Topic 7: Materials

Materials Overview

• Deforming materials: Understanding how materials respond to applied forces.

• Stress and strain: Quantifying the deformation of materials under load.

• Young’s modulus: Measures a material's resistance to stretching.

• Shear modulus: Measures a material's resistance to shear stress.

• Bulk modulus: Measures a material's resistance to uniform compression.

• Thermal expansion: How materials change in size with temperature changes.

• Phases of matter: Solid, liquid, and gas states and their properties.

Applying Forces to Materials

• Examples of forces applied to materials:

  • Stretching a rubber band or bungee rope.

  • Bending sheet metal like a car body panel.

  • Predicting the weight a bridge, lift cable, or floor can withstand.

  • Squeezing rocks to produce layered strata.

• Applying a force results in a measurable deformation.

Deformation of Materials

• Normal Stress: Force applied perpendicularly to an area.

  • Compressional stress: Stress that decreases the volume of the material.

  • Tensional stress: Stress that increases the length of the material.

  • $\sigma = \frac{F}{A}$, where $\sigma$ is stress, $F$ is force, and $A$ is area.

• Shear Stress: Force applied parallel to a surface area.

  • $\text{Shear Stress} = \frac{F}{A}$, where $F$ is the force applied parallel to the surface and $A$ is the area of the surface.

Stress and Strain

• Stress: Tensile (stretching) force $F$ applied to a rope of cross-sectional area $A$.

  • $\text{stress } \sigma = \frac{\text{Force } F}{\text{Area } A}$

  • Units of stress are $N \cdot m^{-2}$ (Pascals, Pa), equivalent to pressure units.

• Strain: Fractional change in length due to stress.

  • $\text{strain } \epsilon = \frac{\text{displacement } e}{\text{Length } l}$

  • Strain is dimensionless.

Young’s Modulus

• Young’s modulus is an inverse measure of stretchiness.

  • A metal cable will undergo less strain than a bungee rope of the same area, length, and force applied.

• For small stresses, stress is proportional to strain:

  • $\sigma \propto \epsilon$

• Constant of proportionality is Young’s modulus, $E$.

  • $\sigma = E \epsilon$

  • $E = \frac{\sigma}{\epsilon}$

Young's Modulus (Continued)

• Young's modulus: Property of a material that indicates how easily it stretches and deforms.

• Key point: relates Force $F$ to fractional-extension $\frac{e}{l}$

  • $E = \frac{\sigma}{\epsilon} = \frac{F/A}{e/l} = \frac{Fl}{eA}$

• Units of Young's modulus are $N \cdot m^{-2} = Pa$, the same as stress.

• Values of Young's modulus:

  • Steel: $21 \times 10^{10} Pa = 210 GPa$ (very low stretch)

  • Copper: $130 GPa$

  • Glass: $70 GPa$

  • Polyethylene: $5 GPa$

  • Rubber: $0.05 GPa$ (very stretchy)

  • Quartz Crystal (SiO2): $76-97 GPa$

  • Diamond (C): $1050-1210 GPa$

  • Graphene and carbon nanotube (C): $+1000 GPa$

Example

• A 100g weight is hung on a 1.5m length of copper wire with a cross sectional area of 0.1$\text{mm}^2$. By how much is the wire stretched? For copper Young’s modulus is 130GPa.

• Use $\sigma = E \epsilon$, so $\epsilon = \frac{\sigma}{E}$

• $\sigma = \frac{F}{A} = \frac{mg}{A} = \frac{0.1 \text{ kg} \times 9.8 \text{ ms}^{-2}}{0.1 \text{ mm}^2 \times 10^{-6} \text{ m}^2/\text{mm}^2} = 9.8 \times 10^6 \text{ Nm}^{-2}$

• $\epsilon = \frac{\sigma}{E} = \frac{9.8 \times 10^6 \text{ Nm}^{-2}}{130 \times 10^9 \text{ Pa}} = 7.5 \times 10^{-5}$

• Extension $e = \epsilon l = 7.5 \times 10^{-5} \times 1.5\text{m} = 1.13 \times 10^{-4} \text{ m}$

Shear Modulus

• Shear modulus is like Young’s modulus but for “shear”

  • Shear Modulus = $\frac{\text{shear stress}}{\text{shear strain}}$

• Shear stress = force $F$ applied to top face of block area $A$ of top face of block.

• Shear strain = $\tan \theta$

• $\frac{\Delta x}{l} = \tan \theta$

Seismic Waves (S- and P-waves)

• Seismic waves are used to study the Earth's internal structure.

• Waves are refracted (bent) as they pass through different layers.

• Wave speed depends on density, allowing mapping of density changes with depth.

• Earth is composed of layers: inner core, outer core, mantle.

• P-waves: refracted at core-mantle boundary, create shadow zones.

• S-waves: do not reappear, indicating liquid outer core.

Seismic waves (S- and P- waves) Continued

• After an earthquake, seismic waves travel through the Earth’s interior.

• Two types: primary (P-waves) and secondary (S-waves).

  • P-waves (Primary):

    • Longitudinal.

    • Speed depends on compressibility of the medium (rock) but faster than S-waves.

    • Can travel in solids and liquids.

  • S-Waves (Secondary):

    • Transverse.

    • Speed depends on different properties of medium (rock elasticity / stiffness).

    • Cannot travel in liquids (rapid diffusion).

    • Not propagated through Earth’s core.

Shear Modulus Continued

• The velocity of a shear wave, $v_s$, depends on the shear modulus

  • $v_s = \sqrt{\frac{G}{\rho}}$; $G$ = shear modulus, $\rho$ = solid’s density

• Typical values for shear modulus

  • Diamond 478 GPa

  • Steel 79 GPa

  • Glass 26 GPa

  • Polyethylene 0.117 GPa

  • Rubber 0.0006 GPa

Bulk Modulus (uniform compression)

• A volume of a substance is compressed by a change in external pressure; volume reduces (e.g. waste material in a landfill site).

• Bulk strain: relative change in volume $\frac{\Delta V}{V}$ (where $V$ is the original volume).

• Bulk stress: increase in external pressure $\Delta p$

• Bulk modulus: $K = \frac{\text{bulk stress}}{\text{bulk strain}} = - \frac{\Delta p}{\frac{\Delta V}{V}} = -V \frac{\Delta p}{\Delta V}$

• Values of bulk modulus:

  • Steel 160 GPa

  • Glass 35-55 GPa (note the minus sign indicates compression)

  • Water 2.2 GPa

Example - Bulk Modulus

• A steel bar with a volume of $10^{-4} \text{ m}^3$ at the surface is taken to a depth of 1 km in the ocean. By how much does its volume change?

• $\Delta p = \rho gh = 1000 \text{ kg m}^{-3} \times 9.8 \text{ ms}^{-2} \times 1000 \text{ m} = 9.8 \times 10^6 \text{ Pa}$

• $\Delta V = -V \frac{\Delta p}{K} = - 10^{-4} \text{ m}^3 \times \frac{9.8 \times 10^6 \text{ Pa}}{160 \times 10^9 \text{ Pa}} = -6.1 \times 10^{-9} \text{ m}^3$

Compressibility

• Compressibility = $\frac{1}{\text{bulk modulus}}$

• If you want to work out how much you can squash into a landfill site, the compressibility of various waste materials is relevant.

• The velocity of a pressure wave, $v_p$, depends on both the bulk modulus $K$ and the shear modulus $G$

  • $v_p = \sqrt{\frac{K + \frac{4}{3} G}{\rho}}$

Stiffness of Materials

• Young's modulus describes the material's response to uniaxial stress.

  • Examples: pulling on the ends of a wire or putting a weight on top of a column.

• Shear modulus describes the material's response to shear stress.

  • Example: cutting it with dull scissors.

• Bulk modulus describes the material's response to uniform pressure.

  • Examples: the pressure at the bottom of the ocean or a deep swimming pool.

Thermal Expansion

• Solids and liquids usually expand when heated.

• Different materials expand by different amounts.

• Expansion occurs in all directions.

• Water is unusual – it expands when it freezes → burst pipes, freeze-thaw weathering

• Physics laws for thermal expansion similar to those we’ve explained for extension of elastic materials

Linear Thermal Expansion (1 dimension)

• Examples: mercury thread in thermometer, metal rod or bar

• $L$ = original length, $\Delta L$ = increase in length, $\Delta T$ = change in temperature

• $\alpha$ is the coefficient of linear thermal expansion.

• Very similar equation to that for Young’s modulus ($E$)

  • tensile strain, $e = \frac{\Delta L}{L} = (\frac{1}{E})\sigma = (\frac{1}{E})(\frac{\text{Force}}{\text{cross-sec-area}})$

  • thermal expansion, $\frac{\Delta L}{L} = \alpha \Delta T$ (change-in-temperature)

  • $\alpha = \frac{\Delta L}{L} \frac{1}{\Delta T}$

Coefficient of linear thermal expansion, $\alpha$

• $\alpha$ has units of K$^{-1}$

• $\Delta L=L_{T}-L_0\rightarrow L_{T}=L_0\left(1+\alpha\Delta T\right)$

  • where L$_{0}$ is the length after a temperature rise $\Delta T$

  • (note: in reality, $\alpha$ varies slightly with temperature but this can be neglected in most applications because small across temperature range in the environment).

  • $\frac{\Delta L}{L} = \alpha \Delta T$

Values of linear thermal expansion (at 20°C)

• Steel 1.1 to 1.3 x 10-5 K$^{-1}$

• Copper 1.7 x 10-5 K$^{-1}$

• Brass 1.9 x 10-5 K$^{-1}$

• Concrete 1.2 x 10-5 K$^{-1}$

• Water 6.9 x 10-5 K$^{-1}$

• Rubber 7.7 x 10-5 K$^{-1}$

• Oak 5.4 x 10-5 K$^{-1}$

• Diamond 1.0 x 10-6 K$^{-1}$

Example of Linear Thermal Expansion

• A 0.5m long brass rod is heated from 20°C to 100°C. By how much does the length of the rod change?

• Use $\Delta L = L \alpha \Delta T$

  • = 0.5 m x 1.9 x 10-5 K$^{-1}$ x 80 K

  • = 7.6 x 10-4 m

Bimetallic strip

• Bimetallic strip made by joining together 2 different metals with different linear thermal expansion coefficients

• Bends when heated

• Uses:

  • Thermometer

  • In thermostats

  • Temperature compensation in clocks

  • Circuit breakers

Coefficient of volume expansion $\gamma$

• $\gamma$ is the fractional increase in volume per unit temperature rise

• $V<em>T = V</em>0 (1 + \gamma \Delta T )$

  • where $V<em>0$ is the original volume and $V</em>T$ is the volume after a temperature rise $\Delta T$

• $\gamma = 3\alpha$

Thermal expansion - sea level rise

• Thermal expansion of water contributes to sea level rise.

• IPCC projections for 21st-century sea level rise include contributions from:

  • Thermal expansion

  • Melting glaciers

  • Melting ice sheets (Greenland and Antarctica)

• Different Representative Concentration Pathways (RCPs) predict varying levels of sea level rise.

Density change with temperature

• A liquid has a density $\rho_0$ at a certain temperature

• If the temperature rises by T degrees then the density decreases to $\rho_{_{\theta}}$

• $\rho_{0_{}}=\rho_{\theta}\left(1+\gamma T\right)$

• density change of a liquid as it expands relates to its coefficient of thermal expansion $\rightarrow$ that’s because density is mass/volume and volume increases as liquid expands.

Expansion of water

• Water has its maximum density at 4ºC

• Cool from 20ºC to 4ºC → contracts

• Cool from 4ºC to 0ºC → expands

• Becomes solid at 0ºC

• Below 0ºC → contracts slightly

• 100 cm3 of water becomes 109 cm3 of ice.

• Densities:

  • ice at 0 °C: 0.917 g/cm3

  • water at 0 °C: 0.9998 g/cm3

  • ice at −180 °C: 0.934 g/cm3

Phases of matter

• Inter-atomic / inter-molecular forces pull atoms / molecules together.

• Motion due to internal energy pushes them apart

• Balance between these two forces determines whether the material is solid, liquid or gas.

Solids, liquids and gases

• Solids

  • Atoms / molecules vibrate about equilibrium positions

  • Well ordered, packed together

• Liquids

  • Atoms / molecules can partly overcome the interatomic / intermolecular forces

  • Move randomly but can't leave the liquid

  • Less ordered but still closely packed

• Gases

  • Atoms / molecules move randomly at high speeds

  • Atoms / molecules are further apart and fill available space

Types of solid

• Crystalline

  • Most solids, all metals, many minerals

  • Regular, repeating structures, ordered throughout the material

  • Polycrystalline solids are a mass of tiny crystals at various angles (e.g. metals)

• Amorphous / glassy

  • More disordered structure

  • Short-range order only

  • E.g. glass

Recycling metals

• Economics of recycling metals depends on:

  • purity of scrap metal

  • cost of extraction of the metal (e.g. Al is very expensive to extract from its ore)

• Recycling is straightforward if there is a main use for a given metal

  • e.g. photographic film for silver

  • e.g. car batteries for lead

Metal alloys

• Pure metals are often alloyed with other metals or non-metals to improve mechanical properties (e.g. 0.002-2% carbon added iron → steel for stronger/tougher metal)

• But alloying can create problems :

  • Some elements in alloys (e.g. Al) can be leached (e.g. environmental hazard from slag heap)

  • Some elements (e.g. Cu and Sn) remain in the alloyed metal which makes re-cycling difficult.

Polymer materials (chained molecular structure)

• Many polymer materials have variable structure some regions semi-crystalline, others amorphous

  • semi-crystalline implies its structure is ordered (where polymer chains in parallel) → rigid, fairly strong material

  • amorphous means it has disordered regions where polymer chains are tangled → here the material is soft or flexible

• For many applications flexibility is beneficial whereas in others it is a potential weakness.

Thermo-plastics and thermoset-plastics

• Thermoplastics are materials that can be softened by heating and re-moulding (e.g. polyethene, PVC)

• By contrast, Thermosetting plastics are chemically cross-linked to avoid re-moulding – retain shape.

  • Strong chemical bonds are introduced between polymer chains (e.g. vulcanise = sulphur in rubber)

  • Cross-linking polymers acts to strengthen material à less temperature softening → prevent remoulding (e.g. ebonite, bakelite, melamine, Formica)

Strategies for Stronger, Stiffer Materials

• Increase degree of crystallinity

• Stretch material to produce more ordered material (untangles and aligns molecules)

• Adjust chain length, chain branching

• Chemical cross-linking

• Produce polymer chains which are themselves stiff (e.g. by using a monomer* with a rigid ring structure)

  • *a molecule that may bind chemically to other molecules to form a polymer

Stronger, stiffer, tougher polymers

• Co-polymers: polymerize different monomers together → co-polymer with different properties from either polymer type

• Polymer blends: mixture of two types of polymer chain (e.g. synthetic rubber with PVC to give a rubber toughened polymer)

• Implications of chemical cross-linking, co-polymers and blends for recycling polymers?

Summary

• Deformation of materials (elastic properties)

  • Stress and strain – Young’s modulus

  • Shear modulus

  • Bulk modulus

• Thermal expansion

• Phases of matter

• Types of solid

• Engineering materials for particular applications

  • Alloying metals and strengthening plastics