Principles of composite material mechanics, third edition (1)

Gibson Materials Science/Mechanical Engineering “Professor Ron Gibson provides a comprehensive textbook to cover the basic concept of anisotropy and inhomogeneity of composites as well as state-of-art issues such as dynamic behavior, fracture and testing The many worked-out examples and homework problems are most useful for readers to understand the basic concepts to be used in practical applications of composites Such knowledge is essential for advanced composite materials to be further applied to lightweight structures successfully.” —Nobuo Takeda, University of Tokyo, Japan Principles of Composite Material Mechanics, Third Edition presents a unique blend of classical and contemporary mechanics of composites technologies While continuing to cover classical methods, this edition also includes frequent references to current state-of-the-art composites technology and research findings New to the Third Edition • • • • Many new worked-out example problems, homework problems, figures, and references An appendix on matrix concepts and operations Coverage of particle composites, nanocomposites, nanoenhancement of conventional fiber composites, and hybrid multiscale composites Expanded coverage of finite element modeling and test methods Easily accessible to students, this popular bestseller incorporates the most worked-out example problems and exercises of any available textbook on mechanics of composite materials It offers a rich, comprehensive, and up-to-date foundation for students to begin their work in composite materials science and engineering K12275 ISBN: 978-1-4398-5005-3 90000 781439 850053 PRINCIPLES OF COMPOSITE MATERIAL MECHANICS PRINCIPLES OF COMPOSITE MATERIAL MECHANICS THIRD EDITION PRINCIPLES OF COMPOSITE MATERIAL MECHANICS THIRD EDITION Ronald F Gibson PRINCIPLES OF COMPOSITE MATERIAL MECHANICS THIRD EDITION MECHANICAL ENGINEERING A Series of Textbooks and Reference Books Founding Editor L L Faulkner Columbus Division, Battelle Memorial Institute and Department of Mechanical Engineering The Ohio State University Columbus, Ohio 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Spring Designer’s Handbook, Harold Carlson Computer-Aided Graphics and Design, Daniel L Ryan Lubrication Fundamentals, J George Wills Solar Engineering for Domestic Buildings, William A Himmelman Applied Engineering Mechanics: Statics and Dynamics, G Boothroyd and C Poli Centrifugal Pump Clinic, Igor J Karassik Computer-Aided Kinetics for Machine Design, Daniel L Ryan Plastics Products Design Handbook, Part A: Materials and Components; Part B: Processes and Design for Processes, edited by Edward Miller Turbomachinery: Basic Theory and Applications, Earl Logan, Jr Vibrations of Shells and Plates, Werner Soedel Flat and Corrugated Diaphragm Design Handbook, Mario Di Giovanni Practical Stress Analysis in Engineering Design, Alexander Blake An Introduction to the Design and Behavior of Bolted Joints, John H Bickford Optimal Engineering Design: Principles and Applications, James N Siddall Spring Manufacturing Handbook, Harold Carlson Industrial Noise Control: Fundamentals and Applications, edited by Lewis H Bell Gears and Their Vibration: A Basic Approach to Understanding Gear Noise, J Derek Smith Chains for Power Transmission and Material Handling: Design and Applications Handbook, American Chain Association Corrosion and Corrosion Protection Handbook, edited by Philip A Schweitzer Gear Drive Systems: Design and Application, Peter Lynwander Controlling In-Plant Airborne Contaminants: Systems Design and Calculations, John D Constance CAD/CAM Systems Planning and Implementation, Charles S Knox Probabilistic Engineering Design: Principles and Applications, James N Siddall 24 Traction Drives: Selection and Application, Frederick W Heilich III and Eugene E Shube 25 Finite Element Methods: An Introduction, Ronald L Huston and Chris E Passerello 26 Mechanical Fastening of Plastics: An Engineering Handbook, Brayton Lincoln, Kenneth J Gomes, and James F Braden 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Pressure Gauge Handbook, AMETEK, U.S Gauge Division, edited by Philip W Harland 41 Fabric Filtration for Combustion Sources: Fundamentals and Basic Technology, R P Donovan 42 Design of Mechanical Joints, Alexander Blake 43 CAD/CAM Dictionary, Edward J Preston, George W Crawford, and Mark E Coticchia 44 Machinery Adhesives for Locking, Retaining, and Sealing, Girard S Haviland 45 Couplings and Joints: Design, Selection, and Application, Jon R Mancuso 46 Shaft Alignment Handbook, John Piotrowski 47 BASIC Programs for Steam Plant Engineers: Boilers, Combustion, Fluid Flow, and Heat Transfer, V Ganapathy 48 Solving Mechanical Design Problems with Computer Graphics, Jerome C Lange 49 Plastics Gearing: Selection and Application, Clifford E Adams 50 Clutches and Brakes: Design and Selection, William C Orthwein 51 Transducers in Mechanical and Electronic Design, Harry L Trietley 52 Metallurgical Applications of Shock-Wave and High-Strain-Rate Phenomena, edited by Lawrence E Murr, Karl P 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Technology: A Practical Guide, Second Edition, Revised and Expanded, Marsbed H Hablanian 112 Geometric Dimensioning and Tolerancing: Workbook and Answerbook, James D Meadows 113 Handbook of Materials Selection for Engineering Applications, edited by G T Murray 114 Handbook of Thermoplastic Piping System Design, Thomas Sixsmith and Reinhard Hanselka 115 Practical Guide to Finite Elements: A Solid Mechanics Approach, Steven M Lepi 116 Applied Computational Fluid Dynamics, edited by Vijay K Garg 117 Fluid Sealing Technology, Heinz K Muller and Bernard S Nau 118 Friction and Lubrication in Mechanical Design, A A Seireg 119 Influence Functions and Matrices, Yuri A Melnikov 120 Mechanical Analysis of Electronic Packaging Systems, Stephen A McKeown 121 Couplings and Joints: Design, Selection, and Application, Second Edition, Revised and Expanded, Jon R Mancuso 122 Thermodynamics: Processes and Applications, Earl Logan, Jr 123 Gear Noise and Vibration, J Derek Smith 124 Practical Fluid Mechanics for Engineering Applications, John J Bloomer 125 Handbook of Hydraulic Fluid Technology, edited by George E Totten 126 Heat Exchanger Design Handbook, T Kuppan 127 Designing for Product Sound Quality, Richard H Lyon 128 Probability Applications in Mechanical Design, Franklin E Fisher and Joy R Fisher 129 Nickel Alloys, edited by Ulrich Heubner 130 Rotating Machinery Vibration: Problem Analysis and Troubleshooting, Maurice L Adams, Jr 131 Formulas for Dynamic Analysis, Ronald L Huston and C Q Liu 132 Handbook of Machinery Dynamics, Lynn L Faulkner and Earl Logan, Jr 133 Rapid Prototyping Technology: Selection and Application, Kenneth G Cooper 134 Reciprocating Machinery Dynamics: Design and Analysis, Abdulla S Rangwala 135 Maintenance Excellence: Optimizing Equipment Life-Cycle Decisions, edited by John D Campbell and Andrew K S Jardine 136 Practical Guide to Industrial Boiler Systems, Ralph L Vandagriff 137 Lubrication Fundamentals: Second Edition, Revised and Expanded, D M 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Testing, R E Little 149 Industrial Boilers and Heat Recovery Steam Generators: Design, Applications, and Calculations, V Ganapathy 150 The CAD Guidebook: A Basic Manual for Understanding and Improving Computer-Aided Design, Stephen J Schoonmaker 151 Industrial Noise Control and Acoustics, Randall F Barron 152 Mechanical Properties of Engineered Materials, Wolé Soboyejo 153 Reliability Verification, Testing, and Analysis in Engineering Design, Gary S Wasserman 154 Fundamental Mechanics of Fluids: Third Edition, I G Currie 155 Intermediate Heat Transfer, Kau-Fui Vincent Wong 156 HVAC Water Chillers and Cooling Towers: Fundamentals, Application, and Operation, Herbert W Stanford III 157 Gear Noise and Vibration: Second Edition, Revised and Expanded, J Derek Smith 158 Handbook of Turbomachinery: Second Edition, Revised and Expanded, edited by Earl Logan, Jr and Ramendra Roy 159 Piping and Pipeline Engineering: Design, Construction, Maintenance, Integrity, and Repair, George A Antaki 160 Turbomachinery: Design and Theory, Rama S R Gorla and Aijaz Ahmed Khan 161 Target Costing: Market-Driven Product Design, M Bradford Clifton, Henry M B Bird, Robert E Albano, and Wesley P Townsend 162 Fluidized Bed Combustion, Simeon N Oka 163 Theory of Dimensioning: An Introduction to Parameterizing Geometric Models, Vijay Srinivasan 164 Handbook of Mechanical Alloy Design, edited by George E Totten, Lin Xie, and Kiyoshi Funatani 165 Structural Analysis of Polymeric Composite Materials, Mark E Tuttle 166 Modeling and Simulation for Material Selection and Mechanical Design, edited by George E Totten, Lin Xie, and Kiyoshi Funatani 167 Handbook of Pneumatic Conveying Engineering, David Mills, Mark G Jones, and Vijay K Agarwal 168 Clutches and Brakes: Design and Selection, Second Edition, William C Orthwein 169 Fundamentals of Fluid Film Lubrication: Second Edition, Bernard J Hamrock, Steven R Schmid, and Bo O Jacobson 170 Handbook of Lead-Free Solder Technology for Microelectronic 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Design and Manufacturing, Wasim Ahmed Khan and Abdul Raouf S.I 197 Maintenance, Replacement, and Reliability: Theory and Applications, Andrew K S Jardine and Albert H C Tsang 198 Finite Element Method: Applications in Solids, Structures, and Heat Transfer, Michael R Gosz 199 Microengineering, MEMS, and Interfacing: A Practical Guide, Danny Banks 200 Fundamentals of Natural Gas Processing, Arthur J Kidnay and William Parrish 201 Optimal Control of Induction Heating Processes, Edgar Rapoport and Yulia Pleshivtseva 202 Practical Plant Failure Analysis: A Guide to Understanding Machinery Deterioration and Improving Equipment Reliability, Neville W Sachs, P.E 203 Shaft Alignment Handbook, Third Edition, John Piotrowski 204 Advanced Vibration Analysis , S Graham Kelly 205 Principles of Composite Materials Mechanics, Second Edition, Ronald F Gibson 206 Applied Combustion, Second Edition, Eugene L Keating 207 Introduction to the Design and Behavior of Bolted Joints, Fourth Edition: Non-Gasketed Joints, John H Bickford 208 Analytical and Approximate Methods in Transport Phenomena, Marcio L de Souza-Santos 209 Design and Optimization of Thermal Systems, Second Edition, Yogesh Jaluria 210 Friction Science and Technology: From Concepts to Applications, Second Edition, Peter J Blau 211 Practical Guide to the Packaging of Electronics, Second Edition: Thermal and Mechanical Design and Analysis, Ali Jamnia 212 Practical Stress Analysis in Engineering Design, Third Edition, Ronald L Huston and Harold Josephs 213 Principles of Biomechanics, Ronald L Huston 214 Mechanical Vibration Analysis, Uncertainties, and Control, Third Edition, Haym Benaroya and Mark L Nagurka 215 Solid Fuels Combustion and Gasification: Modeling, Simulation, and Equipment Operations, Second Edition, Third Edition, Marcio L de Souza-Santos 216 Asset Management Excellence, edited by John D Campbell, Andrew K S Jardine, Joel McGlynn 217 Mechanical Tolerance Stackup and Analysis, Second Edition, Bryan R Fischer 218 Principles of Composite Materials Mechanics, Third Edition, Ronald F Gibson 614 Appendix A: Matrix Concepts and Operations consists of the cofactors Cij arranged in matrix form For example, for the (3 × 3) matrix and associated determinants, C11 [C ] = C21 C31 C13   C23  C33  C12 C22 C32 The adjoint matrix [adj a] of a square matrix [a] is defined as the transpose of the cofactor matrix [C] as [adj a] = [C]T For example, for the (3 × 3) matrix defined above, the adjoint matrix is  adj a  = [C ] T  C11  = C12 C13 C21 C22 C23 C31   C32  C33  The inverse [a]−1 of a square matrix [a] having determinant D and adjoint matrix [adj a] is defined as [ a]−1 =  adj a  D It is important to note that the inverse of a matrix does not exist when its determinant D = 0 Such a matrix is referred to as a singular matrix To demonstrate these operations, it is instructive to find [S]−1, the inverse of the lamina compliance matrix [S] for a specially orthotropic lamina, which is the same as the lamina stiffness matrix [Q] Recalling from Equations 2.24 and 2.25 that the compliance matrix is symmetric, Sij = Sji and S11 [S] = S12  S12 S22 0  0 S66  Expanding the determinant of [S] along the first row, S11 D = S12 S12 S22 ( S22 = S11 S66 = S66 S11S22 − S12 ) S12 − S12 S66 0 S12 +0 S66 S22 615 Appendix A: Matrix Concepts and Operations The minors of the Sij are 0 M11 = S22 S66 = S22S66 M12 = S12 S66 = S12S66 M13 = S12 S22 =0 M21 = S12 0 = S12S66 S66 M22 = S11 0 = S11S66 S66 M23 = S11 S12 =0 M31 = S12 S22 =0 M32 = S11 S12 =0 M33 = S11 S12 S12 = S11S22 − S12 S22 Applying the equation Cij = (−1)i+jMij, the cofactor matrix is found to be  S22S66 [C ] =  −S12S66  −S12S66 S11S66    S11S22 − S12  0 Noting that the cofactor matrix is symmetric for this case, the adjoint matrix is  adj S  = [C ] T  S22S66  = [C ] =  −S12S66  −S12S66 S11S66     S11S22 − S12 0 The inverse of matrix [S] is then [S ] −1 S22   S S − S2 12  11 22  adj S  S12 = = − D S S − S122  11 22   S12 S11S22 − S12 S11 S11S22 − S12 −    Q11    = Q12     S66  Q12 Q22 0    Q66  which was previously given in Equation 2.27 As another example of matrix inversion, recall from Chapter that the  2D  transformation of stresses and strains from the xy-coordinate 616 Appendix A: Matrix Concepts and Operations s­ ystem  to the 12-coordinate system is governed by the transformation matrix  c2  [T ] =  s  −cs  2cs   −2cs  c − s2  s2 c2 cs and the corresponding transformation from 12 to xy coordinates is governed by the inverse matrix [T]−1, where c = cos θ and s = sin θ The determinant of [T] is c2 D = s2 −cs s2 c2 cs 2cs c2 −2cs = c cs c − s2 −2cs s − s c − s2 −cs ( = c + 3c s + 3c s + s6 = c + s ) s2 −2cs cs + c − s2 −cs c2 cs =1 After finding the minors, the cofactor matrix, and the adjoint matrix, the resulting inverse matrix is [T ] −1 c2  adj T   = = s D  cs  s2 c2 −cs −2cs   2cs  c − s2  which was previously given in Equation 2.30 For a numerical example, consider the inverse of the nonsymmetric matrix 5 [ a] =   1  2  The determinant expanded along the first row is D= 1 2 = ( 5) − (4) + (1) = 30 − − = 24 617 Appendix A: Matrix Concepts and Operations The minors of the aij are M11 = 2 = 6, M21 = = 12, M31 = = 6, M12 = 1 M22 = M32 = = 1, M13 = = 14, = 9, 1 M23 = M33 = = −2 5 = −4 =6 Applying the equation Cij = (−1)i+j Mij, the cofactor matrix is found to be  [C ] =  −12  −1 14 −9 −2  4  The adjoint matrix is 6 T  adj a  = [C ] =  −1   −2 −12 14 6  −9  and the inverse of [a] is [ a] −1 6  adj a   = = −1 24  D  −2 −12 14   0.25   −9 =  −0.042   −0.083 −0.5 0.583 0.166 0.25   −0.375 0.25  For large matrices, this inversion method may be cumbersome due to the large number of determinants that is required, and other methods such as Gauss–Jordan elimination may be more computationally efficient The reader is referred to books on numerical methods for more details on such methods This page intentionally left blank Appendix B: Stress Equilibrium Equations For a body under static loading, the stresses at every point in the body must satisfy the static equilibrium conditions, Σ F = and Σ M = 0, where F and M are the resultant external forces and moments, respectively, acting at the point Consider the infinitesimal two-dimensional element of unit thickness shown in Figure B1, where normal stresses are denoted by σ and shear stresses are denoted by τ The fact that the stresses in a body generally vary from point-to-point is accounted for by including differential changes in stresses from one face to another For example, while the normal stress, σx, acts on the left face of the element, the variation of stresses from point-to-point is accounted for by assigning the normal stress, σx + (∂σx/∂x)dx, to the right face of the element (i.e., the stress, σx, changes at the rate ∂σx/∂x over the increment dx along the x direction) Similarly, the other stress components, σy, and τxy, must vary from the left face to the right face, and from the bottom face to the top face of  the element Now, the scalar component of the moment equilibrium ­condition, Σ MO = 0, at the center point O can be written as ∑M O = τ xydy(1)  dx ∂τ xy  dy dx  − τ yxdx(1) +  τ xy + dx dy(1)   ∂x     dy ∂τ yx  −  τ yx + dy  dx(1) =0 ∂y    After dividing each term by dx dy, the remaining terms are τ xy + ∂τ xy dx ∂τ yx dy − τ yx − =0 ∂x ∂y At a point in the body, where dx and dy both approach zero, the result is the proof of symmetry of the shear stresses τxy = τyx and in the general 3D case, τij = τji where i, j = 1, 2, 619 620 Appendix B: Stress Equilibrium Equations y σy ∂σy ∂y dy τyx ∂τyx ∂y τxy σx dy O τxy σx dy ∂τxy ∂x dx ∂σx dx ∂x σy τyx dx x FIGURE B1 Infinitesimal element representing stresses at a point in a body Thus, moment equilibrium requires that the shear stresses be symmetric Recall that symmetry of the stresses is a key assumption leading to the development of the contracted notation in Chapter Now with regard to the force equilibrium requirement, the scalar component of the force equilibrium condition Σ F = along the x direction is ∑F x ∂σ x   = −σ x dy(1) +  σ x + dx dy(1)   ∂x  ∂τ yx  dy  dx(1) = − τ yx dx(1) +  τ yx + ∂y   Simplifying the above equation, writing a similar equilibrium equation ΣFy = 0, and making use of the symmetry condition τxy = τyx, we get the 2D stress equilibrium equations ∂σ x ∂τ yx + =0 ∂x ∂y ∂τ xy ∂σ y + =0 ∂x ∂y Appendix B: Stress Equilibrium Equations 621 A similar derivation for a 3D infinitesimal element including the force equilibrium requirement, ΣFz = 0, leads to the full set of stress equilibrium equations ∂σ x ∂τ yx ∂τ zx + + =0 ∂x ∂y ∂z ∂τ xy ∂σ y ∂τ zy + + =0 ∂x ∂y ∂z ∂τ xz ∂τ yz ∂σ z + + =0 ∂x ∂y ∂z This page intentionally left blank Appendix C: Strain–Displacement Equations The relationships between strains and displacements at a point in a stressed body can be derived by considering the geometry of deformation of an infinitesimal element Figure C1 shows the geometry of deformation for a 2D plane strain condition, where the original undeformed element is denoted by ABCD and the deformed element is denoted by A′B′C′D′ So during deformation, point A deforms to point A′, point B deforms to point B′, and so forth The displacement along the x direction is u, and the corresponding displacement along the y direction is v Using the definition of normal strain (i.e., the change in length per unit length), for small deformations and strains, the normal strain along the x direction can be written as ∂u   dx + ∂x dx  − dx A′B′ − AB ∂u = εx = = AB dx ∂x Similarly, the normal strain along the y direction is given by  ∂v  dy  − dy dy + ∂y  A′D′ − AD ∂v  = = εy = AD dy ∂y Referring to Figure 2.2 and the definition of engineering shear strain, γxy, it is seen that γxy is equal to the difference between original right angle DAB in the undeformed element and the angle D′A′B′ in the deformed ­element If we assume small deformations and a correspondingly small angle between line A′B′ and line AB, and define this small angle as θ, then we can write θ ≅ tan θ = ∂v/∂x dx dx Similarly, if we define the small angle between line A′D′ and line AD as ϕ, we can write φ ≅ tan φ = ∂u/∂y dy dy 623 624 Appendix C: Strain–Displacement Equations Y ϕ π ∂u u dy ∂y C′ D′ ∂ dy ∂y D dy A′ A u y x B′ C θ ∂ dx ∂x B dx γxy u ∂u dx ∂x X FIGURE C1 Geometry of deformation at a point in a stressed body ABCD is undeformed ­element and A′B′C′D′ is deformed element As a result, the shear strain is given by γ xy = (∂v ∂x)dx (∂u ∂y )dy π π ∂v ∂u  −  − θ − φ = θ + φ = + = + 2 dx dy ∂x ∂y  In summary, the 2D strain–displacement equations are εx = ∂u ∂x εy = ∂v ∂y γ xy = ∂v ∂u + ∂x ∂y Appendix C: Strain–Displacement Equations Similarly, the full 3D strain–displacment equations are ∂u ∂x ∂v εy = ∂y ∂w εz = ∂z εx = ∂v ∂u + ∂x ∂y ∂w ∂v = + ∂y ∂z ∂u ∂w = + ∂z ∂x γ xy = γ yz γ zx 625 This page intentionally left blank This page intentionally left blank Gibson Materials Science/Mechanical Engineering “Professor Ron Gibson provides a comprehensive textbook to cover the basic concept of anisotropy and inhomogeneity of composites as well as state-of-art issues such as dynamic behavior, fracture and testing The many worked-out examples and homework problems are most useful for readers to understand the basic concepts to be used in practical applications of composites Such knowledge is essential for advanced composite materials to be further applied to lightweight structures successfully.” —Nobuo Takeda, University of Tokyo, Japan Principles of Composite Material Mechanics, Third Edition presents a unique blend of classical and contemporary mechanics of composites technologies While continuing to cover classical methods, this edition also includes frequent references to current state-of-the-art composites technology and research findings New to the Third Edition • • • • Many new worked-out example problems, homework problems, figures, and references An appendix on matrix concepts and operations Coverage of particle composites, nanocomposites, nanoenhancement of conventional fiber composites, and hybrid multiscale composites Expanded coverage of finite element modeling and test methods Easily accessible to students, this popular bestseller incorporates the most worked-out example problems and exercises of any available textbook on mechanics of composite materials It offers a rich, comprehensive, and up-to-date foundation for students to begin their work in composite materials science and engineering K12275 ISBN: 978-1-4398-5005-3 90000 781439 850053 PRINCIPLES OF COMPOSITE MATERIAL MECHANICS PRINCIPLES OF COMPOSITE MATERIAL MECHANICS THIRD EDITION PRINCIPLES OF COMPOSITE MATERIAL MECHANICS THIRD EDITION Ronald F Gibson ... Second Edition, Bryan R Fischer 218 Principles of Composite Materials Mechanics, Third Edition, Ronald F Gibson This page intentionally left blank PRINCIPLES OF COMPOSITE MATERIAL MECHANICS THIRD EDITION. .. emphasize either the mechanics or the materials science aspects of composites Principles of Composite Material Mechanics is mechanics oriented Composite materials technology is new enough for... sacrificed Principles of Composite Material Mechanics (a) (b) (c) (d) FIGURE 1.4 Types of fiber-reinforced composites (a) Continuous fiber composite, (b) woven composite, (c) chopped fiber composite,