Topic 4: Materials

 1. Density:

• Density is a fundamental bulk property of solids, defined as the mass per unit volume of a substance.
• It’s a measure of how tightly packed the atoms or molecules are in a material.
• Density is typically denoted by the symbol ρ (rho).
• Usually expressed in units of mass per unit volume, such as grams per cubic centimeter (g/cm³) or kilograms per cubic meter (kg/m³).
• Formula:
• [math] \text{Density}(\rho) = \frac{\text{mass (m) }}{\text{volume (V)}} \qquad (1)[/math]
• Liquids and gases expand much more than solids when they are heated, so a fixed mass of fluid occupies a bigger volume than the solid form and so its density is reduced.
• Liquids are generally considered to be incompressible, but gases are readily squeezed (try putting your finger over the outlet of a bicycle pump and pushing in the handle).
• Because of this, the pressure needs to be stated in addition to the temperature when the density of a gas is quoted.
• Table 1 Densities of some fluids
• 

2. Upthrust in fluids:

• Upthrust in fluids is a fundamental concept in physics and engineering. Also known as buoyancy, it’s the upward force exerted by a fluid (such as a liquid or gas) on an object that’s partially or fully submerged in it.
• The magnitude of the upthrust force depends on the:
  1. Density of the fluid
  2. Volume of the displaced fluid (the amount of fluid pushed out of the way by the object)
  3. Acceleration due to gravity (g)
• The buoyancy force is equal to the weight of the displaced fluid. This is known as Archimedes’ Principle.
• Upthrust is responsible for many interesting phenomena, such as:
  – Objects floating or sinking in water
  – Ships and boats able to carry heavy loads
  – Submarines diving and surfacing
  – Helium-filled balloons rising in air
• Consider a cylinder immersed in a liquid, as shown in Figure 1. The upthrust is the difference between the force due to water pressure at the bottom of the cylinder, [math]F_2[/math] and that at the top, [math]F_1 [/math].
• For a fluid of density ρ:
• [math] F_1 = p_1 A = h_1 \rho g A \\
  F_2 = p_2 A = h_2 \rho g A \\
  U = F_2 – F_1 \\
  U = (h_2 – h_1) \rho g A \\
  U = (h_2 – h_1) A \rho g \\
  V = (h_2 – h_1) A [/math]
• So,
• [math] U = Vρg \qquad (2) [/math]
• By equation 1
• [math] ρ = \frac{m}{V} [/math]

• Vρ = m put in equation 2

• [math] U= mg [/math]
• The upthrust is equal to the weight of the displaced fluid.
• 

  Figure 1 Upthrust on a cylinder.

• This result is often stated as Archimedes’ principle.

3. Moving fluids-streamlines and laminar flow:

• Streamlines and laminar flow are two related concepts in fluid dynamics:
• Streamlines:
  – Streamlines are imaginary lines that represent the path of fluid particles in a flow field.
  – They are used to visualize the direction of fluid motion and can be obtained experimentally or computationally.
  – Streamlines are useful for understanding fluid behavior, identifying patterns, and analyzing flow structures.
• Laminar Flow:
  – Laminar flow is a type of fluid flow characterized by smooth, parallel layers of fluid that slide over each other without mixing.
  – In laminar flow, the fluid particles move in straight lines, and the flow is predictable and stable.
  – Laminar flow is often associated with low velocities, high viscosities, and small Reynolds numbers (a dimensionless quantity that predicts flow behavior).
• Key features of laminar flow:
  – Smooth, continuous motion
  – No turbulence or mixing between layers
  – Predictable and stable behavior
  – Low velocities and high viscosities
  – Small Reynolds numbers
• Streamlines in laminar flow:
  – In laminar flow, streamlines are straight and parallel, reflecting the smooth and predictable nature of the flow.
  – Streamlines can be used to visualize the laminar flow pattern and identify any disturbances or deviations from the expected behavior.
• Figure 2b shows that the flow of water in the pipe is laminar at low rates of flow, but that turbulence occurs when the rate reaches a critical level (see Figure 2c).
• This rate of flow depends on the speed of flow, the radius of the tube, the density and viscosity of the fluid.
• Figure 2 Laminar and turbulence flow
• Laminar flow is an important consideration in fluid motion.
• The uplift on an aero plane’s wings is dependent on laminar flow, and passengers experience a rocky ride when turbulent conditions are encountered.
• Similarly, the drag forces on a motor car are affected by turbulence, and wind tunnels are used to observe the nature of the air flow over prototype designs.
• The efficiency of fluid transfer through tubes is greatly reduced if turbulence occurs, so the rate of flow of oil and gas must be controlled so that the critical speed is not exceeded.

• ⇒ Viscosity:

• Viscosity is the measure of a fluid’s resistance to deformation at a given rate. It’s a key property in fluid dynamics, describing the “thickness” or “stickiness” of a fluid. Viscosity is typically denoted by the symbol μ (mu).
• Types of viscosity:
  1. Dynamic viscosity (μ): Measures the resistance to shear stress (tangential force).
  2. Kinematic viscosity (ν): Measures the ratio of dynamic viscosity to density (ν = μ / ρ).
• Units:
  – Dynamic viscosity: Poise (P), Pascal-seconds (Pa·s)
  – Kinematic viscosity: Stokes (St), Square meters per second (m²/s)
• Effects of viscosity:
  1. Resistance to flow
  2. Energy dissipation
  3. Boundary layers
  4. Turbulence

• ⇒ Stokes’ law

• Stokes’ law describes the force (F) required to move a spherical object through a viscous fluid at a constant velocity (v):
• [math] F = 6πμrv [/math]
•  Where:
  – μ is the dynamic viscosity of the fluid
  – r is the radius of the sphere
  – v is the velocity of the sphere
• This law applies to:
  – Creeping flow (low Reynolds number)
  – Spherical objects (e.g., balls, droplets)
  – Viscous fluids (e.g., oil, honey)
• Stokes’ law has many applications:
  1. Fluid dynamics
  2. Particle sedimentation
  3. Viscosity measurements
  4. Drag calculations
  5. Biological systems (e.g., cell movement, blood flow)
• Stokes’ law has limitations:
  1. Small spherical objects: Stokes’ law assumes the object is spherical and much smaller than the characteristic length scale of the flow.
  2. Low speeds: The equation is valid only for low velocities, where inertial effects are negligible.
  3. Laminar flow: Stokes’ law assumes laminar flow, meaning the fluid flows smoothly and predictably, without turbulence.
  4. Temperature dependence: Viscosity (μ) is indeed temperature-dependent, typically decreasing with increasing temperature.
• By recognizing these limitations, you can apply Stokes’ law appropriately and consider additional factors when dealing with more complex scenarios.
• Some extensions and generalizations of Stokes’ law include:
  – Oseen’s correction: Accounts for inertial effects at higher Reynolds numbers.
  – Faxén’s correction: Includes the effects of nearby boundaries.
  – Non-Newtonian fluids: Stokes’ law can be modified for fluids with non-linear stress-strain relationships.

• ⇒Terminal velocity:

• Terminal velocity is the maximum velocity an object reaches as it falls through a fluid (like air or water).
• At this point, the force of gravity pulling the object down is balanced by the frictional force (like air resistance) pushing the object up.
• Terminal velocity depends on:
  1. Mass of the object
  2. Shape and size of the object
  3. Density of the fluid
  4. Viscosity of the fluid
• When an object reaches terminal velocity:
  – The net force acting on the object is zero
  – The object no longer accelerates
  – The velocity remains constant
• Examples of terminal velocity:
  – Skydivers: Reach terminal velocity around 120-140 mph (193-225 kph)
  – Raindrops: Typically around 10-15 mph (16-24 kph)
  – Feather: Much slower, around 0.3-1.5 mph (0.5-2.4 kph)
• Terminal velocity plays a crucial role in various fields, including:
  – Aerodynamics
  – Hydrodynamics
  – Meteorology (study of precipitation)
  – Biology (animal movement and migration)

• ⇒Measuring viscosity using Stokes’ law:

• By measuring the terminal velocity of a sphere falling through a fluid it is possible to determine the coefficient of viscosity of the fluid.
• For a sphere of radius r and density  falling through a fluid of density  and viscosity η with a terminal velocity v, the following equilibrium equation:
• [math] U + W = F [/math]
• where
• U = weight of displaced fluid [math] = m_{f} g = Vρ_f g = \frac{4}{3} πr^3 ρ_f g [/math]
• F = viscous drag [math] = 6πηrv [/math]
• W = weight of sphere [math] = m_s g = Vρ_s g = \frac{4}{3} πr^3 ρ_s g [/math]
• can be written:
• [math] \frac{4}{3} πr^3 ρ_f g + 6πηrv = \frac{4}{3} πr^3 ρ_s g [/math]
• which gives the viscosity as:
• [math] η =\frac{ 2(ρ_s– ρ_f)gr^2 }{9v} [/math]
• This also tells us that the terminal velocity of a falling sphere in a fluid depends on the square of its radius, so very small drops of rain – and the minute droplets from an aerosol – fall slowly through the air.

4. CORE PRACTICAL 4: Use a falling-ball method to determine the viscosity of a liquid.

• In this practical, you’ll use a falling-ball method to determine the viscosity of a liquid. Here’s a step-by-step guide:
• Materials:
  – A tall, clear plastic or glass tube or cylinder
  – A metal or glass ball (e.g., a steel or glass marble)
  – A liquid with unknown viscosity (e.g., vegetable oil, honey, or glycerol)
  – A ruler or measuring tape
  – A stopwatch or timer app
  – A calculator

   

  
  Figure 3 a falling-ball method to determine the viscosity of a liquid.

• Procedure:
  1. Fill the tube with the liquid, leaving about 10-15 cm at the top.
  2. Drop the ball into the liquid, making sure it’s fully submerged.
  3. Measure the time it takes for the ball to fall a certain distance (e.g., 10 cm).
  4. Repeat step 3 several times to ensure accurate results.
  5. Use Stokes’ law to calculate the viscosity (μ) of the liquid:
  6. [math]  \mu = \frac{2}{9} (\rho_{\text{ball}} – \rho_{\text{liquid}}) g R^2  / (9 \eta) [/math]
• where:
  [math] \rho_{ball} [/math]= density of the ball
  [math] \rho_{liquid} [/math]= density of the liquid
  g = acceleration due to gravity ([math] 9.8 m/s^2[/math)
  R = radius of the ball
  η = measured time
• Compare your result with the known viscosity of the liquid (if available).
• Tips and Variations:
  – Use a ball with a known density and size.
  – Ensure the liquid is at room temperature.
  – Measure the time it takes for the ball to fall different distances (e.g., 5 cm, 10 cm, 15 cm).
  – Use a high-speed camera or photogate to measure the ball’s velocity.
  – Investigate how temperature affects viscosity.
• By following this practical, you’ll gain hands-on experience with measuring viscosity and understanding the relationship between viscosity, density, and gravity.

5. Hooke’s law:

• Hooke’s Law is a fundamental principle in physics that describes the relationship between the force applied to a spring and its resulting deformation. Here’s a more detailed explanation:
• Statement:
• “Hooke’s Law states that the force (F) required to stretch or compress a spring by a distance (x) is proportional to the distance of stretching or compressing”.

  
  Figure 4 mass-spring system

• Mathematical Formulation:
• [math] F = k∆x [/math]
• Where:
  – F is the force applied to the spring (in Newtons, N)
  – k is the spring constant (stiffness of the object) (in Newtons per meter, N/m)
  – ∆xis the displacement of the spring from its equilibrium position (in meters, m)
• Assumptions:
  – The spring is ideal, meaning it has no mass, no friction, and no damping.
  – The displacement (x) is small compared to the spring’s length.
  – The spring is subjected to a single force (F) applied along its axis.
• Characteristics:
  – Linearity: The force (F) vs. displacement (x) graph is a straight line.
  – Proportionality: The force (F) is directly proportional to the displacement (x).
  – Spring Constant (k): Depends on the spring’s material, geometry, and stiffness.
• Applications:
  – Mechanical systems (springs, elastic materials)
  – Electrical systems (capacitors, inductors)
  – Biological systems (tissues, cells)
• Limitations:
  – Hooke’s Law only applies to small displacements (x).
  – Real springs may have non-linear behavior, friction, or damping.

6. Stress and strain: the Young modulus:

• ⇒Stress

• Stress is a measure of the force per unit area on an object. It’s a 2nd rank tensor, meaning it has both magnitude and direction. There are two main types of stress:
  1. Normal stress (σ): perpendicular to the surface
  2. Shear stress (τ): parallel to the surface
• Formula:
• [math] \text{Stress} = \frac{\text{Force}}{\text{Area}} \\
  \sigma = \frac{F}{A} [/math]
• Units: Pascals (Pa) or Newtons per square meter (N/m²)
• Stress can cause:
  – Deformation (elastic or plastic)
  – Fracture (material breakage)
  – Failure (loss of functionality)

• ⇒Strain

• Strain is a measure of the deformation of an object, describing how much an object changes shape due to stress. It’s a dimensionless quantity, often expressed as a percentage or decimal.
• Types of strain:
  1. Tensile strain: stretching (positive)
  2. Compressive strain: compressing (negative)
  3. Shear strain: sliding (parallel)
• Strain is calculated as:
• [math]\text{Strain} = \frac{\text{change in length}}{\text{original length}} \\
  \varepsilon = \frac{\Delta l}{l_0} [/math]
• where:
  ε = strain
  [math] \Delta l [/math]= change in length
  [math] l_o [/math]= original length
• Key strain-related concepts:
  – Elastic strain: reversible deformation (returns to original shape)
  – Plastic strain: irreversible deformation (permanent change)
  – Strain rate: rate of change of strain
  – Strain energy: energy stored due to deformation
• Strain is essential in understanding material behavior, designing structures, and analyzing mechanical systems.

• ⇒The Young modulus

• The Young modulus (E) is a measure of a material’s stiffness, describing how much stress (force per unit area) is required to produce a corresponding strain (deformation).
• It’s a fundamental property in mechanics of materials.
• Young’s modulus is defined as:
• [math] \text{Young’s Modulus} = \frac{\text{Stress}}{\text{Strain}} \\
  E = \frac{\sigma}{\varepsilon} [/math]
• where:
  E = Young’s modulus
  σ = stress (force per unit area)
  ε = strain (deformation)
• Units: Pascals (Pa) or Newtons per square meter (N/m²)
• Typical values of Young’s modulus:
  – Steel: 200-210 GPa (gigapascals)
  – Aluminum: 68-78 GPa
  – Copper: 110-120 GPa
  – Wood: 10-15 GPa
• The Young modulus is used to:
  – Predict stress and strain behavior
  – Design structures and components
  – Select materials for specific applications
  – Analyze material properties

7. Force–extension graph:

• It is not always possible to use the spring set-up to investigate the tensile properties of materials.
• Metals, for example, will often require very large forces to produce measurable extensions, and so different arrangements or specialized equipment are needed.

• ⇒Natural rubber

• A force–extension graph for a rubber band can be obtained in a similar manner to the experiment with the spring, but an alternative method is shown in Figure 5 (a).
• Safety note: It is important that safety glasses are worn to reduce the risk of eye damage when the band breaks.
• It is best to use a short, thin rubber band, as it will stretch to several times its original length and thick bands are difficult to break.
• To alter the length of the band, the boss is loosened and moved up the stand. The force at each extension is read off the newton-meter.
• The rubber band stretches very easily at first, reaching a length of three or four times its original value (Figure 5 (b)). It then becomes very stiff and difficult to stretch as it approaches its breaking point.
• Natural rubber is a polymer.
• It contains long chains of atoms that are normally tangled in a disordered fashion (like strands of spaghetti). Relatively small forces are needed to ‘untangle’ these molecules so a large extension is produced for small loads.
• When the chains are fully extended, additional forces need to stretch the bonds between the atoms, so much smaller extensions are produced for a given load; the rubber becomes stiffer.
• It should be noted that the band returns to its original length if the force is removed at any stage prior to breaking – that is, the rubber band is elastic.

Figure 5 (b) Force–extension curve for a rubber band.

• ⇒Force–compression graphs:

• Up to now we have considered only the behavior of materials that have been subjected to stretching – or tensile forces.
• If weights were placed on top of a large rectangular sponge, the sponge would be noticeably squashed and force–compression readings easily measured.
• This is much more difficult for metals.
• Whereas long thin samples of copper wire can be extended by several centimeters with tensile forces of 50N or less, the shorter, thicker samples needed for compression tests require much larger forces to produce measurable compressions.
• In engineering laboratories, large hydraulic presses are used, but a school’s compression testing kit, shown in Figure 6, can demonstrate the effects of compression on a range of sample materials.
• The sample is placed into the press and the screw is tightened to hold it firmly in position.
• Force and position sensors are connected to a display module or to a computer via a data-processing interface and the readings are zeroed.
• The sample is compressed by rotating the lever clockwise.
• Figure 6 Compression testing kit
• A series of values of force and the corresponding compression is taken and a force–compression graph drawn.
• The elastic region is similar to that seen in the tensile tests.
• The bonds are ‘squashed’ as the atoms are pushed together, and the particles move back to their original position when the force is removed.
• Plastic behavior is much more difficult to examine as the samples twist and buckle at failure.
• Some kits allow the sample to be stretched as well as compressed, so that a comparison may be made.

• ⇒ Elastic and plastic behavior during stretching:

• Figure 7 shows a steep linear region followed by a region of large extension with reducing force.

  
  Figure 7 Force–extension curve for copper wire.

• In the initial section (O–A), the extension is proportional to the applied force, so Hooke’s law is obeyed.
• If the load is removed from the wire up to the limit of proportionality, or even a little beyond this, the wire will return to its original length.
• This is known as the elastic region of the extension, in which loading and unloading are reversible. Arrows are drawn on the graph to illustrate the load–unload cycles.
• The atoms in a solid are held together by bonds.
• These behave like springs between the particles and as the copper wire is stretched, the atomic separation increases.
• In the elastic region, the atoms return to their original positions when the deforming force is removed. Point B on the graph is known as the elastic limit.
• Beyond this point the wire ceases to be elastic.
• Although the wire may shorten when the load is removed, it will not return to its original length – it has passed the point of reversibility and has undergone permanent deformation.
• As the load is increased, the wire yields and will not contract at all when the load is removed.
• Beyond this yield point – point C on the graph – the wire is plastic and can be pulled like modelling clay until it breaks.
• If the broken end of the wire is wound around a pencil, the plasticity can be felt when the wire is extended. In the plastic region, the bonds between the atoms are no longer being stretched and layers of atoms slide over each other with no restorative forces.

  
  Figure 8 Repeated loading and unloading of copper wire

• A very strange effect is noticed if the load is removed during the plastic phase and the wire is reloaded: the wire regains its springiness and has the same stiffness as before (Figure 8).
• The ability of some metals to be deformed plastically and then regain their elasticity is extremely important in engineering.
• A mild steel sheet can be pressed into a mould to the shape of a car panel.
• After the plastic deformation, the stiffness and elasticity of the steel is regained and further pressings of the panel are also possible.

8. Stress–strain graphs:

• Stress-strain graphs resemble force-extension graphs in shape, but for large extensions, a decrease in cross-sectional area will lead to an increase in stress for a given load.
• The primary benefit is that the information obtained from the graph pertains to the material’s qualities, not only the characteristics of the specific sample that was tested.
• A stress-strain graph for the copper wire used in the stretching experiment previously may be found in Figure 9.

  
  Figure 9 stress-strain graph for the copper wire

• Note that in order to encompass the whole area of the plastic region, the strain scale has been stretched beyond the first 1%.
• O–A represents the Hooke’s law region.
• Strain is proportional to stress up to this point.
• The Young modulus of copper can be found directly by taking the gradient of the graph in this section.
• B is the elastic limit. If the stress is removed below this value, the wire returns to its original state.
• The stress at C is termed the yield stress. For stresses greater than this, copper will become ductile and deform plastically.
• D is the maximum stress that the copper can endure. It is called the ultimate tensile strength (UTS) or simply the strength of copper.
• E is the breaking point. There may be an increase in stress at this point due to a narrowing of the wire at the position on the wire where it breaks, which reduces the area at that point.
• Figure 9 provides stress–strain graphs to show the comparative properties of high carbon steel, mild steel and copper.
• The early part of the strain axis is extended to show the Hooke’s law region more clearly.
• The graphs illustrate the different behavior of the three materials.
• The gradient of the Hooke’s law region is the same for the steels and is greater than that for copper.
• The steels have a Young modulus of about 200GPa and are stiffer than copper (E ≈ 130GPa). High-carbon steel is the strongest as it has the greatest breaking stress (UTS), but it fractures with very little plastic deformation – it is brittle.
• Quench hardened high-carbon steel is commonly used for cutting tools and drill bits.

9. CORE PRACTICAL 5: Determine the Young modulus of a material

• In this practical, we’ll determine the Young modulus (E) of a material by measuring the stress (σ) and strain (ε) of a specimen under tension.
• Equipment:
  – Tensile testing machine (or a simple setup with weights and a ruler)
  – Specimen material (e.g., metal, plastic, or wood)
  – Calipers or micrometer (for measuring dimensions)
  – Stopwatch or timer (optional)
• Figure 10 Young’s modulus of a material
• Procedure:
  1. Prepare the specimen: Measure its initial length ([math]L_o [/math]) and cross-sectional area (A).
  2. Apply a series of increasing loads ([math] F_1, F_2, …, F_n [/math]) along the specimen’s axis.
  3. Measure the corresponding extensions ([math] ΔL_1, ΔL_2, …, ΔL_n [/math]).
  4. Calculate stress (σ) and strain (ε) for each load:
  5. [math] σ = F / A \\ ε = ΔL / L_0 [/math]
  6. 1. Plot a stress-strain graph (σ vs. ε).
  7. 2. Determine the Young modulus (E) from the slope of the linear elastic region:
  8. [math] E = σ / ε [/math]
• Tips and Variations:
  – Use a tensile testing machine for accurate control and measurement.
  – Ensure the specimen is aligned and securely gripped.
  – Measure the specimen’s dimensions carefully.
  – Consider using a extensometer for more precise strain measurements.
  – Repeat the experiment with different materials or specimen geometries.
• By following this practical, you’ll gain hands-on experience with measuring the Young modulus and understanding the stress-strain behavior of materials.

10. Elastic strain energy:

• This relates to the ability of an object to do work by virtue of its position or state.
• Elastic potential energy – or elastic strain energy – is therefore the ability of a deformed material to do work as it regains its original dimensions.
• The work done stretching the rubber of a catapult (slingshot) is transferred to elastic strain energy in the rubber and then to kinetic energy of the missile on its release.
• The work done during the stretching process is equal to the average force times the distance moved in the direction of the force:
• [math] ΔW = F_{av} Δx [/math]
• The work done on a wire, and hence its elastic strain energy, can be obtained from a force–extension graph.
• Figure 11 Elastic strain energy
• For the Hooke’s law region of the graph (O–A) in Figure 11, the average force is [math] F_{\text{max}}  [/math], so the work done is:
• [math] \Delta W = \frac{1}{2} F_{\text{max}} \Delta x [/math]
• This represents the area between the line and the extension axis – that is, the area of the triangle made by the line and the axis.
• Similarly, the work done when the force is constant (A–B on the graph) will be the area of the rectangle below the line.
• For any force–extension graph, the elastic strain energy is equal to the area under the graph.
• To calculate the energy for non-linear graphs, the work equivalent of each square is calculated and the number of squares beneath the line is counted.
• For estimated values, the shape can be divided into approximate triangular or rectangular regions.