A-level Physics (Advancing Physics)/Stress, Strain & the Young Modulus

Stress

Stress is a measure of how strong a material is. This is defined as how much pressure the material can stand without undergoing some sort of physical change. Hence, the formula for calculating stress is the same as the formula for calculating pressure:

$P=\frac{F}{A}$

where P = stress (in Pascals, commonly abbreviated Pa), F = force (in Newtons, commonly abbreviated N) and A = the cross sectional area of the sample.

Fracture Stress

The fracture stress is the level of stress at which a material will fracture. Fracture stress is also known as tensile strength. If a material fractures by 'crack propagation' (ie. it shatters), the material is brittle.

Yield Stress

The yield stress is the level of stress at which a material will deform permanently. This is also known as yield strength.

Strain

Stress causes strain. Putting pressure on an object causes it to stretch. Strain is a measure of how much an object is being stretched. The formula for strain is:

$\text{Strain }=\frac{\text{Extension}}{\text{Original Length}}$

Both lengths are given in metres. It is important to note that the extension does not included the original length of the sample; it is just the additional length added to the object as a result of stress. Strain has no units, as it is a ratio between two lengths.

Strain can be converted to a percentage by multiplying it by 100.

Young's Modulus

Young's Modulus is a measure of the stiffness of a material. It states how much a material will stretch (ie. how much strain it will undergo) as a result of a given amount of stress. The formula for calculating it is:

$E=\frac{\text{tensile stress}}{\text{tensile strain}}$

The values for stress and strain must be taken at as low a stress level as possible, provided a difference in the length of the sample can be measured. The unit of the Young's Modulus is Pascals, as strain has no units, so the units for stress are retained.

Stress-Strain Graphs

Stress (σ) can be graphed against strain (ξ). The toughness of a material (ie. how much it resists stress, in J m^-3) is equal to the area under the curve, between the y-axis and the fracture point. Graphs such as the one on the right show how stress affects a material. This image shows the stress-strain graph for aluminium. It has three main features:

Elastic Region

In this region (between the origin and point 3), the ratio between stress and strain (Young's modulus) is constant, meaning that the material is obeying Hooke's law, which states that a material is elastic (it will return to its original shape) if force is directly proportional to extension.

Plastic Region

In this region (between points 3 and 4), the rate at which extension is increasing is going up, and the material has passed the elastic limit. It will no longer return to its original shape. After point 1, the amount of stress decreases due to 'necking', so the cross-sectional area is going down.

Fracture Point

At point 4, the material finally breaks in two completely - it fractures, and the graph stops.

Other Typical Graphs

The graph above is typical of most ductile metals. Some may dip between points 2 and 3 due to a brief thinning of the sample as it stops stretching elastically.

In a brittle material, such as glass or ceramics, the stress-strain graph will have an extremely short elastic region, and then will fracture. There is no plastic region on the stress-strain graph of a brittle material.

Questions

1. 10N of force are exerted on a wire with cross-sectional area 0.5mm². How much stress is being exerted on the wire?

2. Another wire has a tensile strength of 70MPa, and breaks under 100N of force. What is the cross-sectional area of the wire just before breaking?

3. What is the strain on a Twix bar (original length 10cm) if it is now 12cm long?

4. What is this strain, expressed as a percentage?

5. When 50N are applied to a wire with a radius of 1mm. The wire was 0.7m long, but is now 0.75m long. What is the Young's Modulus for the material the wire is made of?

6. Glass, a brittle material, fractures at a strain of 0.004 and a stress of 240 MPa. Sketch the stress-strain graph for glass.

7. (Extra nasty question which you won't ever get in an exam) What is the toughness of glass?

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