Teaching yourself phase diagrams 369 Fig. A1.56. precipitates ε and ε precipitates β . Just above 600°C 18 53 (= 34%) ε (65% B) + 35 53 (= 66%) β (12% B). At 600°C all β forms α and ε in eutectoid reaction. Just below 600°C 25 60 (= 42%) ε (65% B) + 35 60 (= 58%) α (5% B). 600°C → 300°C, α precipitates ε and ε precipitates α . Just above 300°C 27 66 (= 41%) ε (69% B) + 39 66 (= 59%) α (3% B). At 300°C all ε and some α form δ in peritectoid reaction. Just below 300°C 27 37 (= 73%) δ (40% B) + 10 37 (= 27%) α (3% B). 300°C → 0°C, amount of α decreases and δ increases. At 0°C 30 35 (= 86%) δ (35% B) + 5 35 (= 14%) α (0% B). 370 Engineering Materials 2 Appendix 2 Symbols and formulae List of principal symbols Symbol Meaning(units) Note: Multiples or sub-multiples of basic units indicate the unit suffixes normally used in materials data. a lattice parameter (nm) a crack length (mm) A availability (J) A 1 eutectoid temperature (°C) A 3 first ferrite temperature (°C) A cm first Fe 3 C temperature (°C) b Burgers vector (nm) c height of c.p.h. unit cell (nm) C concentration (m −3 ) CCR critical cooling rate (°C s −1 ) DP degree of polymerisation (dimensionless) E Young’s modulus of elasticity (GPa) f force (N) F force (N) g acceleration due to gravity on the Earth’s surface (m s −2 ) G shear modulus (GPa) G Gibbs function (J) G c toughness (kJ m −2 ) H hardness (GPa) ∆H latent heat of transformation (J) I second moment of area of structural section (mm 4 ) k ratio of C solid /C liquid on phase diagram (dimensionless) k Boltzmann’s constant (J K −1 ) k shear yield strength (MPa) K IC fracture toughness (MPa m 1/2 ) L liquid phase Symbols and formulae 371 Symbol Meaning(units) m mass (kg) m Weibull modulus (dimensionless) M bending moment (N m) M F martensite finish temperature (°C) M S martensite start temperature (°C) n time exponent for slow crack-growth (dimensionless) p pressure (Pa) P f failure probability (dimensionless) P S survival probability (dimensionless) q activation energy per atom (J) Q activation energy per mole (kJ mol −1 ) r* critical radius for nucleation (nm) R universal gas constant (J K −1 mol −1 ) T absolute temperature (K) T e equilibrium temperature (K) T g glass temperature (K) T m melting temperature (K) ∆T thermal shock resistance (K) ν velocity (m s −1 ) V volume (m 3 ) V volume fraction (dimensionless) W A weight % (dimensionless) W f free work (J) X A mol % (dimensionless) α linear coefficient of thermal expansion (MK −1 ) γ energy of interface (J m −2 ) or tension of interface (N m −1 ) δ elastic deflection (mm) ε true (logarithmic) strain (dimensionless) ε f (nominal) strain after fracture; tensile ductility (dimensionless) ε . ss steady-state tensile strain-rate in creep (s −1 ) η viscosity (P, poise) ν Poisson’s ratio (dimensionless) ρ density (Mg m −3 ) σ true stress (MPa) σ c (nominal) compressive strength (MPa) σ r modulus of rupture (MPa) σ TS (nominal) tensile strength (MPa) σ y (nominal) yield strength (MPa) Greek letters are used to label the phases on phase diagrams. 372 Engineering Materials 2 Summary of principal formulae and magnitudes Chapter 3 and Teaching yourself phase diagrams: phase diagrams Composition is given by W A = weight of A weight of A weight of B+ × 100 in weight %, and by X A = atoms (mols) of A atoms (mols) of A atoms (mols) of B+ × 100 in atom (mol) %. W A + W B = 100%; X A + X B = 100%. Three-phase reactions Eutectic: L a α + β Eutectoid: β a α + γ Peritectic: L + α a β Peritectoid: A + B a δ Chapter 4: Zone refining C s = Ck kx l 0 11 ( )exp .−− −             C s = concentration of impurities in refined solid; C 0 = average impurity concentration; k = C solid /C liquid ; x = distance from start of bar; l = zone length. Chapter 5: Driving forces Driving force for solidification W f = −∆G = −− ∆H T TT m m ( ). ∆H = latent heat of solidification; T m = absolute melting temperature; T = actual tem- perature (absolute). Driving force for solid-state phase change W f = −∆G = −− ∆H T TT e e ( ). ∆H = latent heat of transformation; T e = equilibrium temperature (absolute). Symbols and formulae 373 Chapter 6: Kinetics of diffusive transformations Speed of interface ν ∝ e −q/kT ∆T. q = activation energy per atom; k = Boltzmann’s constant; T = absolute temperature; ∆T = difference between interface temperature and melting or equilibrium temperature. Chapter 7: Nucleation Nucleation of solids from liquids: critical radius for homogeneous and heterogeneous nucleation r* = 2 γ SL T HT T m m ∆ ( ) . − γ SL = solid–liquid interfacial energy; T m = absolute melting temperature; ∆H = latent heat of solidification; T = actual temperature (absolute). Chapter 8: Displacive transformations Overall rate of diffusive transformation ∝ no. of nuclei × speed of interface. Chapter 10: The light alloys Solid solution hardening σ y ∝ ε s C 32 12// . C = solute concentration; ε s = mismatch parameter. Work-hardening σ y ∝ ε n . ε = true strain; n = constant. Chapter 14: Metal processing Forming pressure No friction p f = σ y . 374 Engineering Materials 2 Sticking friction p f = σ y wx d 1 2 () .+ −       / σ y = yield strength; w = width of forging die; x = distance from centre of die face; d = distance between dies. Chapter 17: Ceramic strengths Sample subjected to uniform tensile stress Tensile strength σ TS = K a m IC π . K IC = fracture toughness; a m = size of widest microcrack (crack width for surface crack; crack half-width for buried crack). Modulus of rupture σ r = 6 2 M bd r . M r = bending moment to cause rupture; b = width of beam; d = depth of beam. Compressive strength σ c ≈ 15 σ TS , σ c = CK a IC π . C = constant (≈15); a = average crack size. Thermal shock resistance ∆T = σ TS /E α . E = Young’s modulus; α = linear coefficient of thermal expansion. ˙ exp( ). εσ ss n AQRT=−/ ε . ss = steady-state tensile strain rate; A, n = constants; σ = tensile stress; Q = activation energy for creep; R = universal gas constant; T = absolute temperature. Chapter 18: Statistics of fracture Weibull distribution P s (V ) = exp −                 V V m 00 σ σ Symbols and formulae 375 or ln ln ln ln . 1 00 P V V m s             =+       σ σ P s = survival probability of component; V = volume of component; σ = tensile stress on component; V 0 = volume of test sample; σ 0 = stress that, when applied to test sample, gives P s = 1/e (= 0.37); m = Weibull modulus. Failure probability P f = 1 − P s . Slow crack-growth σ σ TS (test)       = n t t . σ = strength of component after time t; σ TS = strength of component measured over time t(test); n = slow crack-growth exponent. Chapter 19: Ceramics processing Sintering d d / ρ t C a QRT n exp( ).=− ρ = density; t = time; C, n = constants; a = particle size; Q = activation energy for sinter- ing; R = universal gas constant; T = absolute temperature. Glass forming η ∝ exp(Q/RT). η = viscosity; Q = activation energy for viscous flow. Chapter 20: Cements and concretes Hardening rate ∝ exp(–Q/RT). Q = activation energy for hardening reaction; R = universal gas constant; T = absolute temperature. Chapter 23: Mechanical behaviour of polymers Modulus: WLF shift factor log(a T ) = CT T CTT 11 0 210 ( ) . − +− 376 Engineering Materials 2 C 1 , C 2 = constants; T 1 , T 0 = absolute temperatures. Polymer viscosity ηη 10 11 0 210 exp ( ) .= −− +−       CT T CTT Chapter 25: Composites Unidirectional fibre composites E c|| = V f E f + (1 − V f )E m , E V E V E f f f m c⊥ − =+ −           . 1 1 E c|| = composite modulus parallel to fibres; E c⊥ = composite modulus perpendicular to fibres; V f = volume fraction of fibres; E f = Young’s modulus of fibres; E m = Young’s modulus of matrix. σ TS = VV f f f f y m σσ ( ) .+−1 σ TS = tensile strength parallel to fibres; σ f f = fracture strength of fibres; σ y m = yield strength of matrix. Optimum toughness G c = V d f f f s m () . 8 2 σ σ d = fibre diameter; σ s m = shear strength of matrix. Magnitudes of properties The listed properties lie, for most structural materials, in the ranges shown Property Metals Ceramics Polymers Composites (unfoamed) (polymer matrix) Density (Mg m −3 ) 2 to 10 1 to 5 1 to 2 1.5 to 2.0 Young’s modulus (GPa) 50 to 200 10 to 1000 0.01 to 10 10 to 200 Yield strength (MPa) 25 to 1500 3000 to 50,000 – – Tensile strength (MPa) 50 to 2000 1 to 800 5 to 100 100 to 1000 Fracture toughness (MPa m 1/2 ) 5 to 200 0.1 to 10 0.5 to 5 20 to 50 Creep temperature (°C) 50 to 1000 −20 to 2000 0 to 200 0 to 200 Index 377 Index Adhesives 204, 260 Age hardening see Precipitation hardening Alexander Keilland oil platform 136 Alloy 15, 25, 321 Alumina 163, 164, 167 Aluminium-based alloys 8, 12, 100 et seq., 347, 351 Amorphous metals 96 polymers 236 structure 16 Anisotropy 266, 280, 316 Annealing 151 Atactic polymers 231 Austenite 114, 130, 355 Availability 50 Bain strain 84 Bakelite 221 Beryllium 100 Binary alloy 25, 327, 336 Boiler design 133 Bone 164, 165 Borosilicate glass 162, 165 Boundaries 18 Boundary tension 22 Brass 7, 12, 342 Brick 163, 201 Bronze 7, 12, 356 Carbide formers 129 Carbon equivalent 138 Carbon fibres see CFRP Carburising 155 Case studies in ceramics and glasses 190, 303 in design 296 et seq. in phase diagrams 34 et seq. in phase transformations 89 et seq. in steels 133 et seq. Casting 91, 121, 144 Casting defects 144 Cast iron 6, 12, 121 Catalysis 91, 93 C-curves see TTT curves Cellular solids 272 et seq. Cellulose 224, 279 Cement and concrete 163, 207 et seq. chemistry 207 strength 212 structure 210 Cementite 114 et seq., 355 Ceramics 161 et seq. brittle fracture 180, 185 et seq. case studies in 190, 303 cement and concrete 207 et seq. production, forming and joining 194 et seq. properties 164, 177 et seq. structures 167, 174 et seq. Cermets 164, 203 CFRP 164, 263 et seq., 317 Chain-folded crystals 233 Chemical reactions 47 Chemical vapour deposition 198 China 163 Coherent interfaces 20, 83, 107 Cold drawing 248, 249 Columnar crystals 91, 144 Components 22, 25, 321 Composites 165, 203, 215, 263 et seq. case studies in 312 et seq. Composition 25, 321, 336 Compounds 17 Compression moulding 257, 259 Compressive strength 182, 213 Concentration 321 378 Index Constitution 22, 30, 324 Constitution point 27, 336, 337 Continuous casting 145 Conveyor drum design 296 Co-polymers 255 Copper-based alloys 6, 12, 30, 31, 356, 361 Corrosion 129 Cooling curves 333 Covalent ceramics 167, 170 Crazing 248, 250 Creep of ceramics 183 Critical nucleus 69 Cross-linked polymers 221, 226 Crystal growth 91 Crystal structure of ceramics 168 metals 14 polymers 233 Cupronickel 7 Dacron 221 Data for ceramics and glasses 163, 165 composites 265 metals 11 polymers 224, 225 woods 278 Decomposition of polymers 246 Degree of polymerisation 228 Dendrites 65, 92, 352 Density of ceramics and glasses 164 foams 272 metals 12 polymers 224 woods 278 Design-limiting properties 289 Design methodology 291, 292 Diamond 164 Die casting 145 Differential thermal analysis 334 Diffusion bonding 204 Diffusion-controlled kinetics 63 Diffusive transformations 57 et seq. Displacive transformations 76 et seq. Driving force 46 et seq. Duralumin 103 Dynamic equilibrium 61 Elastic constants see Moduli Elastomers 221, 224, 232, 244 Energy-efficient forming 155 Enthalpy 52 Entropy 49 Epoxies 221, 224 Equiaxed crystals 92, 142 Equilibrium 28, 51, 61 Equilibrium diagrams 25 et seq. case studies 34 et seq. teach yourself 326 et seq. Eutectics 35, 42, 114, 346 et seq. Eutectoids 346 et seq. Extrusion 258 Fatigue 298 Ferrite 114 et seq., 355 Ferrous alloys 6, 10 Failure probability 185 Failure analyses 133, 296 Fibres 260, 263 Foams 263, 272 Forging 147 Formica 223 Forming of 194 ceramics and glasses 194 et seq. composites 264 metals 143 et seq. polymers 254 et seq. Formulae 372 et seq. Forsterite 173 Fracture strength of ceramics and glasses 164, 180 composites 267 metals 13 polymers 225, 248 woods 278 Fracture toughness of ceramics and glasses 164, 180 composites 265, 269 metals 13 polymers 225 woods 278 Free work 50 Germanium 39 GFRP 219, 263 et seq., 317 Gibbs’ function 53 Gibbs’ phase rule 341 [...]... implantation 155 Ionic ceramics 167, 168 Iron-based alloys 5, 12 Isotactic polymers 231 Joining of ceramics and glasses 204 of metals 154 of polymers 260 Jominy test 126 Kevlar fibres see KFRP KFRP 219, 271 Kinetics 59 et seq Lead-tin alloys 12, 26, 34, 326 et seq Ledeburite 115 Lever rule 339 Light alloys 100 et seq Lignin 224, 279 Liquid phase sintering 197 Limestone 164 Linear polymers 220, 225 Machining 153 ... Mould design 308 Moulding 200, 257 Mullite 173 Mylar 221 Neoprene 223 Network ceramics and glasses 167 Nickel-based alloys 7, 12 Nitriding 155 Normalisation of steels 113 Nucleation 68, 73, 77, 89 et seq Nylon 222, 224, 255, 312 Offshore structures 303 Optimised materials 264, 270, 275 Pearlite 64, 115 et seq Peritectic 359 Peritectoid 380 Phase boundaries 18, 21 Phase diagrams 26 et seq., 326 case studies... transformation see Kinetics Rayon 254 Reaction bonding 197 Recovery 151 Recrystallisation 55, 151 Resins 221, 224 Reversibility 49 Rolling 150 , 200 Salol 58 Sand casting 145 Sandstone 175 Sandwich panels 272, 318 Scale of structural features 14, 66 Seeding 91, 93 Segregation 93, 144 Semicoherent interfaces 20 Index Semiconduction materials 39, 94 Sialons 163, 164 Silica 94, 167, 170 Silicates 170,... energy and tension 21 Surface engineering 155 Superalloys 7, 12 Symbols 321, 370 et seq Synthesis of polymers 254 Teach yourself phase diagrams 320 et seq Ternary alloy 25, 327 Terylene 221 Thermal properties of ceramics and glasses 165 metals 13 polymers 225 Thermal shock resistance 166, 182 Thermoplastics 220, 224, 230 Thermosets 220, 224 Time-dependent strength 189 Titanium-based alloys 10, 100, 103... Silicon nitride 163, 164, 170 Silver iodide 90 Silver solder 34, 37 Sindiotactic polymers 231 Sintering 194 Slip casting 201 Soda glass 162, 164, 191 Sodium chloride 167 Soft solder see Lead-tin alloys Solder see Lead-tin alloys Solid solutions 16 Solid solution hardening 101, 108, 128 Solidification 51, 57 Spherulites 234 Spinel 173 Stability 50, 110 Stainless steels see Steels State variables 325 Statistics... transition 239 Glassy metals 63, 97 Glaze bonding 204 Glazes 202 GP zones 106 Grain boundaries 18 growth 55, 137 shape 20, 64 size 93 strengthening 153 Grains 20 Granite 164, 175 Graphite 121 Habit plane 83 Hammer design 139 Hardenability 125 Heat 48 Heat-affected zone 137 Heat flow 62 Heterogeneous nucleation 69, 90 Homogeneous nucleation 69 Hot isostatic pressing 196 Hot pressing 196 Hydrogen cracking... Viscoelastic behaviour 242 Viscosity 198, 245 Vitreous ceramics 162, 174 Vulcanisation 247, 257 Weibull statistics 185 et seq Welding 136, 154 , 303 Wheel design 303 Window design 190 Wood 277 et seq case study in 312 properties 280 structure 278 Work 46 Work hardening 110, 152 Working of metals 147 Zinc 12 Zirconia 163, 164, 169, 202, 203 Zone refining 39 et seq 382 Index ... alloys 12, 26, 34, 326 et seq Ledeburite 115 Lever rule 339 Light alloys 100 et seq Lignin 224, 279 Liquid phase sintering 197 Limestone 164 Linear polymers 220, 225 Machining 153 Magnesia 168 Magnesium-based alloys 100 et seq Martensite 83, 118, 134, 137, 140 Mechanical properties of cement and concrete 212 et seq ceramics and glasses 164, 177 et seq composites 265 et seq foams 273 metals 12, 101, 118,... Peritectoid 380 Phase boundaries 18, 21 Phase diagrams 26 et seq., 326 case studies 34 et seq teach yourself 320 et seq Phase reactions 348 Phase transformations 46 et seq., 89 Phases 18, 25, 323 Phenol-formaldehyde 223, 224 Plasticisers 256 Plumbers solder 34, 35 Polyacrylonitrile 221 Polybutadiene 223, 224 Polychloroprene 223, 224 Polyester 223, 224 Polyethylene 222, 224 Polyethyleneteraphthalate 221 . and formulae 371 Symbol Meaning(units) m mass (kg) m Weibull modulus (dimensionless) M bending moment (N m) M F martensite finish temperature (°C) M S martensite start temperature (°C) n time. 11 polymers 224, 225 woods 278 Decomposition of polymers 246 Degree of polymerisation 228 Dendrites 65, 92, 352 Density of ceramics and glasses 164 foams 272 metals 12 polymers 224 woods 278 Design-limiting. 278 Design-limiting properties 289 Design methodology 291, 292 Diamond 164 Die casting 145 Differential thermal analysis 334 Diffusion bonding 204 Diffusion-controlled kinetics 63 Diffusive transformations