3.8. Perturbation Solutions for the Bending of a Composite Material Plate With Mid-Plane Symmetry and No Bending-Twisting Coupling 3.9. Quasi-Isotropic Composite Panels Subjected to a Uniform Lateral Load 3. 10. A Static Analysis of Composite Material Panels Including Transverse Shear Deformation effects 11 3.1l. Boundary Conditions for a Plate Using the Refined Plate Theory Which Includes Transverse Shear Deformation 3. 12. Composite Plates on an Elastic Foundation 3. 13. Solutions for Plates of Composite Materials Including Transverse-Shear Deformation Effects, Simply Supported on All Four Edges l16 3. 14. Dynamic Effects on Panels of Composite Materials 3.15. Natural Flexural Vibrations of Rectangular Plates: Classical Theory 3. 16. Natural Flexural Vibrations of Composite Material Plate Including Transverse-Shear Deformation effects 3. 17. Forced-Vibration Response of a Composite Material Plate Subjected to a Dynamic Lateral Load 3. 18. Buckling of a Rectangular Composite Material Plate-Classical Theory 130 3. 19. Buckling of a Composite Material Plate Including Transverse-Shear Deformation Effects 3.20. Some Remarks on Compo 3. 21. Methods of Analysis for Sandwich Panels With Composite Material Faces, and Their Structural Optimization 3. 22. Governing Equations for a Composite Material Plate With Mid-Plane metry 3. 23. Governing Equations for a Composite Material Plate With Bending Twisting Coupling 3.24. Concluding Remarks 3.26. Problems and Exercises 4. Beams, Columns and Rods of Composite Materials 4.1. Development of Classical Beam Theory 4.2. Some Composite Beam Solutions 43. Composite Beams With Abrupt Changes in Geometry or Load 4.4. Solutions by Green's Functions 17 4.6. Rods osite Beams of Continuously Varying Cross-Section 4.5.C 4.7. Vibration of C ite beams 4.8. Beams With Mid-Plane Asymmetry 4.9. Advanced Beam Theory for Dynamic Loading Including Mid-Plane 184 4.10. Advanced Beam Theory Including Transverse Shear Deformation Effects 19 4. 11. Buckling of Composite Columns 4. 12. References 4. 13. Problen
XII 3.8. 3.9. 3.10. 3.11. 3.12. 3.13. 3.14. 3.15. 3.16. 3.17. 3.18. 3.19. 3.20. 3.21. 3.22. 3.23. 3.24. 3.25. 3.26. Perturbation Solutions for the Bending of a Composite Material Plate With Mid-Plane Symmetry and No Bending-Twisting Coupling Quasi-Isotropic Composite Panels Subjected to a Uniform Lateral Load A Static Analysis of Composite Material Panels Including Transverse Shear Deformation Effects Boundary Conditions for a Plate Using the Refined Plate Theory Which Includes Transverse Shear Deformation Composite Plates on an Elastic Foundation Solutions for Plates of Composite Materials Including Transverse-Shear Deformation Effects, Simply Supported on All Four Edges Dynamic Effects on Panels of Composite Materials Natural Flexural Vibrations of Rectangular Plates: Classical Theory Natural Flexural Vibrations of Composite Material Plate Including Transverse-Shear Deformation Effects Forced-Vibration Response of a Composite Material Plate Subjected to a Dynamic Lateral Load Buckling of a Rectangular Composite Material Plate – Classical Theory Buckling of a Composite Material Plate Including Transverse-Shear Deformation Effects Some Remarks on Composite Structures Methods of Analysis for Sandwich Panels With Composite Material Faces, and Their Structural Optimization Governing Equations for a Composite Material Plate With Mid-Plane Asymmetry Governing Equations for a Composite Material Plate With BendingTwisting Coupling Concluding Remarks References Problems and Exercises 4. Beams, Columns and Rods of Composite Materials 4.1. 4.2. 4.3. 4.4. 4.5. 4.6. 4.7. 4.8. 4.9. 4.10. 4.11. 4.12. 4.13. Development of Classical Beam Theory Some Composite Beam Solutions Composite Beams With Abrupt Changes in Geometry or Load Solutions by Green’s Functions Composite Beams of Continuously Varying Cross-Section Rods Vibration of Composite Beams Beams With Mid-Plane Asymmetry Advanced Beam Theory for Dynamic Loading Including Mid-Plane Asymmetry Advanced Beam Theory Including Transverse Shear Deformation Effects Buckling of Composite Columns References Problems 106 109 111 114 115 116 119 120 122 124 130 132 135 138 138 139 140 141 143 155 155 160 165 171 173 177 179 183 184 193 197 200 200
5. Composite Material Shells 215 5. 1. Introduction 5.2. Analysis of Composite Material Circular Cylindrical Shells 215 5.3. Some Edge Load and Particular Solutions 5.4. A General Solution for Composite Cylindrical Shells Under Axially Symmetric Loads 5.5. Response of a Long Axi-Symmetric Laminated C Shell to an Edge Displacement 5.6. Sample Solutions 5.7. Mid-Plane Asymmetric Circular Cylindrical Shells 5.8. Buckling of Circular Cylindrical Shells of Composite Materials Subjected to various loads 24 5.9. Vibrations of Composite Shells 5.10. Additional Reading On Composite Shells 5. 11. References 5. 12. problems 6. Energy Methods For Composite Material Structures 349 6. 1. Introduction 6. 2. Theorem of Minimum Potential Energy 63. Analysis of a Beam Using the Theorem of Minimum Potential Energy 6.4. Use of Minimum Potential Energy for Designing a Composite Electrical ansmission owe 268 6.5. Minimum Potential Energy for Rectangular Plates 6.6. A Rectangular Composite Material Plate Subjected to Lateral and ygrothermal Loads 6.7. In-Plane Shear Strength Determination of Composite Materials in Laminated Composite Panels 276 6.8. Use of the Theorem of Minimum Potential Energy to Determine Buckling Loads in Composite Plates 6.9. Trial Functions for Various Boundary Conditions for Composite Material 6. 10. Reissner's Variational Theorem and its Applications 6. 11. Static Deformation of Moderately Thick Beams 6. 12. Flexural Vibrations of Moderately Thick Beams 6. 13. Flexural Natural Frequencies of a Simply Supported Beam Including Transverse Shear Deformation and Rotatory Inertia Effects 6. 14. References 6. 15. Problems 7. Strength and Failure Theories 7. 1. Introduction 7. 2. Failure of Monolithic Isotropic Materials 7.3.1. Maximum Stress Theory 310 7.3.2. Maximum Strain Theory
XIII 5. Composite Material Shells 5.1. 5.2. 5.3. 5.4. 5.5. 5.6. 5.7. 5.8. 5.9. Introduction Analysis of Composite Material Circular Cylindrical Shells Some Edge Load and Particular Solutions A General Solution for Composite Cylindrical Shells Under Axially Symmetric Loads Response of a Long Axi-Symmetric Laminated Composite Shell to an Edge Displacement Sample Solutions Mid-Plane Asymmetric Circular Cylindrical Shells Buckling of Circular Cylindrical Shells of Composite Materials Subjected to Various Loads Vibrations of Composite Shells 5.10. 5.11. 5.12. Additional Reading On Composite Shells References Problems 6. Energy Methods For Composite Material Structures 6.1. 6.2. 6.3. 6.4. 6.5. 6.6. 6.7. 6.8. 6.9. Introduction Theorem of Minimum Potential Energy Analysis of a Beam Using the Theorem of Minimum Potential Energy Use of Minimum Potential Energy for Designing a Composite Electrical Transmission Tower Minimum Potential Energy for Rectangular Plates A Rectangular Composite Material Plate Subjected to Lateral and Hygrothermal Loads In-Plane Shear Strength Determination of Composite Materials in Laminated Composite Panels Use of the Theorem of Minimum Potential Energy to Determine Buckling Loads in Composite Plates Trial Functions for Various Boundary Conditions for Composite Material Rectangular Plates 6.10. 6.11. 6.12. 6.13. 6.14. 6.15. Reissner’s Variational Theorem and its Applications Static Deformation of Moderately Thick Beams Flexural Vibrations of Moderately Thick Beams Flexural Natural Frequencies of a Simply Supported Beam Including Transverse Shear Deformation and Rotatory Inertia Effects References Problems 7. Strength and Failure Theories 7.1. 7.2. 7.3. Introduction Failure of Monolithic Isotropic Materials Anisotropic Strength and Failure Theories 7.3.1. 7.3.2. Maximum Stress Theory Maximum Strain Theory 215 215 215 222 228 230 232 239 243 252 253 253 254 259 259 260 261 268 272 274 276 282 285 286 289 293 295 299 299 303 303 306 309 310 310
XIV 7. 3.3. Interactive Failure Theories 311 74. Lamina strength theories 7.5. Laminate Strength Analysis 7. 6. References 331 7.7. Problems 332 8. Joining of Composite Material Structures 333 8.1. General Remarks 8.2. Adhesive bonding 333 8.3. Mechanical Fastening 348 8. 4. Recommended Reading 8.5. References Introduction to Composite Design 9.1. Introduction 9. 2. Structural Composite Design Procedures 93. Engineering Analysis 371 A-1 Micromechanics 375 A-2 Test Standards for Polymer Matrix Composites A-3 Properties of Various Polymer Composites 393 Author index 397 Subject Index 40
XIV 7.3.3. Interactive Failure Theories 311 315 328 331 332 333 333 333 348 354 354 357 361 361 368 371 375 375 391 393 397 401 7.4. 7.5. 7.6. 7.7. Lamina Strength Theories Laminate Strength Analysis References Problems 8. Joining of Composite Material Structures 8.1. 8.2. 8.3. 8.4. 8.5. 8.6. General Remarks Adhesive Bonding Mechanical Fastening Recommended Reading References Problems 9. Introduction to Composite Design 9.1. 9.2. 9.3. Introduction Structural Composite Design Procedures Engineering Analysis Appendices A-1 A-2 A-3 Micromechanics Test Standards for Polymer Matrix Composites Properties of Various Polymer Composites Author Index Subject Index
CHAPTER 1 INTRODUCTION TO COMPOSITE MATERIALS 1.1 General History The combining of materials to form a new material system with enhanced material properties is well documented in history. For example, the ancient Israelite workers during their tenure under the Pharaohs incorporated chopped straw in bricks as a means of enhancing their structural integrity(see Exodus 5). The Japanese Samurai warriors were known to use laminated metals in the forging of their swords to obtain desirable material properties. More recently, in the 20 th century civil engineers placed steel rebars in cement and aggregate to make a well-known composite material, i.e reinforced concrete. One could say that the modern era of composite materials began with fiberglass polymer matrix composites about the time of World War IL. In order to introduce the reader to the subject matter of new high-performance composite materials it is necessary to begin by defining precisely what constitutes such a class of materials. Furthermore. one must also define the level or scale of material characterization to adequately describe such systems for discussion. This is done with the understanding that any definition and classification scheme introduced is somewhat arbitrary For introductory purposes, many workers in the field of composites use a somewhat loose description for defining a composite material as simply being th ombination of two or more materials formed to obtain some useful new material or specific material property. In some cases the addition of the words microscopic and acroscopic are added to describe the level of material characterization The definition posed above is to a large extent broad-based, in that it encompasses any number of material systems for which different levels of characterization must be used to specify the system and for which different analytical tools may be necessary for modeling purposes. As a simplistic example of the definition used above we can consider a beam consisting of clad copper and titanium material elements used in a switching strip. Such a composite system can be considered at the macroscopic level as providing enhanced temperature-dependent material behavior due to the mismatch in oefficients of thermal expansion between the copper and titanium metallic elements This material system, while consisting of two dissimilar materials and falling within the realm of satisfying the definition of a composite material would not be acceptable as being representative of modern definitions of composites for current applications in the aerospace, automotive and other technical areas. A representative list of journals dealing with composite materials is given in Section 1.9
CHAPTER 1 INTRODUCTION TO COMPOSITE MATERIALS 1.1 General History The combining of materials to form a new material system with enhanced material properties is well documented in history. For example, the ancient Israelite workers during their tenure under the Pharaohs incorporated chopped straw in bricks as a means of enhancing their structural integrity (see Exodus 5). The Japanese Samurai warriors were known to use laminated metals in the forging of their swords to obtain desirable material properties. More recently, in the century civil engineers placed steel rebars in cement and aggregate to make a well-known composite material, i.e., reinforced concrete. One could say that the modern era of composite materials began with fiberglass polymer matrix composites about the time of World War II. In order to introduce the reader to the subject matter of new high-performance composite materials it is necessary to begin by defining precisely what constitutes such a class of materials. Furthermore, one must also define the level or scale of material characterization to adequately describe such systems for discussion. This is done with the understanding that any definition and classification scheme introduced is somewhat arbitrary. For introductory purposes, many workers in the field of composites use a somewhat loose description for defining a composite material as simply being the combination of two or more materials formed to obtain some useful new material or specific material property. In some cases the addition of the words microscopic and macroscopic are added to describe the level of material characterization. The definition posed above is to a large extent broad-based, in that it encompasses any number of material systems for which different levels of characterization must be used to specify the system and for which different analytical tools may be necessary for modeling purposes. As a simplistic example of the definition used above we can consider a beam consisting of clad copper and titanium material elements used in a switching strip. Such a composite system can be considered at the macroscopic level as providing enhanced temperature-dependent material behavior due to the mismatch in coefficients of thermal expansion between the copper and titanium metallic elements. This material system, while consisting of two dissimilar materials and falling within the realm of satisfying the definition of a composite material would not be acceptable as being representative of modern definitions of composites for current applications in the aerospace, automotive and other technical areas. A representative list of journals dealing with composite materials is given in Section 1.9
1.2 Composite Material Description In order that agreement may be reached at the outset on a suitable modern day definition for advanced composite materials a structural classification according to the use of the following typical constituent elements is tabulated below STRUCTURAL LEVELS (BASIC/ELEMENTAL Single molecules, crystal cells (D MICROSTRUCTURAL Crystals, Phases, Compounds IID MACROSTRUCTURAL Matrices. Particles. Fibers Of the structural types cited above, Type(IlD, or the Macrostructural type is the most important for further discussion herein. Continuing with this, next consider a further classification within the structural framework adopted. A classification of combinations of materials is described and shown in Table 1.1 TABLE l.L. Classification of Composite Materials Fiber. Either continuous(long or chopped whiskers) suspended in a matrix material Fiber Composite Particulate. Composed of particles suspended in a matrix material Flake. Composed of flakes which have large ratios of platform area to thickness and pended in a matrix material
2 1.2 Composite Material Description In order that agreement may be reached at the outset on a suitable modern day definition for advanced composite materials a structural classification according to the use of the following typical constituent elements is tabulated below. Of the structural types cited above, Type (III), or the Macrostructural type is the most important for further discussion herein. Continuing with this, next consider a further classification within the structural framework adopted. A classification of combinations of materials is described and shown in Table 1.1. Fiber. Either continuous (long or chopped whiskers) suspended in a matrix material Particulate. Composed of particles suspended in a matrix material. Flake. Composed of flakes which have large ratios of platform area to thickness and are suspended in a matrix material