406 Subject index nonlinear behavior of isotropic polymeric nonlinear behavior of a cross-ply nonlinear behavior of an angle-ply nonlinear behavior of composite material nonlinear behavior of elastomers 124 nonlinear constitutive equations 161, 188 nonlinear constitutive theory 124, 142 nonlinear deformation of an anisotropic nonlinear elastic behavior of a unidirec- nonlinear elastic material 133 nonlinear elastic material model 5, 124, nonlinear elastic models 126 nonlinear elastic-plastic behavior of metal nonlinear hereditary theory 331 nonlinear material behavior 5 nonlinear models 124, 142, 161, 166, 188 nonlinear phenomenological model of the nonlinear stress-strain diagram 17, 127, nonlinear viscoelastic behavior 331 nonsymmetric laminate 257 normal deflection 226 normal element 226 normal stress 29, 33, 72 normalized strength 15-1 6 number of layers 225 nylon 6 material 128 layer 166 layer 188 145 unidirectional layer 161 tional layer 143 I62 layers 128 cross-ply layer 174 21 1 oblique plane 33 off-axis compression 278 off-axis tension 278 off-axis test 155-157 off-axis test of a unidirectional one-term exponential approximation 323 one-term exponential approximation of the optimal angles 368 optimal composite structures 365 optimal composite vessel 370 composite 153 creep compliance 326 optimal filament wound shell 381-383, optimal laminate 368-370, 372, 374-377 optimal orientation angles 377-378 optimal structural parameters of the optimal structure of the end closure of optimal structures 369 optimality conditions 367, 376 optimality criterion 365, 373 organic fibers 12 organic resin 21 organosilicone resin 6, 19 orthogonally reinforced orthotropic orthotropic angle-ply *4 layer 187 orthotropic core 250 orthotropic homogeneous material 1 18 orthotropic laminate 239 orthotropic layer 140, 233, 271-272, 309, orthotropic material 4W7, 146,287,290, orthotropic ply 80, 271 385 laminate 374 pressure vessel 387 layer 163 373 327 PA1 6, 19 PAN-based carbon fibers 11 parametric equation for the shell PEEK 6, 19 periodical system of fibers 91 permanent failure 174 petroleum pitch 12 phenol-formaldehyde resin 6 phenolic resin 21 phenomenological homogeneous model of a ply 117 phenomenological model 217 phenomenological model of a composite material 118 phenomenological theory 29 Physics of Solids 59 pitch 21 pitch-based carbon fibers 11 plain weave 209 plane laminated material structure 22, 24 plane stress state 130, 143, 146, 163, 271, meridian 383 372 Subject index 407 plastic behavior 130 plastic deformation 5 plastic Poisson’s ratio 132 plastic potential 145-146 plastic strain 5, 8, 128-130, 137, 145 plasticity criterion of maximum shear stress 293 plasticity theory 130 ply degradation 177, 197 ply interaction 201 ply longitudinal modulus 1 15 ply microstructure 8 1, 83 ply orientation 121 ply stiffness across the fibers 91 ply stiffness in shear 91 ply strength 365 ply thermal conductivity 305 ply transverse stiffness 91 ply waviness 353-355 ply dissipation factor 335 Poisson’s effect 279 Poisson’s ratio 46, 86, 96-97, 123, 208, polar opening 388 polyacrylonitrile (PAN) filaments I 1 polyamide-imide (PAI) 6 polyester 6 polyester resin 6. 17 polyetheretherketone (PEEK) 6 polyethylene 6 polyethylene fibers 7, 12 polyimidc resin 6, 17, 19 polymeric and metal matrices 99 polymeric composites 18, 301, 318 polymeric layer 124 polymeric matrices 17-1 8, 318-3 19 polynomial approximation 125 polynomial criterion 28 1, 284 polyphenylenesulfide (PPS) 6 polypropylene fibers 6 polystyrene 6 polysulfone (PSU) 6, 19 potential energy 3, 43 potential energy of the body power approximation 130-133, 138 power hardening law 132 power law for the plastic potential 128, 130, 137-1 38, ply 55, 121 245, 252 46 146 preimpregnated fiberglass fabric 25 preliminary tension 24, 352 prepreg 22-23 pressure vessel 26, 140, 179,369-370,379, principal axes 34, 36 principal coordinates 34, 36 principal material axes 140,217,255,271, principal material coordinates 147, 388 291, 309, 335 153-154, 161, 163, 166, 211, 218, 260, 285-287, 295, 304, 368, 374, 380 principal planes 35-36 principal strains 373 principal stresses 34-36, 293, 373, 376, principle of minimum strain energy 52, principle of minimum total potential process of the failure 349 processing 2 1 processing temperature 18-1 9 projectile residual velocity 347 projectile striking velocity 347 properties of unidirectional composites proportional loading 138-1 39, 156 protective coatings 21 pultrusion 22-23, 301 pure shear 35, 286, 372 pure shear in a twisted tubular specimen 103 pure transverse shear of a cross-ply layer 165 pyrolysis 11, 13, 21, 26 pyrolytic carbon 21 pyrolytic carbon matrix 24 38 1 91, 169, 234 energy 51,93, 233 96 quadratic approximation strength quartz fibers 6, IO quasi-homogeneous laminates 240 quasi-isotropic laminates 243, 245, 369, quasi-isotropic materials 377 quasi-isotropic properties of the quasi-static test 347 criterion 293 377 laminates 244 408 Subject index radial compression 262 radial displacement 263 radius of polar opening 390 rate of loading 5, 319 rayon organic filaments 11 rectangular pulse 343 reference plane 225, 236, 252 reference surface 267 reference surface of the laminate 314 reflected pulse 343 regular inclusions 90 regular system of cracks reinforced materials 9 relaxation diagram 325 relaxation kernel 322 relaxation modulus 322, 327 residual strain 5, 8, 361 resin transfer molding 26 resultants of the shear stresses 230 rheologic characteristics of structural rheological mechanical model 323 rice fibers 14 Riemannian (curved) space 41 rotation 33, 226 rotation angle 40-41, 194 rule of mixtures 86 Russian Composite Center - Central 17 1 materials 319 Institute of Special Machinery (CRISM) vi safety factor 180 sandwich laminate 230, 250 sandwich structure 239, 249, 252, 344 satin weave 209 secant modulus 5, 7, 131-133, 143 second-order model of a ply 87, 90, 98 second-order polynomial approximation second-order polynomial criterion 285 second-order tensor criterion 286 sewing 215 shear deformation 142, 156 shear failure under compression 105 shear mode of buckling 108 shear mode of fracture 113 shear modulus 46, 96, 123, 208 shear modulus of a ply 81 shear modulus of an angle-ply layer 28 1 188 shear stiITness 142 shear strain 38-39, 69, 85, 294 shear strain in the matrix layer 69 shear strength 365 shear stress 29, 34, 72, 85, 197, 198 shear stress concentration 77 shear-shear coupling coefficient 46, 153 shear-extension coupling 242 shear-extension coupling coefficient 46, shear-twisting coupling 242 short fibers 215 shrinkage characteristics of a unidirectional ply 362 shrinkage strains 360 silicon carbide 13, 21 silicon carbide (Sic) fibers 7, 13 simplest approximation criterion 283 simulation of pure shear in a square single fiber fragmentation test 77 singular approximations 326 sisal fibers 14 sixth-order approximation 143 small strains 39 small variation of displacements 51 small variation of strains 51 space of stresses 271 spatial (3D, 4D) structures 24 spatial material structure 22, 215 spatially reinforced composites 215, 217 spatially reinforced layers 214 spatially reinforced structure 217 special loading effects 301 specific elastic energy 4 specific energy 60 specific heat 302 specific modulus 4, 6-7 specific stiffness 365 specific strain energy 44, 47 specific strength 2, 6-7, 365, 369, 391 specific surface energy 60 specimen buckling I13 specimens of matrix material 82 spider silk fibers 14 square array 58 square fiber distribution 56 stacking sequence 225, 252-253, 344, 365 stacking-sequence notations 240, 247 153 frame 105 Subject index 409 standard types of specimens 95 static boundary conditions 48 static strength 79 statically admissible stress field 49 statically and kinematically admissible stress and strain fields 94 Statics of Solids 230 stiffness 1 stiffness coefficients 45, 150-1 51, 164, 217, 219, 225, 230, 232, 235, 237, 247, 251, 313 layer 236 stiffness coefficients of a homogeneous stiffness coefficientsof a laminate 238 stiffness coefficients of an orthotropic stiffness degradation 339 stiffness matrix 46, 218 stiffness of composite materials 339 stiffnesses of the degraded layer 176 stitching 21 5 strain 2, 37 strain concentration factor 106 strain energy 3, 129, 143, 234 strain energy of fibers 109 strain energy of the matrix strain intensity 129 strain interaction 279 strain transformation 39 strain-displacement equations 38, 48, 53, strength analysis 271, 293 strength criterion 271, 284, 287, strength deviation 63 strength dispersion 64 strength matrix 289 strength of a composite bundle of strength of a dry bundle of fibers 66, 96 strength of a unidirectional ply 64 strength of laminates 271 strength of the optimal laminate 378 strength scatter 63 strength tensor 285-286, 291 strength variation 65 stress 2, 29, 33 stress analysis of the matrix in the vicinity layer 237 109 202, 230 289-290 fibers 66, 96 of fibers 91 stress concentration 77, 171, 201, 205, stress concentration in the matrix 96 stress diffusion 16, 171 stress diffusion in fibers 65 stress diffusion in the vicinity of damaged or broken fibers 181 stress formulation 48, 69 stress function 69 stress intensity stress invariants 126-1 27 stress ratio 338 stress relaxation 319, 325 stress resultants and couples 227-229 stress space 139, 272 stress tensor 285 stress transformation 32 stress wave 343 stress-strain curve 3, 5, 100, 124-125, 144, 162, 200 stress-strain diagram of the cross-ply layer 174 stress-strain diagram 3-4, 7, 10, 12, 17, 64, 79, 97, 99, 133, 182, 209. 319 336-337 128, 130, 137, 292 stresses in laminates 254 stresses in the principal material coordi- stretching ability 209 structural failure criteria 273 structural materials I structural nonlinearity 189 structural strength criteria 274, 281 structure of uniform stress 382 structure of uniform stress and strain 368 sun radiation 3 18 surface cracks 62 surface tractions 29, 31, 42, 51 symmetric f45" angle-ply laminate 369 symmetric kq5 angle-ply laminate 370 symmetric &4 angle-ply layer 3 10 symmetric angle-ply layer 186 symmetric laminates 245, 247 symmetric system of helical ribs symmetry conditions 143 symmetry conditions for shear synthetic fibers 6 nates 164 212 stresses 32 410 Subject index system of circumferential ribs 212 system of cracks parallel to fibers 168 table rolling 22-23 tangent modulus tape orientation angle 384 tape overlap 352 teflon 6 teflon fibers 6 temperature coefficients 312 temperature effects 301 temperature gradient 305 temperature resistance 18 tensile interface stress 344 tensile load cycle 338 tensile stiffnesses 151 tensile strength 96-97, 100, 316 tensile strength of unidirectional tensile test 153 tension across the fibers 95 tension along the fibers 95 tension in two orthogonal directions 368 tension of f45" angle-ply specimen tension of a cross-ply laminate 166 tension of angle-ply specimens 204 tensor criterion 290, 293 tensor strength criterion 290 tensor-polynomial criterion 292 tensor-polynomial strength criteria 29 I, test fixtures for composite rings 101 testing of a microcomposite specimen 83 tex-number 15 textile denier-numbers (den) 15 theorem of proportional loading 139 theoretical density 57 theoretical prediction of material stiffness 197 theoretical strength 58, 60, 62 theoretical strength of glass 62 theory of composite beams 217 theory of elasticity 134 theory of plasticity thermal coefficients of the layer 314 thermal conductivity 301-302, 304 thermal conductivity of the fiber 304 thermal expansion coefficients 308-309 thermal resistance 306, 316 5, 7, 132, 138, 339 composites 78 188 293 128, 145, 292 thermal strains thermal stresses 301, 307 thermoelastic behavior of a unidirectional thermoelastic deformation 31 7 thermoelasticity 301, 307 thennoelasticity constitutive equations thermo-mechanical diagram 18 thermoplastic carbon composite 31 1 thermoplastic matrices 17-19, 183 thermoplastic polymers 6, 121, 184 thermoset matrices 17 thermoset polymeric resins 6 thermostable structures 1 14 thick fiberglass rings 205 thin-walled cylindrical drive shaft 294 thin-walled tubular specimens 273 three-dimensional constitutive three-dimensional stressed state of a three-dimensionally woven or braided three-dimensional weaving 21 5 three-point beam bending test 217 time of relaxation 325 time-dependent behavior of polymeric composites 3I9 Timoshenko energy method 108 titanium carbide fibers 7 titanium matrix 20 Toray T- 1000 carbon fibers total potential energy of the body 51 total shear strain 165 toughness 21 tow 15-16, 55 tow processability 15 tow tex-number 57 translational displacement 226 transverse compression 113, 273 transverse compressive strength 96, 208 transverse conductivity 304 transverse CTE 304 transverse deformability 181 transverse extension failure mode I05 transverse force 157 transverse impact loads 340 301, 307, 309-310, 317, 356 composite ply 307 31 1 equations 217 layer 124 fabrics 215 12 Subject index 41 1 transverse modulus 96, 100, 141, 208 transverse modulus of a ply 81 transverse normal stress transverse shear compliance 249 transverse shear deformation 202, 260, transverse shear forces 230 transverse shear modulus 142, 165 transverse shear modulus of the ply 202 transverse shear stiffness coefficients 247 transverse shear stiffnesses 236, 239, transverse shear 237 transverse shear stress 202, 262, 284 transverse strains 85 transverse strength 365 transverse strength of a unidirectional transverse tensile strain 279 transverse tensile strength 96, 100, 208 transverse tension 84, 92, 97, 102, 168, 273, 323 transverse tension of a ply 91 triaxial braid 210 triaxial reinforcement 2 15 tubular specimen for shear test tubular specimens 274 twill weave 209 twist 361 twisting 227, 229, 237, 356, 359 two-dimensional (2D) braiding 24 two-layered angle-ply laminate 358 two-layered orthotropic cross-ply two-matrix fiberglass composite 162, 182 two-matrix fiberglass unidirectional two-matrix glass-epoxy composite 183, two-stage impregnation 18 1 124, 198, 266, 278, 284 263 249-250 material 99 105 laminate 356 composite 144 189 ultimate elongation 79, 95, 124, 168, 180 ultimate elongation across the fibers 180 ultimate elongation along the fibers 180 ultimate elongations 1 I5 ultimate fiber volume fraction 58 ultimate pressure 296, 391 ultimate strain 16-17 ultimate stress 2, 95, 274, 278 ultimate tensile stress 6-7, 14 ultimate theoretical stress 59 uniaxial longitudinal loading 143 uniaxial tension unidirectional anisotropic layer 147 unidirectional aramid-epoxy composite unidirectional boron-aluminum unidirectional boron-epoxy composite unidirectional carbon-epoxy composite unidirectional carbon-epoxy layer 154 unidirectional carbon-glass epoxy unidirectional composite 76, 142, 144, unidirectional composite ply 80, 303, unidirectional-epoxy composites 376 unidirectional glass4poxy composite unidirectional layer 147, 297, 359 unidirectional orthotropic layer 140 unidirectional ply 5-56, 121, 255, unidirectional tape 55 unidirectional tow (roving) 15 unidirectional two-matrix composite 183 unidirectional weave 209 unidirectionally reinforced plates 96 uniformly stressed composite struc- uniformly stressed fibers 365, 371 universal form of the polynomial criterion universal stress-strain curve 13 1 universal stress-strain diagram 130 upper bound for the actual transverse shear upper bound on the apparent transverse 129, 131, 138, 175, 197 material 99, 338 composite 79 material 100 material 98, 335 composite 1 17 217, 287, 304, 329, 351 327-328, 335, 365 material 97, 322 277-278, 304, 308 tures 391 282 stiffness 233-234 modulus 94 variation of the strain energy 51 variational equations 5 1 variational principle of minimum strain energy 52 41 2 Subject index variational principle of minimum total potential energy 51 variational principles 49, 91 viscoelastic deformation 3 19 viscoelastic material 334 viscoelastisity 319 viscose fibers 6 viscous strain 321 volume deformation 123 volume fraction 55, 57, 86, 95 volume modulus 124 warp direction 207, 221, 276, 375 warp yarns 207 warping 355 warping of laminates 359 water absorption 318 wave interaction 344 wave propagation 341 weave patterns 209 weight-saving factor 378 wet process 22 whiskers 60, 215 winding 351-352, 380 winding of a lattice layer 213 winding of unidirectional composite wood compressed along the fibers 105 wood fibers 14 wool fibers 14 work of external forces 43 work of fracture 80 work of internal forces 43 woven composites 206 woven fabric composites 210 woven structure 206-207 tapes 384 yarn 15 zero thermal expansion 114, 31 1 zero-order model of a ply 83 . polysulfone (PSU) 6, 19 potential energy 3, 43 potential energy of the body power approximation 130 -133 , 138 power hardening law 132 power law for the plastic potential 128, 130 , 137 -1 38, ply 55,. stress-strain curve 13 1 universal stress-strain diagram 130 upper bound for the actual transverse shear upper bound on the apparent transverse 129, 131 , 138 , 175, 197 material 99, 338 composite. 99, 133 , 182, 209. 319 336-337 128, 130 , 137 , 292 stresses in laminates 254 stresses in the principal material coordi- stretching ability 209 structural failure criteria 273 structural materials