xi试 Contents 5.4 Strength theories 256 5.4.1 Rule-of-mixtures 257 5.4.2 Probabilistic initial failure strength 257 5.4.3 Probabilistic ultimate failure strength 262 5.5 Softening strips 273 5.6 Mechanical properties 275 5.7 Property optimization analysis 279 5.7.1 Constitutive relations 279 5.7.2 Graphical illustration of performance optimization 282 6 Two-dimensional textile structural composites 285 6.1 Introduction 285 6.2 Textile preforms 287 6.2.1 Wovens 288 6.2.2 Knits 292 6.2.3 Braids 294 6.3 Methodology of analysis 300 6.4 Mosaic model 302 6.5 Crimp(fiber undulation)model 308 6.6 Bridging model and experimental confirmation 314 6.7 Analysis of the knee behavior and summary of stiffness and strength modeling 319 6.8 In-plane thermal expansion and thermal bending coefficients 327 6.9 Hybrid fabric composites:mosaic model 335 6.9.1 Definitions and idealizations 336 6.9.2 Bounds of stiffness and compliance constants 340 6.9.2.1 Iso-strain 341 6.9.2.2 Iso-stress 343 6.9.3 One-dimensional approximation 344 6.9.4 Numerical results 345 6.10 Hybrid fabric composites:crimp and bridging models 348 6.10.1 Crimp model 349 6.10.2 Bridging model 352 6.10.3 Numerical results and summary of thermoelas- tic properties 354 6.11 Triaxial woven fabric composites 356 6.11.1 Geometrical characteristics 356 6.11.2 Analysis of thermoelastic behavior 358 6.11.3 Biaxial non-orthogonal woven fabric composites 365
xii Contents 5.4 Strength theories 256 5.4.1 Rule-of-mixtures 257 5.4.2 Probabilistic initial failure strength 257 5.4.3 Probabilistic ultimate failure strength 262 5.5 Softening strips 273 5.6 Mechanical properties 275 5.7 Property optimization analysis 279 5.7.1 Constitutive relations 279 5.7.2 Graphical illustration of performance optimization 282 6 Two-dimensional textile structural composites 285 6.1 Introduction 285 6.2 Textile preforms 287 6.2.1 Wovens 288 6.2.2 Knits 292 6.2.3 Braids 294 6.3 Methodology of analysis 300 6.4 Mosaic model 302 6.5 Crimp (fiber undulation) model 308 6.6 Bridging model and experimental confirmation 314 6.7 Analysis of the knee behavior and summary of stiffness and strength modeling 319 6.8 In-plane thermal expansion and thermal bending coefficients 327 6.9 Hybrid fabric composites: mosaic model 335 6.9.1 Definitions and idealizations 336 6.9.2 Bounds of stiffness and compliance constants 340 6.9.2.1 Iso-strain 341 6.9.2.2 Iso-stress 343 6.9.3 One-dimensional approximation 344 6.9.4 Numerical results 345 6.10 Hybrid fabric composites: crimp and bridging models 348 6.10.1 Crimp model 349 6.10.2 Bridging model 352 6.10.3 Numerical results and summary of thermoelastic properties 354 6.11 Triaxial woven fabric composites 356 6.11.1 Geometrical characteristics 356 6.11.2 Analysis of thermoelastic behavior 358 6.11.3 Biaxial non-orthogonal woven fabric composites 365
Contents xiii 6.12 Nonlinear stress-strain behavior 366 6.13 Mechanical properties 368 6.13.1 Friction and wear behavior 368 6.13.2 Notched strength 371 7 Three-dimensional textile structural composites 374 7.1 Introduction 374 7.2 Processing of textile preforms 376 7.2.1 Braiding 377 7.2.1.1 2-step braiding 377 7.2.1.2 4-step braiding 379 7.2.1.3 Solid braiding 382 7.2.2 Weaving 382 7.2.2.1 Angle-interlock multi-layer weaving 383 7.2.2.2 Orthogonal weaving 387 7.2.3 Stitching 387 7.2.4 Knitting 389 7.3 Processing windows for 2-step braids 389 7.3.1 Packing of fibers and yarn cross-sections 390 7.3.2 Unit cell of the preform 395 7.3.3 Criterion for yarn jamming 398 7.4 Yarn packing in 4-step braids 402 7.4.1 Unit cell of the preform 402 7.4.2 Criterion for yarn jamming 403 7.5 Analysis of thermoelastic behavior of composites 405 7.5.1 Elastic strain-energy approach 406 7.5.2 Fiber inclination model 407 7.5.3 Macro-cell approach 414 7.5.3.1 Geometric relations 414 7.5.3.2 Elastic constants 416 7.6 Structure-performance maps of composites 419 7.7 Mechanical properties of composites 428 7.7.1 Tensile and compressive behavior 428 7.7.2 Shear behavior 431 7.7.3 Fracture behavior 432 7.7.3.1 In-plane fracture 432 7.7.3.2 Interlaminar fracture 435 7.7.4 Impact 440 8 Flexible composites 443 8.1 Introduction 443 8.2 Cord/rubber composites 445
Contents xiii 6.12 Nonlinear stress-strain behavior 366 6.13 Mechanical properties 368 6.13.1 Friction and wear behavior 368 6.13.2 Notched strength 371 7 Three-dimensional textile structural composites 374 7.1 Introduction 374 7.2 Processing of textile preforms 376 7.2.1 Braiding 377 7.2.1.1 2-step braiding 377 7.2.1.2 4-step braiding 379 7.2.1.3 Solid braiding 382 7.2.2 Weaving 382 7.2.2.1 Angle-interlock multi-layer weaving 383 7.2.2.2 Orthogonal weaving 387 7.2.3 Stitching 387 7.2.4 Knitting 389 7.3 Processing windows for 2-step braids 389 7.3.1 Packing of fibers and yarn cross-sections 390 7.3.2 Unit cell of the preform 395 7.3.3 Criterion for yarn jamming 398 7.4 Yarn packing in 4-step braids 402 7.4.1 Unit cell of the preform 402 7.4.2 Criterion for yarn jamming 403 7.5 Analysis of thermoelastic behavior of composites 405 7.5.1 Elastic strain-energy approach 406 7.5.2 Fiber inclination model 407 7.5.3 Macro-cell approach 414 7.5.3.1 Geometric relations 414 7.5.3.2 Elastic constants 416 7.6 Structure-performance maps of composites 419 7.7 Mechanical properties of composites 428 7.7.1 Tensile and compressive behavior 428 7.7.2 Shear behavior 431 7.7.3 Fracture behavior 432 7.7.3.1 In-plane fracture 432 7.7.3.2 Interlaminar fracture 435 7.7.4 Impact 440 8 Flexible composites 443 8.1 Introduction 443 8.2 Cord/rubber composites 445
Xiv Contents 8.2.1 Rubber and cord properties 446 8.2.2 Unidirectional composites 447 8.2.3 Laminated composites 449 8.2.4 Cord loads in tires 453 8.3 Coated fabrics 456 8.4 Nonlinear elastic behavior-incremental analysis 459 8.4.1 Geometry of wavy fibers 460 8.4.2 Axial tensile behavior 462 8.4.2.1 Iso-phase model 462 8.4.2.2 Random-phase model 465 8.4.2.3 Nonlinear tensile stress-strain behavior 467 8.4.3 Transverse tensile behavior 471 8.4.3.1 Iso-phase model 471 8.4.3.2 Random-phase model 472 9 Nonlinear elastic finite deformation of flexible composites 474 9.1 Introduction 474 9.2 Background 478 9.2.1 Tensor notation 478 9.2.2 Lagrangian and Eulerian descriptions 480 9.3 Constitutive relations based on the Lagrangian description 485 9.3.1 Finite deformation of a composite lamina 485 9.3.2 Constitutive equations for a composite lamina 487 9.3.2.1 Strain-energy function 487 9.3.2.2 General constitutive equations for a unidirectional lamina 488 9.3.2.3 Pure homogeneous deformation 490 9.3.2.4 Simple shear 492 9.3.2.5 Simple shear superposed on simple extension 495 9.3.3 Constitutive equations of flexible composite laminates 499 9.3.3.1 Constitutive equations 499 9.3.3.2 Homogeneous deformation 500 9.3.3.3 Simple extension of a symmetric composite laminate 502 9.3.4 Determination of elastic constants 505 9.3.4.1 Tensile properties 505 9.3.4.2 Shear properties 506
xiv Contents 8.2.1 Rubber and cord properties 446 8.2.2 Unidirectional composites 447 8.2.3 Laminated composites 449 8.2.4 Cord loads in tires 453 8.3 Coated fabrics 456 8.4 Nonlinear elastic behavior - incremental analysis 459 8.4.1 Geometry of wavy fibers 460 8.4.2 Axial tensile behavior 462 8.4.2.1 Iso-phase model 462 8.4.2.2 Random-phase model 465 8.4.2.3 Nonlinear tensile stress-strain behavior 467 8.4.3 Transverse tensile behavior 471 8.4.3.1 Iso-phase model 471 8.4.3.2 Random-phase model 472 9 Nonlinear elastic finite deformation of flexible composites 474 9.1 Introduction 474 9.2 Background 478 9.2.1 Tensor notation 478 9.2.2 Lagrangian and Eulerian descriptions 480 9.3 Constitutive relations based on the Lagrangian description 485 9.3.1 Finite deformation of a composite lamina 485 9.3.2 Constitutive equations for a composite lamina 487 9.3.2.1 Strain-energy function 487 9.3.2.2 General constitutive equations for a unidirectional lamina 488 9.3.2.3 Pure homogeneous deformation 490 9.3.2.4 Simple shear 492 9.3.2.5 Simple shear superposed on simple extension 495 9.3.3 Constitutive equations of flexible composite laminates 499 9.3.3.1 Constitutive equations 499 9.3.3.2 Homogeneous deformation 500 9.3.3.3 Simple extension of a symmetric composite laminate 502 9.3.4 Determination of elastic constants 505 9.3.4.1 Tensile properties 505 9.3.4.2 Shear properties 506
Contents XV 9.4 Constitutive relations based on the Eulerian description 508 9.4.1 Stress-energy function 509 9.4.2 General constitutive equations 511 9.4.3 Pure homogeneous deformation 514 9.4.4 Simple shear superposed on simple extension 515 9.4.5 Determination of elastic compliance constants 517 9.5 Elastic behavior of flexible composites reinforced with wavy fibers 519 9.5.1 Introduction 519 9.5.2 Longitudinal elastic behavior based on the Lagrangian approach 520 9.5.3 Longitudinal elastic behavior based on the Eulerian approach 522 References 526 Author index 556 Subject index 563
Contents xv 9.4 Constitutive relations based on the Eulerian description 508 9.4.1 Stress-energy function 509 9.4.2 General constitutive equations 511 9.4.3 Pure homogeneous deformation 514 9.4.4 Simple shear superposed on simple extension 515 9.4.5 Determination of elastic compliance constants 517 9.5 Elastic behavior of flexible composites reinforced with wavy fibers 519 9.5.1 Introduction 519 9.5.2 Longitudinal elastic behavior based on the Lagrangian approach 520 9.5.3 Longitudinal elastic behavior based on the Eulerian approach 522 References 526 Author index 556 Subject index 563
Preface The science and technology of composite materials are based on a design concept which is fundamentally different from that of conventional structural materials.Metallic alloys,for instance, generally exhibit a uniform field of material properties;hence,they can be treated as homogeneous and isotropic.Fiber composites,on the other hand,show a high degree of spacial variation in their microstructures,resulting in non-uniform and anisotropic pro- perties.Furthermore,metallic materials can be shaped into desired geometries through secondary work (e.g.rolling,extrusion,etc.); the macroscopic configuration and the microscopic structure of a metallic component are related through the processing route it undergoes.With fiber composites,the co-relationship between microstructure and macroscopic configuration and their dependence on processing technique are even stronger.As a result,composites technology offers tremendous potential to design materials for specific end uses at various levels of scale. First,at the microscopic level,the internal structure of a component can be controlled through processing.A classical example is the molding of short-fiber composites,where fiber orientation,fiber length and fiber distribution may be controlled to yield the desired local properties.Other examples can be found in the filament winding of continuous fibers,hybridization of fibers, and textile structural forms based upon weaving,braiding,knitting, etc.In all these cases,the desired local stiffness,strength,toughness and other prespecified properties may be achieved by controlling the fiber type,orientation,and volume fraction throughout the structural component. Second,the external geometrical shape of a structural component can also be designed.Advances in the technology of filament winding enable the automated production of components with complex contours.It is now also feasible to fabricate three- dimensional fiber preforms using advanced textile technology.As the ability to fabricate larger and more integrated structural components of net shape is further enhanced,the need to handle and join a large number of small parts,as is currently done with metallic materials,diminishes
Preface The science and technology of composite materials are based on a design concept which is fundamentally different from that of conventional structural materials. Metallic alloys, for instance, generally exhibit a uniform field of material properties; hence, they can be treated as homogeneous and isotropic. Fiber composites, on the other hand, show a high degree of spacial variation in their microstructures, resulting in non-uniform and anisotropic properties. Furthermore, metallic materials can be shaped into desired geometries through secondary work (e.g. rolling, extrusion, etc.); the macroscopic configuration and the microscopic structure of a metallic component are related through the processing route it undergoes. With fiber composites, the co-relationship between microstructure and macroscopic configuration and their dependence on processing technique are even stronger. As a result, composites technology offers tremendous potential to design materials for specific end uses at various levels of scale. First, at the microscopic level, the internal structure of a component can be controlled through processing. A classical example is the molding of short-fiber composites, where fiber orientation, fiber length and fiber distribution may be controlled to yield the desired local properties. Other examples can be found in the filament winding of continuous fibers, hybridization of fibers, and textile structural forms based upon weaving, braiding, knitting, etc. In all these cases, the desired local stiffness, strength, toughness and other prespecified properties may be achieved by controlling the fiber type, orientation, and volume fraction throughout the structural component. Second, the external geometrical shape of a structural component can also be designed. Advances in the technology of filament winding enable the automated production of components with complex contours. It is now also feasible to fabricate threedimensional fiber preforms using advanced textile technology. As the ability to fabricate larger and more integrated structural components of net shape is further enhanced, the need to handle and join a large number of small parts, as is currently done with metallic materials, diminishes