Matthias Honhon
Formules MatTech, te kennen of af te leiden
HOOFDSTUK 2
2.3
Net force between two atoms FN (N)
𝐹𝑁 = 𝐹𝐴 + 𝐹𝑅
2.4
In state of equilibrium
𝐹𝐴 + 𝐹𝑅 = 0
2.5 a
+
2.5 b
Relation between energy E (J) and
force F (N)
 FA = attractive force (N)
 FR = repulsive force (N)
Centers of atoms are separated by
equilibrium spacing r0
𝐸 = ∫ 𝐹 𝑑𝑟
𝑑𝐸
𝑑𝑟
𝐹=
∞
𝐸𝑁 = ∫ 𝐹𝑁 𝑑𝑟
2.6
+
2.7
+
2.8 a
2.8 b
𝑟
Net energy EN (J)
∞
𝐸𝑁 = ∫ 𝐹𝐴 𝑑𝑟 + ∫ 𝐹𝑅 𝑑𝑟
𝑟
Relation between net force FN (N)
and net energy EN (J)
∞
𝑟
𝐸𝑁 = 𝐸𝐴 + 𝐸𝑅
𝑑𝐸𝐴 𝑑𝐸𝑅
𝐹𝑁 =
+
𝑑𝑟
𝑑𝑟
 EA = attractive energy (N)
 ER = repulsive energy (N)
Minimum of net energy curve corresponds
to the equilibrium spacing r0 (energy at this
point: bonding energy E0)
HOOFDSTUK 4
4.1
Cube edge length for
FCC a (m)
𝑎 = 2𝑅√2

R = atomic radius (m)
4.2
Number of atoms per cubic
unit cell N
𝑁𝑓 𝑁𝑐
𝑁 = 𝑁𝑖 +
+
2
8



Ni = number of interior atoms
Nf = number of face atoms
Nc = number of corner atoms
4.3
Atomic packing factor APF
4.4
Cube edge length for
BCC a (m)
4.5
4.6
4.7 a
4.7 b
Number of atoms per
hexagonal unit cell N
Volume of cubic unit cell
Vc (m³)
Volume of hexagonal unit
cell Vc (m³)
4.8
Density of metals ρ (g/cm³)
4.9
Density of ceramics
ρ (g/cm³)
𝐴𝑃𝐹 =
𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑎𝑡𝑜𝑚𝑠 𝑖𝑛 𝑎 𝑢𝑛𝑖𝑡 𝑐𝑒𝑙𝑙 𝑉𝑆
=
𝑡𝑜𝑡𝑎𝑙 𝑢𝑛𝑖𝑐 𝑐𝑒𝑙𝑙 𝑣𝑜𝑙𝑢𝑚𝑒
𝑉𝐶
4𝑅
𝑎=
√3
𝑁𝑓 𝑁𝑐
𝑁 = 𝑁𝑖 +
+
2
6
3
𝑉𝑐 = 𝑎
3𝑎2 𝑐√3
𝑉𝑐 =
2
2
𝑉𝑐 = 6𝑅 𝑐√3
𝜌=
𝑛𝐴
𝑉𝑐 𝑁𝐴
𝑛′ (∑ 𝐴𝐶 + ∑ 𝐴𝐴 )
𝜌=
𝑉𝐶 𝑁𝐴
assuming the atomic hard-sphere model

R = atomic radius (m)

a = cube edge (m)


a = short unit cell dimension (m)
c = long unit cell dimension (m)
With a = 2R
 n = number of atoms associated
with each unit cell
 A = atomic weight (g/mol)
 VC = volume of unit cell (m³)
 NA = Avogadro’s number
(atoms/mol)
 n’ = number of formula units within
the unit cell (all the ions included in
the chemical formula unit)
 ∑ 𝑨𝑪 = sum of the atomic weights of
all the cations in the formula unit
(g/mol)
 ∑ 𝑨𝑨 = sum of the atomic weights of
all the anions in the formula unit
(g/mol)
 𝑽𝑪 = unit cell volume (m³)
 𝑵𝑨 = Avogadro’s number (atoms/mol)
2
4.11
Linear density LD
4.12
[110] linear density for
FCC
4.13
Planar density PD
4.14
(110) planar density for
FCC
𝐿𝐷 =
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑎𝑡𝑜𝑚𝑠 𝑐𝑒𝑛𝑡𝑒𝑟𝑒𝑑 𝑜𝑛 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑣𝑒𝑐𝑡𝑜𝑟
𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑣𝑒𝑐𝑡𝑜𝑟
The number of atoms per unit length whose
centers lie on the direction vector for a
specific crystallographic direction
2 𝑎𝑡𝑜𝑚𝑠
1
=
4𝑅
2𝑅
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑎𝑡𝑜𝑚𝑠 𝑐𝑒𝑛𝑡𝑒𝑟𝑒𝑑 𝑜𝑛 𝑎 𝑝𝑙𝑎𝑛𝑒 The number of atoms per unit area that are
centered on a particular crystallographic
𝑃𝐷 =
𝑎𝑟𝑒𝑎 𝑜𝑓 𝑝𝑙𝑎𝑛𝑒
plane
2 𝑎𝑡𝑜𝑚𝑠
1
𝑃𝐷110 =
=
8𝑅2 √2
4𝑅2 √2
𝐿𝐷110 =
HOOFDSTUK 5

extra
̅ 𝐧 (g/mol)
Number-average molecular weight 𝐌
̅𝑛 = ∑ 𝑥𝑖 𝑀𝑖
𝑀
5.6
Degree of polymerization DP
̅𝑛
𝑀
𝐷𝑃 =
𝑚
Mi = mean (middle)
molecular weight of size
range i (g/mol)
 xi = fraction of the total
number of chains within the
corresponding size range
̅ 𝐧 = number-average
 𝐌
molecular weight (g/mol)
 m = repeat unit molecular
weight (g/mol)
Average number of repeat units in a
chain
HOOFDSTUK 6
6.1
Equilibrium number of vacancies Nv
(vacancies/m³)
𝑄𝑣
𝑁𝑣 = N ∗ exp (− )
𝑘𝑇


N = total number of atomic sites (atoms/m³)
Qv =energy required for the formation of a vacancy
(J/mol)
3


𝑁𝐴 𝜌
𝐴
6.2
Number of atomic sites per cubic meter N
(atoms/m³)
𝑁=
6.3
equilibrium number of Frenkel defects Nfr
(defects/m³)
𝑁𝑓𝑟 = 𝑁 exp (−
6.4
equilibrium number of Schottky defects Ns
(defects/m³)
𝑁𝑠 = 𝑁 exp (−
𝑄𝑓𝑟
)
2𝑘𝑇
𝑄𝑠
)
2𝑘𝑇
T = absolute temperature (K)
k = gas or Boltzmann’s constant= 1.38*1023
J/atom*K or 8.62*10-5 eV/atom*K
 NA =Avogadro’s number (atoms/mol)
 ρ = density (g/cm³)
 A = atomic weight (g/mol)
 Qfr = energy required for the formation of each
Frenkel defect (eV/defect)
 k = Boltzmann’s constant = 8.62*10-5 eV/K
 T = absolute temperature (K)
 Qs = energy required for the formation of each
Scottky defect (eV/defect)
 k = Boltzmann’s constant = 8.62*10-5 eV/K
 T = absolute temperature (K)
In an AX-type compound
HOOFDSTUK 7

7.1
Diffusion flux J (kg/m²*s or
atoms/m²*s)
7.2
Fick’s first law
𝐽=
𝐽 = −𝐷
𝑀
𝐴𝑡
𝑑𝐶
𝐶𝐴 − 𝐶𝐵
= −𝐷
𝑑𝑥
𝑥𝐴 − 𝑥𝐵
M = mass or number of atoms
diffusing through (g or atoms)
 A = area across which diffusion is
occurring (m²)
 t = elapsed diffusion time (s)
 D = diffusion coefficient (m2/s)
 C = concentration (kg/m³)
 x = position within the solid
(m)
for constant concentrations or pressures
of the diffusing species on both surfaces
of the plate (CA > CB)
4
7.3
Concentration gradient
∆𝐂
∆𝐱
(kg/m4)
7.4 a
Fick’s second law
7.4 b
7.7
7.8
For diffusion situations where time and
temperature are variables and
composition remains constant at some
value of x
Diffusion coefficient D (m²/s)
𝑑𝐶 ∆𝐶 𝐶𝐴 − 𝐶𝐵
=
=
𝑑𝑥 ∆𝑥 𝑥𝐴 − 𝑥𝐵
𝜕𝐶 𝜕
𝜕𝐶
= (𝐷 )
𝜕𝑡 𝑥
𝜕𝑥
2
𝜕𝐶
𝜕 𝐶
=𝐷 2
𝜕𝑡
𝜕𝑥
𝐷𝑡 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝑄𝑑
𝐷 = 𝐷0 exp (− )
𝑅𝑇


If the diffusion coefficient is independent
of composition


D = diffusion coefficient (m²/s)
t = elapsed diffusion time (s)

D0 = temperature-independent
preexponential (m²/s)
Qd = activation energy for
diffusion (J/mol or eV/atom)
R = gas constant = 8,81 J/mol*K
or 8.62*10-5 eV/atom*K
T = absolute temperature (K)
PM = permeability coefficient
((cm³ STP)(cm)/(cm²*s*Pa))
Δx = membrane thickness (m)
ΔP = difference in pressure of the
gas across the membrane (Pa)
D = diffusion coefficient (m²*s)
S = solubility of the diffusing
species in the polymer
((cm³ STP)/Pa*cm³)




7.14
Diffusion flux for a polymer membrane
J (kg/m²*s)
7.15
Permeability coefficient for small
molecules in nonglassy polymers PM
((cm³ STP)(cm)/(cm²*s*Pa))
𝐽 = −𝑃𝑀
∆𝑃
∆𝑥
𝑃𝑀 = 𝐷𝑆
C = concentration (kg/m³)
x = position within the solid
(m)




5
HOOFDSTUK 8

𝐹
𝐴0
8.1
Engineering stress or stress σ (106 N/m²
or MPa)
8.2
Engineering strain ϵ (unitless or m/m or
inches/inch)
8.3
Shear stress τ (MPa)
extra
shear strain γ
γ = tan θ
8.5
Hooke’s law
𝜎 = 𝐸𝜖
𝜖=
𝑙𝑖 − 𝑙0 ∆𝑙
=
𝑙0
𝑙0
𝜏=
𝐸∝(
8.6
8.7
relation shear stress τ and shear strain γ
8.8
Poisson’s ratio υ
8.11
𝜎=
Percent elongation %EL
𝐹
𝐴0
𝑑𝐹
)
𝑑𝑟 𝑟0
𝜏 = 𝐺𝛾
𝜖𝑦
𝜖𝑥
=−
𝜖𝑧
𝜖𝑧
𝑙𝑓 − 𝑙0
%𝐸𝐿 = (
) ∗ 100
𝑙0
𝜐=−
F = instantaneous load applied
perpendicular to the specimen cross
section (N)
 A0 = original cross-sectional area
before any load is applied (m²)
 l0 = original length before any load
is applied (m or inch)
 li = instantaneous length (m or inch)
 Δl = deformation elongation (m or
inch)
 F = load of force imposed parallel to
the upper and lower faces (N)
 A0 = area of upper and lower faces
(m²)
 θ = strain angle (° or radians)
 ϵ = engineering strain
 E = modulus of elasticity or
Young’s modulus (GPa / gigapascal)
the modulus of elasticity is proportional to
the slope of the interatomic force-separation
curve at the equilibrium spacing
 G = shear modulus, slope of the
linear elastic region of the shear
stress-strain curve (MPa)
ratio of the lateral and axial strains
 l0 = original gauge length (m)
 lf = fracture length (m)
percentage of plastic strain at fracture
6
8.12
Percent reduction in area %RA
𝐴0 − 𝐴𝑓
%𝑅𝐴 = (
) ∗ 100
𝐴0
𝜖𝑦
𝑈𝑟 = ∫ 𝜎 𝑑𝜖
8.13 a
0
8.13 b
Modulus of resilience Ur (J/m³ or Pa)
8.14
8.15
8.16
True stress σT (MPa)
True strain ϵT
8.17
8.18 a
8.18 b
relation true stress σT (MPa),
engineering stress σ (MPa) and
engineering strain ϵ
relation true strain ϵT and engineering
strain ϵ
1
𝑈𝑟 = 𝜎𝑦 𝜖𝑦
2
𝜎𝑦
𝜎𝑦2
1
1
𝑈𝑟 = 𝜎𝑦 𝜖𝑦 = 𝜎𝑦 ( ) =
2
2
𝐸
2𝐸
𝐹
𝜎𝑇 =
𝐴𝑖
𝑙𝑖
𝜖 𝑇 = 𝑙𝑛
𝑙0
𝐴𝑖 𝑙𝑖 = 𝐴0 𝑙0
𝜎𝑇 = 𝜎(1 + 𝜖)

A0 = original cross-sectional area
(m²)
 Af = cross-sectional area at the
point of fracture (m²)
strain energy per unit volume required to
stress a material from an unloaded state up
to the point of yielding
 ϵy = strain at yielding
assuming a linear elastic region




if no volume change occurs during
deformation
𝜖 𝑇 = ln(1 + 𝜖)

8.19
true stress σT - true strain ϵT relationship
in the plastic region of deformation (to
the point of necking
F = load (N)
Ai = instantaneous cross-sectional
area over which deformation is
occurring (m²)
l0 = original length before any load
is applied (m)
li = instantaneous length (m)
𝜎𝑇 = 𝐾𝜖 𝑇𝑛

K = varies from alloy to alloy and
depends on the condition of the
material (MPa)
n = strain-hardening exponent
(constant < 1)
7
HOOFDSTUK 9
9.1 a
Burgers vector for FCC b
9.1 b
Burgers vector for BCC b
9.1 c
Burgers vector for HCP b
𝑎
⟨110⟩
2
𝑎
𝒃(𝐵𝐶𝐶) = ⟨111⟩
2
𝑎
𝒃(𝐻𝐶𝑃) = ⟨112̅0⟩
3
𝒃(𝐹𝐶𝐶) =
Resolved shear stress τR (MPa)
𝜏𝑅 = 𝜎 cos 𝜙 cos 𝜆
9.3
Largest resolved shear stress
τR(max) (MPa)
𝜏𝑅 (max) = 𝜎(cos 𝜙 cos 𝜆)𝑚𝑎𝑥
extra
Critical resolved shear stress τcrss
(MPa)
if 𝜏𝑅 (max) = 𝜏𝑐𝑟𝑠𝑠 then yielding occurs
9.4
Yield strength σy (MPa)
9.6
The angle between directions 1
and 2 in cubic unit cells θ (°)
𝜃 = cos −1 [
𝜏𝑐𝑟𝑠𝑠
(cos 𝜙 cos 𝜆)𝑚𝑎𝑥
+
𝑣12
+
𝑤12 )(𝑢22

a = unit cell edge length (m)

a = unit cell edge length (m)
σ = applied stress
ϕ = angle between the normal
to the slip plane and direction
within that plane (° or radians)
 λ = angle between the slip and
stress directions (° or radians)
 with most favorably oriented
slip system
minimum shear stress required to
initiate slip
property of the material that
determines when yielding occurs
 τcrss = critical resolved shear
stress (MPa)
 ϕ = angle between the normal
to the slip plane and direction
within that plane (° or radians)
 λ = angle between the slip and
stress directions (° or radians)
𝑢1 𝑢2 + 𝑣1 𝑣2 + 𝑤1 𝑤2
√(𝑢12
a = unit cell edge length (m)


9.2
𝜎𝑦 =

+
𝑣22
+
𝑤22 )
]


direction 1 = [u1v1w1]
direction 2 = [u2v2w2]
8
HOOFDSTUK 10
extra
Critical stress for crack propagation σc (MPa)
2𝐸𝛾𝑠
𝜎𝑐 = √
𝜋𝑎




10.4
Fracture toughness Kc (MPa√m)
𝐾𝑐 = 𝑌𝜎𝑐 √𝜋𝑎



10.5
Plain strain fracture toughness KIc (MPa√m)
𝐾𝐼𝑐 = 𝑌𝜎√𝜋𝑎



10.6
Critical stress σc (MPa)
𝐾
𝜎𝑐 = 𝑙𝑐
𝑌√𝜋𝑎


E = modulus of elasticity (Pa)
γs = specific surface energy
(J/m²)
a = one-half of the length of an
internal crack (m)
Y = dimensionless parameter
that depends on both crack and
specimen sizes and geometries
as well as on the manner of load
application
σc = crack propagation (MPa)
a = crack length (m)
Y = dimensionless parameter
that depends on both crack and
specimen sizes and geometries
as well as on the manner of load
application
σ = stress (MPa)
a = crack length (m)
KIc = plain strain fracture
toughness (MPa√m)
Y = dimensionless parameter
that depends on both crack and
specimen sizes and geometries
as well as on the manner of load
application
a = crack length (m)
9

1 𝐾𝐼𝑐 2
𝑎𝑐 = ( )
𝜋 𝜎𝑌
10.7
Maximum allowable flaw size ac (m)
10.8
Circumferential wall stress σ (MPa)
10.9
Plain strain fracture toughness KIc (MPa√m)
𝜎
𝐾𝐼𝑐 = 𝑌 ( 𝑦 ) √𝜋𝑎𝑐
𝑁
critical crack length ac (m)
𝑁 2 𝐾𝑙𝑐
𝑎𝑐 = 2 ( )
𝑌 𝜋 𝜎𝑦
𝜎=
𝑝𝑟
2𝑡
2
10.10
KIc = plain strain fracture
toughness (MPa√m)
 σ = stress (MPa)
 Y = dimensionless parameter
that depends on both crack and
specimen sizes and geometries
as well as on the manner of load
application
 p = pressure in the vessel (Pa)
 r = radius (m)
 t = wall thickness (m)
in pressurized spherical tank
 Y = dimensionless parameter
that depends on both crack and
specimen sizes and geometries
as well as on the manner of load
application
 σy = stress (MPa)
 N = factor of safety
 ac = critical crack length (m)
in pressurized spherical tank
 N = factor of safety
 Y = dimensionless parameter
that depends on both crack and
specimen sizes and geometries
as well as on the manner of load
application
 KIc = plain strain fracture
toughness (MPa√m)
 σy = stress (MPa)
in pressurized spherical tank
10

10.11
Plain strain fracture toughness KIc (MPa√m)
10.12
Wall thickness t (m)
10.13
Pressure in the vessel p (Pa)
𝐾𝐼𝑐 = 𝑌𝜎√𝜋𝑡
𝑡=
𝑝𝑟
2𝜎
2
𝐾𝐼𝑐2
𝑝= 2 ( )
𝑌 𝜋𝑟 𝜎𝑦
Y = dimensionless parameter
that depends on both crack and
specimen sizes and geometries
as well as on the manner of load
application
 σ = stress (MPa)
 t = wall thickness (m)
leak-before-break criterion : when onehalf the internal crack length =
thickness of the pressure vessel
in pressurized spherical tank
 p = pressure in the vessel (Pa)
 r = radius (m)
 σ = stress (MPa)
leak-before-break criterion : when onehalf the internal crack length =
thickness of the pressure vessel
in pressurized spherical tank
 Y = dimensionless parameter
that depends on both crack and
specimen sizes and geometries
as well as on the manner of load
application
 r = radius (m)
 KIc = plain strain fracture
toughness (MPa√m)
 σy = stress (MPa)
in pressurized spherical tank
11
HOOFDSTUK 11

11.1 a
+
11.1 b
Mass fraction of the liquid
phase (in the α-L phase)
𝑊𝐿 =

𝑊𝐿 =
11.2 a
+
11.2 b
Mass fraction of the α phase
(in the α-L phase)
𝑆
𝑅+𝑆
𝐶𝛼 − 𝐶0
𝐶𝛼 − 𝐶𝐿
𝑊𝛼 =
𝑊𝛼 =

C0 = overall alloy composition

Distance to the other phase
(here the L phase) at the same
temperature, divided by the
distance between the two
phases at the same temperature

C0 = overall alloy composition

Wα = mass fraction of the
alpha-phase
WL = mass fraction of the
liquid phase
Because only two phases are
present, the sum of the mass
fractions must be equal to 1
𝑅
𝑅+𝑆
𝐶0 − 𝐶𝐿
𝐶𝛼 − 𝐶𝐿

𝑊𝛼 + 𝑊𝐿 = 1
11.3
𝑊𝛼 𝐶𝛼 + 𝑊𝐿 𝐶𝐿 = 𝐶0
11.4
11.5
Volume fraction of the α
phase (in an alloy consisting
of α and β phases)
𝑣𝛼
𝑉𝛼 =
𝑣𝛼 + 𝑣𝛽
Distance to the other phase
(here the α phase) at the same
temperature, divided by the
distance between the two
phases at the same temperature
At the same temperature =
horizontal line



vα = volume of the α-phase in
the alloy
vβ = volume of the β-phase in
the alloy
12
11.8
Eutectic reaction
11.9
Eutectic reaction in a leadtin alloy
11.10
Fraction of the eutectic
microconstituent We
11.11
Fraction of primary α, Wα’
11.12
Fraction of total α, Wα
11.13
Fraction of total β, Wβ
11.14
Eutectoid reaction
11.15
Peritectic reaction
11.18
Eutectic reaction in the ironiron carbide system
11.19
Eutectoid reaction in the
iron-iron carbide system
𝑐𝑜𝑜𝑙𝑖𝑛𝑔
𝐿(𝐶𝐸 ) ⇌ 𝛼(𝐶𝛼𝐸 ) + 𝛽(𝐶𝛽𝐸 )
ℎ𝑒𝑎𝑡𝑖𝑛𝑔




CE = eutectic composition
TE = eutectic temperature
CαE = composition of the α
phase at TE
CβE = composition of the β
phase at TE
𝑐𝑜𝑜𝑙𝑖𝑛𝑔
𝐿(61.9 𝑤𝑡% 𝑆𝑛) ⇌ 𝛼(18.3 𝑤𝑡% 𝑆𝑛) + 𝛽(97.8 𝑤𝑡% 𝑆𝑛)
ℎ𝑒𝑎𝑡𝑖𝑛𝑔
𝑃
𝑃+𝑄
𝑄
𝑊𝛼′ =
𝑃+𝑄
𝑄+𝑅
𝑊𝛼 =
𝑃+𝑄+𝑅
𝑃
𝑊𝛽 =
𝑃+𝑄+𝑅
𝑊𝑒 = 𝑊𝐿 =


WL = fraction of liquid from
which We transforms
Fraction of the α phase that
existed prior to the eutectic
transformation

Total α = both primary and
eutectic α

Total β = both primary and
eutectic β
𝑐𝑜𝑜𝑙𝑖𝑛𝑔
⇌ 𝛾+𝜖
ℎ𝑒𝑎𝑡𝑖𝑛𝑔

𝑐𝑜𝑜𝑙𝑖𝑛𝑔
⇌ 𝜖
ℎ𝑒𝑎𝑡𝑖𝑛𝑔
𝑐𝑜𝑜𝑙𝑖𝑛𝑔
𝐿 ⇌ 𝛾 + 𝐹𝑒3 𝐶
ℎ𝑒𝑎𝑡𝑖𝑛𝑔
𝑐𝑜𝑜𝑙𝑖𝑛𝑔
𝛾(0.76 𝑤𝑡% 𝐶) ⇌ 𝛼(0.022 𝑤𝑡% 𝐶) + 𝐹𝑒3 𝐶(6.7 𝑤𝑡% 𝐶)
ℎ𝑒𝑎𝑡𝑖𝑛𝑔

Upon cooling, a solid phase δ
transforms into two other solid
phases (γ and ϵ)
Upon heating, a solid phase ϵ
transforms into a liquid phase
L and another solid phase δ
𝛿
𝛿+𝐿

Liquid solidifies to form
austenite and cementite phases

Solid γ phase is transformed
into α-iron and cementite
13
11.20
Fraction of pearlite Wp
11.21
Fraction of proeutectoid
α, Wα
11.22
Fraction of pearlite Wp
11.23
Fraction of proeutectoid
cementite WFe3C’
𝑇
𝑇+𝑈
𝑈
𝑊𝛼′ =
𝑇+𝑈
𝑋
𝑊𝑝 =
𝑉+𝑋
𝑉
𝑊𝐹𝑒3𝐶′ =
𝑉+𝑋
𝑊𝑝 =
HOOFDSTUK 12
12.1
Total free energy change
ΔG
4
Δ𝐺 = 𝜋𝑟 3 Δ𝐺𝑣 + 4𝜋𝑟 2 𝛾
3
12.2
𝑑(Δ𝐺) 4
= 𝜋Δ𝐺𝑣 (3𝑟 2 ) + 4𝜋𝛾(2𝑟) = 0
𝑑𝑟
3
12.3
2𝛾
𝑟∗ = −
Δ𝐺𝑣
Critical radius r*
12.4
Critical free energy ΔG*
16𝜋𝛾 3
Δ𝐺 =
3(Δ𝐺𝑣 )2
12.12
Interfacial energy γIL
𝛾𝐼𝐿 = 𝛾𝑆𝐼 + 𝛾𝑆𝐿 𝑐𝑜𝑠𝜃
∗








ΔGv = volume free energy, the
free energy change between
the solid and liquid phase
γ =Surface free energy
ΔGv = volume free energy, the
free energy change between
the solid and liquid phase
γ =Surface free energy
ΔGv = volume free energy, the
free energy change between
the solid and liquid phase
γ =Surface free energy
Activation free energy, free
energy required for the
formation of a stable nucleus
γSL and γSI = interfacial
energies at two-phase
14
12.13
𝑟∗ = −
Critical radius r*
2𝛾𝑆𝐿
𝛥𝐺𝑣
12.14
Critical free energy ΔG*
3
16𝜋𝛾𝑆𝐿
∗
Δ𝐺 = (
) 𝑆(𝜃)
3Δ𝐺𝑣2
12.19
Iron-iron carbide eutectoid
reaction
cooling
𝛾(0.76 𝑤𝑡% 𝐶) ⇌ 𝛼(0.022 𝑤𝑡% 𝐶) + 𝐹𝑒3 𝐶(6.70 𝑤𝑡% 𝐶)
heating
12.20
Formation of tempered
martensite
𝑚𝑎𝑟𝑡𝑒𝑛𝑠𝑖𝑡𝑒 (𝐵𝐶𝑇, 𝑠𝑖𝑛𝑔𝑙𝑒 𝑝ℎ𝑎𝑠𝑒)
→ 𝑡𝑒𝑚𝑝𝑒𝑟𝑒𝑑 𝑚𝑎𝑟𝑡𝑒𝑛𝑠𝑖𝑡𝑒 (𝛼 + 𝐹𝑒3 𝐶 𝑝ℎ𝑎𝑠𝑒𝑠)

boundaries
θ = wetting angle, angle
between the γSI and γSL vectors

γSL = interfacial energy


γSL = interfacial energy
S(θ) = function only of θ,
shape of the nucleus, value
between 0 and 1
HOOFDSTUK 14

14.1
fracture toughness Klc
𝐾𝑙𝑐 = 𝑌𝜎√𝜋𝑎


Y = dimensionless
parameter or function that
depends on both crack
geometries
σ = applied stress
a = length of surface crack
or half the length of internal
crack
HOOFDSTUK 15
15.1
Relaxation modulus Er(t)
𝐸𝑟 (𝑡) =
𝜎(𝑡)
𝜖0


σ(t) = measured timedependent stress
ϵ0 = strain level
15
15.2
Creep modulus Ec(t)
15.4
Vulcanization
𝐸𝑐 (𝑡) =
time-dependent elastic
modulus for viscoelastic
polymers
 σ0 = constant applied
stress
 ϵ(t) = time-dependent
strain
𝜎0
𝜖(𝑡)
HOOFDSTUK 19


19.1
Ohm’s law
𝑉 = 𝐼𝑅

V = voltage (volts (V) or J/C)
I = current, time rate of charge
passage (ampere (A) or C/s)
R = resistance of the material
through which the current is
passing (ohms or V/A)
16

19.2
+
19.3
𝜌=
Electrical resistivity ρ
𝑅𝐴
𝑙


19.4
Electrical conductivity σ
19.5
Current density J
19.6
Electric field intensity E
𝑉𝐴
𝜌=
𝐼𝑙
1
𝜎=
𝜌
𝐽 = 𝜎E
E=
𝑉
𝑙


ρ = electrical resistivity



E = electric field intensity
σ = electrical conductivity
V = voltage difference between
two points (volts (V) or J/C)
l = distance between the two points
at which the voltage is measured
ρt = individual thermal resistivity
contribution
ρi = individual impurity resistivity
contribution
ρd = individual deformation
resistivity contribution
Q = quantity of charge stored on
either plate
V = voltage applied across the
capacitor (volts (V) or J/C)


19.9
Matthiessen’s rule
Total resistivity of a metal ρtotal
𝜌𝑡𝑜𝑡𝑎𝑙 =𝜌𝑡 + 𝜌𝑖 + 𝑝𝑑


19.24
Capacitance C (farads (F) or C/V)
𝑄
𝐶=
𝑉
R = resistance of the material
through which the current is
passing (ohms or V/A)
A = cross-sectional area
perpendicular to the direction of
the current
l = distance between the two points
at which the voltage is measured
I = current, time rate of charge
passage (ampere (A) or C/s)


17
𝐴
𝐶 = 𝜖0
𝑙
19.25
Capacitance C of a parallel-plate capacitor with a
vacuum in the region between the plates
19.26
Capacitance C when a dielectric material is inserted
into the region within the plates
𝐶=𝜖
19.27
Dielectric constant or relative permittivity ϵr
𝜖𝑟 =
𝐴
𝑙
𝜖
𝜖0









19.28
Dipole moment p
𝑝 = 𝑞𝑑


19.29
surface charge density D0 when vacuum is present
𝐷0 = 𝜖0 E
𝐷 = 𝜖E
19.30
+
19.31




Dielectric displacement or surface density D

𝐷 = 𝜖0 E + 𝑃

19.32
Polarization P
𝑃 = 𝜖0 (𝜖𝑟 − 1)E

ϵ0 = permittivity of a vacuum =
8.85 *10-12 F/m
A =area of the plates
l = distance between the plates
ϵ = permittivity of the dielectric
medium
A =area of the plates
l = distance between the plates
ϵ = permittivity of the dielectric
medium
ϵ0 = permittivity of a vacuum =
8.85 *10-12 F/m
q = magnitude of each dipole
charge
d = distance of separation between
them
ϵ0 = permittivity of a vacuum =
8.85 *10-12 F/m
E = electric field intensity
ϵ = permittivity of the dielectric
medium
E = electric field intensity
ϵ0 = permittivity of a vacuum =
8.85 *10-12 F/m
P = polarization, increase in
charge density above that for a
vacuum because of the presence of
a dielectric
ϵ0 = permittivity of a vacuum =
8.85 *10-12 F/m
ϵr = dielectric constant
18
19.33
Magnitude of dipole moment for each ion pair pi
𝑝𝑖 = 𝑞𝑑𝑖
19.34
Total polarization P
𝑃 = 𝑃𝑒 + 𝑃𝑖 + 𝑃𝑜






E = electric field intensity
q = charge on each ion
di = relative displacement
Pe = electronic polarization
Pi = ionic polarization
Po = orientation polarization
HOOFDSTUK 20
20.1
20.3 a
+
20.3 b
Heat capacity C
(J/mol*K or cal/mol*K)
Change of length with
temperature for a solid
material
𝚫𝒍
𝒍𝟎
Change of volume with
20.4
𝚫𝑽
temperature 𝚫𝑽
𝟎
20.5
Heat flux or heat flow q
(W/m²)
20.8
Magnitude of the stress
resulting from a
temperature change σ
(MPa)
𝐶=
𝑑𝑄
𝑑𝑇
𝑙𝑓 − 𝑙0
= 𝛼𝑙 (𝑇𝑓 − 𝑇0 )
𝑙0
Δ𝑙
= 𝛼𝑙 Δ𝑇
𝑙0
Δ𝑉
= 𝛼𝑣 Δ𝑇
Δ𝑉0
𝑞 = −𝑘
𝑑𝑇
𝑑𝑥
𝜎 = 𝐸𝛼𝑙 (𝑇0 − 𝑇𝑓 ) = 𝐸𝛼𝑙 Δ𝑇

















dQ = energy required to produce a dT temperature
change
l0 = initial length (m)
lf = final length (m)
T0 = initial temperature (K)
Tf = final temperature (K)
αl = linear coefficient of thermal expansion (K-1)
ΔV = volume change (m³)
V0 = original volume (m³)
αv = volume coefficient of thermal expansion (K-1)
ΔT = change in absolute temperature (K)
k = thermal conductivity (W/m*K)
𝒅𝑻
= temperature gradient through the conducting
𝒅𝒙
medium (K/m)
E = modulus of elasticity (GPa)
αl = linear coefficient of thermal expansion (K-1)
T0 = initial temperature (K)
Tf = final temperature (K)
ΔT = change in absolute temperature (K)
19