Ultimate tensile strength (UTS)

Materials will break under sufficiently high tensile forces.

This breaking point is defined by two key terms:

• Ultimate tensile strength (UTS) - This is the maximum stress a material can withstand before it breaks.
• Breaking stress - The stress level at which a material fractures.

UTS values are crucial for engineers designing safe structures.

Elastic Potential Energy

When materials deform, they store energy. This stored energy is known as elastic potential energy.

For a force (F) causing an extension (x), the elastic potential energy is calculated as:

• Work done = average force x extension = $\frac{F}{2}\times x$
• Elastic potential energy (E) = $\frac{Fx}{2}$

Worked example - Calculating elastic potential energy in a stretched spring

A spring is stretched by 4 cm using a force of 20 N. Calculate the elastic potential energy stored in the spring.

Step 1: Formula

E = $\frac{Fx}{2}$

Step 2: Convert cm to m

to convert from cm to m, divide by 100

4 cm = 0.04 m

Step 3: Substitution and correct evaluation

E = $\frac{20\times 0.04}{2}\text{ = 0.4 J}$

Young's modulus

Young's modulus is the ratio of stress to strain, and it's constant up to the material's limit of proportionality.

$E=\frac{\sigma }{ϵ}=\frac{\text{F l}}{\text{A x}}$

Where:

E = Young's modulus (N m${}^{-2}$)

F = applied force (N)

A = cross-sectional area (m${}^2$)

l = original length (m)

x = extension (m)

Young's modulus indicates how stiff a material is, helping engineers choose materials capable of withstanding certain stresses.

Determining Young's modulus experimentally

To find Young's modulus, stress-strain data for an extending wire is plotted.

Method:

1. Attach a thin wire sample of known length (l) and cross-sectional area (A).
2. Gradually add weights to the wire.
3. Measure the extension distance (x) for each weight.
4. Calculate stress ($\sigma$) and strain ($ϵ$) values.
5. Plot the stress-strain data.
6. The gradient of the linear part equals Young's modulus (E).

Worked example - Determining Young's modulus for a material

A 2-meter long wire with a cross-sectional area of 1 x 10^-4 m² stretches by 1 cm under a force of 1,000 N. Calculate Young's modulus for this material.

Step 1: Formula for Young's modulus

$\text{Youngs modulus (E) =}\frac{\sigma }{ϵ}=\frac{Fl}{Ax}$

Step 2: Convert cm to m

to convert from cm to m, divide by 100

1 cm = 0.01 m

Step 3: Calculate stress (σ)

$\sigma =\frac{F}{A}=\frac{1000}{1\times {10}^{-4}}=10\times {10}^6$

Step 4: Calculate strain (ε)

$ϵ=\frac{x}{l}=\frac{0.01}{2}=5\times {10}^{-3}$

Step 5: Calculate Young's modulus (E)

$E=\frac{\sigma }{ϵ}=\frac{10\times {10}^6}{5\times {10}^{-3}}=2\times {10}^9\text{ Pa = 2 GPa}$