Review for Materials Science and Engineering
Introduction
Metals:
• Composed of one or more metallic elements and can include small amounts of
nonmetallic elements.
• Useful due to:
-
Being good conductors of heat and electricity
-
Malleability and ductility make them easy to shape
-
Can withstand a lot of force before breaking (tensile strength)
-
Some cannot be easily cut or deformed (hardness)
Polymers:
• Organic compounds that are chemically based on carbon, hydrogen and other nonmetallics. Commonly referred to as plastics and rubber.
• Useful due to:
-
Being very lightweight
-
Can be highly deformable, we can make complex shapes
-
Chemically inert and unreactive
Ceramics:
• Compounds between metallic and nonmetallic. Includes oxides, carbides, and
nitrides.
• Useful due to:
-
Being very stiff, strong, and hard
-
Insulative to heat and electricity
Composites:
• A mix between two or more of the previous materials. The goal is to achieve a
combination of properties not available to a single type of material.
What is Materials Science & Engineering?
• Materials Science: the study of the relationship between a material’s structure and
its properties.
• Materials Engineering: designing or engineering the structure of a material to
produce a predetermined set of properties.
As mechanical engineers, we care about PSPP:
• Process
• Structure
• Property
• Performance
Mechanical Properties
How a material responds to an external force is dictated by the material’s mechanical
properties
Properties:
• Mechanical: Relate deformation to an applied load or force (stiffness, hardness,
ductility).
• Thermal: Related to changes in temperature or temperature gradients across a
material (thermal expansion, heat capacity).
• Electrical: Stimulus is an applied electric field (electrical conductivity, dielectric
constant).
• Optical: Stimulus is electromagnetic or light radiation (index of refraction,
reflectivity).
• Magnetic: Response to a magnetic field (magnetic susceptibility, magnetization).
• Deteriorative: Chemical reactivity of materials (corrosion, biocompatibility, wear).
Forces or loads can be applied in one of the following ways:
If a load is static or changes relatively slowly with time and is applied uniformly over a cross
section or surface of a member, we can determine the mechanical properties with a simple
stress-strain test.
Test Procedure: We apply a load and record the displacement until the sample breaks.
Engineering Stress
Engineering Strain
• The material’s internal resistance to
• A material’s relative deformation
forces applied externally.
𝜎=
when a force is applied.
𝐹
𝐴0
Where:
𝜖=
𝛥𝑙
𝑙0
Where:
𝑁
σ – engineering stress [ 2 (Pa) or psi]
ϵ – engineering stress [dimensionless]
F – axial load or force [N or lb]
Δl – change in length
𝐴0 – original cross section [m2 or in2]
𝑙0 – original gauge length
𝑚
• In a Force vs Displacement Test, the material’s geometry matters.
• In a Stress-Strain Test, geometry does not matter, only the material itself matters.
SS (Stress-Strain) Curve
Properties from SS Curve:
• Yield Strength (𝜎𝑦 )
• Ultimate Tensile Strength (𝜎𝑈𝑇𝑆 )
• Modulus of Elasticity (𝐸)
• Ductility (%E)
• Toughness (𝑈𝑡 )
• Resilience (𝑈𝑟 )
Elastic Deformation:
• The material deforms but not permanently.
• The region of elasticity on the SS curve is linear.
-
This follows as 𝑦 = 𝑚𝑥 + 𝑏 → 𝜎 = 𝐸𝜖 + 0 , where E is the Modulus of Elasticity
(Young’s Modulus)
• The limit of this region is the Yield Strength.
Hooke’s Law:
• Stress and Strain are proportional at low stress levels.
𝜎 = 𝐸𝜖
To find the yield strength in an SS graph, we use the 0.2% approach:
• Throw a parallel line with the same E and 𝜎 while adding 0.2% to 𝜖.
• Where this new line intersects the original is the yield strength.
Plastic Deformation and Region:
When the load goes beyond the yield strength, the material is in the plastic regime. Here,
the material deforms permanently.
• Strain Hardening: plastic deformation increases the strength of the material.
𝐹
• Ultimate Tensile Strength: 𝜎𝑈𝑇𝑆 = 𝑚𝑎𝑥
𝐴0
• Necking: Reduction in cross-sectional area. As necking progresses, the material’s
ability to carry load decreases, eventually leading to fracture.
Elastic Strain Recovery:
• If we were to remove the load after the yield point, a fraction of the total deformation
is recovered as Elastic Strain.
• Establishes a new 𝜎𝑦 where the load was removed.
• The resulting E is identical to the original.
Ductility:
• Measure of how much plastic deformation has occurred when a material fractures.
-
It is the measure of the degree of plastic strain that has been sustained at fracture.
• Reported as Percent Elongation (%EL) or Reduction Area (%RA).
%𝐸𝐿 =
(𝑙𝑓 −𝑙0 )
𝑙0
× 100
%𝑅𝐴 =
(𝐴0 −𝐴𝑓 )
𝐴0
× 100
Resilience (𝑈𝑟 )
Toughness (𝑈𝑡 )
• The ability of a material to absorb
• The ability of a material to absorb
energy when elastically deformed
energy before a fracture occurs.
and release it when unloaded.
𝜖
𝑈𝑟 = ∫0 𝑦𝑖𝑒𝑙𝑑 𝜎𝑑𝜖
𝜖𝑓𝑟𝑎𝑐𝑡𝑢𝑟𝑒
𝑈𝑡 = ∫
𝜎𝑑𝜖
0
Resilience:
• In the elastic region, the area under the curve (the integral) can be approximated as
a triangle. → 𝑈𝑟 =
𝜎𝑦2
2𝐸
The birth of Fracture Mechanics occurred during WWII, when the SS Schenectady had a
catastrophic failure in the middle of the night during winter after only 16 days of operation.
DBTT:
• Ductile to brittle transition temperature (DBTT) is the temperature at which a material
transitions from ductile to brittle behavior
-
T < TDBTT: Brittle behavior
-
T > TDBTT: Ductile behavior
True Stress – True Strain:
Because a material stretches and changes under stress, Engineering Stress and
Engineering Strain aren’t as accurate.
True Stress
𝜎𝑇 =
𝐹
𝐴𝑡
True Strain
𝑙
𝜖 𝑇 = ln 𝑖
𝑙0
Corrected plots account for complex stress state within the neck region (since stresses
occur in 3 dimensions)
• 𝜎𝑇 = 𝜎(1 + 𝜖)
• 𝜖 𝑇 = ln(1 + 𝜖)
For some materials, the onset of plastic deformation up to necking can be approximated
by 𝜎𝑇 = 𝐾𝜖 𝑇𝑛
Hardness:
• Hardness is the resistance to localized plastic deformation.
• It is NOT a mechanical property.
Indentation Hardness Testing:
• Nondestructive
• Relatively inexpensive
• Scales:
-
Macroindentation: Load > 200 gf
-
Microindentation: Load < 200 gf
-
Conversion: 1 lbf = 453 gf
• Brinell Test (HB)
-
Steel sphere is pressed against a metal surface for a specified time
-
ASTM E10 Standard Conditions:
➢ Ball Diameter: 10 mm
➢ Load: 3000 kgf
➢ Time: 10 to 15 s
-
Naming Convention:
➢ 360 HB (standard conditions)
➢ 63 HB 10/500/30
▪ Ball Diameter: 10
▪ Load: 500 kgf
▪ Time: 30 s
-
𝐻𝐵 =
𝑃
𝜋𝐷 × 𝑑𝑒𝑝𝑡ℎ
, P → Load
• Rockwell Test (HR)
-
The most popular hardness test
-
No need to measure the size of the indentation optically
-
Preload and Applied Load
-
Scales: A, C, D, N
-
Naming Convention
➢ #-HR-Scale
➢ 64HRC
▪ Result: 64
▪ Scale: C
• Vickers/Knoop (HV)
-
Uses pyramid indenter with a square base made of diamond
-
Microindentation – traditionally smaller loads than Brinell and Rockwell
-
Main advantage is in measuring the hardness of microscopic details in your
sample!
-
Naming Convention:
➢ #-HV-Load
➢ 440HV30
▪ Result: 440
▪ Load: 30 kgf
-
𝐻𝑉 =
1.8544𝑃
𝑑2
The measurement of hardness won’t always be the same for a given material, but the
mechanical properties should always be the same for a given material.
Isotropic vs Anisotropic:
• Isotropic Material: Mechanical properties are the same in all directions.
• Anisotropic Material: Mechanical properties depend on the orientation.
Crystal Structures:
How a material responds to an external force is dictated by the material’s mechanical
properties, which depend on the internal structure of the material.
Atomic Arrangement of Materials:
• Crystalline Material: Regularity in the arrangement of the atoms across the entire
material. Commonly referred to as “single crystal”.
• Polycrystalline Material: Atoms arranged in regular order across small areas.
• Amorphous Material: No long-range atomic order.
Crystal Structure:
• Crystal Structure: Defines the arrangement of atoms in the crystalline or
polycrystalline materials in the solid state.
• Unit Cell: The smallest repeating unit that has the full symmetry of the crystal
structure. We use xyz coordinate system.
• Lattice: Used to define the type of crystal structure.
• Lattice Parameters:
-
Edge Length: a, b, c
-
Interaxial Angles: α, ß, γ
• Simple Cubic: Atoms are located at the vertices of the cube. This is the simplest
atomic arrangement.
-
Polonium (Po) is the only element that crystallizes as simple cubic since this
structure is very inefficient (it has a lot of space without atoms).
-
Lattice Parameters:
➢ a=b=c
➢ α = ß = γ = 90°
• Alternative Locations:
-
Corner → Simple Cubic
-
Face → Face Centered Cubic (FCC)
-
Center → Body Centered Cubic (BCC)
• Bravais Lattices: Combination of lattice parameters and atom locations results in 14
unique structures.
• Seven (7) Crystal Structures:
-
Cubic
-
Tetragonal
-
Orthorhombic
-
Monoclinic
-
Triclinic
-
Rhombohedral
-
Hexagonal
Total # of Atoms
• Total amount of atoms inside the
lattice.
• More atoms mean more ductile.
𝑁 = 𝑁𝑖 +
𝑁𝑓 𝑁𝑐
+
2
8
𝑁𝑖 = # of interior atoms
𝑁𝑓 = # of face atoms
𝑁𝑐 = # of corner atoms
Theoretical Density
𝜌=
𝑛𝐴
𝑉𝐶 𝑁𝐴
𝑛 = number of atoms
𝐴 = atomic weight
𝑁𝐴 = Avogadro’s number
𝑉𝐶 = total unit cell volume
Atomic Packing Factor (APF)
𝐴𝑃𝐹 =
# 𝑎𝑡𝑜𝑚𝑠 × 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑎𝑡𝑜𝑚
𝑡𝑜𝑡𝑎𝑙 𝑢𝑛𝑖𝑡 𝑐𝑒𝑙𝑙 𝑣𝑜𝑙𝑢𝑚𝑒 (𝑉𝐶 )
• APF is the fraction of volume in a crystal structure that is occupied by its elements.
• Higher APF means higher ductility.
BCC:
• Number of atoms 𝑁 = 𝑁𝑖 +
𝑁𝑓
2
+
𝑁𝑐
8
8
=1+0+ =2
8
• Edge Length (a) = 4𝑟⁄
√3
•
3
2(4𝜋𝑟 ⁄3)
𝐴𝑃𝐹 = 4𝑟 3 = 0.68 𝑜𝑟 68%
( ⁄ )
√3
• Common BCC Materials: Chromium, Iron (α), Molybdenum, Tantalum, Tungsten
FCC:
• Number of atoms 𝑁 = 4
• Edge Length (a) = 2𝑟√2
• 𝐴𝑃𝐹 = 0.74 𝑜𝑟 74%
• Common FCC Materials: Aluminum, Copper, Gold, Lead, Nickle, Platinum, Silver
HCP:
• Number of atoms 𝑁 = 6
• Edge Length (a) = 2R
• 𝐴𝑃𝐹 = 0.74 𝑜𝑟 74%
• Common HCP Materials: Cadmium, Cobalt, Titanium (α), Zinc
Crystallographic Points:
• Sometimes we need to specify a location/point within a unit cell.
• Lattice Position Coordinates are associated with the x, y, z axes.
• Point Indices: q, r, s
-
Indices are fractional lengths along their respective axes.
-
Axes are normalized (i.e. max length is 1)
𝑃𝑥 = 𝑞𝑎
𝑃𝑦 = 𝑟𝑏
𝑃𝑧 = 𝑠𝑐
Crystallographic Directions:
• It is a vector.
• Right hand rule for the x-y-z coordinate system.
• Directions [𝑢𝑣𝑤]
• Steps:
1. Identify vector head point
2. Identify vector tail point
3. Subtract points
4. Clear the fraction
5. Values enclosed in brackets: [uvw]
➢ If negative, place a line over the value: [𝑢̅𝑣̅ 𝑤
̅]
𝑢 = 𝑥2 − 𝑥1
𝑣 = 𝑦2 − 𝑦1
𝑤 = 𝑧2 − 𝑧1
• Crystallographically Equivalent: The spacing of atoms along each direction is the
same.
• Family Direction: <𝑢𝑣𝑤>
-
Cubic: [100], [010], [001] are all the same <100>
• Miller Indices (hkl): Used to identify a plane within cell.
• Steps:
1. Identify the coordinate intercepts of the plane.
2. If the plane is parallel to an axis, the intercept is infinity.
3. If the plane passes through the origin, choose a parallel plane.
4. Take the reciprocal of the intercepts.
5. Clear the fractions but do not reduce the lowest integers.
6. Cite plane in parentheses ( ), with bars over negative indices.
• Additional notes:
-
Corchetes {} indicates a family of planes.
-
{100} in a cubic system includes:
➢ (100)
➢ (1̅00)
➢ (010)
➢ (01̅0)
➢ (001)
➢ (001̅)
-
In cubic crystals, a plane is orthogonal to the direction with the same indices (i.e.
the direction is normal to the plane).
• Notation Summary:
-
Points: hkl or (h,k,l)
➢ We use negative signs (-) and do not reduce fractions
-
Specific Direction: [uvw]
-
Family of Directions: <uvw>
-
Specific Plane: (hkl)
-
Family of Planes {hkl}
Linear & Planar Density:
• Linear Density (LD):
-
# 𝑜𝑓 𝑎𝑡𝑜𝑚𝑠 𝑐𝑒𝑛𝑡𝑒𝑟𝑒𝑑 𝑜𝑛 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑣𝑒𝑐𝑡𝑜𝑟
𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑣𝑒𝑐𝑡𝑜𝑟
Units: nm-1
• Planar Density (PD) or Area Density (AD):
-
# 𝑜𝑓 𝑎𝑡𝑜𝑚𝑠 𝑐𝑒𝑛𝑡𝑒𝑟𝑒𝑑 𝑜𝑛 𝑝𝑙𝑎𝑛𝑒
𝑎𝑟𝑒𝑎𝑜𝑓 𝑝𝑙𝑎𝑛𝑒
Units: nm-2
• LD and PD are important relative to the process of slip
-
The mechanism by which metals plastically deform.
-
Slip occurs on the most densely packed crystallographic planes and, in those
planes, along directions having the greatest atomic packing factor.
Atomic Arrangement:
• The atomic arrangement for a crystallographic plane depends on the crystal
structure.
• Stacking Sequence:
-
HCP: ABABABAB…
-
FCC [along (111)]: ABCABCABC…
Crystalline Materials:
• A crystalline material has an uninterrupted pattern.
• These are known as Single Crystal.
• They are difficult to obtain.
• They are very strong.
• Superior creep and thermal fatigue resistance.
Polycrystalline Materials:
• Polycrystalline materials have the same repeated structure with periodic
interruptions.
• Most materials behave this way.
• These individual crystals are known as Grains.
• The spaces in between are known as Grain Boundaries.
• Samples are prepared in the lab to look at with a microscope and produce
micrographs.