Materials under load (Topic 4B)

Forces don’t just make objects move; they can also change their size and shape. This effect is called deformation.

When an object deforms, it either stretches or compresses from its natural length.

Different materials respond differently to deformation: some deform elastically and return to their natural length, while others deform plastically and remain permanently deformed.

When two forces pull outwards on an object in opposite directions, they create tension, leading to tensile deformation. As a result, the object extends.

An example of this is a stretched spring

• When a load is applied to a spring, the force pulls down due to gravity.
• The spring stretches because the load applies a tensile force.
• The greater the load, the more the extension.

Other examples of tensile forces include:

• A stretched rubber band.
• A rope in a tug-of-war, where both teams pull in opposite directions.
• A metal wire in a suspension bridge holds up weight as it is stretched.

When two forces push inwards on an object from opposite directions, they create compression, leading to compressive deformation. Under this force, the object shortens or squashes.

An example of this is a compressed Spring:

• When a force is applied from above, the spring pushes back but shortens in response.
• This is an example of a compressive force at work.
• The greater the load, the more the spring will shorten.

Other examples of compressive forces:

• Standing on a sponge – your weight compresses it.
• A chair leg pressing on the floor – the force from the chair applies compression.
• A car tyre on the ground – the tyre compresses under the car’s weight.

The greater the force applied to a spring or elastic material, the greater the extension (if tensile) or compression (if compressive).

Remember, extension is the amount the length of the spring changes. It is not the final length of the spring.

There are several cases worth being familiar with where Hooke’s law is and is not obeyed:

A force-extension graph illustrates how a material deforms when a force is applied.

Different materials (e.g., brittle or ductile materials) produce unique graphs based on their mechanical properties.

These graphs help identify whether a material follows Hooke’s Law and determine its elastic and plastic behaviour.

A material obeys Hooke’s Law if the graph is a straight line through the origin.

This means that extension is directly proportional to force

The gradient of a force-extension graph can be used to determine the force constant (). The force constant measures a material’s stiffness — it tells us how much force is needed to produce a certain extension. It’s often called the spring constant when describing springs, but we generally use force constant for any material or object, not just springs.

The limit of proportionality represents the point where Hooke’s Law ceases to be valid—beyond this, extension is no longer proportional to force. Beyond this point, increasing force results in larger, non-uniform extensions

The elastic limit is the maximum extension a material can experience before returning to its natural length once the force is removed.

If stretched beyond the elastic limit, the material undergoes plastic deformation, meaning it will not return to its original length.

The elastic limit is always beyond the limit of proportionality on the graph.

As more force is applied beyond the limit of proportionality, the force-extension graph curves instead of remaining linear. The material exhibits non-linear behaviour and no longer obeys Hooke’s Law.

The force-extension relationship becomes non-linear, and the material may eventually fail or break if the force continues to increase.

It is important to note that force-extension graphs are commonly illustrated with force on the -axis and extension on the -axis!

Always check the axes before calculating the force constant

• If force () is on the y-axis and extension () is on the -axis, then the gradient gives the force constant
• If force is on the -axis and extension is on the y-axis, the force constant is =1/gradient instead.

Stress can be referred to as compressive or tensile, depending on the direction of the applied force:

• An object extends under tensile stress.
• An object contracts under compressive stress.

Strain is defined as the change in length of an object per unit of its original length, under an applied force

Strain has no unit and is sometimes written as a percentage. As a percentage, the strain can be found from the equation:

Tensile strain is positive: it results from the extension of an object under tensile stress.

Compressive strain is negative: it results from the contraction of an object under compressive stress.

The Young’s modulus of a material is defined as the ratio of stress to strain:

The Young’s modulus of an object depends only on the material. It is a measure of the material’s ‘stiffness’, independent of the shape and size of the material.

The stress-strain graph for a material deformed in the elastic limit has a constant gradient. The gradient equals the Young’s modulus of the material.

Young’s modulus is also called the elastic modulus.

Experimental procedure for determining the Young’s modulus of a metal wire:

1. Find the diameter of the wire using a micrometre. The measurement should be taken in multiple places and averaged.
2. From the diameter, calculate the cross-sectional area of the wire.
3. Clamp the wire at one end and suspend it over a table using a pulley.
4. Place a ruler parallel to the wire with a tape marker to measure the wire’s extension.

1. Apply different masses onto the wire and calculate the force applied to the wire. using where is the total mass hanging from the wire and is acceleration due to gravity.
2. Plot data points of stress () against strain () for at least 5 different masses and calculate the gradient.
3. The gradient is the Young’s modulus of the metal.

The force-extension graph for an unknown-metal wire is shown below.

This graph shows that the wire:

• Follows Hooke’s Law in the initial elastic region, meaning force and extension are proportional.
• Elastic deformation occurs up to the elastic limit, where the material returns to its original shape upon unloading.
• Beyond the elastic limit, the wire undergoes plastic deformation, meaning it is permanently stretched.
• The unloading curve has the same gradient as the loading curve*, but the material now has a longer final length due to plastic deformation.

It is useful to know that the term curve is sometimes used broadly in physics to refer to any plotted relationship, even if it appears linear as a straight line.

The force-extension graph for a rubber band is shown below.

• The rubber band does not obey Hooke’s Law, as its force-extension graph is curved rather than linear.
• Elastic deformation only—it returns to its original shape after unloading.
• The loading and unloading curves do not overlap, forming a hysteresis loop.
• The hysteresis loop area represents the energy lost as thermal energy, meaning that not all the work done in stretching the material is recovered.
• The curve for contraction is always below the curve for stretching because some energy is dissipated.

The force-extension graph for a polyethene strip is shown below.

• Polyethene strips do not obey Hooke’s Law and behave very differently from metals or rubber.
• Plastic deformation occurs immediately—even a small force permanently stretches the material.
• This makes polymeric materials easy to reshape but difficult to return to their original form.

To validate Hooke’s Law, we may want to investigate the relationship between force and extension for:

A metal spring (expected to follow Hooke’s Law),
A rubber band (expected to form a hysteresis loop),
A polythene strip (expected to show immediate plastic deformation).

In this experiment, our variables are:

Independent variable: Force () (applied using weights).
Dependent variable: Extension ().
Control variables:

• Same measuring equipment.
• Same starting length for each material.
• Consistent increments of force are applied.

Equipment List

• Clamp stand, boss, and clamp.
• Metre ruler (resolution = ).
• Set square (to ensure accurate ruler alignment).
• Mass hanger and slotted masses.
• Fiducial marker (for measuring extension accurately).

Safety Precautions

• Risk of snapping materials → Wear eye protection to prevent injury.
• Falling weights hazard → Place a box or soft mat beneath to catch falling masses.
• Clamp stability → Ensure the stand is securely fixed to prevent tipping.

Materials can be deformed from their natural shape.

When a material is deformed within its elastic limit, the work done on the material is stored as elastic potential energy.

On a force–extension graph, a linear relationship shows that the material is following Hooke’s law:

where:

• is the force applied in
• is the extension of the material from its natural length in and
• is the spring constant in units of

A force–extension graph can be used to find the work done to deform a material elastically.

The work done on the material equals the area under the force–extension graph, which is a triangle for extension within the elastic limit.

The triangular area under the graph representing the work done can be found by using the equation:

and is measured in units of which is equivalent to

Elastic deformation is a reversible change of the shape of an object under stress. Removing the force allows the material to return to its original shape.

Elastic deformation occurs within the elastic region, where Hooke’ law holds.

The work done on an object within the elastic region is stored as elastic potential energy.

Examples of materials that significantly elastically deform include springs and elastic bands.

When a relatively small force is applied to a material, the bonds between atoms stretch but return to their initial state when the force is relaxed.

Plastic deformation is the permanent change of an object under stress. The object does not return to its original shape after the force is removed.

A significant rearrangement of the material’s atoms occurs in the plastic region. This requires energy, meaning not all the work done is converted to elastic potential energy.

When an applied force on a material is great enough to cause plastic deformation, the bonds between atoms stretch and planes shear. Deformation remains after the applied force is relaxed. Plastic deformation can be useful, for instance, in metal shaping.

Tensile testing is a common method to find the relationship between the stress acting on a material and the strain it causes.

Stress-strain curves have characteristic shapes dependent on the type of material.

There are five significant points of interest on this stress-strain curve:

1. the proportionality limit – the final point on the curve where the graph is linear.
2. the elastic limit – removing the force below this point will allow the material to return to its original shape. Lower values of strain correspond to the elastic region, and higher strains correspond to the plastic region
3. the yield point – any additional stress above this point, and the strain will begin to increase rapidly during plastic deformation. An object does not return to its original length after plastic deformation.
4. the ultimate tensile strength – the maximum stress value on the curve. Beyond this point, the material will begin necking, which is when a weak point of the material rapidly becomes thinner.
5. the fracture point – the material will break.

Glass, cast iron, and ceramics are examples of brittle materials.

Brittle materials follow stress-strain curves with the shape shown below.

Brittle materials can not withstand strain beyond the ultimate tensile strength and do not undergo plastic deformation. The fracture point of the material coincides with the maximum stress point at

Ductile materials include metals like copper and aluminium.

They follow the stress-strain curves shown below.

The plastic region extends to large strain values – ductile materials can withstand large plastic deformation (stretching) before breaking.

Polymeric materials are made of polymers, such as rubber or nylon. They are naturally stretchy materials and have a stress-strain curve, as shown below.

The curve is non-linear. Polymeric materials can undergo large elastic deformation but still return to their original shape when force is removed.

The yield point can vary depending on the material. Some polymeric materials have no yield point at all.

The stress-strain curves of ductile, brittle and polymeric materials have distinct shapes, summarised in the table below.

Note that linear elastic region indicates that the material obeys Hooke’ law.

The work done on an object within the elastic limit is stored completely as elastic potential energy, , and released when the object is allowed to return to its original length.

Force–extension graphs, when deformed within the elastic limit, are linear. The area under the graph is the work done, which is equal to the elastic potential energy stored in the material:

Within the elastic limit, Hooke’s law is obeyed:

Substituting this into the expression for elastic potential energy gives:

The gradient of a force–extension graph (in the elastic limit) is equal to the spring constant, which indicates the stiffness of a material.

A stretchy material like a rubber band has a spring constant whereas a stiff material like a metal rod can have a spring constant of