This episode introduces the mechanical properties of materials: often an important consideration when choosing a material for a particular application. It focuses on the strength and stiffness of materials when subject to linear tensile or compressive forces (i.e. forces acting along a line in one dimension). Shearing forces (acting in a plane or two dimensions) or bulk forces (acting throughout a volume or in three dimensions) are not covered here.

Lesson Summary

• Demonstration and discussion: Hooke’s law (20 minutes)
• Student experiment: Stretching fibres (30 minutes)
• Demonstration: How stiffness depends on physical dimensions (20 minutes)

The topic of Hooke’s law is likely to be a revision of existing knowledge.

Discussion and demonstrations: Hooke’s law

Start with a recap of Hooke’s law. As a visual aid add 1 N weights (i.e. 100 g masses) to a suitable spring. If you also have a compression spring then so much the better. It should be apparent that

extension (or compression) Δ x  ∝  load

(Note: although many text books simply use x for extension when discussing Hooke’s Law, it is helpful to use Δ x to avoid confusion later with the original length x when discussing the Young Modulus.)

Note that a pair of forces are involved when applying a tension or compression. If only one force acted on the sample, it would be unbalanced, which implies the sample would accelerate. If you suspend a spring from a support and hang a weight from it, the weight is one force; the other is the upward force provided by the support.

Episode 227-1: Strengths of some materials (Word, 36 KB)

Tension (in the sample) F  ∝  extension

i.e. F = k ×  Δ x

This is a mathematical statement of Hooke’s law , where k is the stiffness , equal to

the tension per unit extension = F Δ x , with units N m^-1. (The opposite of stiffness is the compliance .)

The stiffness k can be found from the slope of the linear part of the F

Δ x graph – taking the slope averages all the individual data points.

Student experiment: Stretching fibres

Students can perform some careful experiments to see if fibres (rather than springs) obey Hooke’s law.

Episode 227-2: Tension and extension (Word, 42 KB)

All materials will show Hooke’s Law behaviour up to a point. This point is sometimes called the elastic limit. However, this is a simplification. Technically:

The linear part of the graph ends at the limit of proportionality.

The elastic part of the behaviour ends at the elastic limit.

These two points do not necessarily coincide. Behaviour is described as plastic beyond the elastic limit.

Loading many types of material (such as metals) simply stretches the bonds between the atoms from which it is made. The linear relationship between extension and load is simply a reflection of the fact that the bond force is varying linearly with the separation of the atoms. Plastic behaviour results from the permanent displacement of the atoms from their original positions.

Many polymers (e.g. rubber) show very large extensions – far too large to be explained by bond stretching. Initially it is the uncoiling and/or straightening out of the long molecules that takes place. When they are all pulled straight, the sample gets much stiffer as bond stretching takes over.

Episode 227-3: Explaining stiffness and elasticity (Word, 353 KB)

Episode 227-4: Plasticity in polythene (Word, 44 KB)

Episode 227-5: Elastic behaviour in rubber (Word, 38 KB)

Demonstration

Stiffness k obviously depends upon the actual material; it also depends on the dimensions of the sample. Thicker samples stretch less per newton than thinner ones. Imagine two identical samples in parallel – twice the cross-sectional area A implies half the extension for given load, as each sample supports half the load, so k for the system as a whole is doubled.

What about two identical samples in series? Both are subjected to the same load, thus each stretches the by the same amount, so the total stretch is doubled, so the stiffness for the system as a whole is halved.

Check the above conclusions in a demonstration using (identical) springs in parallel and in series.

Episode 227-6: More than one spring (Word, 52 KB)

The Young modulus is often regarded as the quintessential material property, and students can learn to measure it. It is a measure of the stiffness of a material; however, in practice, other properties of materials, scientists and engineers are often interested in, such as yield stress, have more influence on the selection of materials for a particular purpose.

Lesson Summary

• Discussion: Defining the Young modulus (20 minutes)
• Student activity: Studying data (20 minutes)
• Student experiment: Measuring the Young modulus (60 minutes)
• Student experiment: An alternative approach using a cantilever (30 minutes)
• Discussion: Comparing experimental approaches (10 minutes)
• Student questions: Involving the Young modulus (30 minutes)

Discussion: Defining the Young modulus

A typical value of k might be 60 N m^-1.

What does this mean? (60 N will stretch the sample 1 m.) What would happen in practice if you did stretch a sample by 1 m? (It will probably snap!)

A measure of stiffness that is independent of the particular sample of a substance is the Young modulus E.

Recall other examples you have already met of sample independent properties that only depend upon the substance itself:

• density = massvolume
• electrical resistivity = resistance  ×  arealength
• specific heating capacity = energy transferredmass  ×  temperature change
• thermal conductivity = power  ×  lengtharea  ×  temperature difference

We need to correct k for sample shape and size (i.e. length and surface area).

Episode 228-1: The Young modulus (Word, 53 KB)

Note the definitions, symbols and units used:

Quantity	Definition	Symbol	Units
Stress	tensionarea=FA	σ (sigma)	N m-2={Pa
Strain	extension per original length= Δ xx	ε (epsilon)	No units (because it’s a ratio of two lengths)
Young modulus	stressstrain	E	N m-2 = Pa

Strains can be quoted in several ways: as a %, or decimal. E.g. a 5% strain is 0.05.

Episode 228-2: Hooke's law and the Young modulus (Word, 75 KB)

Student activity: Studying data

It is helpful if students can learn to find their way around tables of material properties. Give your students a table and ask them to find values of the Young modulus. Note that values are often given in GPa (1 × 10⁹ Pa).

Some interesting values of E :

• DNA ~ 10⁸ Pa
• spaghetti (dry) ~ 10⁹ Pa
• cotton thread ~ 10¹⁰ Pa
• plant cell walls ~ 10¹¹ Pa
• carbon fullerene nanotubes ~ 10¹² Pa

Episode 228-3: Materials database (Word, 115 KB)

Student experiment: Measuring the Young modulus

You can make measuring the Young modulus E a more interesting lab exercise than one which simply follows a recipe. Ask students to identify the quantities to be measured, how they might be measured, and so on. At the end, you could show the standard version of this experiment (with Vernier scale etc.) and point out how the problems have been minimized.

What needs to be measured? Look at the definition: we need to measure load (easy), cross-sectional area A , original length x₀ (so make it reasonably long), and extension Δ x .

Problems? Original length – what does this correspond to for a particular experimental set up? Cross-sectional area: introduce the use of micrometer and/or vernier callipers. Is the sample uniform? If sample gets longer, won’t it get thinner? Extension – won’t it be quite small?

Should the sample be arranged vertically or horizontally?

Divide the class up into pairs and brainstorm possible methods of measuring the quantities above, including the pros and cons of their methods.

Some possibilities for measuring Δ x :

Attach a pointers to the wire

• Pro: measures Δ x directly
• Con: may affect the sample; only moves a small distance

Attach a pointer to the load

• Pro: measures Δ x directly, does not effect the sample
• Con: only moves a small distance

Attach a pulley wheel

• Pro: amplifies the Δ x
• Con: need to convert angular measure to linear measure, introduces friction

Attach a pointer to the pulley wheel

• Pro: amplifies the Δ x even more
• Con: need to convert angular measure to linear measure, introduces friction

Exploit an optical level

• Pro: a frictionless pointer, amplifies the Δ x even more
• Con: need to convert angular measure to linear measure, more tricky to setup?

Illuminate the pointer etc to produce a magnified shadow of the movement

• Pro: easy to see movement
• Con: need to calculate magnification, can be knocked out of place

use a lever system to amplify or diminish the load and provide a pointer

• Pro: useful for more delicate or stiff samples; can use smaller loads
• Con: fixing the sample so it doesn’t slip, need to convert angular measure to linear measure

Different groups could try the different ideas they come up with. Depending upon the time available, it may be worth having some of the ideas already set up.

Give different groups different materials, cut to different sizes, for example: metal wires (copper, manganin, constantan etc), nylon (fishing line), human hair (attach in a loop using Sellotape), rubber. Note that in the set up above, the sample is at an angle to the ruler – a source of systematic error.

Safety

Students should wear eye protection, provide safe landing for the load should sample break, e.g. a box containing old cloth. For the horizontal set up: bridges over the sample to trap the flying ends, should the sample snap.

Good experimental practice: measure extension when adding to the load and when unloading, to check for any plastic behaviour.

Episode 228-4: Measuring the stiffness of a material (Word, 59 KB)

Episode 228-5: Stress–strain graph for mild steel (Word, 68 KB)

Information about the use of precision instruments (micrometer screw gauge, Vernier callipers and Vernier microscope).

Episode 228-6: Measure for measure (Word, 82 KB)

Student experiment: An alternative approach using a cantilever

An alternative approach to measuring the Young modulus is to bend a cantilever. (Potential engineering students will benefit greatly from this.)

For samples too stiff to extend easily (e.g. wooden or plastic rulers, spaghetti, glass fibres) the deflection y of a cantilever is often quite easy to measure and is directly related to its Young modulus E .

If the weight of the cantilever itself is mg, and the added load is Mg and L is the length of the cantilever (the distance from where the cantilever is supported to where the load is applied):

For a rectangular cross section, dimension in the direction of the load is d , other dimension is b

• y = 4 (Mg + 5mg/16) L³
• E b d³

(for square cross-section d = b )

For a circular cross-section radius r

• y = 4 (Mg + 5mg/16) L³
• 3 π r⁴ E

Discussion: Comparing experimental approaches

Finish with a short plenary session to compare the pros and cons of the different experimental approaches.

Student questions: Involving the Young modulus

Questions involving stress, strain and the Young modulus, including data-handling.

Episode 228-7: Calculations on stress, strain and the Young modulus (Word, 59 KB)

Episode 228-8: Stress, strain and the Young modulus (Word, 26 KB)

A stress-strain graph includes a great deal of information about a material. Your students may need to learn to extract this information, and to compare one material with another by comparing their graphs.

Lesson Summary

• Discussion: Interpreting stress-strain graphs (15 minutes)
• Student questions: On stress-strain graphs (30 minutes)
• Discussion: Energy stored in a stretched material (20 minutes)
• Student activity: Calculating energy stored (15 minutes)
• Discussion: Correct terminology (20 minutes)

Discussion: Interpreting stress-strain graphs

So far, we have considered elastic behaviour, as characterized by stiffness and the Young modulus. Now we go on to plastic behaviour and fracture (breaking).

Show a selection of stress-strain graphs and discuss various examples to show how different materials behave, and to identify yield point, breaking stress. Your syllabus may refer to three different measures of the strength of a material:

• Yield stress: the stress at the yield point
• Breaking stress: the stress at fracture (the end of the graph)
• Ultimate tensile stress: the maximum stress withstood by a material (the highest point on the graph)

Student questions: On stress-strain graphs

Graphs for five different materials, for interpretation.

Episode 229-1: Analysis of tensile testing experiments (Word, 63 KB)

Discussion:Energy stored in a stretched material

The stress-strain graph can tell you about the energy stored in a stretched wire.

Work is done on the sample as it is stretched. When under tension, this energy is stored elastically. When the load is released, energy can be recovered. If the behaviour is elastic, no energy is stored thermally after stretching. If the behaviour is plastic, some energy is stored thermally. If the sample shows hysteresis , it returns to its original shape, but does not follow the same load extension graph when being unloaded).

Work done = force × distance

Work done = load × extension

Work done = area under a load-extension graph

This result is valid for any shape of force extension graph. For Hookean materials the area required is a triangle.

Area of a triangle = 12 × base  ×  height

Work done = 12  ×  load  ×  extension

so,

Work done = 12 F Δ x

or,

Work done = 12 k Δ x²

For a material, energy stored per unit volume = 12stress  ×  strain

Student activity: Calculating energy stored

Students can calculate values for the materials used in their measurements of the Young modulus.

Discussion: Correct terminology

Since much of the terminology used in this topic can cause confusion, it is worth ensuring that students have grasped the different concepts. You could ask them to define the following terms in their own words: stiff, strong, weak, elastic and plastic.

Extend this to include the following:

brittle (cracks or breaks without plastic deformation); ductile (can be deformed under tension); malleable (hammerable – can be deformed under compression).

Finish by looking at some charts which summarize how various properties of materials are related to one another, and which are used by engineers in selecting materials for particular applications.

Episode 229-2: Introduction to materials selection chart (Word, 262 KB)