of composite laminates using different traditional methods. Pagano 113 developed an exact three – dimensional (3 – D) elasticity solution for static analysis of rectangular bi – directional composites and sandwich plates. Noor 114 presented a solution for stability of multi – layered composite plates based on 3 – D elasticity theory by solving equations with finite difference method. Also, 3 – D elasticity solutions are presented by Gu and Chattopadhyay 115 for the buckling of simply supported orthotropic composite plates. When the problem is reduced from a three – dimensional one (3 – D) to a two-dimensional case to contemplate more efficiently the computational analysis of plate composite structures, the displacement based theories and the corresponding finite element models receive the most attention 116.

Bifurcation buckling of laminated structures has been investigated by many researchers without considering the flatness before buckling 117. This point was first clarified for laminated composite plates for some boundary conditions and for some lamina configurations by Leissa 117. Qatu and Leissa 118 applied this result to identify true buckling behavior of composite plates. Elastic bifurcation of plates has been extensively studied and well documented in standard texts e.g. 33 and 119, research monographs {120 – 122} and journal papers {123 – 126}.

It is important to recognize that, with the advent of composite media, certain new material imperfections can be found in composite structures in addition to the better – known imperfections that one finds in metallic structures. Thus, broken fibers, delaminated regions, cracks in the matrix material, as well as holes, foreign inclusions and small voids constitute material and structural imperfections that can exist in composite structures. Imperfections have always existed and their effect on the structural response of a system has been very significant in many cases. These imperfections can be classified into two broad categories: initial geometrical imperfections and material or constructional imperfections.

The first category includes geometrical imperfections in the structural configuration (such as a local out of roundness of a circular cylindrical shell, which makes the cylindrical shell non – circular; a small initial curvature in a flat plate or rod, which makes the structure non – flat, etc.), as well as imperfections in the loading mechanisms (such as load eccentricities; an axially loaded column is loaded at one end in such a manner that a bending moment exists at that end). The effect of these imperfections on the response of structural systems has been investigated by many researchers and the result of these efforts can be easily found in books 3, as well in published papers 127 – 144.

The second class of imperfections is equally important, but has not received as much attentions as the first class; especially as far as its effect on the buckling response characteristics is concerned. For metallic materials, one can find several studies which deal with the effect of material imperfections on the fatigue life of the structural component. Moreover, there exist a number of investigations that deal with the effect of cut – outs and holes on the stress and deformation response of thin plates. Another material imperfection is the rigid inclusion. The effect of rigid inclusions on the stress field of the medium in the neighborhood of the inclusion has received limited attention. The interested reader is referred to the bibliography of Professor Naruoka 127.

There exist two important classes of material and constructional – type imperfections, which are very important in the safe design, especially of aircraft and spacecraft. These classes consist of fatigue cracks or cracks in general and delamination in systems that employ laminates (i.e. fiber – reinforced composites). There is considerable work in the area of stress concentration at crack tips and crack propagation. Very few investigations are cited, herein, for the sake of brevity. These include primarily those dealing with plates and shells and non – isotropic construction. Some deal with cracks in metallic plates and shells {145 – 148}. Others deal with non – isotropic construction and investigate the effects of non – isotropy {149 – 154}. In all of these studies, there is no mention of the effect of the crack presence on the overall stability or instability of the system.

Finally, delamination is one of the most commonly found defects in laminated structural components. Most of the work found in the literature deals with flat configurations.

Composite structures often contain delamination. Causes of delamination are many and include tool drops, bird strikes, runway debris hits and manufacturing defects. Moreover, in some cases, especially in the vicinity of holes or close to edges in general, delamination starts because of the development of interlaminar stresses. Several analyses have been reported on the subject of edge delamination and its importance in the design of laminated structures. A few of these works are cited {155 – 161}. These and their cited references form a good basis for the interested reader. The type of delamination that comprises the basic and primary treatise is the one that is found to be present away from the edges (internal). This delaminating could be present before the laminate is loaded or it could develop after loading because of foreign body (birds, micrometer, and debris) impact. This is an extremely important problem especially for laminated structures that are subject to destabilizing loads (loads that can induce instability in the structure and possibly cause growth of the delamination; both of these phenomena contribute to failure of the laminate). The presence of delamination in these situations may cause local buckling and / or trigger global buckling and therefore induce a reduction in the overall load – bearing capacity of the laminated structure. The problem, because of its importance, has received considerable attention.

In the present study, the composite media are assumed free of imperfections i.e. initial geometrical imperfections due to initial distortion of the structure, and material and / or constructional imperfections such as broken fibers, delaminated regions, cracks in the matrix material, foreign inclusions and small voids which are due to inconvenient selection of fibers / matrix materials and manufacturing defects. Therefore, the fibers and matrix are assumed perfectly bonded.

2 CHAPTER TWO. FIBER REINFORCED LAMINA

2.1 Introduction

Composites were first considered as structural materials a little more than three quarters of a century ago. From that time to now, they have received increasing attention in all aspects of material science, manufacturing technology, and theoretical analysis.

The term composite could mean almost anything if taken at face value, since all materials are composites of dissimilar subunits if examined at close enough details. But in modern materials engineering, the term usually refers to a matrix material that is reinforced with fibers. For instance, the term "FRP" which refers to Fiber Reinforced Plastic usually indicates a thermosetting polyester matrix containing glass fibers, and this particular composite has the lion's share of today commercial market.

Many composites used today are at the leading edge of materials technology, with performance and costs appropriate to ultra-demanding applications such as space crafts. But heterogeneous materials combining the best aspects of dissimilar constituents have been used by nature for millions of years. Ancient societies, imitating nature, used this approach as well: The book of Exodus speaks of using straw to reinforce mud in brick making, without which the bricks would have almost no strength. Here in Sudan, people from ancient times dated back to Meroe civilization, and up to now used zibala (i.e. animals’ dung) mixed with mud as a strong building material.

As seen in Table (2.1) below, which is cited by David Roylance 1, Stephen et al. 2 and Turvey et al. 3, the fibers used in modern composites have strengths and stiffnesses far above those of traditional structural materials. The high strengths of the glass fibers are due to processing that avoids the internal or external textures flaws which normally weaken glass, and the strength and stiffness of polymeric aramid fiber is a consequence of the nearly perfect alignment of the molecular chains with the fiber axis.

Table (2.1) Properties of Composite Reinforcing Fibers

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Where E is Young's modulus, is the breaking stress, is the breaking strain, and is the mass density.

These materials are not generally usable as fibers alone, and typically they are impregnated by a matrix material that acts to transfer loads to the fibers, and also to protect the fibers from abrasion and environmental attack. The matrix dilutes the properties to some degree, but even so very high specific (weight – adjusted) properties are available from these materials. Polymers are much more commonly used, with unsaturated styrene – hardened polyesters having the majority of low to medium performance applications and Epoxy or more sophisticated thermosets having the higher end of the market. Thermoplastic matrix composites are increasingly attractive materials, with processing difficulties being perhaps their principal limitation.

Recently, composite materials are increasingly used in many mechanical, civil, and aerospace engineering applications due to two desirable features: the first one is their high specific stiffness (i.e. stiffness per unit density) and high specific strength (i.e. strength per unit density), and the second is their properties that can be tailored through variation of the fiber orientation and stacking sequence which gives the designers a wide spectrum of flexibility. The incorporation of high strength, high modulus and low-density filaments in a low strength and a low modulus matrix material is known to result in a structural composite material with a high strength to weight ratio. Thus, the potential of a two-material composite for use in aerospace, under-water, and automotive structures has stimulated considerable research activities in the theoretical prediction of the behavior of these materials. One commonly used composite structure consists of many layers bonded one on top of another to form a high-strength laminated composite plate. Each lamina is fiber reinforced along a single direction, with adjacent layers usually having different filament orientations. For these reasons, composites are continuing to replace other materials used in structures such as conventional materials. In fact, composites are the potential structural materials of the future as their cost continues to decrease due to the continuous improvements in production techniques and the expanding rate of sales.

2.2 Structure of Composites

There are many situations in engineering where no single material will be suitable to meet a particular design requirement. However, two materials in combination may possess the desired properties and provide a feasible solution to the materials selection problem. A composite can be defined as a material that is composed of two or more distinct phases, usually a reinforced material supported in a compatible matrix, assembled in prescribed amounts to achieve specific physical and chemical properties.

In order to classify and characterize composite materials, distinction between the following two types is commonly accepted; see Vernon 4, Jan Stegmann and Erik Lund 5, and David Roylance 1.

1. Fibrous composite materials: Which are composed of high strength fibers embedded in a matrix. The functions of the matrix are to bond the fibers together to protect them from damage, and to transmit the load from one fiber to another. {See Figure (2.1)}.
2. Particulate composite materials: These are composed of particles encased within a tough matrix, e.g. powders or particles in a matrix like ceramics.

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Figure (2.1) Structure of a Fibrous Composite

In this study the focus will be on fiber reinforced composite materials, as they are the basic building element of a rectangular laminated plate structure. Typically, such a material consists of stacks of bonded-together layers (i.e. laminas or plies) made from fiber reinforced material. The layers will often be oriented in different directions to provide specific and directed strengths and stiffnesses of the laminate. Thus, the strengths and stiffnesses of the laminated fiber reinforced composite material can be tailored to the specific design requirements of the structural element being built.

2.3 Mechanical Properties of a Fiber Reinforced Lamina

Composite materials have many mechanical characteristics, which are different from those of conventional engineering materials such as metals. More precisely, composite materials are often both inhomogeneous and non-isotropic. Therefore, and due to the inherent heterogeneous nature of composite materials, they can be studied from a micromechanical or a macro mechanical point of view. In micromechanics, the behavior of the inhomogeneous lamina is defined in terms of the constituent materials; whereas in macro mechanics the material is presumed homogeneous and the effects of the constituent materials are detected only as averaged apparent macroscopic properties of the composite material. This approach is generally accepted when modeling gross response of composite structures. The micromechanics approach is more convenient for the analysis of the composite material because it studies the volumetric percentages of the constituent materials for the desired lamina stiffnesses and strengths, i.e. the aim of micromechanics is to determine the moduli of elasticity and strength of a lamina in terms of the moduli of elasticity, and volumetric percentage of the fibers and the matrix. To explain further, both the fibers and the matrix are assumed homogeneous, isotropic and linearly elastic.

2.3.1 Stiffness and Strength of a Lamina

The fibers may be oriented randomly within the material, but it is also possible to arrange for them to be oriented preferentially in the direction expected to have the highest stresses. Such a material is said to be anisotropic (i.e. different properties in different directions), and control of the anisotropy is an important means of optimizing the material for specific applications. At a microscopic level, the properties of these composites are determined by the orientation and distribution of the fibers, as well as by the properties of the fiber and matrix materials.

Consider a typical region of material of unit dimensions, containing a volume fraction, Vf of fibers all oriented in a single direction. The matrix volume fraction is then, . This region can be idealized by gathering all the fibers together, leaving the matrix to occupy the remaining volume. If a stress is applied along the fiber direction, the fiber and matrix phases act in parallel to support the load. In these parallel connections the strains in each phase must be the same, so the strain in the fiber direction can be written as:

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(Where: the subscripts l, f and m denote the lamina, fibers and matrix respectively).

The forces in each phase must add to ba­lance the total load on the material. Since the forces in each phase are the phase stresses times the area (here numerically equal to the volume fraction), we have

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The stiffness in the fiber direction is found by dividing the stress by the strain:

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(Where: E is the longitudinal Young's modulus)

This relation is known as a rule of mixtures prediction of the overall modulus in terms of the moduli of the constituent phases and their volume fractions.

Rule of mixtures estimates for strength proceed along lines similar to those for stiffness. For instance, consider a unidirectional reinforced composite that is strained up to the value at which the fiber begins to fracture. If the matrix is more ductile than the fibers, then the ultimate tensile strength of the lamina in equation (2.2) will be transformed to:

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Where the superscript u denotes an ultimate value, and is the matrix stress when the fibers fracture as shown in Figure (2.2).

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It is clear that if the fiber volume fraction is very small, the behavior of the lamina is controlled by the matrix.

This can be expressed mathematically as follows:

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Figure (2.2) Stress-strain relationships for fiber and matrix

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If the lamina is assumed to be useful in practical applications, then there is a minimum fiber volume fraction that must be added to the matrix. This value is obtained by equating equations (2.4) and (2.5) i.e.

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The variation of the strength of the lamina with the fiber volume fraction is illustrated in Figure (2.3). It is obvious that when the strength of the lamina is dominated by the matrix deformation which is less than the matrix strength. But when the fiber volume fraction exceeds a critical value (i.e. Vf > VCritical), Then the lamina gains some strength due to the fiber reinforcement.

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Figure (2.3) Variation of Unidirectional Lamina Strength with the Fiber Volume Fraction

The micromechanical approach is not responsible for the many defects which may arise in fibers, matrix, or lamina due to their manufacturing. These defects, if they exist include misalignment of fibers, cracks in matrix, non-uniform distribution of the fibers in the matrix, voids in fibers and matrix, delaminated regions, and initial stresses in the lamina as a result of its manufacture and further treatment. The above-mentioned defects tend to propagate as the lamina is loaded causing an accelerated rate of failure. The experimental and theoretical results in this case tend to differ. Hence, due to the limitations necessary in the idealization of the lamina components, the properties estimated on the basis of micromechanics should be proved experimentally. The proof includes a very simple physical test in which the lamina is considered homogeneous and orthotropic. In this test, the ultimate strength and modulus of elasticity in a direction parallel to the fiber direction can be determined experimentally by loading the lamina longitudinally. When the test results are plotted, as in Figure (2.4) below, the required properties may be evaluated as follows: -

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Figure (2.4) Unidirectional Lamina Loaded in the Fiber-Direction

Similarly, the properties of the lamina in a direction perpendicular to the fiber direction can be evaluated in the same procedure.

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