section epub:type=”appendix”>
Symbols and Formulae
Contents
Principal Symbols
| Symbol | Meaning (units) |
|---|---|
| Note: | Multiples or submultiples of basic units indicate the unit suffixes typically used with materials data. |
| a | side of cubic or hexagonal unit cell (nm) |
| a | crack length (mm) |
| A | constant in fatigue crack-growth law |
| A | constant in creep law  |
| b | Burgers vector (nm) |
| c | concentration (m−3) |
| D | diffusion coefficient (m2 s−1) |
| D0 | pre-exponential constant in diffusion coefficient (m2 s−1) |
| E | Young’s modulus of elasticity (GN m−2) |
| f | force acting on unit length of dislocation line (N m−1) |
| F | force (N) |
| g | acceleration due to gravity on the Earth’s surface (m s−2) |
| G | shear modulus (GN m−2) |
| Gc | toughness (or critical strain energy release rate) (kJ m−2) |
| H | hardness (MN m−2 or kg mm−2) |
| J | diffusion flux (m−2 s−1) |
| k | shear yield strength (MN m−2) |
| k | Boltzmann’s constant  |
| K | bulk modulus (GN m−2) |
| K | stress intensity factor (MN m−3/2) |
| Kc | fracture toughness (critical stress intensity factor) (MN m−3/2) |
| ΔK | K range in fatigue cycle (MN m−3/2) |
| m | constant in fatigue crack growth law (dimensionless) |
| n | creep exponent in  |
| N | number of fatigue cycles |
| NA | Avogadro’s number (mol−1) |
| Nf | number of fatigue cycles leading to failure (dimensionless) |
| Q | activation energy per mole (kJ mol−1) |
| r0 | equilibrium interatomic distance (nm) |
| R | universal gas constant (J K−1 mol−1) |
| S0 | bond stiffness at r=r0 (N m−1) |
| tf | time-to-failure (s) |
| T | line tension of dislocation (N) |
| T | absolute temperature (K) |
| TM | absolute melting temperature (K) |
| Uel | elastic strain energy (J) |
| γ | (true) engineering shear strain (dimensionless) |
| Δ | dilatation (dimensionless) |
| ɛ | true (logarithmic) strain (dimensionless) |
| ɛf | (nominal) strain after fracture; tensile ductility (dimensionless) |
| ɛn | nominal (linear) strain (dimensionless) |
| ɛ0 | permittivity of free space (F m−1) |
 | steady-state tensile strain-rate in creep (s−1) |
| µk | coefficient of kinetic friction (dimensionless) |
| µs | coefficient of static friction (dimensionless) |
| ν | Poisson’s ratio (dimensionless) |
| ρ | density (Mg m−3) |
| σ | true stress (MN m−2) |
| σn | nominal stress (MN m−2) |
| σTS | (nominal) tensile strength (MN m−2) |
| σy | (nominal) yield strength (MN m−2) |
 | ideal strength (GN m−2) |
| Δσ | stress range in fatigue (MN m−2) |
| τ | shear stress (MN m−2) |
Other Symbols
| Symbol | Meaning (units) |
|---|---|
| a | true contact area (mm2) |
| a | side of square cross section (mm) |
| a | external diameter of thick-walled tube (mm) |
| a | atomic weight of element (g mol− 1, kg kmol− 1) |
| A | area (mm2) |
| A0 | initial area (mm2) |
| b | breadth or width of cross section (mm) |
| b | internal diameter of thick-walled tube (mm) |
| b | constant in fatigue equations (dimensionless) |
| c | height of hexagonal unit cell (nm) |
| c | maximum distance from neutral axis of cross section (mm) |
| c | constant in fatigue equations (dimensionless) |
| C | specific heat (J kg− 1 K− 1) |
| C | consumption rate (ton year− 1) |
| d | depth of cross section (mm) |
| d | minor diameter of shouldered shaft (mm) |
| d | grain size (μm) |
| D | major diameter of shouldered shaft (mm) |
| Ec | Young’s modulus of composite material (GN m− 2) |
| Ef | Young’s modulus of fibers/reinforcement (GN m− 2) |
| Em | Young’s modulus of matrix (GN m− 2) |
| Et | tangent modulus in plastic buckling (MN m− 2) |
| f | frequency (s− 1, Hz) |
| Fcr | critical buckling force (N) |
| Fs | shear component of force (N) |
| g | throat dimension for Class W weld (mm) |
| gS | “standard” value of g (9.807 m s− 2) |
| h | Planck’s constant (J s) |
| h | vertical distance (m) |
| i | electrical current density (μA cm− 2) |
| I | electrical current (A) |
| I | second moment of area of cross section (mm4) |
| kL | constant in linear oxidation equation (g s− 1) |
| kP | constant in parabolic oxidation equation (g2 s− 1) |
| K | thermal conductivity (W m− 1 K− 1) |
| L | length, spacing (mm) |
| L0 | initial length (mm) |
| m | mass (kg) |
| m | Weibull modulus (dimensionless) |
| m | (with subscripts 1, 2, b, etc.) materials index (various units) |
| Δm | mass gain in oxidation (g) |
| M | bending moment (N m) |
| Mp | fully plastic bending moment (N m) |
| Mr | bending moment at rupture (N m) |
| n | number (dimensionless) |
| p | pressure (kN m− 2) |
| p | probability (dimensionless) |
| P | contact force (N) |
| Pf | failure probability (dimensionless) |
| Ps | survival probability (dimensionless) |
| q | heat flux (W m− 2 s− 1) |
| q | ionic charge (C) |
| Q | heat (J) |
| r | growth rate (percent per year) |
| r | radius (mm) |
| r | radius of plastic zone at crack tip (mm) |
| r | distance between atoms/ions (nm) |
| rD | dissociation distance between atoms/ions (nm) |
| r0 | equilibrium distance between atoms/ions (nm) |
| R | electrical resistance (Ω) |
| S | notch sensitivity factor (dimensionless) |
| S | safety factor (dimensionless) |
| S | stiffness (N m− 1) |
| SCF | stress concentration factor (dimensionless) |
| SCFeff | effective stress concentration factor (dimensionless) |
| t | time (s, hour, year) |
| t | thickness (mm) |
| tD | doubling time (s, hour, year) |
| T | torque (N m) |
| TD | Debye temperature (K) |
| TG | glass transition temperature (K) |
| ΔT | temperature difference (K) |
| u | small displacement; extension/compression (mm) |
| U | energy; energy of deformation (J) |
| V | electrical potential difference (V) |
| V | volume (mm3) |
| V | volume fraction (dimensionless) |
| Vf | volume fraction of fibers/reinforcement (dimensionless) |
| Vm | volume fraction of matrix (dimensionless) |
| V0 | specimen volume in Weibull equation (mm3) |
| w | small displacement (mm) |
| w | width (mm) |
| w | weight (kg) |
| W | work (J) |
| x | distance (mm) |
| Y | crack stress intensity factor (dimensionless) |
| z | distance from neutral axis of cross section (mm) |
| z | number of electrons exchanged in unit reaction (dimensionless) |
| α | thermal expansion coefficient (10− 6 K− 1) |
| αc | thermal expansion coefficient of composite (10− 6 K− 1) |
| αf | thermal expansion coefficient of fiber (10− 6 K− 1) |
| αm | thermal expansion coefficient of matrix (10− 6 K− 1) |
| β | constant in Hall-Petch equation (MN m− 3/2) |
| δ | deflection (mm) |
| Δɛ | strain range in fatigue cycle (dimensionless) |
| ɛ′f | true fracture strain (dimensionless) |
| ɛ1,2,3 | principal strain components (dimensionless) |
| λ | thermal diffusivity (m2 s− 1) |
| ν | vibration frequency (s− 1) |
| ρf | density of fibers/reinforcement (Mg m− 3) |
| ρm | density of matrix (Mg m− 3) |
| σf | nominal fracture stress (MN m− 2) |
| σ′f | true fracture stress (MN m− 2) |
| σm | tensile mean stress in fatigue cycle (MN m− 2) |
| σr | modulus of rupture (MN m− 2) |
| σ0 | normalizing stress in Weibull equation (MN m− 2) |
| σ1,2,3 | principal stress components (MN m− 2) |
| σ0.1% | 0.1% proof stress (MN m− 2) |
| Δσ | stress range in fatigue cycle (MN m− 2) |
| τy | dislocation (shear) yield strength (MN m− 2) |
| ω | angular velocity (s− 1) |
| Ω | atomic volume (nm3) |
| (− − −) | individual plane in cubic system (Miller indices) |
| {− − −} | family of symmetry-related planes in cubic system |
| [− − −] | individual direction in cubic system (direction indices) |
| 〈− − −〉 | family of symmetry-related directions in cubic system |
Principal Formulae
Chapter 2
Exponential growth
- C = consumption rate (ton per year)
- r = growth rate (percent per year)
- t = time
- tD= doubling time
Chapter 3
Stress, strain, Poisson’s ratio, elastic moduli
when v =1/3, K = E, and G = (3/8)E
- F(Fs) = normal (shear) component of force
- A = area
- u(w) = normal (shear) component of displacement
- σ(ɛn) = true tensile stress (nominal tensile strain)
- τ(γ) = true shear stress (true engineering shear strain)
- p(Δ) = external pressure (dilatation)
- ν = Poisson’s ratio
- E = Young’s modulus
- G = shear modulus
- K = bulk modulus.
Chapter 6
Young’s modulus (longitudinal) of unidirectional composite
Young’s modulus (transverse) of unidirectional composite
