Formula Sheet: Structure and Properties | Engineering Materials - Mechanical Engineering PDF Download

 Page 1


F ormula Sheet: Structure of Engineering Materials
Crystal Structures
• Lattice Par ameters : Defined b y lattice constants ( a,b,c ) and angles (a,ß,? ).
• Atomic P acking F actor (APF) :
APF=
V olume of atoms in unit cell
V olume of unit cell
• Common C rystal Structures :
– BCC: APF = 0.68, Coordination number = 8
– FCC: APF = 0.74, Coordination number = 12
– HCP: APF = 0.74, Coordination number = 12
• Density of Crystal :
?=
n·A
V
c
·N
A
wheren = number of atoms per unit cell,A = atomic weight,V
c
= volume of
unit cell,N
A
= A vogadro’ s number (6.022×10
23
mol
-1
).
• Miller Indices for Planes : (hkl) defined b y intercepts with axes, recipro-
cals, and reduc tion to smallest integers.
• Interplanar Spacing :
d
hkl
=
a
v
h
2
+k
2
+l
2
(for cubic crystals)
Imperfections in Solids
• Point Def ects :
– V acancy: Missing atom at lattice site.
– Interstitial: Extr a atom in interstitial site.
– Schottky Defect: V acancy pair in ionic crystals.
– Frenk el Defect: Atom displaced to interstitial site.
• Line Defect s (Dislocations) :
– Burgers V ector (
?
b ): Magnitude and direction of lattice distortion.
– Edge Dislocation:
?
b? dislocation line.
– Screw Dislocation:
?
b? dislocation line.
• Surface De fects : Gr ain boundaries, twin boundaries.
• Dislocation Density :
?
d
=
T otal dislocation length
V olume
1
Page 2


F ormula Sheet: Structure of Engineering Materials
Crystal Structures
• Lattice Par ameters : Defined b y lattice constants ( a,b,c ) and angles (a,ß,? ).
• Atomic P acking F actor (APF) :
APF=
V olume of atoms in unit cell
V olume of unit cell
• Common C rystal Structures :
– BCC: APF = 0.68, Coordination number = 8
– FCC: APF = 0.74, Coordination number = 12
– HCP: APF = 0.74, Coordination number = 12
• Density of Crystal :
?=
n·A
V
c
·N
A
wheren = number of atoms per unit cell,A = atomic weight,V
c
= volume of
unit cell,N
A
= A vogadro’ s number (6.022×10
23
mol
-1
).
• Miller Indices for Planes : (hkl) defined b y intercepts with axes, recipro-
cals, and reduc tion to smallest integers.
• Interplanar Spacing :
d
hkl
=
a
v
h
2
+k
2
+l
2
(for cubic crystals)
Imperfections in Solids
• Point Def ects :
– V acancy: Missing atom at lattice site.
– Interstitial: Extr a atom in interstitial site.
– Schottky Defect: V acancy pair in ionic crystals.
– Frenk el Defect: Atom displaced to interstitial site.
• Line Defect s (Dislocations) :
– Burgers V ector (
?
b ): Magnitude and direction of lattice distortion.
– Edge Dislocation:
?
b? dislocation line.
– Screw Dislocation:
?
b? dislocation line.
• Surface De fects : Gr ain boundaries, twin boundaries.
• Dislocation Density :
?
d
=
T otal dislocation length
V olume
1
Phase Diagr ams
• Lever Ru le (for two-phase region):
Fr action of phasea =
C
ß
-C
C
ß
-C
a
, Fr action of phaseß =
C-C
a
C
ß
-C
a
whereC = allo y composition,C
a
,C
ß
= compositions of phases.
• Gibbs Phase Rule :
P +F =C +2
whereP = number of phases,F = degrees of freedom,C = number of com-
ponents.
• Eutectic Point : Composition and temper ature where liquid tr ansforms to
two solid ph ases.
Mechanical Properties
• Stress :
s =
F
A
whereF = force,A = cross-sectional area.
• Str ain :
?=
?L
L
0
where?L = change in len gth,L
0
= origina l length.
• Y oung’ s Mod ulus (Elastic Modulus):
E =
s
?
• Poisson’ s Ratio :
? =-
?
later al
?
longitudinal
• Shear Mod ulus :
G=
E
2(1+?)
• Yield Stre ngth : Stress at which plastic deformation begins.
• Hardness : Resistance to indentation, e.g., Brinell Hardness Number (BHN):
BHN=
2P
pD(D-
v
D
2
-d
2
)
whereP = load,D = indenter diameter ,d = indentation diameter .
2
Page 3


F ormula Sheet: Structure of Engineering Materials
Crystal Structures
• Lattice Par ameters : Defined b y lattice constants ( a,b,c ) and angles (a,ß,? ).
• Atomic P acking F actor (APF) :
APF=
V olume of atoms in unit cell
V olume of unit cell
• Common C rystal Structures :
– BCC: APF = 0.68, Coordination number = 8
– FCC: APF = 0.74, Coordination number = 12
– HCP: APF = 0.74, Coordination number = 12
• Density of Crystal :
?=
n·A
V
c
·N
A
wheren = number of atoms per unit cell,A = atomic weight,V
c
= volume of
unit cell,N
A
= A vogadro’ s number (6.022×10
23
mol
-1
).
• Miller Indices for Planes : (hkl) defined b y intercepts with axes, recipro-
cals, and reduc tion to smallest integers.
• Interplanar Spacing :
d
hkl
=
a
v
h
2
+k
2
+l
2
(for cubic crystals)
Imperfections in Solids
• Point Def ects :
– V acancy: Missing atom at lattice site.
– Interstitial: Extr a atom in interstitial site.
– Schottky Defect: V acancy pair in ionic crystals.
– Frenk el Defect: Atom displaced to interstitial site.
• Line Defect s (Dislocations) :
– Burgers V ector (
?
b ): Magnitude and direction of lattice distortion.
– Edge Dislocation:
?
b? dislocation line.
– Screw Dislocation:
?
b? dislocation line.
• Surface De fects : Gr ain boundaries, twin boundaries.
• Dislocation Density :
?
d
=
T otal dislocation length
V olume
1
Phase Diagr ams
• Lever Ru le (for two-phase region):
Fr action of phasea =
C
ß
-C
C
ß
-C
a
, Fr action of phaseß =
C-C
a
C
ß
-C
a
whereC = allo y composition,C
a
,C
ß
= compositions of phases.
• Gibbs Phase Rule :
P +F =C +2
whereP = number of phases,F = degrees of freedom,C = number of com-
ponents.
• Eutectic Point : Composition and temper ature where liquid tr ansforms to
two solid ph ases.
Mechanical Properties
• Stress :
s =
F
A
whereF = force,A = cross-sectional area.
• Str ain :
?=
?L
L
0
where?L = change in len gth,L
0
= origina l length.
• Y oung’ s Mod ulus (Elastic Modulus):
E =
s
?
• Poisson’ s Ratio :
? =-
?
later al
?
longitudinal
• Shear Mod ulus :
G=
E
2(1+?)
• Yield Stre ngth : Stress at which plastic deformation begins.
• Hardness : Resistance to indentation, e.g., Brinell Hardness Number (BHN):
BHN=
2P
pD(D-
v
D
2
-d
2
)
whereP = load,D = indenter diameter ,d = indentation diameter .
2
Diffusion
• Fick’ s Firs t Law :
J =-D
dC
dx
whereJ = flux, D = diffusion coefficient,
dC
dx
= concentr ation gr adient.
• Fick’ s Sec ond Law :
?C
?t
=D
?
2
C
?x
2
• Arrhenius Eq uation for Diffusion :
D =D
0
exp
(
-
Q
RT
)
whereD
0
= pre-exponential factor ,Q = activation energy ,R = gas constant,
T = absolute temper ature.
3