Materials science

Mechanical properties

Stress

Stress (\sigma) is the measure of force applied to an object per unit of cross-sectional area. If the cross-sectional area changes along the force axis, the smallest area is used. The unit of stress is \N\m^{-2} or \unit{Pa}.

Strain

Strain (\epsilon) is the measure of deformation in a material, expressed as the relative change in length (absolute change in length \Delta L over the original length L_0). It is dimensionless.

Deformation

Elastic deformation is a temporary change to the shape of a material due to applied stress. The material returns to its original shape once the stress is removed.

Plastic deformation is a permanent change to the shape of a material. The material does not return to its original shape, even after the stress is removed. The amount of stress required before plastic deformation occurs is called the yield stress, or \sigma_y. The yield stress of titanium is 820\unit{GPa} and of aluminium is 240\unit{MPa}.

Poisson’s ratio

Poisson’s ratio \nu is a measure of how much deformation a material experiences perpendicular to the applied force (transverse deformation) when stretched or compressed (axial deformation). The \nu of most metals is around 0.3, and of most polymers is around 0.5.

Strength

The strength of a material is the amount of stress it can withstand before some limit is reached.

Yield strength is the amount of stress a material can withstand before it experiences plastic deformation. This is equal to the stress at the yield point. For most metals a precise yield point cannot be determined, so the 0.2% proof stress is used instead (found by tracing a line upwards from the 0.2% strain point with gradient E).

Ultimate tensile strength (UTS) is the amount of stress a material can withstand before fracturing. This is equal to the stress at the apex of a stress-strain graph. In a brittle material, this is also equal to the yield strength, because brittle materials will fracture instead of plastically deforming.

Toughness

The toughness of a material is the total amount of energy (work) per unit volume that it can dissipate/absorb before fracturing. Toughness is given by the total area underneath a stress-strain graph between the origin and the fracture point, and is measured in joules.

Ductility

Ductility is the amount of plastic deformation that occurs in a material before fracturing, measured as either percentage elongation or percentage reduction in area. Ductile materials tend to be tough, but not strong, and will form a ‘cup and cone’ shape when fractured.

A material is brittle if it experiences no plastic deformation before fracture. The resultant fragments can be collected and slotted back together without any gaps. Examples of brittle materials are ceramics and glass. Brittle materials tend to be strong, but not tough.

Fracture

Fracture is the point where one piece of material becomes many. A gap is formed in the material at the point of fracture due to sudden recovery of elastic deformation.

Stress-strain curve

Stress is tightly related to strain. A stress-strain graph plots the amount of strain experienced by a material as an increasing tensile stress is applied.

Young’s modulus

The initial strain in a material is due to elastic deformation. This elastic portion of the graph is a straight line, with steep slopes indicating a stiff material (steel, glass) and shallow slopes indicating a flexible material (polyethylene). The value of this slope is called Young’s modulus (E), and is measured in pascals. The lower the number, the more ‘elastic’ the material. The Young’s modulus of titanium is 100\unit{GPa}, and of aluminium is 70\unit{GPa}.

Yield point

A change in the gradient of the stress-strain graph indicates the yield point of the material, the point where the material stops undergoing further elastic deformation and begins to undergo plastic deformation. The stress at this point is called the yield stress (\sigma_y). If a precise yield point cannot be determined, the 0.2% proof stress is used instead (found by tracing a line upwards from the 0.2% strain point with gradient E).

Work hardening

Continuing to strain the material past this point will ‘work harden’ the material (yield strength of a material increases with plastic deformation). If stress is removed after this point, the graph will descend with gradient E as the elastic portion of deformation is recovered. The point at which stress was removed is the new yield point of the material, and the distance between the original L_0 and the new length at rest measures permanent deformation. The curve will continue along the original trajectory once stress is reapplied.

Discontinuous yield

Discontinuous yield is a phenomenon where the curve directly following the yield point is generally flat with a lot of random noise. After this point the curve will become smooth again, even after the material is unloaded and reloaded.

Ultimate tensile stress

The stress at the peak of the curve is called the ultimate tensile stress, or UTS. Necking (a localised reduction in cross-sectional area) occurs past this point before fracture. This explains the downwards slope past the UTS, where the reduction in cross-sectional area leads to an increase in strain (due to the Poisson effect).

Since the stress on the graph is measured based on the original cross-sectional area (engineering stress), this appears incorrectly on the graph as a reduction in stress when in reality the reduction in cross-sectional area and a lack of reduction in force would lead to an increase in stress.

Engineering stress vs true stress

Stress-strain curves show engineering stress, which is calculated using the original cross-sectional area of the material. In contrast, true stress is calculated using the real cross-sectional area (which changes during the test). Engineering stress is used because it’s simpler to calculate, and engineering projects should never come close to exceeding the yield stress of a material anyway (as this would permanently damage the material, impacting the integrity of the project).

Crystal structures

Crystallinity refers to the degree of long-range structural order in the atoms of a material. Solid metals are generally crystalline solids, with atoms arranged in regular repeating patterns.

Unit cells

We can represent a crystal lattice using a unit cell, the smallest tilable portion of the lattice.

The three most common unit cells for metals are the body-centered cubic (BCC), face-centered cubic (FCC), and hexagonal close packed (HCP). FCC and HCP are close-packed structures, because they have the greatest possible atomic packing factor.

Polymorphic materials can exist in more than one structure. Iron is BCC at room temperature, but FCC at temperatures exceeding 912°C.

• Atoms per cell
  The (fractional) number of atoms contained in the cell.
• Coordination number
  the number of atoms bonded to any atom in the lattice.
• Unit cell dimensions
  Edge length a of the unit cell relative to the radius R of the atoms. To calculate, choose a close-packed vector in the cell, count the number of radii along that edge and divide by the edge length (a\sqrt{2} for a planar diagonal, a\sqrt{3} for a cubic diagonal).
• Atomic packing factor
  The spherical packing ratio, given by the total atom volume over the cell volume, or atoms per cell times \frac{4}{3}\pi R^3 over a^3.

Ceramics

Ceramics are crystalline solids consisting of both metallic and non-metallic atoms and joined by a combination of ionic and covalent bonds. The crystalline structure is determined by the balance of charges (the resulting solid is electrically neutral) and by the relative sizes of ions (cations are positive and small, anions are negative and large). Cations can only bond with anions.

The two crystalline structures we’re looking at for ceramics are the rock salt structure (used by MgO, FeO, NaCl) and the silicate structure (used by rock, soil, clay, sand).

In the rock salt structure, each of the two atoms forms a separate face-centered cubic lattice, with the two lattices interpenetrating to form a three-dimensional checker pattern. The interstitial sites of one lattice are filled with the atoms of the other. Coordination number is 6 (each cation only touches six anions, along cardinal directions), unit cell dimension in a = 2(R_a + R_c). Materials are thermally and electrically insulating, probably due to a lack of free electrons.

The silicate structure is based on the \ce{SiO4^{4-}} tetrahedra (\ce{Si} cation is +4, \ce{O} anions are -2). \ce{SiO4^{4-}} is not stable, but multiple of them can link together into a crystal lattice and sharing oxygen atoms (called ‘bridging oxygens’) to create \ce{SiO2} (silicon dioxide / silica), which has a neutral charge. Examples are quartz and amethyst (quartz with iron impurities).

Glass is an example of an amorphous silicate. Whether a silicate structure becomes crystalline or glass depends on how fast it cools from the liquid state. Rapid cooling doesn’t give the atoms time to arrange into a stable lattice, and so creates glass.

Miller indices

Miller indices are a notation for describing a plane intersecting a unit cell. Each index is the reciprocal of a distance measured along an axis from an origin, where 0 means that the plane is parallel to that axis, -1 means that the intersection is one cell over, and \infty means that the plane intersects the origin on that axis (I think?). Planes repeat on a period of 1, so a negative index has the effect of reflecting the plane left-to-right along that axis (across the plane which bisects that axis in the cell). An index of 2 intersects the cell only halfway along the axis.

Planes are denoted (h\ k\ l). Families of planes are denoted \{h\ k\ l\}, containing all planes which can be transformed into (h\ k\ l) by a symmetry operation that would leave the crystal unchanged (these planes are ‘crystallographically equivalent’). Since we’re working with cubic cells, this means any combination of reflections in all three axial planes. More specifically, for a cubic crystal we just need to find all permutations of the indices including the positive and negative of each index. The family \{1\ 1\ 0\} includes the 12 planes \{\pm 1, \pm 1, 0\}, \{\pm 1, 0, \pm 1\}, \{0, \pm 1, \pm 1\}. For lattices with lower symmetry, the planes in the family would be more constrained.

A second similar notation is used for describing a ‘direction’ inside a unit cell (essentially a vector). Each index gives the distance along an axis from an origin where the direction intersects with the bounds of the cell, where 0 means that the direction is perpendicular to that axis. Indices are conventionally scaled to be integers (in other words, such that the greatest common divisor is 1), so the direction [1\ 0\ \frac{1}{2}] would become [2\ 0\ 1].

Directions within a cell are denoted [u\ v\ w]. Families of directions are denoted ⟨u\ v\ w⟩, containing all crystallographically equivalent directions in the same way as for planes.

Negative indices are denoted with an overbar, as in \{\bar{1}\ 0\ 1\}.

Planes in unit cells

We can draw how a plane intersects with a unit cell in order to see how closely-packed the cell is along that plane. Only draw atoms where the center lies on that plane. The close-packed directions within that plane are those where the atoms are touching all along the line. A close-packed plane is one where all directions are close-packed, or in other words, where there is no gap between adjacent atoms.

If the atomic arrangement is the same on multiple planes, those planes are crystallographically equivalent.

In FCC, \{1\ 0\ 0\} is close-packed in ⟨1\ 1\ 0⟩ directions. In BCC, \{1\ 1\ 0\} is most closely packed in ⟨1\ 1\ 1⟩ directions, but doesn’t have a true close-packed plane.

Lattice deformations

With elastic deformation, the bonds between the atoms stretch without breaking. This is analogous to spring deformation. The Young’s modulus E is an innate property of the material.

With plastic deformation, the bonds between the atoms break and the planes in the lattice slide across one another, settling and re-bonding at the next closest stable position (this is easier in metals because there are a lot of free electrons). This is called slip. The model of slip that we use in the course is called box slip, where we look at unit cells in the lattice as cubes in a grid. In order for block slip to occur, every atom on the plane has to break and slip simultaneously, which requires a lot of energy.

On a close-packed plane, the distance to the next stable position is small, so the amount of energy/work (and therefore force) required to slip is small. This results in a weaker, more ductile material along that plane. In comparison, non-close-packed planes result in stronger, more brittle materials along that plane. This means that FCC materials tend to be more ductile, and BCC materials tend to be more brittle. An HCP structure has all close-packed planes lying in one orientation, so HCP materials are very strong and brittle (slip is very difficult).

The main slip system of a lattice is the planes and directions in which slip (and therefore deformation) can most easily occur (these are the closest-packed planes and directions).

• The main slip system of an FCC lattice is \{1\ 1\ 1\}\ ⟨1\ 1\ 0⟩.
• The main slip system of a BCC lattice is \{1\ 1\ 0\}\ ⟨1\ 1\ 1⟩.

For rock-salt structures, plastic deformation will cause cations to move adjacent to other cations, causing planes to repulse one another and the material to fracture. Plastic deformation of this structure is improbable.

Defects

The theoretical shear strength of pure elemental iron (based on the energy required to break bonds) is 10000\unit{MPa}, but in practice the strength is around 20\unit{MPa}. This is due to defects in the material (notated as \perp).

One dimensional defects / point defects:

• Vacancies
  An atom is missing from the crystal lattice, leaving an unfilled gap.
• Substitutional atoms
  An atom is replaced with an atom of a different element in the lattice.
• Interstitial atoms
  A smaller atom of another element is jammed into the interstitial space of a lattice.

Two dimensional defects / planar defects:

• Edge dislocation
  An extra half-plane is inserted into the lattice.
• Screw dislocation
  The lattice splits and twists such that the top and bottom interfaces of the split no longer align.

Dislocations in a lattice reduce the amount of energy required to slip because there are now fewer atoms on the plane, meaning that there are fewer bonds that need to break before slip can occur. Less work is required to slip (corresponding directly to yield strength), and the material is less strong. Slip happens most easily on close-packed planes in close-packed directions.

Dislocations cannot occur in ceramics due to the strict structure enforced by bonds between cations and anions.

Grains

Most metals are crystalline, but are formed of many smaller interlocking lattices of different orientations (polycrystalline) rather than one single lattice. These individual lattices are called grains. Grains that are roughly spherical in shape are ‘equiax’, and grains that are longer and narrower are ‘elongated’.

Grains are formed when a metal is cooled from a molten state (molten metal is amorphous, atoms move freely past one another). When the metal begins to cool, some atoms cling to one another to form nuclei crystals, and as the metal cools further these crystals attract more atoms, eventually forming a voronoi pattern of grains.

Grain nucleation falls into two categories. Homogeneous nucleation is when atoms spontaneously cling to one another in the molten material, forming the basis for a spherical grain. This is very rare. Heterogeneous nucleation is where atoms nucleate on a surface (called a mould) creating a flatter grain across the surface, or on an atom of a different element. This is far more common. We’re told that this has to do with thermodynamics.

Grain boundaries are a form of imperfection. Atoms inside a grain are called bulk atoms, and atoms at a grain boundary are called surface atoms. All of the bonds of a bulk atom are formed (linked into the lattice on all sides), but a surface atom has ‘free’ bonds. This makes a grain boundary a high-energy region due to these free atomic bonds, which can bond with other atoms to reach a lower energy state.

Grain boundaries inhibit the movement of dislocations, because the lattices on each side of the boundary have different orientations (‘discontinuity in planes’). This means that materials with many smaller grains are strong and brittle (‘lower grain boundary density’), and materials with fewer larger grains are weak and ductile (dislocations can move much further).

Grain formation

An example used in class was the cross-section of a cast aluminium cylinder. To prepare the sample, molten 99.5% aluminium at 720°C (60°C above melting point) was poured into a mould made of 1” steel plate at room temperature and left to cool.

First, small equiax grains formed on the mould walls due to heterogeneous nucleation (called chill crystals). Next, columnar grains extended from the equiax grains, pointing towards the center of the mould. Finally, larger equiax grains formed in the center of the sample, occupying roughly 50% of the cross-sectional surface area. The aluminium contracted as it cooled, creating a large depression in what was the top surface.

There are multiple situations that can occur after the formation of columnar grains along the sample edges:

• Equiax grains form in the center, bounded by the walls of columnar grains (most desirable outcome)
• The walls of columnar grains keep extending until they meet in the center
• Holes or pores form in the center (least desirable outcome)
• Dendrites (columnar grains branch into tree structures) are formed

All of these situations are still non-optimal because the columnar grains make for an anisotropic material (a material that has different behaviour along different planes). What we most want is an isotropic material, one consisting entirely of equiax grains. This can be achieved by annealing the metal (heating then controlled cooling) or by using grain refiners (sprinkling Ti or Cd into Al to act as nucleation sites).

Metallography

Metallography is used to reveal the grain boundaries in a polycrystalline sample so that they can be seen by the naked eye.

• Slice a thin sample of the material
• Grind with silicon carbide sandpaper to flatten the surface (creating a scratched surface)
• Polish with diamond paste to remove scratches (creating a mirror finish)
• Etch with an acid or alkali to selectively corrode grain boundaries (we used 3% nital, nitric acid in ethanol). This corrodes the grain boundaries with a redox reaction.

Strengthening mechanisms

• Make grains smaller (dislocations inhibited by grain boundaries)

Hall-Petch equation

The Hall-Petch equation describes the relation between grain size and the yield strength of a material. \sigma_0 is the friction stress, the minimum stress required for dislocation movement to occur (or resistance of the lattice to dislocation motion), k is the strengthening coefficient or Hall-Petch constant, and d is the average grain diameter in millimeters. \sigma_0 and k are specific to each material.

When graphed, the x axis is \frac{1}{\sqrt{d}} (ranging from 0 to 0.3, smaller grains to the right), and the y axis is \sigma_y, with \sigma_0 being the y intercept. The gradient of the line is k and is positive. If we get smaller grains, we get an increase in yield stress.

Grain deformation

When a polycrystalline material is plastically deformed, the individual grains are also permanently deformed. Running a material through rollers will flatten equiax grains into elongated grains. The grain volumes do not change.

When dislocations start slipping, we get also get multiplication of dislocations. Dislocation density increases significantly. Dislocations inhibit the movement of other dislocations, so the material gets way stronger. At a point it becomes so strong that it has no ductility, and becomes brittle and breaks.

Cold working is when a material is plastically deformed at a temperature below its recrystallisation temperature. The amount of cold working that a material has experienced is given as a percentage (‘percentage of cold work’). This can be measured either as a change in thickness (when rolling a material) or a change in cross-sectional area (when drawing a material). t_0 is the original thickness, t_f is the final thickness, and then the equivalent for cross-sectional area. These equations rely on the assumption that the width of the material doesn’t change much.

Hardness

Hardness is the resistance of a material to localised plastic deformation.

The indentation test is used to determine the hardness of a material. To perform the test, an indenter of a harder material than the one being tested is pressed into the material sample using a known force, and then the indentation created is measured. The smaller the indent, the harder the material.

Hardness is related to strength because the material must be plastically deformed in order to leave an indentation, and plastic deformation only occurs once the yield strength of the material is exceeded. Hardness is usually proportional to strength for isotropic materials.

There are many different scales used for hardness testing, using different indenters and different loads. Each scale measures from 0 to 100. Measured values can’t be compared between scales. Geologists use the Moh’s scale, which measures from 1 to 10.

Electrical properties

Resistivity \rho is a material property that affects how easily electrons can move through the material. Resistivity depends on the crystal structure.

Conductivity \sigma is the reciprocal of resistivity. It shares the same symbol as stress, which is a little confusing.

Metallic crystal lattices are formed from a lattice of positive ion cores surrounded by a cloud of delocalised electrons that can move freely through the lattice. When a potential difference / electric field is placed across the lattice, these free valence electrons ratchet from ion core to ion core, moving towards the higher potential. As these valence electrons move, they accelerate to the next ion core, accelerating and then colliding and slowing down (losing energy and generating heat). The electrons don’t hit every ion core along the path: if the temperature of the material is lower, the ion cores will vibrate slower, and the paths travelled by each electron will be clearer.

The velocity of an electron moving through a metallic lattice can be modelled with a saw-wave graph, mapping electron drift velocity to time. Peaks on the graph indicate maximum electron velocity. Average drift velocity V_d is given by electron mobility \mu (a function of collision rate, affected by atomic spacing and temperature) and electric field strength E:

Resistivity is non-linear with temperature.

Ceramics and glasses are very good insulators. There is no ‘sea of electrons’, all electrons in the lattice are tied up in covalent and ionic bonds.