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Symbols and Formulae

Contents

Principal Symbols
Other Symbols
Principal Formulae
Magnitudes of Properties

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 ɛ˙ss=Aσne–Q/RT
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)
G_c	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 R/NA(J K–1)
K	bulk modulus (GN m⁻²)
K	stress intensity factor (MN m^−3/2)
K_c	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 ɛ˙ss=Aσne–Q/RT
N	number of fatigue cycles
N_A	Avogadro’s number (mol⁻¹)
N_f	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)
t_f	time-to-failure (s)
T	line tension of dislocation (N)
T	absolute temperature (K)
T_M	absolute melting temperature (K)
U^el	elastic strain energy (J)
γ	(true) engineering shear strain (dimensionless)
Δ	dilatation (dimensionless)
ɛ	true (logarithmic) strain (dimensionless)
ɛ_f	(nominal) strain after fracture; tensile ductility (dimensionless)
ɛ_n	nominal (linear) strain (dimensionless)
ɛ₀	permittivity of free space (F m⁻¹)
ɛ˙ss	steady-state tensile strain-rate in creep (s⁻¹)
µ_k	coefficient of kinetic friction (dimensionless)
µ_s	coefficient of static friction (dimensionless)
ν	Poisson’s ratio (dimensionless)
ρ	density (Mg m⁻³)
σ	true stress (MN m⁻²)
σ_n	nominal stress (MN m⁻²)
σ_TS	(nominal) tensile strength (MN m⁻²)
σ_y	(nominal) yield strength (MN m⁻²)
σ˜	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)
E_c	Young’s modulus of composite material (GN m^− 2)
E_f	Young’s modulus of fibers/reinforcement (GN m^− 2)
E_m	Young’s modulus of matrix (GN m^− 2)
E_t	tangent modulus in plastic buckling (MN m^− 2)
f	frequency (s^− 1, Hz)
F_cr	critical buckling force (N)
F_s	shear component of force (N)
g	throat dimension for Class W weld (mm)
g_S	“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)
k_L	constant in linear oxidation equation (g s^− 1)
k_P	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)
M_p	fully plastic bending moment (N m)
M_r	bending moment at rupture (N m)
n	number (dimensionless)
p	pressure (kN m^− 2)
p	probability (dimensionless)
P	contact force (N)
P_f	failure probability (dimensionless)
P_s	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)
r_D	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)
SCF_eff	effective stress concentration factor (dimensionless)
t	time (s, hour, year)
t	thickness (mm)
t_D	doubling time (s, hour, year)
T	torque (N m)
T_D	Debye temperature (K)
T_G	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)
V_f	volume fraction of fibers/reinforcement (dimensionless)
V_m	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)
α_c	thermal expansion coefficient of composite (10^− 6 K^− 1)
α_f	thermal expansion coefficient of fiber (10^− 6 K^− 1)
α_m	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)
ɛ_1,2,3	principal strain components (dimensionless)
λ	thermal diffusivity (m² s^− 1)
ν	vibration frequency (s^− 1)
ρ_f	density of fibers/reinforcement (Mg m^− 3)
ρ_m	density of matrix (Mg m^− 3)
σ_f	nominal fracture stress (MN m^− 2)
σ′_f	true fracture stress (MN m^− 2)
σ_m	tensile mean stress in fatigue cycle (MN m^− 2)
σ_r	modulus of rupture (MN m^− 2)
σ₀	normalizing stress in Weibull equation (MN m^− 2)
σ_1,2,3	principal stress components (MN m^− 2)
σ_0.1%	0.1% proof stress (MN m^− 2)
Δσ	stress range in fatigue cycle (MN m^− 2)
τ_y	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

dCdt=r100C,tD=100rIn2

si6_e

C = consumption rate (ton per year)
r = growth rate (percent per year)
t = time
t_D= doubling time

Chapter 3

Stress, strain, Poisson’s ratio, elastic moduli

σ=FA,τ=FsA,p=–FA,v=–lateral straintensile strainɛn=uL,γ=wL,Δ=ΔVVσ=Eɛn,τ=Gγ,p=–KΔ

si7_e

when v =1/3, K = E, and G = (3/8)E

F(F_s) = normal (shear) component of force
A = area
u(w) = normal (shear) component of displacement
σ(ɛ_n) = 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

Ec=VfEf+(1–Vf)Em

si8_e

Young’s modulus (transverse) of unidirectional composite

Ec=1/VfEf+(1–Vf)Em

Subscripts f and m = reinforcement and matrix, respectively
V = volume fraction

Chapter 7

Stresses and strains in three dimensions

ɛ1=σ1E–vσ2E–vσ3Eɛ2=σ2E–vσ1E–vσ3Eɛ3=σ3E–vσ1E–vσ2EΔ=ɛ1+ɛ2+ɛ3

si10_e

Stresses in thin-walled tube under internal pressure

Hoop stress

σ=prt

Axial stress (when tube carries longitudinal pressure loading)

σ=pr2t

Bending stress in beam

σ=MzI

Bending deflection of beams

δ=C1FL3EI

Vibration frequency of beams

f=C2EIML3

Buckling load of beams

Fcr=C3EIL2

ɛ₁, ɛ₂, ɛ₃ (σ₁, σ₂, σ₃) = principal strain (stress) components in 1, 2, and 3 directions
r = tube radius
t = wall thickness
M = bending moment
z = distance from neutral axis
I = second moment of area of beam cross section
C₁ = constant depending on support and loading geometry
L = length of beam
C₂ = constant depending on mass distribution and loading geometry
M = vibrating mass
C₃ = constant depending on support and loading geometry

Chapter 9

Nominal/true stress–strain

σn=FA0,σ=FA,ɛn=uL0=L–L0L0,ɛ=∫L0LdLL=lnLL0

si17_e

A₀L₀ = AL for plastic deformation; or for elastic or elastic/plastic deformation when ν = 0.5. Thus,

σ=σn1+ɛn

Also

ɛ=ln1+ɛn

Work of deformation, per unit volume

U=∫ɛn,1ɛn,2σndɛn=∫ɛ1ɛ2σdɛ

For linear-elastic deformation only

U=σn22E

Hardness

H=F/A

σ_n = nominal stress
A₀(L₀) = initial area (length)
A(L) = current area (length)
ɛ = true strain

Chapters 10 and 11

The dislocation yield-strength

τy=2TbL≈GbLσy=3τy

si23_e

Grain-size effect

τy=βd–1/2

T = line tension (about Gb²/2)
b = Burgers vector
L = obstacle spacing
σ_y = yield strength
d = grain size
β = constant

Chapter 12

Shear yield stress

k=σy/2

si25_e

Hardness

H≈3σTS

Necking starts when

dσdɛ=σ

Fully plastic bending moments of cross-sections

Mp=σya3/4 (square, side a)

Mp=σybd2/4 (rectangle, breadth b, depth d)

Mp=4σyr3/3 (round bar, radius r)

Mp=4σyr13–r23/3 (tube, external and internal radii r₁ and r₂)

Mp=4σytr2 (thin-walled tube)

Shear-yielding torques

T=2πkr3/3 (round bar)

T=2πkr13–r23/3 (tube)

T=2πkr2t (thin-walled tube)

Chapters 14 and 15

The stress intensity

K=Yσπa

si36_e

Fast fracture occurs when

K=Kc=EGc

a = crack length
Y = dimensionless constant
K_c = critical stress intensity or fracture toughness
G_c = critical strain energy release rate or toughness

Chapter 16

Tensile strength of brittle material

σTS=Kcπam

si38_e

Modulus of rupture

σr=6Mrbd2=3FL2bd2σTS=σr2(m+1)21/m

Weibull equation

Ps(V)=exp–VV0σσ0m(constant stress)Ps(V)=exp–1σ0mV0∫VσmdV(varying stress)Ps=1–Pf=1–(j–0.375)/(n+0.25)

K_c = fracture toughness
a_m = size of longest microcrack (crack depth for surface crack, crack half-length for buried crack)
M_r = bending moment to cause fracture
b, d = width and depth of beam
F,L = fracture load and span of beam
P_s = survival probability of component
P_f = failure probability of component
V = volume of component
V₀ = volume of test specimen
σ = tensile stress in component
σ₀ = normalizing stress (P_s = 1/e = 0.37)
m = Weibull modulus
n = number of test results
j = rank

Chapters 18 and 19

Relation between total strain amplitude and number of reversals to failure, 2N_f

Δɛtot2≈σ′f(2Nf)bE+ɛ′f(2Nf)c

si41_e

Basquin’s law (high cycle)

Δσ2≈σ′f2Nfb

Effect of tensile mean stress (strain approach)

Δɛtot2≈σ′f–σm2NfbE+ɛ′f2Nfc

Effect of tensile mean stress (stress approach)

Δσσm2≈Δσσ′f–σm2σ′f

Crack growth law

ΔK=YΔσπadadN=A(ΔK)m

Failure by crack growth

Nf=∫a0afdaA(ΔK)mSCFeff=S(SCF–1)+1

σ′fɛ′f = true fracture stress (strain)
b,c = constants b≈–0.05to–0.12,c≈–0.5to–0.7
Δσ = stress range (tensile for crack growth law)
σ_m = tensile mean stress
ΔK = stress intensity range
A, m = constants
SCF_eff = effective stress concentration factor
SCF = stress concentration factor
S = notch sensitivity factor

Chapter 21

Creep rate

ɛ˙ss=Aσne–Q/RT

si49_e

ɛ˙ss = steady-state tensile strain-rate
Q = activation energy
R = universal gas constant
T = absolute temperature
A, n = constants

Chapter 22

Fick’s law

J=–Ddcdx

si51_e

Arrhenius’s law

Rate∝e–Q/RT

Diffusion coefficient

D=D0e–Q/RT

Diffusion distance

x≈Dt

J = diffusive flux
D = diffusion coefficient
c = concentration
x = distance
D₀ = pre-exponential factor
t = time

Chapter 23

Rate of diffusion creep

ɛ˙ss=C′σe–Q/RTd2C′=constant;d=grain size

si55_e

Chapter 25

Linear growth law for oxidation

Δm=kLt;kL=ALe–QL/RT

si56_e

Parabolic growth law for oxidation

(Δm)2=kpt;kp=Ape–QP/RT

Δm = mass gain per unit area
k_L, k_P A_L, A_P = constants

Chapter 27

Uniform loss rate in µm/year

87,600(w/ρAt)=3.27(ai/zρ)

si58_e

w = weight metal lost in mg
ρ = metal density in Mg m⁻³ or g cm⁻³
A = corroding area in cm²
t = exposure time in hours
a = atomic weight of metal
i = current density at corroding surface in µA cm⁻²
z = number electrons lost per metal atom

Chapter 29

True contact area a≈P/σy si59_e

P = contact force

Chapter 31

Linear thermal strain ɛ=L–L0L0=αΔT si60_e

Volume strain Δ = 3αΔT

α = linear coefficient of thermal expansion

ΔT = temperature interval

L₀ = original length

L = final length

Expansion coefficient (transverse) of unidirectional composite

α_c ≈ V_fα_f + (1 – V_f)α_m

Expansion coefficient (longitudinal) of unidirectional composite

αc≈VfEfαf+1–VfEmαmVfEf+1–VfEm si61_e

Subscripts f and m = reinforcement and matrix, respectively

V = volume fraction

Chapter 32

Heat flow equation (unidirectional, steady state)

q=–KdTdx si62_e

dT/dx = temperature gradient
q = heat flux
K = thermal conductivity

Heat flow equation (unidirectional, unsteady state)

∂T∂t=λ∂2T∂x2 si63_e

t = time
λ = K/ρC = thermal diffusivity
ρ = density
C = specific heat

Specific heat

dQ = CdT
dT = temperature increase
dQ = heat input

Heat-flow distance

x≈λt
x = distance

Magnitudes of Properties

The listed properties for most structural materials have the ranges shown in this table.

Modulus of elasticity, E	2–200 GN m⁻²
Density, ρ	1–10 Mg m⁻³
Yield strength, σ_y	20–200 MN m⁻²
Toughness, G_c	0.2–200 kJ m⁻²
Fracture toughness, K_c	0.2–200 MN m^−3/2

Tags: Engineering materials an introduction to properties applications and design

Aug 9, 2021 | Posted by admin in General Engineer | Comments Off

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Symbols and Formulae

Symbols and Formulae

Principal Symbols

Other Symbols

Principal Formulae

Chapter 2

Chapter 3

Chapter 6

Chapter 7