Chapter 1. INRODUCTION
1 .1 Historical Perspective
Materials are so important in the development of civilization that we associate Ages
with them. In the origin of human life on Earth, the Stone Age, people used only
natural materials, like stone, clay, skins, and wood. When people found copper and
how to make it harder by alloying, the Bronze Age started about 3000 BC. The use of
iron and steel, a stronger material that gave advantage in wars started at about 1200
BC. The next big step was the discovery of a cheap process to make steel around
1850, which enabled the railroads and the building of the modern infrastructure of the
industrial world.
1.2 Materials Science and Engineering
Understanding of how materials behave like they do, and why they differ in properties
was only possible with the atomistic understanding allowed by quantum mechanics,
that first explained atoms and then solids starting in the 1930s. The combination of
physics, chemistry, and the focus on the relationship between the properties of a
material and its microstructure is the domain of Materials Science. The development
of this science allowed designing materials and provided a knowledge base for the
engineering applications (Materials Engineering).
Structure:
· At the atomic level: arrangement of atoms in different ways. (Gives different
properties for graphite than diamond both forms of carbon.)
· At the microscopic level: arrangement of small grains of material that can be
identified by microscopy. (Gives different optical properties to transparent vs.
frosted glass.)
Properties are the way the material responds to the environment. For instance, the
mechanical, electrical and magnetic properties are the responses to mechanical,
electrical and magnetic forces, respectively. Other important properties are thermal
(transmission of heat, heat capacity), optical (absorption, transmission and scattering
of light), and the chemical stability in contact with the environment (like corrosion
resistance).
Processing of materials is the application of heat (heat treatment), mechanical forces,
etc. to affect their microstructure and, therefore, their properties
1.3 Why Study Materials Science and Engineering?
· To be able to select a material for a given use based on considerations of cost
and performance.
· To understand the limits of materials and the change of their properties with
use.
· To be able to create a new material that will have some desirable properties.
All engineering disciplines need to know about materials. Even the most
"immaterial", like software or system engineering depend on the development of new
materials, which in turn alter the economics, like software-hardware trade-offs.
Increasing applications of system engineering are in materials manufacturing
(industrial engineering) and complex environmental systems.
1.4 Classification of Materials
Like many other things, materials are classified in groups, so that our brain can handle
the complexity. One could classify them according to structure, or properties, or use.
The one that we will use is according to the way the atoms are bound together:
Metals: valence electrons are detached from atoms, and spread in an 'electron sea'
that "glues" the ions together. Metals are usually strong, conduct electricity and heat
well and are opaque to light (shiny if polished). Examples: aluminum, steel, brass,
gold.
Semiconductors: the bonding is covalent (electrons are shared between atoms). Their
electrical properties depend extremely strongly on minute proportions of
contaminants. They are opaque to visible light but transparent to the infrared.
Examples: Si, Ge, GaAs.
Ceramics: atoms behave mostly like either positive or negative ions, and are bound
by Coulomb forces between them. They are usually combinations of metals or
semiconductors with oxygen, nitrogen or carbon (oxides, nitrides, and carbides).
Examples: glass, porcelain, many minerals.
Polymers: are bound by covalent forces and also by weak van der Waals forces, and
usually based on H, C and other non-metallic elements. They decompose at moderate
temperatures (100 – 400 C), and are lightweight. Other properties vary greatly.
Examples: plastics (nylon, Teflon, polyester) and rubber.
Other categories are not based on bonding. A particular microstructure identifies
composites, made of different materials in intimate contact (example: fiberglass,
concrete, wood) to achieve specific properties. Biomaterials can be any type of
material that is biocompatible and used, for instance, to replace human body parts.
1.5 Advanced Materials
Materials used in "High-Tec" applications, usually designed for maximum
performance, and normally expensive. Examples are titanium alloys for supersonic
airplanes, magnetic alloys for computer disks, special ceramics for the heat shield of
the space shuttle, etc.
1.6 Modern Material's Needs
· Engine efficiency increases at high temperatures: requires high temperature
structural materials
· Use of nuclear energy requires solving problem with residues, or advances in
nuclear waste processing.
· Hypersonic flight requires materials that are light, strong and resist high
temperatures.
· Optical communications require optical fibers that absorb light negligibly.
· Civil construction – materials for unbreakable windows.
· Structures: materials that are strong like metals and resist corrosion like
plastics.
Chapter 2. ATOMIC STRUCTURE AND BONDING
2.2 Fundamental Concepts
Atoms are composed of electrons, protons, and neutrons. Electron and protons are
negative and positive charges of the same magnitude, 1.6 × 10-19 Coulombs.
The mass of the electron is negligible with respect to those of the proton and the
neutron, which form the nucleus of the atom. The unit of mass is an atomic mass unit
(amu) = 1.66 × 10-27 kg, and equals 1/12 the mass of a carbon atom. The Carbon
nucleus has Z=6, and A=6, where Z is the number of protons, and A the number of
neutrons. Neutrons and protons have very similar masses, roughly equal to 1 amu. A
neutral atom has the same number of electrons and protons, Z.
A mole is the amount of matter that has a mass in grams equal to the atomic mass in
amu of the atoms. Thus, a mole of carbon has a mass of 12 grams. The number of
atoms in a mole is called the Avogadro number, Nav = 6.023 × 1023. Note that Nav = 1
gram/1 amu.
Calculating n, the number of atoms per cm3 in a piece of material of density d (g/cm3).
n = Nav × d / M
where M is the atomic mass in amu (grams per mol). Thus, for graphite (carbon) with
a density d = 1.8 g/cm3, M =12, we get 6 × 1023 atoms/mol × 1.8 g/cm3 / 12 g/mol) =
9 × 1022 C/cm3.
For a molecular solid like ice, one uses the molecular mass, M(H2O) = 18. With a
density of 1 g/cm3, one obtains n = 3.3 × 1022 H2O/cm3. Note that since the water
molecule contains 3 atoms, this is equivalent to 9.9 × 1022 atoms/cm3.
Most solids have atomic densities around 6 × 1022 atoms/cm3. The cube root of that
number gives the number of atoms per centimeter, about 39 million. The mean
distance between atoms is the inverse of that, or 0.25 nm. This is an important number
that gives the scale of atomic structures in solids.
2.3 Electrons in Atoms
The forces in the atom are repulsions between electrons and attraction between
electrons and protons. The neutrons play no significant role. Thus, Z is what
characterizes the atom.
The electrons form a cloud around the neutron, of radius of 0.05 – 2 nanometers.
Electrons do not move in circular orbits, as in popular drawings, but in 'fuzzy' orbits.
We cannot tell how it moves, but only say what is the probability of finding it at some
distance from the nucleus. According to quantum mechanics, only certain orbits are
allowed (thus, the idea of a mini planetary system is not correct). The orbits are
identified by a principal quantum number n, which can be related to the size, n = 0 is
the smallest; n = 1, 2 .. are larger. (They are "quantized" or discrete, being specified
by integers). The angular momentum l is quantized, and so is the projection in a
specific direction m. The structure of the atom is determined by the Pauli exclusion
principle, only two electrons can be placed in an orbit with a given n, l, m – one for
each spin. Table 2.1 in the textbook gives the number of electrons in each shell (given
by n) and subshells (given by l).
2.4 The Periodic Table
Elements are categorized by placing them in the periodic table. Elements in a column
share similar properties. The noble gases have closed shells, and so they do not gain
or lose electrons near another atom. Alkalis can easily lose an electron and become a
closed shell; halogens can easily gain one to form a negative ion, again with a closed
shell. The propensity to form closed shells occurs in molecules, when they share
electrons to close a molecular shell. Examples are H2, N2, and NaCl.
The ability to gain or lose electrons is termed electronegativity or electropositivity, an
important factor in ionic bonds.
2.5 Bonding Forces and Energies
The Coulomb forces are simple: attractive between electrons and nuclei, repulsive
between electrons and between nuclei. The force between atoms is given by a sum of
all the individual forces, and the fact that the electrons are located outside the atom
and the nucleus in the center.
When two atoms come very close, the force between them is always repulsive,
because the electrons stay outside and the nuclei repel each other. Unless both atoms
are ions of the same charge (e.g., both negative) the forces between atoms is always
attractive at large internuclear distances r. Since the force is repulsive at small r, and
attractive at small r, there is a distance at which the force is zero. This is the
equilibrium distance at which the atoms prefer to stay.
The interaction energy is the potential energy between the atoms. It is negative if the
atoms are bound and positive if they can move away from each other. The interaction
energy is the integral of the force over the separation distance, so these two quantities
are directly related. The interaction energy is a minimum at the equilibrium position.
This value of the energy is called the bond energy, and is the energy needed to
separate completely to infinity (the work that needs to be done to overcome the
attractive force.) The strongest the bond energy, the hardest is to move the atoms, for
instance the hardest it is to melt the solid, or to evaporate its atoms.
2.6 Primary Interatomic Bonds
5
Ionic Bonding
This is the bond when one of the atoms is negative (has an extra electron) and another
is positive (has lost an electron). Then there is a strong, direct Coulomb attraction. An
example is NaCl. In the molecule, there are more electrons around Cl, forming Cl- and
less around Na, forming Na+. Ionic bonds are the strongest bonds. In real solids, ionic
bonding is usually combined with covalent bonding. In this case, the fractional ionic
bonding is defined as %ionic = 100 × [1 – exp(-0.25 (XA – XB)2], where XA and XB
are the electronegativities of the two atoms, A and B, forming the molecule.
Covalent Bonding
In covalent bonding, electrons are shared between the molecules, to saturate the
valency. The simplest example is the H2 molecule, where the electrons spend more
time in between the nuclei than outside, thus producing bonding.
Metallic Bonding
In metals, the atoms are ionized, loosing some electrons from the valence band. Those
electrons form a electron sea, which binds the charged nuclei in place, in a similar
way that the electrons in between the H atoms in the H2 molecule bind the protons.
2.7 Secondary Bonding (Van der Waals)
Fluctuating Induced Dipole Bonds
Since the electrons may be on one side of the atom or the other, a dipole is formed:
the + nucleus at the center, and the electron outside. Since the electron moves, the
dipole fluctuates. This fluctuation in atom A produces a fluctuating electric field that
is felt by the electrons of an adjacent atom, B. Atom B then polarizes so that its outer
electrons are on the side of the atom closest to the + side (or opposite to the – side) of
the dipole in A. This bond is called van der Waals bonding.
Polar Molecule-Induced Dipole Bonds
A polar molecule like H2O (Hs are partially +, O is partially – ), will induce a dipole
in a nearby atom, leading to bonding.
Permanent Dipole Bonds
This is the case of the hydrogen bond in ice. The H end of the molecule is positively
charged and can bond to the negative side of another dipolar molecule, like the O side
of the H2O dipole.
2.8 Molecules
If molecules formed a closed shell due to covalent bonding (like H2, N2) then the
interaction between molecules is weak, of the van der Waals type. Thus, molecular
solids usually have very low melting points
Chapter-3: STRUCTURE OF CRYSTALS
3.2 Fundamental Concepts
Atoms self-organize in crystals, most of the time. The crystalline lattice, is a periodic
array of the atoms. When the solid is not crystalline, it is called amorphous. Examples
of crystalline solids are metals, diamond and other precious stones, ice, graphite.
Examples of amorphous solids are glass, amorphous carbon (a-C), amorphous Si,
most plastics
To discuss crystalline structures it is useful to consider atoms as being hard spheres,
with well-defined radii. In this scheme, the shortest distance between two like atoms
is one diameter.
3.3 Unit Cells
The unit cell is the smallest structure that repeats itself by translation through the
crystal. We construct these symmetrical units with the hard spheres. The most
common types of unit cells are the faced-centered cubic (FCC), the body-centered
cubic (FCC) and the hexagonal close-packed (HCP). Other types exist, particularly
among minerals. The simple cube (SC) is often used for didactical purpose, no
material has this structure.
3.4 Metallic Crystal Structures
Important properties of the unit cells are
· The type of atoms and their radii R.
· cell dimensions (side a in cubic cells, side of base a and height c in HCP) in
terms of R.
· n, number of atoms per unit cell. For an atom that is shared with m adjacent
unit cells, we only count a fraction of the atom, 1/m.
· CN, the coordination number, which is the number of closest neighbors to
which an atom is bonded.
· APF, the atomic packing factor, which is the fraction of the volume of the cell
actually occupied by the hard spheres. APF = Sum of atomic volumes/Volume
of cell.
Unit Cell n CN a/R APF
SC 1 6 2 0.52
BCC 2 8 4Ö 3 0.68
FCC 4 12 2Ö 2 0.74
HCP 6 12 0.74
The closest packed direction in a BCC cell is along the diagonal of the cube; in a FCC
cell is along the diagonal of a face of the cube.
3.5 Density Computations
The density of a solid is that of the unit cell, obtained by dividing the mass of the
atoms (n atoms x Matom) and dividing by Vc the volume of the cell (a3 in the case of a
cube). If the mass of the atom is given in amu (A), then we have to divide it by the
Avogadro number to get Matom. Thus, the formula for the density is:
3.6 Polymorphism and Allotropy
Some materials may exist in more than one crystal structure, this is called
polymorphism. If the material is an elemental solid, it is called allotropy. An example
of allotropy is carbon, which can exist as diamond, graphite, and amorphous carbon.
3.11 Close-Packed Crystal Structures
The FCC and HCP are related, and have the same APF. They are built by packing
spheres on top of each other, in the hollow sites (Fig. 3.12 of book). The packing is
alternate between two types of sites, ABABAB.. in the HCP structure, and alternates
between three types of positions, ABCABC… in the FCC crystals.
Crystalline and Non-Crystalline Materials
3.12 Single Crystals
Crystals can be single crystals where the whole solid is one crystal. Then it has a
regular geometric structure with flat faces.
3.13 Polycrystalline Materials
A solid can be composed of many crystalline grains, not aligned with each other. It is
called polycrystalline. The grains can be more or less aligned with respect to each
other. Where they meet is called a grain boundary.
3.14 Anisotropy
Different directions in the crystal have a different packing. For instance, atoms along
the edge FCC crystals are more separated than along the face diagonal. This causes
anisotropy in the properties of crystals; for instance, the deformation depends on the
direction in which a stress is applied.
3.15 X-Ray Diffraction Determination of Crystalline Structure – not covered
3.16 Non-Crystalline Solids
In amorphous solids, there is no long-range order. But amorphous does not mean
random, since the distance between atoms cannot be smaller than the size of the hard
spheres. Also, in many cases there is some form of short-range order. For instance, the
tetragonal order of crystalline SiO2 (quartz) is still apparent in amorphous SiO2 (silica
glass.)
Chapter-4: IMPERFECTIONS\
Imperfections in Solids
4.1 Introduction
Materials are often stronger when they have defects. The study of defects is divided
according to their dimension:
0D (zero dimension) – point defects: vacancies and interstitials. Impurities.
1D – linear defects: dislocations (edge, screw, mixed)
2D – grain boundaries, surfaces.
3D – extended defects: pores, cracks.
Point Defects
4.2 Vacancies and Self-Interstitials
A vacancy is a lattice position that is vacant because the atom is missing. It is created
when the solid is formed. There are other ways of making a vacancy, but they also
occur naturally as a result of thermal vibrations.
An interstitial is an atom that occupies a place outside the normal lattice position. It
may be the same type of atom as the others (self interstitial) or an impurity atom.
In the case of vacancies and interstitials, there is a change in the coordination of atoms
around the defect. This means that the forces are not balanced in the same way as for
other atoms in the solid, which results in lattice distortion around the defect.
The number of vacancies formed by thermal agitation follows the law:
NV = NA × exp(-QV/kT)
where NA is the total number of atoms in the solid, QV is the energy required to form a
vacancy, k is Boltzmann constant, and T the temperature in Kelvin (note, not in oC or
oF).
When QV is given in joules, k = 1.38 × 10-23 J/atom-K. When using eV as the unit of
energy, k = 8.62 × 10-5 eV/atom-K.
Note that kT(300 K) = 0.025 eV (room temperature) is much smaller than typical
vacancy formation energies. For instance, QV(Cu) = 0.9 eV/atom. This means that
NV/NA at room temperature is exp(-36) = 2.3 × 10-16, an insignificant number. Thus, a
high temperature is needed to have a high thermal concentration of vacancies. Even
so, NV/NA is typically only about 0.0001 at the melting point.
4.3 Impurities in Solids
All real solids are impure. A very high purity material, say 99.9999% pure (called 6N
– six nines) contains ~ 6 × 1016 impurities per cm3.
Impurities are often added to materials to improve the properties. For instance, carbon
added in small amounts to iron makes steel, which is stronger than iron. Boron
impurities added to silicon drastically change its electrical properties.
Solid solutions are made of a host, the solvent or matrix) which dissolves the solute
(minor component). The ability to dissolve is called solubility. Solid solutions are:
· homogeneous
· maintain crystal structure
· contain randomly dispersed impurities (substitutional or interstitial)
Factors for high solubility
· Similar atomic size (to within 15%)
· Similar crystal structure
· Similar electronegativity (otherwise a compound is formed)
· Similar valence
Composition can be expressed in weight percent, useful when making the solution,
and in atomic percent, useful when trying to understand the material at the atomic
level.
Miscellaneous Imperfections
4.4 Dislocations—Linear Defects
Dislocations are abrupt changes in the regular ordering of atoms, along a line
(dislocation line) in the solid. They occur in high density and are very important in
mechanical properties of material. They are characterized by the Burgers vector,
found by doing a loop around the dislocation line and noticing the extra interatomic
spacing needed to close the loop. The Burgers vector in metals points in a close
packed direction.
Edge dislocations occur when an extra plane is inserted. The dislocation line is at the
end of the plane. In an edge dislocation, the Burgers vector is perpendicular to the
dislocation line.
Screw dislocations result when displacing planes relative to each other through shear.
In this case, the Burgers vector is parallel to the dislocation line.
4.5 Interfacial Defects
The environment of an atom at a surface differs from that of an atom in the bulk, in
that the number of neighbors (coordination) decreases. This introduces unbalanced
forces which result in relaxation (the lattice spacing is decreased) or reconstruction
(the crystal structure changes).
The density of atoms in the region including the grain boundary is smaller than the
bulk value, since void space occurs in the interface.
Surfaces and interfaces are very reactive and it is usual that impurities segregate there.
Since energy is required to form a surface, grains tend to grow in size at the expense
of smaller grains to minimize energy. This occurs by diffusion, which is accelerated at
high temperatures.
Twin boundaries: not covered
4.6 Bulk or Volume Defects
A typical volume defect is porosity, often introduced in the solid during processing. A
common example is snow, which is highly porous ice.
4.7 Atomic Vibrations
Atomic vibrations occur, even at zero temperature (a quantum mechanical effect) and
increase in amplitude with temperature. Vibrations displace transiently atoms from
their regular lattice site, which destroys the perfect periodicity we discussed in
Chapter 3.
Chapter-5: DIFUSSION
5.1 Introduction
Many important reactions and processes in materials occur by the motion of atoms in
the solid (transport), which happens by diffusion.
Inhomogeneous materials can become homogeneous by diffusion, if the temperature
is high enough (temperature is needed to overcome energy barriers to atomic motion.
5.2 Diffusion Mechanisms
Atom diffusion can occur by the motion of vacancies (vacancy diffusion) or
impurities (impurity diffusion). The energy barrier is that due to nearby atoms which
need to move to let the atoms go by. This is more easily achieved when the atoms
vibrate strongly, that is, at high temperatures.
There is a difference between diffusion and net diffusion. In a homogeneous material,
atoms also diffuse but this motion is hard to detect. This is because atoms move
randomly and there will be an equal number of atoms moving in one direction than in
another. In inhomogeneous materials, the effect of diffusion is readily seen by a
change in concentration with time. In this case there is a net diffusion. Net diffusion
occurs because, although all atoms are moving randomly, there are more atoms
moving in regions where their concentration is higher.
5.3 Steady-State Diffusion
The flux of diffusing atoms, J, is expressed either in number of atoms per unit area
and per unit time (e.g., atoms/m2-second) or in terms of mass flux (e.g., kg/m2-
second).
Steady state diffusion means that J does not depend on time. In this case, Fick’s first
law holds that the flux along direction x is:
J = – D dC/dx
Where dC/dx is the gradient of the concentration C, and D is the diffusion constant.
The concentration gradient is often called the driving force in diffusion (but it is not a
force in the mechanistic sense). The minus sign in the equation means that diffusion is
down the concentration gradient.
5.4 Nonsteady-State Diffusion
This is the case when the diffusion flux depends on time, which means that a type of
atoms accumulates in a region or that it is depleted from a region (which may cause
them to accumulate in another region).
5.5 Factors That Influence Diffusion
As stated above, there is a barrier to diffusion created by neighboring atoms that need
to move to let the diffusing atom pass. Thus, atomic vibrations created by temperature
assist diffusion. Also, smaller atoms diffuse more readily than big ones, and diffusion
is faster in open lattices or in open directions. Similar to the case of vacancy
formation, the effect of temperature in diffusion is given by a Boltzmann factor: D =
D0 × exp(–Qd/kT).
5.6 Other Diffusion Paths
Diffusion occurs more easily along surfaces, and voids in the material (short circuits
like dislocations and grain boundaries) because less atoms need to move to let the
diffusing atom pass. Short circuits are often unimportant because they constitute a
negligible part of the total area of the material normal to the diffusion flux. .
Chapter-6: Mechanical Properties of Metals
1. Introduction
Often materials are subject to forces (loads) when they are used.
Mechanical engineers calculate those forces and material scientists
how materials deform (elongate, compress, twist) or break as a
function of applied load, time, temperature, and other conditions.
Materials scientists learn about these mechanical properties by testing
materials. Results from the tests depend on the size and shape of
material to be tested (specimen), how it is held, and the way of
performing the test. That is why we use common procedures, or
standards, which are published by the ASTM.
2. Concepts of Stress and Strain
To compare specimens of different sizes, the load is calculated per unit
area, also called normalization to the area. Force divided by area is
called stress. In tension and compression tests, the relevant area is that
perpendicular to the force. In shear or torsion tests, the area is
perpendicular to the axis of rotation.
s = F/A0 tensile or compressive stress
t = F/A0 shear stress
The unit is the Megapascal = 106 Newtons/m2.
There is a change in dimensions, or deformation elongation, DL as a
result of a tensile or compressive stress. To enable comparison with
specimens of different length, the elongation is also normalized, this
time to the length L. This is called strain, e.
e = DL/L
The change in dimensions is the reason we use A0 to indicate the initial
area since it changes during deformation. One could divide force by
the actual area, this is called true stress (see Sec. 6.7).
For torsional or shear stresses, the deformation is the angle of twist, q
(Fig. 6.1) and the shear strain is given by:
g = tg q
3. Stress—Strain Behavior
Elastic deformation. When the stress is removed, the material returns
to the dimension it had before the load was applied. Valid for small
strains (except the case of rubbers).
Deformation is reversible, non permanent
Plastic deformation. When the stress is removed, the material does
not return to its previous dimension but there is a permanent,
irreversible deformation.
In tensile tests, if the deformation is elastic, the stress-strain
relationship is called Hooke's law:
s = E e
That is, E is the slope of the stress-strain curve. E is Young's modulus
or modulus of elasticity. In some cases, the relationship is not linear so
that E can be defined alternatively as the local slope:
E = ds/de
Shear stresses produce strains according to:
t = G g
where G is the shear modulus.
Elastic moduli measure the stiffness of the material. They are related to
the second derivative of the interatomic potential, or the first derivative
of the force vs. internuclear distance (Fig. 6.6). By examining these
curves we can tell which material has a higher modulus. Due to
thermal vibrations the elastic modulus decreases with temperature. E is
large for ceramics (stronger ionic bond) and small for polymers (weak
covalent bond). Since the interatomic distances depend on direction in
the crystal, E depends on direction (i.e., it is anisotropic) for single
crystals. For randomly oriented policrystals, E is isotropic. .
4. Anelasticity
Here the behavior is elastic but not the stress-strain curve is not
immediately reversible. It takes a while for the strain to return to zero.
The effect is normally small for metals but can be significant for
polymers.
5. Elastic Properties of Materials
Materials subject to tension shrink laterally. Those subject to
compression, bulge. The ratio of lateral and axial strains is called the
Poisson's ratio n.
n = elateral/eaxial
The elastic modulus, shear modulus and Poisson's ratio are related by
E = 2G(1+n)
6. Tensile Properties
Yield point. If the stress is too large, the strain deviates from being
proportional to the stress. The point at which this happens is the yield
point because there the material yields, deforming permanently
(plastically).
Yield stress. Hooke's law is not valid beyond the yield point. The
stress at the yield point is called yield stress, and is an important
measure of the mechanical properties of materials. In practice, the
yield stress is chosen as that causing a permanent strain of 0.002 (strain
offset, Fig. 6.9.)
The yield stress measures the resistance to plastic deformation.
The reason for plastic deformation, in normal materials, is not that the
atomic bond is stretched beyond repair, but the motion of dislocations,
which involves breaking and reforming bonds.
Plastic deformation is caused by the motion of dislocations.
Tensile strength. When stress continues in the plastic regime, the
stress-strain passes through a maximum, called the tensile strength
(sTS) , and then falls as the material starts to develop a neck and it
finally breaks at the fracture point (Fig. 6.10).
Note that it is called strength, not stress, but the units are the same,
MPa.
For structural applications, the yield stress is usually a more
important property than the tensile strength, since once the it is
passed, the structure has deformed beyond acceptable limits.
Ductility. The ability to deform before braking. It is the opposite of
brittleness. Ductility can be given either as percent maximum
elongation emax or maximum area reduction.
%EL = emax x 100 %
%AR = (A0 - Af)/A0
These are measured after fracture (repositioning the two pieces back
together).
Resilience. Capacity to absorb energy elastically. The energy per unit
volume is the
area under the strain-stress curve in the elastic region.
Toughness. Ability to absorb energy up to fracture. The energy per
unit volume is the total area under the strain-stress curve. It is
measured by an impact test (Ch. 8).
7. True Stress and Strain
When one applies a constant tensile force the material will break after
reaching the tensile strength. The material starts necking (the
transverse area decreases) but the stress cannot increase beyond sTS.
The ratio of the force to the initial area, what we normally do, is called
the engineering stress. If the ratio is to the actual area (that changes
with stress) one obtains the true stress.
8. Elastic Recovery During Plastic Deformation
If a material is taken beyond the yield point (it is deformed plastically)
and the stress is then released, the material ends up with a permanent
strain. If the stress is reapplied, the material again responds elastically
at the beginning up to a new yield point that is higher than the original
yield point (strain hardening, Ch. 7.10). The amount of elastic strain
that it will take before reaching the yield point is called elastic strain
recovery (Fig. 6. 16).
9. Compressive, Shear, and Torsional Deformation
Compressive and shear stresses give similar behavior to tensile
stresses, but in the case of compressive stresses there is no maximum
in the s-e curve, since no necking occurs.
10. Hardness
Hardness is the resistance to plastic deformation (e.g., a local dent or
scratch). Thus, it is a measure of plastic deformation, as is the tensile
strength, so they are well correlated. Historically, it was measured on
an empirically scale, determined by the ability of a material to scratch
another, diamond being the hardest and talc the softer. Now we use
standard tests, where a ball, or point is pressed into a material and the
size of the dent is measured. There are a few different hardness tests:
Rockwell, Brinell, Vickers, etc. They are popular because they are easy
and non-destructive (except for the small dent).
11. Variability of Material Properties
Tests do not produce exactly the same result because of variations in
the test equipment, procedures, operator bias, specimen fabrication,
etc. But, even if all those parameters are controlled within strict limits,
a variation remains in the materials, due to uncontrolled variations
during fabrication, non homogenous composition and structure, etc.
The measured mechanical properties will show scatter, which is often
distributed in a Gaussian curve (bell-shaped), that is characterized by
the mean value and the standard deviation (width).
12. Design/Safety Factors
To take into account variability of properties, designers use, instead of
an average value of, say, the tensile strength, the probability that the
yield strength is above the minimum value tolerable. This leads to the
use of a safety factor N > 1 (typ. 1.2 - 4). Thus, a working value for the
tensile strength would be sW = sTS / N.
Not tested: true stress-true stain relationships, details of the different types of
hardness tests, but should know that hardness for a given material correlates with
tensile strength. Variability of material properties
Chapter 7. DISLOCATIONS AND STRENGTHENING MECHANISM
1. Introduction
The key idea of the chapter is that plastic deformation is due to the
motion of a large number of dislocations. The motion is called slip.
Thus, the strength (resistance to deformation) can be improved by
putting obstacles to slip.
2. Basic Concepts
Dislocations can be edge dislocations, screw dislocations and exist in
combination of the two (Ch. 4.4). Their motion (slip) occurs by
sequential bond breaking and bond reforming (Fig. 7.1). The number
of dislocations per unit volume is the dislocation density, in a plane
they are measured per unit area.
3. Characteristics of Dislocations
There is strain around a dislocation which influences how they interact
with other dislocations, impurities, etc. There is compression near the
extra plane (higher atomic density) and tension following the
dislocation line (Fig. 7.4)
Dislocations interact among themselves (Fig. 7.5). When they are in
the same plane, they repel if they have the same sign and annihilate if
they have opposite signs (leaving behind a perfect crystal). In general,
when dislocations are close and their strain fields add to a larger value,
they repel, because being close increases the potential energy (it takes
energy to strain a region of the material).
The number of dislocations increases dramatically during plastic
deformation. Dislocations spawn from existing dislocations, and from
defects, grain boundaries and surface irregularities.
4. Slip Systems
In single crystals there are preferred planes where dislocations move
(slip planes). There they do not move in any direction, but in preferred
crystallographic directions (slip direction). The set of slip planes and
directions constitute slip systems.
The slip planes are those of highest packing density. How do we
explain this? Since the distance between atoms is shorter than the
average, the distance perpendicular to the plane has to be longer than
average. Being relatively far apart, the atoms can move more easily
with respect to the atoms of the adjacent plane. (We did not discuss
direction and plane nomenclature for slip systems.)
BCC and FCC crystals have more slip systems, that is more ways for
dislocation to propagate. Thus, those crystals are more ductile than
HCP crystals (HCP crystals are more brittle).
5. Slip in Single Crystals
A tensile stress s will have components in any plane that is not perpendicular to the
stress. These components are resolved shear stresses. Their magnitude depends on
orientation (see Fig. 7.7).
tR = s cos f cos l
If the shear stress reaches the critical resolved shear stress tCRSS, slip (plastic
deformation) can start. The stress needed is:
sy = tCRSS / (cos f cos l)max
at the angles at which tCRSS is a maximum. The minimum stress needed for yielding is
when f = l = 45 degrees: sy = 2tCRSS. Thus, dislocations will occur first at slip planes
oriented close to this angle with respect to the applied stress (Figs. 7.8 and 7.9).
6. Plastic Deformation of Polycrystalline Materials
Slip directions vary from crystal to crystal. When plastic deformation
occurs in a grain, it will be constrained by its neighbors which may be
less favorably oriented. As a result, polycrystalline metals are stronger
than single crystals (the exception is the perfect single crystal, as in
whiskers.)
7. Deformation by Twinning
This topic is not included.
Mechanisms of Strengthening in Metals
General principles. Ability to deform plastically depends on ability of
dislocations to move. Strengthening consists in hindering dislocation
motion. We discuss the methods of grain-size reduction, solid-solution
alloying and strain hardening. These are for single-phase metals. We
discuss others when treating alloys. Ordinarily, strengthening reduces
ductility.
8. Strengthening by Grain Size Reduction
This is based on the fact that it is difficult for a dislocation to pass into
another grain, especially if it is very misaligned. Atomic disorder at the
boundary causes discontinuity in slip planes. For high-angle grain
boundaries, stress at end of slip plane may trigger new dislocations in
adjacent grains. Small angle grain boundaries are not effective in
blocking dislocations.
The finer the grains, the larger the area of grain boundaries that
impedes dislocation motion. Grain-size reduction usually improves
toughness as well. Usually, the yield strength varies with grain size d
according to:
sy = s0 + ky / d1/2
Grain size can be controlled by the rate of solidification and by plastic
deformation.
9. Solid-Solution Strengthening
Adding another element that goes into interstitial or substitutional
positions in a solution increases strength. The impurity atoms cause
lattice strain (Figs. 7.17 and 7.18) which can "anchor" dislocations.
This occurs when the strain caused by the alloying element
compensates that of the dislocation, thus achieving a state of low
potential energy. It costs strain energy for the dislocation to move
away from this state (which is like a potential well). The scarcity of
energy at low temperatures is why slip is hindered.
Pure metals are almost always softer than their alloys.
10. Strain Hardening
Ductile metals become stronger when they are deformed plastically at
temperatures well below the melting point (cold working). (This is
different from hot working is the shaping of materials at high
temperatures where large deformation is possible.) Strain hardening
(work hardening) is the reason for the elastic recovery discussed in Ch.
6.8.
The reason for strain hardening is that the dislocation density increases
with plastic deformation (cold work) due to multiplication. The
average distance between dislocations then decreases and dislocations
start blocking the motion of each one.
The measure of strain hardening is the percent cold work (%CW),
given by the relative reduction of the original area, A0 to the final value
Ad :
%CW = 100 (A0–Ad)/A0
Recovery, recrystallization and Grain Growth
Plastic deformation causes 1) change in grain size, 2) strain hardening,
3) increase in the dislocation density. Restoration to the state before
cold-work is done by heating through two processes: recovery and
recrystallization. These may be followed by grain growth.
11. Recovery
Heating à increased diffusion à enhanced dislocation motion à relieves
internal strain energy and reduces the number of dislocation. The
electrical and thermal conductivity are restored to the values existing
before cold working.
12. Recrystallization
Strained grains of cold-worked metal are replaced, upon heating, by
more regularly-spaced grains. This occurs through short-range
diffusion enabled by the high temperature. Since recrystallization
occurs by diffusion, the important parameters are both temperature and
time.
The material becomes softer, weaker, but more ductile (Fig. 7.22).
Recrystallization temperature: is that at which the process is
complete in one hour. It is typically 1/3 to 1/2 of the melting
temperature. It falls as the %CW is increased. Below a "critical
deformation", recrystallization does not occur.
13. Grain Growth
The growth of grain size with temperature can occur in all polycrystalline materials. It
occurs by migration of atoms at grain boundaries by diffusion, thus grain growth is
faster at higher temperatures. The "driving force" is the reduction of energy, which is
proportional to the total area. Big grains grow at the expense of the small ones.
Chapter 8. FAILURE
1. Introduction
Failure of materials may have huge costs. Causes included improper
materials selection or processing, the improper design of components,
and improper use.
2. Fundamentals of Fracture
Fracture is a form of failure where the material separates in pieces due to stress, at
temperatures below the melting point. The fracture is termed ductile or brittle
depending on whether the elongation is large or small.
Steps in fracture (response to stress):
· track formation
· track propagation
Ductile vs. brittle fracture
Ductile Brittle
deformation extensive little
track propagation slow, needs stress fast
type of materials most metals (not too cold) ceramics, ice, cold metals
warning permanent elongation none
strain energy higher lower
fractured surface rough smoother
necking yes no
· Ductile Fracture
Stages of ductile fracture
· Initial necking
· small cavity formation (microvoids)
· void growth (elipsoid) by coalescence into a crack
· fast crack propagation around neck. Shear strain at 45o
· final shear fracture (cup and cone)
The interior surface is fibrous, irregular, which signify plastic deformation.
· Brittle Fracture
There is no appreciable deformation, and crack propagation is very fast. In most
brittle materials, crack propagation (by bond breaking) is along specific
crystallographic planes (cleavage planes). This type of fracture is transgranular
(through grains) producing grainy texture (or faceted texture) when cleavage direction
changes from grain to grain. In some materials, fracture is intergranular.
5. Principles of Fracture Mechanics
Fracture occurs due to stress concentration at flaws, like surface
scratches, voids, etc. If a is the length of the void and r the radius of
curvature, the enhanced stress near the flaw is:
sm » 2 s0 (a/r)1/2
where s0 is the applied macroscopic stress. Note that a is 1/2 the length
of the flaw, not the full length for an internal flaw, but the full length
for a surface flaw. The stress concentration factor is:
Kt = sm/s0 » 2 (a/r)1/2
Because of this enhancement, flaws with small radius of curvature are
called stress raisers.
6. Impact Fracture Testing
Normalized tests, like the Charpy and Izod tests measure the impact energy required
to fracture a notched specimen with a hammer mounted on a pendulum. The energy is
measured by the change in potential energy (height) of the pendulum. This energy is
called notch toughness.
Ductile to brittle transition occurs in materials when the temperature is dropped
below a transition temperature. Alloying usually increases the ductile-brittle
transition temperature (Fig. 8.19.) For ceramics, this type of transition occurs at much
higher temperatures than for metals.
Fatigue
Fatigue is the catastrophic failure due to dynamic (fluctuating) stresses. It can happen
in bridges, airplanes, machine components, etc. The characteristics are:
· long period of cyclic strain
· the most usual (90%) of metallic failures (happens also in ceramics and
polymers)
· is brittle-like even in ductile metals, with little plastic deformation
· it occurs in stages involving the initiation and propagation of cracks.
· Cyclic Stresses
These are characterized by maximum, minimum and mean stress, the
stress amplitude, and the stress ratio (Fig. 8.20).
· The S—N Curve
S—N curves (stress-number of cycles to failure) are obtained using
apparatus like the one shown in Fig. 8.21. Different types of S—N
curves are shown in Fig. 8.22.
Fatigue limit (endurance limit) occurs for some materials (like some
ferrous and Ti allows). In this case, the S—N curve becomes horizontal
at large N . This means that there is a maximum stress amplitude (the
fatigue limit) below which the material never fails, no matter how large
the number of cycles is.
For other materials (e.g., non-ferrous) the S—N curve continues to fall
with N.
Failure by fatigue shows substantial variability (Fig. 8.23).
Failure at low loads is in the elastic strain regime, requires a large
number of cycles (typ. 104 to 105). At high loads (plastic regime), one
has low-cycle fatigue (N < 104 - 105 cycles).
· Crack Initiation and Propagation
Stages is fatigue failure:
I. crack initiation at high stress points (stress raisers)
II. propagation (incremental in each cycle)
III. final failure by fracture
Nfinal = Ninitiation + Npropagation
Stage I - propagation
· slow
· along crystallographic planes of high shear stress
· flat and featureless fatigue surface
Stage II - propagation
crack propagates by repetive plastic blunting and sharpening of the crack tip. (Fig.
8.25.)
· . Crack Propagation Rate (not covered)
· . Factors That Affect Fatigue Life
· Mean stress (lower fatigue life with increasing smean).
· Surface defects (scratches, sharp transitions and edges). Solution:
· polish to remove machining flaws
· add residual compressive stress (e.g., by shot peening.)
· case harden, by carburizing, nitriding (exposing to appropriate gas at high
temperature)
· . Environmental Effects
· Thermal cycling causes expansion and contraction, hence thermal stress, if
component is restrained. Solution:
o eliminate restraint by design
o use materials with low thermal expansion coefficients.
· Corrosion fatigue. Chemical reactions induced pits which act as stress raisers.
Corrosion also enhances crack propagation. Solutions:
o decrease corrosiveness of medium, if possible.
o add protective surface coating.
o add residual compressive stresses.
Creep
Creep is the time-varying plastic deformation of a material stressed at high
temperatures. Examples: turbine blades, steam generators. Keys are the time
dependence of the strain and the high temperature.
· . Generalized Creep Behavior
At a constant stress, the strain increases initially fast with time (primary or transient
deformation), then increases more slowly in the secondary region at a steady rate
(creep rate). Finally the strain increases fast and leads to failure in the tertiary region.
Characteristics:
· Creep rate: de/dt
· Time to failure.
· . Stress and Temperature Effects
Creep becomes more pronounced at higher temperatures (Fig. 8.37).
There is essentially no creep at temperatures below 40% of the melting
point.
Creep increases at higher applied stresses.
The behavior can be characterized by the following expression, where
K, n and Qc are constants for a given material:
de/dt = K sn exp(-Qc/RT)
· . Data Extrapolation Methods (not covered.)
· . Alloys for High-Temperature Use
These are needed for turbines in jet engines, hypersonic airplanes, nuclear reactors,
etc. The important factors are a high melting temperature, a high elastic modulus and
large grain size (the latter is opposite to what is desirable in low-temperature
materials).
Some creep resistant materials are stainless steels, refractory metal alloys (containing
elements of high melting point, like Nb, Mo, W, Ta), and superalloys (based on Co,
Ni, Fe.)
Chapter-9: PHASE DIAGRAMS
9.1 Introduction
Definitions
Component: pure metal or compound (e.g., Cu, Zn in Cu-Zn alloy, sugar,
water, in a syrup.)
Solvent: host or major component in solution.
Solute: dissolved, minor component in solution.
System: set of possible alloys from same component (e.g., iron-carbon
system.)
Solubility Limit: Maximum solute concentration that can be dissolved at a
given temperature.
Phase: part with homogeneous physical and chemical characteristics
9.2 Solubility Limit
Effect of temperature on solubility limit. Maximum content: saturation.
Exceeding maximum content (like when cooling) leads to precipitation.
9.3 Phases
One-phase systems are homogeneous. Systems with two or more phases are
heterogeneous, or mixtures. This is the case of most metallic alloys, but also
happens in ceramics and polymers.
A two-component alloy is called binary. One with three components, ternary.
9.4 Microstructure
The properties of an alloy do not depend only on concentration of the phases but how
they are arranged structurally at the microscopy level. Thus, the microstructure is
specified by the number of phases, their proportions, and their arrangement in space.
A binary alloy may be
a. a single solid solution
b. two separated, essentially pure components.
c. two separated solid solutions.
d. a chemical compound, together with a solid solution.
The way to tell is to cut the material, polish it to a mirror finish, etch it a weak acid
(components etch at a different rate) and observe the surface under a microscope.
9.5 Phase Equilibria
Equilibrium is the state of minimum energy. It is achieved given sufficient
time. But the time to achieve equilibrium may be so long (the kinetics is so
slow) that a state that is not at an energy minimum may have a long life and
appear to be stable. This is called a metastable state.
A less strict, operational, definition of equilibrium is that of a system that does
not change with time during observation.
Equilibrium Phase Diagrams
Give the relationship of composition of a solution as a function of
temperatures and the quantities of phases in equilibrium. These diagrams do
not indicate the dynamics when one phase transforms into another. Sometimes
diagrams are given with pressure as one of the variables. In the phase
diagrams we will discuss, pressure is assumed to be constant at one
atmosphere.
9.6 Binary Isomorphous Systems
This very simple case is one complete liquid and solid solubility, an isomorphous
system. The example is the Cu-Ni alloy of Fig. 9.2a. The complete solubility occurs
because both Cu and Ni have the same crystal structure (FCC), near the same radii,
electronegativity and valence.
The liquidus line separates the liquid phase from solid or solid + liquid phases. That
is, the solution is liquid above the liquidus line.
The solidus line is that below which the solution is completely solid (does not contain
a liquid phase.)
Interpretation of phase diagrams
Concentrations: Tie-line method
a. locate composition and temperature in diagram
b. In two phase region draw tie line or isotherm
c. note intersection with phase boundaries. Read compositions.
Fractions: lever rule
a. construct tie line (isotherm)
b. obtain ratios of line segments lengths.
Note: the fractions are inversely proportional to the length to the boundary for the
particular phase. If the point in the diagram is close to the phase line, the fraction of
that phase is large.
Development of microstructure in isomorphous alloys
a) Equilibrium cooling
Solidification in the solid + liquid phase occurs gradually upon cooling from the
liquidus line. The composition of the solid and the liquid change gradually during
cooling (as can be determined by the tie-line method.) Nuclei of the solid phase form
and they grow to consume all the liquid at the solidus line.
b) Non-equilibrium cooling
Solidification in the solid + liquid phase also occurs gradually. The composition of
the liquid phase evolves by diffusion, following the equilibrium values that can be
derived from the tie-line method. However, diffusion in the solid state is very slow.
Hence, the new layers that solidify on top of the grains have the equilibrium
composition at that temperature but once they are solid their composition does not
change. This lead to the formation of layered (cored) grains (Fig. 9.14) and to the
invalidity of the tie-line method to determine the composition of the solid phase (it
still works for the liquid phase, where diffusion is fast.)
9.7 Binary Eutectic Systems
Interpretation: Obtain phases present, concentration of phases and their fraction (%).
Solvus line: limit of solubility
Eutectic or invariant point. Liquid and two solid phases exist in equilibrium at the
eutectic composition and the eutectic temperature.
Note:
· the melting point of the eutectic alloy is lower than that of the components
(eutectic = easy to melt in Greek).
· At most two phases can be in equilibrium within a phase field.
· Single-phase regions are separated by 2-phase regions.
Development of microstructure in eutectic alloys
Case of lead-tin alloys, figures 9.9–9.14. A layered, eutectic structure develops when
cooling below the eutectic temperature. Alloys which are to the left of the eutectic
concentration (hipoeutectic) or to the right (hypereutectic) form a proeutectic phase
before reaching the eutectic temperature, while in the solid + liquid region. The
eutectic structure then adds when the remaining liquid is solidified when cooling
further. The eutectic microstructure is lamellar (layered) due to the reduced diffusion
distances in the solid state.
To obtain the concentration of the eutectic microstructure in the final solid solution,
one draws a vertical line at the eutectic concentration and applies the lever rule
treating the eutectic as a separate phase (Fig. 9.16).
9.8 Equilibrium Diagrams Having Intermediate Phases or Compounds
A terminal phase or terminal solution is one that exists in the extremes of
concentration (0 and 100%) of the phase diagram. One that exists in the
middle, separated from the extremes, is called an intermediate phase or solid
solution.
An important phase is the intermetallic compound, that has a precise chemical
compositions. When using the lever rules, intermetallic compounds are treated
like any other phase, except they appear not as a wide region but as a vertical
line.
9.9 Eutectoid and Peritectic Reactions
The eutectoid (eutectic-like) reaction is similar to the eutectic reaction but
occurs from one solid phase to two new solid phases. It also shows as V on top
of a horizontal line in the phase diagram. There are associated eutectoid
temperature (or temperature), eutectoid phase, eutectoid and proeutectoid
microstructures.
Solid Phase 1 à Solid Phase 2 + Solid Phase 3
The peritectic reaction also involves three solid in equilibrium, the transition is
from a solid + liquid phase to a different solid phase when cooling. The
inverse reaction occurs when heating.
Solid Phase 1 + liquid à Solid Phase 2
9.10 Congruent Phase Transformations
Another classification scheme. Congruent transformation is one where there is
no change in composition, like allotropic transformations (e.g., a-Fe to g-Fe)
or melting transitions in pure solids.
9.13 The Iron–Iron Carbide (Fe–Fe3C) Phase Diagram
This is one of the most important alloys for structural applications. The diagram Fe—
C is simplified at low carbon concentrations by assuming it is the Fe—Fe3C diagram.
Concentrations are usually given in weight percent. The possible phases are:
· a-ferrite (BCC) Fe-C solution
· g-austenite (FCC) Fe-C solution
· d-ferrite (BCC) Fe-C solution
· liquid Fe-C solution
· Fe3C (iron carbide) or cementite. An intermetallic compound.
The maximum solubility of C in a- ferrite is 0.022 wt%. d-ferrite is only stable at high
temperatures. It is not important in practice. Austenite has a maximum C
concentration of 2.14 wt %. It is not stable below the eutectic temperature (727 C)
unless cooled rapidly (Chapter 10). Cementite is in reality metastable, decomposing
into a-Fe and C when heated for several years between 650 and 770 C.
For their role in mechanical properties of the alloy, it is important to note that:
Ferrite is soft and ductile
Cementite is hard and brittle
Thus, combining these two phases in solution an alloy can be obtained with
intermediate properties. (Mechanical properties also depend on the microstructure,
that is, how ferrite and cementite are mixed.)
9.14 Development of Microstructures in Iron—Carbon Alloys
The eutectoid composition of austenite is 0.76 wt %. When it cools slowly it
forms perlite, a lamellar or layered structure of two phases: a-ferrite and
cementite (Fe3C).
Hypoeutectoid alloys contain proeutectoid ferrite plus the eutectoid perlite.
Hypereutectoid alloys contain proeutectoid cementite plus perlite.
Since reactions below the eutectoid temperature are in the solid phase, the
equilibrium is not achieved by usual cooling from austenite. The new
microstructures that form are discussed in Ch. 10.
9.15 The Influence of Other Alloying Elements
As mentioned in section 7.9, alloying strengthens metals by hindering the motion of
dislocations. Thus, the strength of Fe–C alloys increase with C content and also with
the addition of other elements.
Chapter-10: Phase Transformations in Metals
10.1 Introduction
The goal is to obtain specific microstructures that will improve the mechanical
properties of a metal, in addition to grain-size refinement, solid-solution
strengthening, and strain-hardening.
10.2 Basic Concepts
Phase transformations that involve a change in the microstructure can occur through:
· Diffusion
· Maintaining the type and number of phases (e.g., solidification of a pure
metal, allotropic transformation, recrystallization, grain growth.
· Alteration of phase composition (e.g., eutectoid reactions, see 10.5)
· Diffusionless
· Production of metastable phases (e.g., martensitic transformation, see 10.5)
10.3 The Kinetics of Solid-State Reactions
Change in composition implies atomic rearrangement, which requires diffusion.
Atoms are displaced by random walk. The displacement of a given atom, d, is not
linear in time t (as would be for a straight trajectory) but is proportional to the square
root of time, due to the tortuous path: d = c(Dt) 1/2 where c is a constant and D the
diffusion constant. This time-dependence of the rate at which the reaction (phase
transformation) occurs is what is meant by the term reaction kinetics.
D is called a constant because it does not depend on time, but it depends on
temperature as we have seen in Ch. 5. Diffusion occurs faster at high temperatures.
Phase transformation requires two processes: nucleation and growth. Nucleation
involves the formation of very small particles, or nuclei (e.g., grain boundaries,
defects). This is similar to rain happening when water molecules condensed around
dust particles. During growth, the nuclei grow in size at the expense of the
surrounding material.
The kinetic behavior often has the S-shape form of Fig. 10.1, when plotting percent of
material transformed vs. the logarithm of time. The nucleation phase is seen as an
incubation period, where nothing seems to happen. Usually the transformation rate
has the form r = A e-Q/RT (similar to the temperature dependence of the diffusion
constant), in which case it is said to be thermally activated.
10.4 Multiphase Transformations
To describe phase transformations that occur during cooling, equilibrium phase
diagrams are inadequate if the transformation rate is slow compared to the cooling
rate. This is usually the case in practice, so that equilibrium microstructures are
seldom obtained. This means that the transformations are delayed (e.g., case of
supercooling), and metastable states are formed. We then need to know the effect of
time on phase transformations.
Microstructural and Property Changes in Fe-C Alloys
10.5 Isothermal Transformation Diagrams
We use as an example the cooling of an eutectoid alloy (0.76 wt% C) from the
austenite (g- phase) to pearlite, that contains ferrite (a) plus cementite (Fe3C or iron
carbide). When cooling proceeds below the eutectoid temperature (727 oC) nucleation
of pearlite starts. The S-shaped curves (fraction of pearlite vs. log. time, fig. 10.3) are
displaced to longer times at higher temperatures showing that the transformation is
dominated by nucleation (the nucleation period is longer at higher temperatures) and
not by diffusion (which occurs faster at higher temperatures).
The family of S-shaped curves at different temperatures can be used to construct the
TTT (Time-Temperature-Transformation) diagrams (e.g., fig. 10.4.) For these
diagrams to apply, one needs to cool the material quickly to a given temperature To
before the transformation occurs, and keep it at that temperature over time. The
horizontal line that indicates constant temperature To intercepts the TTT curves on the
left (beginning of the transformation) and the right (end of the transformation); thus
one can read from the diagrams when the transformation occurs. The formation of
pearlite shown in fig. 10.4 also indicates that the transformation occurs sooner at low
temperatures, which is an indication that it is controlled by the rate of nucleation. At
low temperatures, nucleation occurs fast and grain growth is reduced (since it occurs
by diffusion, which is hindered at low temperatures). This reduced grain growth leads
to fine-grained microstructure (fine pearlite). At higher temperatures, diffusion allows
for larger grain growth, thus leading to coarse pearlite.
At lower temperatures nucleation starts to become slower, and a new phase is formed,
bainite. Since diffusion is low at low temperatures, this phase has a very fine
(microscopic) microstructure.
Spheroidite is a coarse phase that forms at temperatures close to the eutectoid
temperature. The relatively high temperatures caused a slow nucleation but enhances
the growth of the nuclei leading to large grains.
A very important structure is martensite, which forms when cooling austenite very
fast (quenching) to below a maximum temperature that is required for the
transformation. It forms nearly instantaneously when the required low temperature is
reached; since no thermal activation is needed, this is called an athermal
transformation. Martensite is a different phase, a body-centered tetragonal (BCT)
structure with interstitial C atoms. Martensite is metastable and decomposes into
ferrite and pearlite but this is extremely slow (and not noticeable) at room
temperature.
In the examples, we used an eutectoid composition. For hypo- and hypereutectoid
alloys, the analysis is the same, but the proeutectoid phase that forms before cooling
through the eutectoid temperature is also part of the final microstructure.
10.6 Continuous Cooling Transformation Diagrams - not covered
10.7 Mechanical Behavior of Fe-C Alloys
The strength and hardness of the different microstructures is inversely related to the
size of the microstructures. Thus, spheroidite is softest, fine pearlite is stronger than
coarse pearlite, bainite is stronger than pearlite and martensite is the strongest of all.
The stronger and harder the phase the more brittle it becomes.
10.8 Tempered Martensite
Martensite is so brittle that it needs to be modified in many practical cases. This is
done by heating it to 250-650 oC for some time (tempering) which produces tempered
martensite, an extremely fine-grained and well dispersed cementite grains in a ferrite
matrix.
Chapter 11. Thermal Processing of Metal Alloys
Annealing Processes
11.1 Introduction
Annealing is a heat treatment where the material is taken to a high temperature, kept
there for some time and then cooled. High temperatures allow diffusion processes to
occur fast. The time at the high temperature (soaking time) is long enough to allow
the desired transformation to occur. Cooling is done slowly to avoid the distortion
(warping) of the metal piece, or even cracking, caused by stresses induced by
differential contraction due to thermal inhomogeneities. Benefits of annealing are:
· relieve stresses
· increase softness, ductility and toughness
· produce a specific microstructure
11.2 Process Annealing
Deforming a piece that has been strengthened by cold working requires a lot of
energy. Reverting the effect of cold work by process annealing eases further
deformation. Heating allows recovery and recrystallization but is usually limited to
avoid excessive grain growth and oxidation.
11.3 Stress Relief
Stresses resulting from machining operations of non-uniform cooling can be
eliminated by stress relief annealing at moderately low temperatures, such that the
effect of cold working and other heat treatments is maintained.
11.4 Annealing of Ferrous Alloys
Normalizing (or austenitizing) consists in taking the Fe-C alloy to the austenitic phase
which makes the grain size more uniform, followed by cooling in air.
Full anneal involves taking hypoeutectoid alloys to the austenite phase and
hypereutectoid alloys over the eutectoid temperature (Fig. 11.1) to soften pieces
which have been hardened by plastic deformation, and which need to be machined.
Spheroidizing consists in prolongued heating just below the eutectoid temperature,
which results in the soft spheroidite structure discussed in Sect. 10.5. This achieves
maximum softness that minimizes the energy needed in subsequent forming
operations.
Heat Treatment of Steels
1.5 Hardenability
To achieve a full conversion of austenite into hard martensite, cooling needs to be fast
enough to avoid partial conversion into perlite or bainite. If the piece is thick, the
interior may cool too slowly so that full martensitic conversion is not achieved. Thus,
the martensitic content, and the hardness, will drop from a high value at the surface to
a lower value in the interior of the piece. Hardenability is the ability of the material to
be hardened by forming martensite.
Hardenability is measured by the Jominy end-quench test (Fig. 11.2). Hardenability
is then given as the dependence of hardness on distance from the quenched end. High
hardenability means that the hardness curve is relatively flat.
11.6 Influence of Quenching Medium, Specimen Size, and Geometry
The cooling rate depends on the cooling medium. Cooling is fastest using water, then
oil, and then air. Fast cooling brings the danger of warping and formation of cracks,
since it is usually accompanied by large thermal gradients.
The shape and size of the piece, together with the heat capacity and heat conductivity
are important in determining the cooling rate for different parts of the metal piece.
Heat capacity is the energy content of a heated mass, which needs to be removed for
cooling. Heat conductivity measures how fast this energy is transported to the colder
regions of the piece.
Precipitation Hardening
Hardening can be enhanced by extremely small precipitates that hinder dislocation
motion. The precipitates form when the solubility limit is exceeded. Precipitation
hardening is also called age hardening because it involves the hardening of the
material over a prolonged time.
11.7 Heat Treatments
Precipitation hardening is achieved by:
a) solution heat treatment where all the solute atoms are dissolved to form a singlephase
solution.
b) rapid cooling across the solvus line to exceed the solubility limit. This leads to a
supersaturated solid solution that remains stable (metastable) due to the low
temperatures, which prevent diffusion.
c) precipitation heat treatment where the supersaturated solution is heated to an
intermediate temperature to induce precipitation and kept there for some time (aging).
If the process is continued for a very long time, eventually the hardness decreases.
This is called overaging.
The requirements for precipitation hardening are:
· appreciable maximum solubility
· solubility curve that falls fast with temperature composition of the alloy that is
less than the maximum solubility
11.8 Mechanism of Hardening
Strengthening involves the formation of a large number of microscopic nuclei, called
zones. It is accelerated at high temperatures. Hardening occurs because the
deformation of the lattice around the precipitates hinder slip. Aging that occurs at
room temperature is called natural aging, to distinguish from the artificial aging
caused by premeditated heating.
11.9 Miscellaneous Considerations
Since forming, machining, etc. uses more energy when the material is hard, the steps
in the processing of alloys are usually:
· solution heat treat and quench
· do needed cold working before hardening
· do precipitation hardening
Exposure of precipitation-hardened alloys to high temperatures may lead to loss of
strength by overaging.
Chapter 12. Ceramics - Structures and Properties
12.1 Introduction
Ceramics are inorganic and non-metallic materials that are commonly electrical and
thermal insulators, brittle and composed of more than one element (e.g., two in Al2O3)
Ceramic Structures
12.2 Crystal Structures
Ceramic bonds are mixed, ionic and covalent, with a proportion that depends on the
particular ceramics. The ionic character is given by the difference of electronegativity
between the cations (+) and anions (-). Covalent bonds involve sharing of valence
electrons. Very ionic crystals usually involve cations which are alkalis or alkalineearths
(first two columns of the periodic table) and oxygen or halogens as anions.
The building criteria for the crystal structure are two:
· maintain neutrality
· charge balance dictates chemical formula
· achieve closest packing
the condition for minimum energy implies maximum attraction and minimum
repulsion. This leads to contact, configurations where anions have the highest number
of cation neighbors and viceversa.
The parameter that is important in determining contact is the ratio of cation to anion
radii, rC/rA. Table 13.2 gives the coordination number and geometry as a function of
rC/rA. For example, in the NaCl structure (Fig. 13.2), rC = rNa = 0.102 nm, rA =rCl.=
0.181 nm, so rC/rA.= 0.56. From table 13.2 this implies coordination number = 6, as
observed for this rock-salt structure.
Other structures were shown in class, but will not be included in the test.
12.3 Silicate Ceramics
Oxygen and Silicon are the most abundant elements in Earth’s crust. Their
combination (silicates) occur in rocks, soils, clays and sand. The bond is weekly ionic,
with Si4+ as the cation and O2- as the anion. rSi = 0.04 nm, rO.= 0.14 nm, so rC/rA =
0.286. From table 13.2 this implies coordination number = 4, that is tetrahedral
coordination (Fig. 13.9).
The tetrahedron is charged: Si4+ + 4 O2- Þ (Si O4)4-. Silicates differ on how the
tetrahedra are arranged. In silica, (SiO2), every oxygen atom is shared by adjacent
tetrahedra. Silica can be crystalline (e.g., quartz) or amorphous, as in glass.
Soda glasses melt at lower temperature than amorphous SiO2 because the addition of
Na2O (soda) breaks the tetrahedral network. A lower melting point makes it easy to
form glass to make, for instance, bottles.
12.4 Carbon
Carbon is not really a ceramic, but an allotropic form, diamond, may be thought as a
type of ceramic. Diamond has very interesting and even unusual properties:
· diamond-cubic structure (like Si, Ge)
· covalent C-C bonds
· highest hardness of any material known
· very high thermal conductivity (unlike ceramics)
· transparent in the visible and infrared, with high index of refraction
· semiconductor (can be doped to make electronic devices)
· metastable (transforms to carbon when heated)
Synthetic diamonds are made by application of high temperatures and pressures or by
chemical vapor deposition. Future applications of this latter, cheaper production
method include hard coatings for metal tools, ultra-low friction coatings for space
applications, and microelectronics.
Graphite has a layered structure with very strong hexagonal bonding within the planar
layers (using 3 of the 3 bonding electrons) and weak, van der Waals bonding between
layers using the fourth electron. This leads to easy interplanar cleavage and
applications as a lubricant and for writing (pencils). Graphite is a good electrical
conductor and chemically stable even at high temperatures. Applications include
furnaces, rocket nozzles, electrodes in batteries.
A recently (1985) discovered formed of carbon is the C60 molecule, also known as
fullerene or bucky-ball (after the architect Buckminster Fuller who designed the
geodesic structure that C60 resembles.) Fullerenes and related structures like buckyonions
amd nanotubes are exceptionally strong. Future applications are as a structural
material and possibly in microelectronics, due to the unusual properties that result
when fullerenes are doped with other atoms.
12.5 Imperfections in Ceramics
Imperfections include point defects and impurities. Their formation is strongly
affected by the condition of charge neutrality (creation of unbalanced charges requires
the expenditure of a large amount of energy.
Non-stoichiometry refers to a change in composition so that the elements in the
ceramic are not in the proportion appropriate for the compound (condition known as
stoichiometry). To minimize energy, the effect of non-stoichiometry is a
redistribution of the atomic charges (Fig. 13.1).
Charge neutral defects include the Frenkel and Schottky defects. A Frenkel-defect is
a vacancy- interstitial pair of cations (placing large anions in an interstitial position
requires a lot of energy in lattice distortion). A Schottky-defect is the a pair of nearby
cation and anion vacancies.
Introduction of impurity atoms in the lattice is likely in conditions where the charge is
maintained. This is the case of electronegative impurities that substitute a lattice
anions or electropositive substitutional impurities. This is more likely for similar ionic
radii since this minimizes the energy required for lattice distortion. Defects will
appear if the charge of the impurities is not balanced.
12.6 Ceramic Phase Diagrams (not covered)
12.7 Brittle Fracture of Ceramics
The brittle fracture of ceramics limits applications. It occurs due to the unavoidable
presence of microscopic flaws (micro-cracks, internal pores, and atmospheric
contaminants) that result during cooling from the melt. The flaws need to crack
formation, and crack propagation (perpendicular to the applied stress) is usually
transgranular, along cleavage planes. The flaws cannot be closely controlled in
manufacturing; this leads to a large variability (scatter) in the fracture strength of
ceramic materials.
The compressive strength is typically ten times the tensile strength. This makes
ceramics good structural materials under compression (e.g., bricks in houses, stone
blocks in the pyramids), but not in conditions of tensile stress, such as under flexure.
Plastic deformation in crystalline ceramics is by slip, which is difficult due to the
structure and the strong local (electrostatic) potentials. There is very little plastic
deformation before fracture.
Non-crystalline ceramics, like common glass deform by viscous flow (like very highdensity
liquids). Viscosity decreases strongly with increases temperature.
Chapter 13. Ceramics - Applications and Processing
13.1 Introduction
Ceramics properties that are different from those of metals lead to different uses. In
structures, designs must be done for compressive loads. The transparency to light of
many ceramics leads to optical uses, like in windows, photographic cameras,
telescopes and microscopes. Good thermal insulation leads to use in ovens, the
exterior tiles of the Shuttle orbiter, etc. Good electrical isolation are used to support
conductors in electrical and electronic applications. The good chemical inertness
shows in the stability of the structures thousands of years old.
13.2 Glass Properties
A special characteristic of glasses is that solidification is gradual, through a viscous
stage, without a clear melting temperature. The specific volume does not have an
abrupt transition at a temperature but rather shows a change in slope at the glasstransition
temperature (Fig. 14.3).
The melting point, working point, softening point and annealing point are defined in
terms of viscosity, rather than temperature (Fig. 14.4), and depend on glass
composition..
13.4 Heat Treating Glasses
Similar to the case of metals, annealing is used at elevated temperatures is used to
remove stresses, like those caused by inhomogeneous temperatures during cooling.
Strengthening by glass tempering is done by heating the glass above the glass
transition temperature but below the softening point and then quenched in an air jet or
oil bath. The interior, which cools later than the outside, tries to contract while in a
plastic state after the exterior has become rigid. This causes residual compressive
stresses on the surface and tensile stresses inside. To fracture, a crack has first to
overcome the residual compressive stress, making tempered glass less susceptible to
fracture. This improvement leads to use in automobile windshields, glass doors,
eyeglass lenses, etc.
Chapter 15. Polymer Structures
14.1 Introduction
Polymers are common in nature, in the form of wood, rubber, cotton, leather, wood,
silk, proteins, enzymes, starches, cellulose. Artificial polymers are made mostly from
oil. Their use has grown exponentially, especially after WW2. The key factor is the
very low production cost and useful properties (e.g., combination of transparency and
flexibility, long elongation).
14.2 Hydrocarbon Molecules
Most polymers are organic, and formed from hydrocarbon molecules. These
molecules can have single, double, or triple carbon bonds. A saturated hydrocarbon
is one where all bonds are single, that is, the number of atoms is maximum (or
saturated). Among this type are the paraffin compounds, CnH2n+2 (Table 15.1). In
contrast, non-saturated hydrocarbons contain some double and triple bonds.
Isomers are molecules that contain the same molecules but in a different
arrangement. An example is butane and isobutane.
14.3 Polymer Molecules
Polymer molecules are huge, macromolecules that have internal covalent bonds. For
most polymers, these molecules form very long chains. The backbone is a string of
carbon atoms, often single bonded.
Polymers are composed of basic structures called mer units. A molecule with just one
mer is a monomer.
14.4 The Chemistry of Polymer Molecules
Examples of polymers are polyvinyl chloride (PVC), poly-tetra-chloro-ethylene
(PTFE or Teflon), polypropylene, nylon and polystyrene. Chains are represented
straight but in practice they have a three-dimensional, zig-zag structure (Fig. 15.1b).
When all the mers are the same, the molecule is called a homopolymer. When there is
more than one type of mer present, the molecule is a copolymer.
14.5 Molecular Weight
The mass of a polymer is not fixed, but is distributed around a mean value, since
polymer molecules have different lengths. The average molecular weight can be
obtained by averaging the masses with the fraction of times they appear (numberaverage)
or with the mass fraction of the molecules (called, improperly, a weight
fraction).
The degree of polymerization is the average number of mer units, and is obtained by
dividing the average mass of the polymer by the mass of a mer unit.
Polymers of low mass are liquid or gases, those of very high mass (called highpolymers,
are solid). Waxes, paraffins and resins have intermediate masses.
14.6 Molecular Shape
Polymers are usually not linear; bending and rotations can occur around single C-C
bonds (double and triple bonds are very rigid) (Fig. 15.5). Random kings and coils
lead to entanglement, like in the spaghetti structure shown in Fig. 15.6.
14.7 Molecular Structure
Typical structures are (Fig. 15.7):
· linear (end-to-end, flexible, like PVC, nylon)
· branched
· cross-linked (due to radiation, vulcanization, etc.)
· network (similar to highly cross-linked structures).
14.8 Molecular Configurations
The regularity and symmetry of the side-groups can affect strongly the properties of
polymers. Side groups are atoms or molecules with free bonds, called free-radicals,
like H, O, methyl, etc.
If the radicals are linked in the same order, the configuration is called isostatic
In a stereoisomer in a syndiotactic configuration, the radical groups alternative sides
in the chain.
In the atactic configuration, the radical groups are positioned at random.
14.9 Copolymers
Copolymers, polymers with at least two different types of mers can differ in the way
the mers are arranged. Fig. 15.9 shows different arrangements: random, alternating,
block, and graft.
14.10 Polymer Crystallinity
Crystallinity in polymers is more complex than in metals (fig. 15.10). Polymer
molecules are often partially crystalline (semicrystalline), with crystalline regions
dispersed within amorphous material. .
Chain disorder or misalignment, which is common, leads to amorphous material since
twisting, kinking and coiling prevent strict ordering required in the crystalline state.
Thus, linear polymers with small side groups, which are not too long form crystalline
regions easier than branched, network, atactic polymers, random copolymers, or
polymers with bulky side groups.
Crystalline polymers are denser than amorphous polymers, so the degree of
crystallinity can be obtained from the measurement of density.
14.11 Polymer Crystals
Different models have been proposed to describe the arrangement of molecules in
semicrytalline polymers. In the fringed-micelle model, the crystallites (micelles) are
embedded in an amorphous matrix (Fig.15.11). Polymer single crystals grown are
shaped in regular platelets (lamellae) (Fig. 15.12). Spherulites (Fig. 15.4) are chainfolded
crystallites in an amorphous matrix that grow radially in spherical shape
“grains”.
15.1 Introduction
15.2 Stress-Strain Behavior
The description of stress-strain behavior is similar to that of metals, but a very
important consideration for polymers is that the mechanical properties depend on the
strain rate, temperature, and environmental conditions.
The stress-strain behavior can be brittle, plastic and highly elastic (elastomeric or
rubber-like), see Fig. 16. 1. Tensile modulus (modulus) and tensile strengths are
orders of magnitude smaller than those of metals, but elongation can be up to 1000 %
in some cases. The tensile strength is defined at the fracture point (Fig. 16.2) and can
be lower than the yield strength.
Mechanical properties change dramatically with temperature, going from glass-like
brittle behavior at low temperatures (like in the liquid-nitrogen demonstration) to a
rubber-like behavior at high temperatures (Fig. 16.3).
In general, decreasing the strain rate has the same influence on the strain-strength
characteristics as increasing the temperature: the material becomes softer and more
ductile.
15.3 Deformation of Semicrystalline Polymers
Many semicrystalline polymers have the spherulitic structure and deform in the
following steps (Fig. 16.4):
· elongation of amorphous tie chains
· tilting of lamellar chain folds towards the tensile direction
· separation of crystalline block segments
· orientation of segments and tie chains in the tensile direction
The macroscopic deformation involves an upper and lower yield point and necking.
Unlike the case of metals, the neck gets stronger since the deformation aligns the
chains so increasing the tensile stress leads to the growth of the neck. (Fig. 16.5).
15.4 Factors that Influence the Mechanical Properties of Polymers
The tensile modulus decreases with increasing temperature or diminishing strain rate.
Obstacles to the steps mentioned in 16.4 strengthen the polymer. Examples are crosslinking
(aligned chains have more van der Waals inter-chain bonds) and a large mass
(longer molecules have more inter-chain bonds). Crystallinity increases strength as
the secondary bonding is enhanced when the molecular chains are closely packed and
parallel. Pre-deformation by drawing, analogous to strain hardening in metals,
increases strength by orienting the molecular chains. For undrawn polymers, heating
increases the tensile modulus and yield strength, and reduces the ductility - opposite
of what happens in metals.
15.5 Crystallization, Melting, and Glass Transition Phenomena
Crystallization rates are governed by the same type of S-curves we saw in the case of
metals (Fig. 16.7). Nucleation becomes slower at higher temperatures.
The melting behavior of semicrystalline polymers is intermediate between that of
crystalline materials (sharp density change at a melting temperature) and that of a
pure amorphous material (slight change in slope of density at the glass-transition
temperature). The glass transition temperature is between 0.5 and 0.8 of the melting
temperature.
The melting temperature increases with the rate of heating, thickness of the lamellae,
and depends on the temperature at which the polymer was crystallized.
Melting involves breaking of the inter-chain bonds, so the glass and melting
temperatures depend on:
· chain stiffness (e.g., single vs. double bonds)
· size, shape of side groups
· size of molecule
· side branches, defects
· cross-linking
Rigid chains have higher melting temperatures.
15.6 Thermoplastic and Thermosetting Polymers
Thermoplastic polymers (thermoplasts) soften reversibly when heated (harden when
cooled back)
Thermosetting polymers (thermosets) harden permanently when heated, as crosslinking
hinder bending and rotations. Thermosets are harder, more dimensionally
stable, and more brittle than thermoplasts.
15.7 Viscoelasticity
At low temperatures, amorphous polymers deform elastically, like glass, at small
elongation. At high temperatures the behavior is viscous, like liquids. At intermediate
temperatures, the behavior, like a rubbery solid, is termed viscoelastic.
Viscoelasticity is characterized by the viscoelastic relaxation modulus
Er = s(t)/e0.
If the material is strained to a value e0.it is found that the stress needs to be reduced
with time to maintain this constant value of strain (see figs. 16.11 and 16.12).
In viscoelastic creep, the stress is kept constant at s0 and the change of deformation
with time e(t) is measured. The time-dependent creep modulus is given by
Ec = s0/e(t).
15.8 Deformation and Elastomers
Elastomers can be deformed to very large strains and the spring back elastically to the
original length, a behavior first observed in natural rubber. Elastic elongation is due to
uncoiling, untwisting and straightening of chains in the stress direction.
To be elastomeric, the polymer needs to meet several criteria:
· must not crystallize easily
· have relatively free chain rotations
· delayed plastic deformation by cross-linking (achieved by vulcanization).
· be above the glass transition temperature
15.9 Fracture of Polymers
As other mechanical properties, the fracture strength of polymers is much lower than
that of metals. Fracture also starts with cracks at flaws, scratches, etc. Fracture
involves breaking of covalent bonds in the chains. Thermoplasts can have both brittle
and ductile fracture behaviors. Glassy thermosets have brittle fracture at low
temperatures and ductile fracture at high temperatures.
Glassy thremoplasts often suffer grazing before brittle fracture. Crazes are associated
with regions of highly localized yielding which leads to the formation of
interconnected microvoids (Fig. 16.15). Crazing absorbs energy thus increasing the
fracture strength of the polymer.
15.10 Miscellaneous Characteristics
Polymers are brittle at low temperatures and have low impact strengths (Izod or
Charpy tests), and a brittle to ductile transition over a narrow temperature range.
Fatigue is similar to the case of metals but at reduced loads and is more sensitive to
frequency due to heating which leads to softening.
15.11 Polymerization
Polymerization is the synthesis of high polymers from raw materials like oil or coal. It
may occur by:
· addition (chain-reaction) polymerization, where monomer units are attached
one at a time
· condensation polymerization, by stepwise intermolecular chemical reactions
that produce the mer units.
15.12 Elastomers
In vulcanization, crosslinking of the elastomeric polymer is achieved by an
irreversible chemical reaction usually at high temperatures (hence ‘vulcan’), and
usually involving the addition of sulfur compounds. The S atoms are the ones that
form the bridge cross-links. Elastomers are thermosetting due to the cross-linking.
Rubbers become harder and extend less with increasing sulfur content. For
automobile applications, synthetic rubbers are strengthened by adding carbon black.
In silicone rubbers, the backbone C atoms are replaced by a chain of alternating
silicon and oxygen atoms. These elastomers are also cross-linked and are stable to
higher temperatures than C-based elastomers.
Chapter 16. Composites
16.1 Introduction
The idea is that by combining two or more distinct materials one can engineer a new
material with the desired combination of properties (e.g., light, strong, corrosion
resistant). The idea that a better combination of properties can be achieved is called
the principle of combined action.
New - High-tech materials, engineered to specific applications
Old - brick-straw composites, paper, known for > 5000 years.
A type of composite that has been discussed is perlitic steel, which combines hard,
brittle cementite with soft, ductile ferrite to get a superior material.
Natural composites: wood (polymer-polymer), bones (polymer-ceramics).
Usual composites have just two phases:
· matrix (continuous)
· dispersed phase (particulates, fibers)
Properties of composites depend on
· properties of phases
· geometry of dispersed phase (particle size, distribution, orientation)
· amount of phase
Classification of composites: three main categories:
· particle-reinforced (large-particle and dispersion-strengthened)
· fiber-reinforced (continuous (aligned) and short fibers (aligned or random)
· structural (laminates and sandwich panels)
Particle-reinforced composites
These are the cheapest and most widely used. They fall in two categories depending
on the size of the particles:
· large-particle composites, which act by restraining the movement of the
matrix, if well bonded.
· dispersion-strengthened composites, containing 10-100 nm particles, similar to
what was discussed under precipitation hardening. The matrix bears the major
portion of the applied load and the small particles hinder dislocation motion,
limiting plastic deformation.
16.2 Large-Particle Composites
Properties are a combination of those of the components. The rule of mixtures
predicts that an upper limit of the elastic modulus of the composite is given in terms
of the elastic moduli of the matrix (Em) and the particulate (Ep) phases by:
Ec = EmVm + EpVp
where Vm and Vp are the volume fraction of the two phases. A lower bound is given
by:
Ec = EmEp / (EpVm + EmVp)
Fig. 17.3 - modulus of composite of WC particles in Cu matrix vs. WC concentration.
Concrete
The most common large-particle composite is concrete, made of a cement matrix that
bonds particles of different size (gravel and sand.) Cement was already known to the
Egyptians and the Greek. Romans made cement by mixing lime (CaO) with volcanic
ice.
In its general from, cement is a fine mixture of lime, alumina, silica, and water.
Portland cement is a fine powder of chalk, clay and lime-bearing minerals fired to
1500o C (calcinated). It forms a paste when dissolved in water. It sets into a solid in
minutes and hardens slowly (takes 4 months for full strength). Properties depend on
how well it is mixed, and the amount of water: too little - incomplete bonding, too
much - excessive porosity.
The advantage of cement is that it can be poured in place, it hardens at room
temperature and even under water, and it is very cheap. The disadvantages are that it
is weak and brittle, and that water in the pores can produce crack when it freezes in
cold weather.
Concrete is cement strengthened by adding particulates. The use of different size
(stone and sand) allows better packing factor than when using particles of similar size.
Concrete is improved by making the pores smaller (using finer powder, adding
polymeric lubricants, and applying pressure during hardening.
Reinforced concrete is obtained by adding steel rods, wires, mesh. Steel has the
advantage of a similar thermal expansion coefficient, so there is reduced danger of
cracking due to thermal stresses. Pre-stressed concrete is obtained by applying tensile
stress to the steel rods while the cement is setting and hardening. When the tensile
stress is removed, the concrete is left under compressive stress, enabling it to sustain
tensile loads without fracturing. Pre-stressed concrete shapes are usually
prefabricated. A common use is in railroad or highway bridges.
Cermets
are composites of ceramic particles (strong, brittle) in a metal matrix (soft, ductile)
that enhances toughness. For instance, tungsten carbide or titanium carbide ceramics
in Co or Ni. They are used for cutting tools for hardened steels.
Reinforced rubber
is obtained by strengthening with 20-50 nm carbon-black particles. Used in auto tires.
16.3 Dispersion-Strengthened Composites
Use of very hard, small particles to strengthen metals and metal alloys. The effect is
like precipitation hardening but not so strong. Particles like oxides do not react so the
strengthening action is retained at high temperatures.
Fiber-reinforced composites
In many applications, like in aircraft parts, there is a need for high strength per unit
weight (specific strength). This can be achieved by composites consisting of a lowdensity
(and soft) matrix reinforced with stiff fibers.
The strength depends on the fiber length and its orientation with respect to the stress
direction.
The efficiency of load transfer between matrix and fiber depends on the interfacial
bond.
16.4 Influence of Fiber Length
Normally the matrix has a much lower modulus than the fiber so it strains more. This
occurs at a distance from the fiber. Right next to the fiber, the strain is limited by the
fiber. Thus, for a composite under tension, a shear stress appears in the matrix that
pulls from the fiber. The pull is uniform over the area of the fiber. This makes the
force on the fiber be minimum at the ends and maximum in the middle, like in the
tug-of-war game.
To achieve effective strengthening and stiffening, the fibers must be larger than a
critical length lc, defined as the minimum length at which the center of the fiber
reaches the ultimate (tensile) strength sf, when the matrix achieves the maximum
shear strength tm:
lc = sf d /2 tm
Since it is proportional to the diameter of the fiber d, a more unified condition for
effective strengthening is that the aspect ratio of the fiber is l/d > sf /2 tm.
16.5 Influence of Fiber Orientation
The composite is stronger along the direction of orientation of the fibers and weakest
in a direction perpendicular to the fiber. For discontinuous, random fibers, the
properties are isotropic.
16.6 Polymer Matrix Composites
Largest and most diverse use of composites due to ease of fabrication, low cost and
good properties.
Glass-fiber reinforced composites (GFRC) are strong, corrosion resistant and
lightweight, but not very stiff and cannot be used at high temperatures. Applications
include auto and boat bodies, aircraft components.
Carbon-fiber reinforced composites (CFRC) use carbon fibers, which have the highest
specific module (module divided by weight). CFRC are strong, inert, allow high
temperature use. Applications include fishing rods, golf clubs, aircraft components.
Kevlar, and aremid-fiber composite (Fig. 17.9) can be used as textile fibers.
Applications include bullet-proof vests, tires, brake and clutch linings.
Wood:
This is one of the oldest and the most widely used structural material. It is a
composite of strong and flexible cellulose fibers (linear polymer) surrounded and held
together by a matrix of lignin and other polymers. The properties are anisotropic and
vary widely among types of wood. Wood is ten times stronger in the axial direction
than in the radial or tangential directions.
Chapter 17. Electrical Properties
Electrical Conduction
17.2 Ohm’s Law
When an electric potential V is applied across a material, a current of magnitude I
flows. In most metals, at low values of V, the current is proportional to V, according to
Ohm's law:
I = V/R
where R is the electrical resistance. R depends on the intrinsic resistivity r of the
material and on the geometry (length l and area A through which the current passes).
R = rl/A
17.3 Electrical Conductivity
The electrical conductivity is the inverse of the resistivity: s = 1/r.
The electric field in the material is E=V/l, Ohm's law can then be expressed in terms
of the current density j = I/A as:
j = s E
The conductivity is one of the properties of materials that varies most widely, from
107 (W-m) typical of metals to 10-20 (W-m) for good electrical insulators.
Semiconductors have conductivities in the range 10-6 to 104 (W-m).
17.4 Electronic and Ionic Conduction
In metals, the current is carried by electrons, and hence the name electronic
conduction. In ionic crystals, the charge carriers are ions, thus the name ionic
conduction (see Sect. 19.15).
17.5 Energy Band Structures in Solids
When atoms come together to form a solid, their valence electrons interact due to
Coulomb forces, and they also feel the electric field produced by their own nucleus
and that of the other atoms. In addition, two specific quantum mechanical effects
happen. First, by Heisenberg's uncertainty principle, constraining the electrons to a
small volume raises their energy, this is called promotion. The second effect, due to
the Pauli exclusion principle, limits the number of electrons that can have the same
property (which include the energy). As a result of all these effects, the valence
electrons of atoms form wide valence bands when they form a solid. The bands are
separated by gaps, where electrons cannot exist. The precise location of the bands and
band gaps depends on the type of atom (e.g., Si vs. Al), the distance between atoms in
the solid, and the atomic arrangement (e.g., carbon vs. diamond).
In semiconductors and insulators, the valence band is filled, and no more electrons
can be added, following Pauli's principle. Electrical conduction requires that electrons
be able to gain energy in an electric field; this is not possible in these materials
because that would imply that the electrons are promoted into the forbidden band gap.
In metals, the electrons occupy states up to the Fermi level. Conduction occurs by
promoting electrons into the conduction band, that starts at the Fermi level, separated
by the valence band by an infinitesimal amount.
17.6 Conduction in Terms of Band and Atomic Bonding Models
Conduction in metals is by electrons in the conduction band. Conduction in insulators
is by electrons in the conduction band and by holes in the valence band. Holes are
vacant states in the valence band that are created when an electron is removed.
In metals there are empty states just above the Fermi levels, where electrons can be
promoted. The promotion energy is negligibly small so that at any temperature
electrons can be found in the conduction band. The number of electrons participating
in electrical conduction is extremely small.
In insulators, there is an energy gap between the valence and conduction bands, so
energy is needed to promote an electron to the conduction band. This energy may
come from heat, or from energetic radiation, like light of sufficiently small
wavelength.
A working definition for the difference between semiconductors and insulators
is that in semiconductors, electrons can reach the conduction band at ordinary
temperatures, where in insulators they cannot. The probability that an electron reaches
the conduction band is about exp(-Eg/2kT) where Eg is the band gap and kT has the
usual meaning. If this probability is, say, < 10-24 one would not find a single electron
in the conduction band in a solid of 1 cubic centimeter. This requires Eg/2kT > 55. At
room temperature, 2kT = 0.05 eV; thus Eg > 2.8 eV can be used as the condition for an
insulator.
Besides having relatively small Eg, semiconductors have covalent bond, whereas
insulators usually are partially ionic bonded.
17.7 Electron Mobility
Electrons are accelerated in an electric field E, in the opposite direction to the field
because of their negative charge. The force acting on the electron is -eE, where e is
the electric charge. This force produces a constant acceleration so that, in the absence
of obstacles (in vacuum, like inside a TV tube) the electron speeds up continuously in
an electric field. In a solid, the situation is different. The electrons scatter by collisions
with atoms and vacancies that change drastically their direction of motion. Thus
electrons move randomly but with a net drift in the direction opposite to the electric
field. The drift velocity is constant, equal to the electric field times a constant called
the mobility m,
vd= – me E
which means that there is a friction force proportional to velocity. This friction
translates into energy that goes into the lattice as heat. This is the way that electric
heaters work.
The electrical conductivity is:
s = n |e| me
where n is the concentration of electrons (n is used to indicate that the carriers of
electricity are negative particles).
17.8 Electrical Resistivity of Metals
The resistivity then depends on collisions. Quantum mechanics tells us that electrons
behave like waves. One of the effects of this is that electrons do not scatter from a
perfect lattice. They scatter by defects, which can be:
o atoms displaced by lattice vibrations
o vacancies and interstitials
o dislocations, grain boundaries
o impurities
One can express the total resistivity rtot by the Matthiessen rule, as a sum of
resistivities due to thermal vibrations, impurities and dislocations. Fig. 19.8 illustrates
how the resistivity increases with temperature, with deformation, and with alloying.
17.9 Electrical Characteristics of Commercial Alloys
The best material for electrical conduction (lower resistivity) is silver. Since it is very
expensive, copper is preferred, at an only modest increase in r. To achieve low r it is
necessary to remove gases occluded in the metal during fabrication. Copper is soft so,
for applications where mechanical strength is important, the alloy CuBe is used,
which has a nearly as good r. When weight is important one uses Al, which is half as
good as Cu. Al is also more resistant to corrosion.
When high resistivity materials are needed, like in electrical heaters, especially those
that operate at high temperature, nichrome (NiCr) or graphite are used.
17.10 Intrinsic Semiconduction
Semiconductors can be intrinsic or extrinsic. Intrinsic means that electrical
conductivity does not depend on impurities, thus intrinsic means pure. In extrinsic
semiconductors the conductivity depends on the concentration of impurities.
Conduction is by electrons and holes. In an electric field, electrons and holes move in
opposite direction because they have opposite charges. The conductivity of an
intrinsic semiconductor is:
s = n |e| me + p |e| mh
where p is the hole concentration and mh the hole mobility. One finds that electrons
move much faster than holes:
me > mh
In an intrinsic semiconductor, a hole is produced by the promotion of each electron to
the conduction band. Thus:
n = p
Thus, s = 2 n |e| (me + mh) (only for intrinsic semiconductors).
17.11 Extrinsic Semiconduction
Unlike intrinsic semiconductors, an extrinsic semiconductor may have different
concentrations of holes and electrons. It is called p-type if p>n and n-type if n>p. They
are made by doping, the addition of a very small concentration of impurity atoms.
Two common methods of doping are diffusion and ion implantation.
Excess electron carriers are produced by substitutional impurities that have more
valence electron per atom than the semiconductor matrix. For instance phosphorous,
with 5 valence electrons, is an electron donor in Si since only 4 electrons are used to
bond to the Si lattice when it substitutes for a Si atom. Thus, elements in columns V
and VI of the periodic table are donors for semiconductors in the IV column, Si and
Ge. The energy level of the donor state is close to the conduction band, so that the
electron is promoted (ionized) easily at room temperature, leaving a hole (the ionized
donor) behind. Since this hole is unlike a hole in the matrix, it does not move easily
by capturing electrons from adjacent atoms. This means that the conduction occurs
mainly by the donated electrons (thus n-type).
Excess holes are produced by substitutional impurities that have fewer valence
electrons per atom than the matrix. This is the case of elements of group II and III in
column IV semiconductors, like B in Si. The bond with the neighbors is incomplete
and so they can capture or accept electrons from adjacent silicon atoms. They are
called acceptors. The energy level of the acceptor is close to the valence band, so that
an electron may easily hop from the valence band to complete the bond leaving a hole
behind. This means that conduction occurs mainly by the holes (thus p-type).
17.12 The Temperature Variation of Conductivity and Carrier Concentration
Temperature causes electrons to be promoted to the conduction band and from donor
levels, or holes to acceptor levels. The dependence of conductivity on temperature is
s = A exp(–Eg/2kT)
where A is a constant (the mobility varies much more slowly with temperature).
Plotting ln s vs. 1/T produces a straight line of slope Eg/2k from which the band gap
energy can be determined. Extrinsic semiconductors have, in addition to this
dependence, one due to the thermal promotion of electrons from donor levels or holes
from acceptor levels. The dependence on temperature is also exponential but it
eventually saturates at high temperatures where all the donors are emptied or all the
acceptors are filled.
This means that at low temperatures, extrinsic semiconductors have larger
conductivity than intrinsic semiconductors. At high temperatures, both the impurity
levels and valence electrons are ionized, but since the impurities are very low in
number and they are exhausted, eventually the behavior is dominated by the intrinsic
type of conductivity.
17.14 Semiconductor Devices
A semiconductor diode is made by the intimate junction of a p-type and an n-type
semiconductor (an n-p junction). Unlike a metal, the intensity of the electrical current
that passes through the material depends on the polarity of the applied voltage. If the
positive side of a battery is connected to the p-side, a situation called forward bias, a
large amount of current can flow since holes and electrons are pushed into the
junction region, where they recombine (annihilate). If the polarity of the voltage is
flipped, the diode operates under reverse bias. Holes and electrons are removed from
the region of the junction, which therefore becomes depleted of carriers and behaves
like an insulator. For this reason, the current is very small under reverse bias. The
asymmetric current-voltage characteristics of diodes (Fig. 19.20) is used to convert
alternating current into direct current. This is called rectification.
A p-n-p junction transistor contains two diodes back-to-back. The central region is
very thin and is called the base. A small voltage applied to the base has a large effect
on the current passing through the transistor, and this can be used to amplify electrical
signals (Fig. 19.22). Another common device is the MOSFET transistor where a gate
serves the function of the base in a junction transistor. Control of the current through
the transistor is by means of the electric field induced by the gate, which is isolated
electrically by an oxide layer.
17.15 Conduction in Ionic Materials
In ionic materials, the band gap is too large for thermal electron promotion. Cation
vacancies allow ionic motion in the direction of an applied electric field, this is
referred to as ionic conduction. High temperatures produce more vacancies and higher
ionic conductivity.
At low temperatures, electrical conduction in insulators is usually along the surface,
due to the deposition of moisture that contains impurity ions.
17.16 Electrical Properties of Polymers
Polymers are usually good insulators but can be made to conduct by doping. Teflon is
an exceptionally good insulator.
Dielectric Behavior
A dielectric is an electrical insulator that can be made to exhibit an electric dipole
structure (displace the negative and positive charge so that their center of gravity is
different).
17.17 Capacitance
When two parallel plates of area A, separated by a small distance l, are charged by
+Q, –Q, an electric field develops between the plates
E = D/ee0
where D = Q/A. e0 is called the vacuum permittivity and e the relative permittivity, or
dielectric constant (e = 1 for vacuum). In terms of the voltage between the plates, V =
E l,
V = Dl/ee0 = Q l/Aee0 = Q / C
The constant C= Aee0/l is called the capacitance of the plates.
17.18 Field Vectors and Polarization
The dipole moment of a pair of positive and negative charges (+q and –q) separated at
a distance d is p = qd. If an electric field is applied, the dipole tends to align so that
the positive charge points in the field direction. Dipoles between the plates of a
capacitor will produce an electric field that opposes the applied field. For a given
applied voltage V, there will be an increase in the charge in the plates by an amount Q'
so that the total charge becomes Q = Q' + Q0, where Q0 is the charge of a vacuum
capacitor with the same V. With Q' = PA, the charge density becomes D = D0 E + P,
where the polarization P = e0 (e–1) E .
19.19 Types of Polarization
Three types of polarization can be caused by an electric field:
· Electronic polarization: the electrons in atoms are displaced relative to the
nucleus.
· Ionic polarization: cations and anions in an ionic crystal are displaced with
respect to each other.
· Orientation polarization: permanent dipoles (like H2O) are aligned.
17.20 Frequency Dependence of the Dielectric Constant
Electrons have much smaller mass than ions, so they respond more rapidly to a
changing electric field. For electric field that oscillates at very high frequencies (such
as light) only electronic polarization can occur. At smaller frequencies, the relative
displacement of positive and negative ions can occur. Orientation of permanent
dipoles, which require the rotation of a molecule can occur only if the oscillation is
relatively slow (MHz range or slower). The time needed by the specific polarization
to occur is called the relaxation time.
17.21 Dielectric Strength
Very high electric fields (>108 V/m) can free electrons from atoms, and accelerate
them to such high energies that they can, in turn, free other electrons, in an avalanche
process (or electrical discharge). This is called dielectric breakdown, and the field
necessary to start the is called the dielectric strength or breakdown strength.
17.22 Dielectric Materials
Capacitors require dielectrics of high e that can function at high frequencies (small
relaxation times). Many of the ceramics have these properties, like mica, glass, and
porcelain). Polymers usually have lower e.
17.23 Ferroelectricity
Ferroelectric materials are ceramics that exhibit permanent polarization in the
absence of an electric field. This is due to the asymmetric location of positive and
negative charges within the unit cell. Two possible arrangements of this asymmetry
results in two distinct polarizations, which can be used to code "0" and "1" in
ferroelectric memories. A typical ferroelectric is barium titanate, BaTiO3, where the
Ti4+ is in the center of the unit cell and four O2- in the central plane can be displaced
to one side or the other of this central ion (Fig. 19.33).
17.24 Piezoelectricity
In a piezolectric material, like quartz, an applied mechanical stress causes electric
polarization by the relative displacement of anions and cations.
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no one fig. for study
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