Sources and References

These notes synthesize the standard undergraduate materials science curriculum using Askeland and Wright’s The Science and Engineering of Materials (the recommended text for ME 115), together with Callister and Rethwisch’s Materials Science and Engineering: An Introduction, Shackelford’s Introduction to Materials Science for Engineers, and Smith and Hashemi’s Foundations of Materials Science and Engineering. Additional perspective is drawn from publicly available open-course materials such as MIT’s 3.091 (Introduction to Solid-State Chemistry) and 3.032 (Mechanical Behavior of Materials), the Cambridge University Materials Tripos curriculum, and Stanford’s MATSCI 157. Hello Joshua; the treatment below follows the Askeland chapter numbering used by the course.

Chapter 1: The Role of Materials in Engineering

Materials science sits at the intersection of chemistry, physics, and engineering. Its central claim is that the macroscopic properties of a part, whether stiffness, strength, toughness, conductivity, or corrosion resistance, are determined by microstructure: the arrangement of atoms into phases, grains, and defects at length scales from angstroms to millimetres. A designer who changes a heat treatment, an alloying addition, or a cooling rate is changing microstructure, and through it the property envelope the part can deliver.

Engineering materials are grouped into four broad families. Metals are characterized by delocalized (metallic) bonding, high stiffness and density, ductility, and high thermal and electrical conductivity. Ceramics, whether oxides, carbides, or nitrides, are held together by ionic or mixed ionic-covalent bonds, giving high stiffness and melting point but low fracture toughness. Polymers consist of long covalent chains joined by weak secondary bonds and span a huge range of behaviour from rubbery elastomers to glassy plastics. Composites combine two or more of the above to exploit the stiffness of one phase and the toughness of another, of which fibre-reinforced plastics and reinforced concrete are everyday examples. Semiconductors form a fifth category distinguished by their electronic properties rather than their mechanical ones.

A guiding idea in this course is the processing-structure-properties-performance tetrahedron. The engineer selects a composition, imposes a process (casting, rolling, extrusion, heat treatment), which yields a microstructure, which fixes the properties, which in service delivers performance. Understanding any one corner requires the other three.

Chapter 2: Atomic Structure and Bonding

Atomic bonding explains why different material families behave so differently. Electrons in atoms occupy quantized orbitals, and when atoms approach each other the valence electrons rearrange to lower the total energy. The interatomic potential can be approximated by a sum of attractive and repulsive terms, for example the Lennard-Jones form

\[ U(r) = -\frac{A}{r^{m}} + \frac{B}{r^{n}}, \qquad n > m. \]

The equilibrium spacing \( r_0 \) occurs where \( dU/dr = 0 \), and the curvature \( d^{2}U/dr^{2}\big|_{r_0} \) is proportional to the stiffness of the bond. Steep, deep wells produce high Young’s modulus, high melting point, and low coefficient of thermal expansion; shallow wells do the opposite. This single picture rationalizes why diamond is stiffer than copper and why polyethylene expands so much on heating.

Four idealized bonding mechanisms are recognized. Ionic bonding arises from electron transfer between electropositive and electronegative species, yielding strong, non-directional Coulomb attraction (NaCl, MgO). Covalent bonding involves shared electron pairs, is strongly directional, and produces hard, brittle solids (diamond, Si, SiC). Metallic bonding is described as positive ion cores immersed in a delocalized electron sea; it is non-directional but not saturable, so metals deform plastically by shearing planes of ions. Secondary bonds, including van der Waals forces and hydrogen bonds, are weak and short-ranged but control the physics of polymers, molecular solids, and biological materials.

Thermal expansion follows directly from the asymmetry of the interatomic potential. Because the well is wider on the outside than on the inside of \( r_0 \), raising the vibrational amplitude (i.e. temperature) shifts the mean separation outward, so

\[ \alpha = \frac{1}{L}\frac{dL}{dT} \]

is positive, larger for shallow wells and smaller for deep ones.

Chapter 3: Crystal Structure and Atomic Arrangements

Most metals and many ceramics are crystalline: their atoms occupy the points of a periodic lattice. A crystal structure is specified by a unit cell, the smallest repeating parallelepiped, and a basis of atoms attached to each lattice point. The three cubic structures that dominate the metallic world are simple cubic (rare), body-centred cubic (BCC: \( \alpha\text{-Fe} \), Cr, W, Mo), and face-centred cubic (FCC: Cu, Al, Ni, \( \gamma\text{-Fe} \), Ag, Au). The hexagonal close-packed (HCP) structure completes the set for Mg, Zn, Ti, and Co.

Two quantities summarize a cell. The coordination number counts nearest neighbours: 8 for BCC, 12 for FCC and HCP. The atomic packing factor (APF) is the volume fraction occupied by atoms modelled as hard spheres,

\[ \text{APF} = \frac{n_{\text{atoms}} \cdot \tfrac{4}{3}\pi r^{3}}{V_{\text{cell}}}. \]

For BCC, with \( a\sqrt{3} = 4r \), one finds APF \( \approx 0.68 \); for FCC and HCP, APF \( \approx 0.74 \), which is the densest packing of identical spheres in three dimensions. The theoretical density follows from

\[ \rho = \frac{n A}{V_{\text{cell}} N_{A}}, \]

where \( n \) is atoms per cell, \( A \) is atomic mass, and \( N_{A} \) is Avogadro’s number.

Directions and planes in a crystal are labelled with Miller indices. A direction is the set of smallest integers proportional to the components of a vector from the origin, written \( \left[uvw\right] \). A plane is labelled by taking the reciprocals of its axis intercepts, clearing fractions, and enclosing the result in parentheses \( \left(hkl\right) \). Families of crystallographically equivalent directions and planes are denoted \( \langle uvw\rangle \) and \( \{hkl\} \). The linear and planar densities,

\[ \text{LD} = \frac{\text{atoms on direction vector}}{\text{length of vector}}, \qquad \text{PD} = \frac{\text{atoms on plane}}{\text{area of plane}}, \]

identify the close-packed directions and planes that carry plastic flow. In FCC the close-packed planes are \( \{111\} \), in BCC the close-packed (though not truly densest) planes are \( \{110\} \), and in HCP the basal plane \( (0001) \) plays that role.

Not every solid is crystalline. Amorphous solids, including silica glass, most oxide glasses, and quenched polymers, lack long-range order. They show a glass transition temperature \( T_{g} \) rather than a sharp melting point, and their mechanical response is governed by free-volume arguments rather than by dislocations.

Chapter 4: Imperfections and Dislocations

Real crystals are never perfect, and it is the imperfections that make them engineeringly useful. Defects are classified by dimensionality.

Point defects include vacancies, self-interstitials, and substitutional or interstitial foreign atoms. Vacancies are always present at equilibrium because their configurational entropy lowers the free energy; their concentration follows

\[ \frac{N_{v}}{N} = \exp\!\left(-\frac{Q_{v}}{k_{B}T}\right), \]

rising exponentially with temperature. Solute atoms distort the surrounding lattice and create stress fields that pin dislocations, which is the microscopic origin of solid-solution strengthening.

Line defects, or dislocations, are the carriers of plastic deformation in crystalline solids. An edge dislocation is an extra half-plane of atoms terminating inside the crystal; a screw dislocation is a spiral ramp of atomic planes; mixed dislocations are linear combinations. Each is characterized by a Burgers vector \( \mathbf{b} \) whose magnitude is of order one atomic spacing and whose direction is the smallest lattice translation. Dislocations move by slip on specific slip systems consisting of a close-packed plane and a close-packed direction lying in that plane. FCC offers twelve \( \{111\}\langle 110\rangle \) slip systems, which is why FCC metals are so ductile. BCC has many more potential systems but they are not truly close-packed, so BCC behaviour is more temperature-sensitive. HCP has only a handful of easy basal systems and is therefore typically less ductile at room temperature.

Slip begins when the resolved shear stress on a slip system exceeds a critical value. Schmid’s law gives

\[ \tau_{R} = \sigma \cos\phi \cos\lambda, \]

where \( \phi \) is the angle between the tensile axis and the slip-plane normal and \( \lambda \) is the angle between the tensile axis and the slip direction. Yield begins on the system with the largest Schmid factor \( \cos\phi\cos\lambda \).

Planar defects include free surfaces, grain boundaries, and twin boundaries. Grain boundaries interrupt the slip of dislocations and are responsible for Hall-Petch strengthening,

\[ \sigma_{y} = \sigma_{0} + \frac{k_{y}}{\sqrt{d}}, \]

where \( d \) is the mean grain diameter. Fine-grained metals are therefore stronger and, usefully, also tougher than coarse-grained ones of the same composition.

The principal strengthening mechanisms for crystalline metals can now be named as a coherent set: grain-size refinement (Hall-Petch), solid-solution hardening, strain hardening, and precipitation hardening. Each works by placing obstacles in the path of dislocations so that a larger applied stress is required to make them glide.

Chapter 6: Mechanical Properties, Part 1

Mechanical testing under uniaxial tension produces the data from which most design allowables are drawn. Engineering stress is defined on the original cross-section and engineering strain on the original gauge length,

\[ \sigma = \frac{F}{A_{0}}, \qquad \varepsilon = \frac{\ell - \ell_{0}}{\ell_{0}}. \]

True stress and true strain are based on instantaneous dimensions,

\[ \sigma_{T} = \frac{F}{A_{i}}, \qquad \varepsilon_{T} = \ln\!\left(\frac{\ell_{i}}{\ell_{0}}\right), \]

and, assuming constant volume in the plastic range, the two pairs are related by \( \sigma_{T} = \sigma(1+\varepsilon) \) and \( \varepsilon_{T} = \ln(1+\varepsilon) \).

The elastic portion of the curve obeys Hooke’s law, \( \sigma = E\varepsilon \), where the Young’s modulus \( E \) is set by interatomic bonding and is therefore nearly insensitive to alloying or heat treatment. The lateral-to-axial strain ratio is Poisson’s ratio \( \nu \), typically 0.25 to 0.35 for metals. Departure from linearity marks the yield point; when yielding is gradual it is conventionally located by the 0.2 % offset method. Beyond yield, strain hardening raises the flow stress until the ultimate tensile strength is reached, after which a local neck forms and load falls to fracture. Ductility is reported as percent elongation \( \left(\ell_{f}-\ell_{0}\right)/\ell_{0} \) or percent reduction in area. The area under the engineering curve up to fracture is the toughness, a measure of energy absorbed per unit volume; the area up to yield is the modulus of resilience.

Hardness tests (Brinell, Vickers, Rockwell, Knoop) measure resistance to local plastic indentation. For most metals hardness correlates empirically with ultimate tensile strength, and because the tests are fast and nearly non-destructive they are the industrial workhorse for quality control.

Temperature and strain-rate sensitivity matter. Metals generally soften and become more ductile as temperature rises and as strain rate falls, because dislocation motion is thermally activated. Polymers are still more sensitive: above \( T_{g} \) they become rubbery, and at high rates they behave glassily.

Chapter 7: Mechanical Properties, Part 2 — Fracture, Fatigue, and Creep

Real parts rarely fail by reaching the tensile strength in a pristine specimen. They fail because of flaws. Fracture mechanics quantifies how a pre-existing crack of length \( a \) raises the local stress and triggers unstable propagation. For a through-thickness crack in a wide plate, the stress-intensity factor is

\[ K_{I} = Y\sigma\sqrt{\pi a}, \]

where \( Y \) is a geometry factor of order unity. Fast fracture occurs when \( K_{I} \) reaches the material’s plane-strain fracture toughness \( K_{IC} \), a true material property with units of \( \mathrm{MPa}\sqrt{\mathrm{m}} \). Rearranging gives the critical flaw size

\[ a_{c} = \frac{1}{\pi}\!\left(\frac{K_{IC}}{Y\sigma}\right)^{\!2}, \]

the largest crack a component can tolerate at a given stress. Ductile metals show dimpled, fibrous fracture surfaces and generous plastic zones at the crack tip; brittle materials show cleavage or intergranular facets and almost no plasticity. The Charpy and Izod impact tests capture the ductile-to-brittle transition of BCC steels, in which toughness collapses below a transition temperature — a lesson taught historically by Liberty-ship failures.

Under cyclic loading most metals fail by fatigue long before the static strength is reached. The classic S-N curve plots stress amplitude against cycles to failure. Ferrous alloys often exhibit a true endurance limit below which life is effectively infinite, whereas aluminium alloys do not and must be designed to finite life. Fatigue crack growth in the stable regime follows Paris’s law,

\[ \frac{da}{dN} = C\,(\Delta K)^{m}, \]

with \( \Delta K \) the stress-intensity range and \( C,m \) material constants. Surface finish, residual stress, and stress concentrations dominate fatigue life, so shot peening, fillet radii, and polishing are powerful remedies.

Creep is the slow, time-dependent plastic deformation that occurs at temperatures above roughly \( 0.4\,T_{m} \) (with \( T_{m} \) the absolute melting point). A creep curve shows primary, secondary (steady-state), and tertiary stages. The steady-state rate typically obeys

\[ \dot{\varepsilon}_{s} = A\,\sigma^{n}\exp\!\left(-\frac{Q_{c}}{RT}\right), \]

where \( Q_{c} \) is an activation energy close to that of self-diffusion. Turbine blades, boiler tubes, and solder joints are designed against creep rather than yield.

Chapter 8: Strain Hardening and Annealing

Cold working, which is plastic deformation below about \( 0.4\,T_{m} \), multiplies the dislocation density and makes further dislocation motion harder. The flow stress rises and ductility falls; a cold-rolled sheet is stronger and harder but less formable than its annealed predecessor. The percent cold work is defined as the fractional reduction in cross-sectional area.

Annealing restores ductility in three overlapping stages. Recovery relieves stored elastic energy as dislocations rearrange into low-energy cell walls, with only modest softening and little change in grain shape. Recrystallization nucleates strain-free grains that consume the deformed matrix; it takes place above a recrystallization temperature that is typically 0.3 to 0.5 of \( T_{m} \) and that depends on prior cold work. After full recrystallization, grain growth proceeds by curvature-driven boundary migration, with large grains eating small ones. Hot working exploits simultaneous deformation and recrystallization so that large strains can be imposed without work hardening — the basis of forging, hot rolling, and extrusion.

These ideas explain why an alloy’s service history matters as much as its nominal composition: the same chemistry can deliver very different strength-ductility combinations depending on the path through processing space.

Chapter 19: Electronic Properties and Semiconductors

Electrical conduction in solids is understood through band theory. Discrete atomic levels broaden into bands when atoms are brought together, and the filling of those bands at absolute zero determines the character of the solid. In metals the highest occupied band is partially filled, so an infinitesimal electric field sets electrons in motion and the conductivity is high. In insulators a completely filled valence band is separated from an empty conduction band by a large forbidden gap \( E_{g} \). In semiconductors the gap is small (roughly 0.5 to 3 eV for common cases), so thermal excitation generates a useful population of mobile electrons in the conduction band and equally mobile holes in the valence band.

The intrinsic carrier concentration rises steeply with temperature as

\[ n_{i} \propto T^{3/2}\exp\!\left(-\frac{E_{g}}{2k_{B}T}\right), \]

and conductivity is the sum of electron and hole contributions,

\[ \sigma = n e \mu_{e} + p e \mu_{h}, \]

where \( \mu_{e} \) and \( \mu_{h} \) are the electron and hole mobilities. Doping adds impurities that sit just below the conduction band (donors, giving n-type silicon when Si is doped with P or As) or just above the valence band (acceptors, giving p-type silicon when doped with B). Very dilute doping, parts per million, can change conductivity by many orders of magnitude, which is what makes semiconductor electronics possible. Bringing n- and p-type material into contact forms a p-n junction whose asymmetric current-voltage relation is the basis of diodes, bipolar transistors, photovoltaic cells, and light-emitting diodes. In metals, by contrast, resistivity rises with temperature as phonon scattering intensifies — the opposite sign from semiconductors, and a useful diagnostic when classifying an unknown sample.

Laboratory Themes

The hands-on component of the course reinforces four experimental ideas. Measuring the deflection of cantilever beams lets students extract Young’s modulus from first principles and confront measurement uncertainty using calipers, micrometers, and dial gauges. Tensile testing of metals produces engineering and true stress-strain curves from which yield, ultimate strength, ductility, and resilience are read off. Fracture and impact experiments connect the classroom fracture-mechanics equations to the real morphology of broken surfaces and to the ductile-to-brittle transition of ferritic steels. Finally, thermal-treatment experiments show how cold work, annealing, and quenching shift hardness and microstructure, closing the loop between processing, structure, and properties that the course set out to establish. Good lab practice — careful documentation, honest uncertainty, and clear written reporting — is as much a learning outcome as any equation in these notes.