Useful Materials Notes
Chapter 1:
Types of Materials: Metals
o High toughnesso Conductiveo Crystallineo Metallic bond
Ceramics & Glasseso Chemically stableo High temperature resistanceo Brittleo Low toughnesso Hardo Crystalline or amorphouso Ionic/covalent bond
Polymerso Synthetico Lightweighto Elastico Low melting pointo Covalent and secondary
Compositeso Combinations of metals, ceramics & glasses, and polymerso Covalent or covalent/ionic
Semiconductorso Chemical purity determines electronic properties
Chapter 2:
Types of Bonding: Primary (strong, high melting points)
o Ionic Non-directional S-shells
o Covalent Directional P-shells and d-shells Low CN
o Metallic Non-directional S-shells High CN Closely packed Electron gas
Secondary (weak, low melting points)o Hydrogen
Attraction of permanent dipoleso Van der Waals
Attraction of temporary dipoles
Coulombic Attraction
FC=k q1q2r2
Repulsive Force
FR=λe−rρ
Where λ and ρ are experimentally determined constants
Net ForceF=FC+F R
Chapter 3:
Crystalline/Amorphous: Crystalline = regular repeating pattern Amorphous = irregular
Unit Cell Smallest repeating structure
Crystalline Structures 7 crystal systems (axial systems) 14 crystal lattices (Bravais lattices) Mainly SC (Simple Cubic), BCC (Body-Centred Cubic), and FCC (Face-Centred Cubic)
SC BCC FCC HPCAtoms/Unit Cell 1 2 4 2
APF 0.52 0.68 0.74 0.74CN 6 8 12 12a 2R 4r/sqrt(3) 2sqrt(2)rd 2Sqrt(3)r 4r 2sqrt(6)r
Pattern AAAAAA AAAAAA ABCABC ABABAB
Density
ρ= nAV C N A
ρ is density (g/cm3)
n is atoms/unit cell (atoms)
A is atomic weight (g/mol)
VC is volume/unit cell (cm3)
NA is 6.02x1023 (atoms/mol)
Ionic Packing Factor APF but when there are two different ions
Indices (bar on top for negative) Lattice Points: [111] Family of Directions <111>
o [111], [1(bar), 1,1], [1,1(bar), 1],… Lattice Planes: (111)
o Miller indices are the reciprocals of the intercepts Miller-Bravais Indices: (hkil)
o i = -(h+k)o used for hexagonal systems
Family of Planes {1,1,1}
a ∙b=¿a∨¿b∨cosθ
Where theta is the angle between the vectors
Linear Density 1/r where r is the repeat distance between adjacent atoms
Bragg’s Lawn λ=2dsinθ
n is the diffraction order
λ is the wavelength (nm)
d is the spacing between adjacent crystal planes (nm)
2ϴ is the diffraction angle
For a cubic system:
d= a
√h2+k 2+l2
Diffraction Occurs Diffraction Does not OccurBCC h+k+l is even h+k+l is oddFCC h,k,l are all even or all odd h,k,l are a mix of even and oddHPC (h+2k) = 3n (where n is an
integer), and l is oddThe rest of the time
Chapter 4:
Types of Materials: 0-D: point defects
o Impurities Substitutionals
Different atom in a crystal Interstitials
Small atom that fits into the crystalo Vacancies
Hole in the crystal 1-D: linear defects
o Edge dislocations Extra half plane of atoms
o Screw dislocations Spiral stacking of crystal planes
o Mixed dislocations Edge dislocation on one edge, screw on another
2-D: planar defectso Grain boundaries
Atoms on boundaries (surface) are not in equilibrium Tilt boundary
Low angle Caused by edge dislocations
Twist boundary Low angle Caused by screw dislocations
Twin Boundary Two crystal structures are mirror images
Stacking Faults: Annealing Twins Fault in stacking of planes ie. ABCABCAB ABCABC FCC
Mechanical Twins Part of the crystal structure bends and then bends back
Deformation Elastic
o Reversible
o Bonds are stretched but not broken Plastic
o Permanento Caused by dislocations moving through the substance
Burger’s Vector (b) Vector required to close the loop of an edge dislocation Perpendicular to the edge dislocation Parallel to screw dislocation Consistent with each dislocation (edge and screw) for a mixed dislocation
Hume-Rothery rules for complete miscibilityIn order for complete solid solubility to occur between 2 solid metals:
1. Similar atomic radii (~15%)2. Same crystal structure3. Similar electronegativites (~10%)4. Same valence
Chapter 6 (1):
Engineering Stress
σ= PA0
σ is engineering stress (Pa)
P is load (N)
A0 is initial cross-sectional area (m2)
Engineering Strain
ε=l−l0l0
=∆ ll0
ε is engineering strain (unit-less)
l0 is initial gauge length (m)
l is final gauge length (m)
Relationship between stress and strain
ε= σE
Stress-Strain Graph Yield Strength
o Point between elastic and plastic phaseo At the top of the linear part of the stress-strain graph
Ultimate Tensile Strengtho Top of the stress –strain curve
E, Young Moduluso Slope of the plastic phase
Fracture Stresso Stress at which it breaks
Ductilityo the strain that corresponds to the intersection of a line with E as its slope traced back
from the fracture pointo % elongation or % decrease in area
Toughnesso Integral of the whole curve
Modulus of Resilienceo Integral of the elastic phase (energy it can take before being deformed)
Specific Strength Strength/Density
Strain hardening (a.k.a. Work Hardening) Second bump in stress-strain curve Pulled to that bump on graph then let return so that it will be stronger
Necking Middle of test specimen starts to become thinner and longer
True Stress
σ T=PA
σT is true stress (Pa)
P is load (N)
A is cross-sectional area (m2)
True Strain
ε T= ln (A0A
)
εT is true strain (unit-less)
A0 is initial cross-sectional area (m2)
A is cross-sectional area (m2)
Poisson’s Ratio Contraction perpendicular to the extension under tensile stress
v=−ε x
ε z
εx is negative strain (strain from change in diameter)
εz is positive strain (from young’s modulus)
Shear Stress
τ=PS
AS
PS is the load on the sample
AS is the area of the sample
Shear StrainShear stress produces an angular displacement α, with the shear strain γ
γ=tanα=∆ yz0
≈a a→0
Shear Modulus
G= τγ
Relationship between moduli E and GE=2G(1+v )
Three Point Bend Test (modulus of rupture)
MOR= 3 FL2bh2
F is fracture force (N)
L is the distance between the two bottom supports (m)
b is the width of the beam (m)
h is the thickness of the beam (m)
Three Point Bend Test (flexural modulus)
EFlex=L3m4bh3
m is slope of the tangent of the linear part of the load –deflection curve (which is F/γ, where γ is the vertical deflection from the original horizontal position) (N/m)
L is the distance between the two bottom supports (m)
b is the width of the beam (m)
h is the thickness of the beam (m)
Griffith Crack Model
σ=12σm√ ρ
c
σ is the stress applied (Pa)
σm is the maximum stress of the material (Pa)
ρ is the radius of the tip (m)
c is the length of the crack (m)
Chapter 6 (2):
Plastic Deformation: Bonds are broken and reformed Does not return to its original position Mechanisms
o Crystalline structure – Slip mechanism Atoms slide over each Dislocations increase slipping In perfect crystal high stress because all the atoms slide at once, in imperfect
crystal low stress because one atom at a time slideso Non-crystalline structure
Viscous flow mechanism
Critical Shear Stress G (Shear Modulus) is 10x larger than τC (critical resolved shear stress)
Resolved Shear Stress
τ=σ cos λcosϕ= FAcos λcos ϕ
τ is the resolved shear stress (Pa)
σ is the applied stress (Pa)
λ is the angle between the tensile axis and the slip direction (°)
ϕ is the angle between the normal of the slip plane and the tensile axis (°)
F is the tensile force (N)
A is the cross-sectional area of the wire normally (not the shear plane)
Schmid’s Law
τC=σC cos λ cosϕ=FAcos λcos ϕ
τC is the critical resolved shear stress (Pa) (when the plastic stage starts)
σC is the critical applied stress (Pa)
λ is the angle between the tensile axis and the slip direction (°)
ϕ is the angle between the normal of the slip plane and the tensile axis (°)
F is the tensile force (N)
A is the cross-sectional area of the wire normally (not the shear plane)
Methods to Increase Yield Strength Use Defects
o Dislocation movement starts in the plane with the highest resolved shear stresso Hinder dislocation movement = increased yield strengtho Dislocations have to change directions to move beyond defects
Smaller Grainso Grain boundaries impede dislocation movement
o σ y=σ 0+K √ 1d
σy is yield strength (Pa) σ0 is a constant for the metal (Pa) K is a constant of the metal d is average grain diameter
Cold workingo Causes work hardening
Solution Hardeningo Impurity atomso Interstitialso Larger Substitutionals
Dispersion Strengthening (Second Phase Strengthening)
Hardness Resistance to indentation Not a fundamental property (depends on many factors)
Brinell Hardness (ball)
BHN= 2P
πD(D−√(D 2−d2 ))
P is load (in kgf “kilogram-force”) ( 1kgf = 9.81N)
D is the maximum diameter of the ball (mm)
d is the diameter of indentation
Creep Plastic Deformation at high temperatures under constant stress Metals ceramics polymers
o Can be caused by vacancy climb
o ε̇=∆ε∆ t ε (with a dot) is creep ε is strain t is time
Exhibits Arrhenius behaviour (faster when hotter)
o dεdt
=Ce−QRT
s is strain (m) C is a constant Q is the activation energy (J/mol) R is the universal gas constant (8.314 J/mol) T is temperature (K)
Stress Relaxation Decreasing stress over time for constant strain polymers
Grain Boundary Sliding During Creep creates voids at an inclusion trapped at the grain boundary creates a void at the triple point of where three grains meet
Lowering Creep Rates1. Keep temperatures low2. Keep loads low (i.e. low σ)3. Use high melting point alloys
4. Add thermally stable particles that impede dislocation mobility5. Minimize grain boundary sliding by using columnar grains or single crystals
Larson-Miller Parameter
L=T (A+Blnt)1000
L is the Larson-Miller Parameter
T is temperature (K)
A and B are experimental constants
T is time (hours)
Chapter 7, 9 and 10:
Relative Thermal Expansion∆ ll0
=α∗∆T
l is length (m)
α is the coefficient of thermal expansion (1/K)
T is temperature (K)
Thermal Conductivity (Fourier’s Law)
k=−( dQ
dt)
A (dTdx
)
K is thermal conductivity (W/K)
Q is the heat (J)
T is temperature (K)
A is area (m^2)
x is distance (m)
Gibbs Phase RuleWhen pressure and temperature are variable:
F=C−P+2
When only pressure or temperature is variable:
F=C−P+1
F is degrees of freedom
C is number of components
P is the number of phases
Phase Diagrams
Eutectic - points down - liquid to solid
Peritectic - points up - liquid to solid
Eutectoid - points down - solid to solid
Peritectoid - points up - solid to solid
Tie Line and Lever Rule
Draw a line at a given temperature
The percent composition of each component is the fraction of the distance away over the whole length of the line
Cold Working
%CW=A0−A f
A0∗100%
A0 is the area before plastic deformation (m2)
Af is the area after plastic deformation (m2)
Chapter 8:
Fracture ToughnessK=Yσ √ πa
K is stress intensity factor (MPa sqrt(m))
Y is geometric factor
σ is stress (MPa)
a is depth of crack(m)
****internal then sub in half of the length given
KC is the critical stress intensity factor
Fracture Toughness of Brittle MaterialsFor a Vickers hardness indentation
K IC=α 0√ EH
∗P
d32
K is stress intensity factor (MPa sqrt(m))
α0 is geometric factor (0.016???)
E is elastic modulus (N/m2)
H is Vickers hardness (N/m2)
P is load (N)
d is distance from tip of crack to centre
Paris’ LawdadN
=A(∆ K )m
K is stress intensity factor (MPa sqrt(m))
N is number of load cycles
A is a constant
m is a constant
a is crack length
Weibull
f=1−e−( σ
σ 0)m
f is probability of failure
σ is stress applied
σ0 is reference stress
m is a constant
Chapter 13 and 15:
Resistivity
ρ=RAl
ρ is resistivity (Ωm)
R is resistance (Ω)
A is area (m2)
l is length (m)
Conductivity
σ=1ρ
σ is conductivity (1/ Ωm)
ρ is resistivity (Ωm)
R is resistance (Ω)
Conductivity and Resistivity Depend on:
1. Defects2. Temperature3. Atomic Arrangement (lattice type, degree of crystallinity)4. Electron structure and atomic bonding
Current Density (Flux)
J= IA
=Eσ=|e|nv
J is current density (A/m2)
I is current (A)
A is area (m2)
E is electric field strength (V/m)
σ is conductivity (1/ Ωm)
e is the charge of an electron (C)
n is the number of electrons per unit volume (1/m3)
v is velocity (average) (m/s)
Temperature Dependence of Resistivityρ (∆T )=ρ ref (1+αref ∆T )
ρ resistivity (Ωm)
T is temperature (K or °C)
αref is the temperature coefficient of resistivity (TCR)
Effect of an Impurity on Resistivityρ=ρ0(1+βx )
ρ0 resistivity of pure metal (Ωm)
B is a material constant
x is the amount of impurity addition
Effect of Temperature on Carriers
σ=σ 0e−Eg
2kT
σ is conductivity (1/ Ωm)
Eg is bandgap energy (J)
K is boltzmann’s constant (1.38 x 10-23 J/K)
T is temperature (K)
Chapter 14:
Magnetism
1. Diamagnetica. μ slightly < 1
2. Paramagnetic a. Ferromagnetic “true” Ferromagnetic
i. μ significantly > 1b. Ferrimagnetic
i. μ slightly > 1c. Antiferromagnetic
Magnetic Field
B⃗=μ H⃗
B is induction (T)
μ is the magnetic permeability of the material
H is magnetic field (T)
M⃗=(μr−1) H⃗
M is magnetization (T) – Volume of magnetic dipoles
μr is the relative permeability (μr/ μ) where μ = 4π x 107 (H/m)
μr - 1 is the magnetic susceptibility
H is magnetic field (T)
General
ASTM Grain Size (G)n=2G−1
n is the number of grains at 100x per sq. in
G is average grain size
Count the number of full grains + half of the number of partial grains in the image to get n
Conversion between other magnifications
n100=(mag100
)2
∗nmag
Hall-Petch Equation
σ y=σ 0+k y
√d
σy is yield stress
σ0 is a material constant
ky is a material constant
d is the ASTM average grain diameter