Changing Shape

Deformation of Materials

When a force is applied to an object, it can change shape. This change in shape is called deformation. There are two main types of deformation:

• Elastic deformation: The object returns to its original shape and size once the force is removed.
• Plastic deformation: The object does not return to its original shape after the force is removed, resulting in a permanent shape change.

If the deformation is elastic, the material behaves like a spring and recovers its shape. If the deformation is plastic, the material has been stretched or compressed beyond its elastic limit, causing permanent change.

For example, a rubber band stretches elastically when pulled gently but if stretched too far, it may not return to its original size, showing plastic deformation.

Hooke's Law

Hooke's Law states that, within the elastic limit, the force applied to stretch or compress a spring is directly proportional to the extension or compression produced:

Force $\propto$ Extension

This can be written as the formula:

$$F = k \times e$$

where:

• $F$ is the force applied (in newtons, N)
• $e$ is the extension or compression (in metres, m)
• $k$ is the spring constant (in newtons per metre, N/m), which measures the stiffness of the spring

The relationship holds true only up to the limit of proportionality. Beyond this point, the force is no longer proportional to extension, and Hooke's Law no longer applies.

A linear force-extension graph shows a straight line through the origin up to the limit of proportionality. Beyond this, the graph curves away.

For instance, if a spring stretches by 0.02 m when a force of 4 N is applied, the spring constant is:

$$k = \frac{F}{e} = \frac{4}{0.02} = 200 \text{ N/m}$$

Types of Forces Causing Shape Change

Different forces can cause objects to change shape by deforming them:

• Tension: A force that pulls and stretches an object. For example, a rope in a tug of war is under tension.
• Compression: A force that pushes and squashes an object. For example, pressing down on a spring compresses it.
• Shear: A force that causes parts of an object to slide past each other in opposite directions. For example, scissors apply shear force to cut paper.

Each of these forces changes the shape of materials in different ways, depending on the material’s properties and the force’s direction.

Elastic and Plastic Behaviour

Materials behave differently depending on how much force is applied and their properties:

• Elastic limit: The maximum force or extension a material can withstand and still return to its original shape. Beyond this limit, the material will deform permanently.
• Recoverable deformation: Deformation that disappears when the force is removed (elastic deformation).
• Permanent deformation: Deformation that remains after the force is removed (plastic deformation).

For example, a metal wire stretched gently will return to its original length (elastic behaviour). If stretched too far, it will be permanently longer (plastic behaviour).

The elastic limit is often close to the limit of proportionality but they are not exactly the same. The limit of proportionality is where Hooke’s Law stops applying, while the elastic limit is where permanent deformation begins.

Learning Example

A spring with a spring constant of 150 N/m is stretched by 0.04 m. Calculate the force applied and state whether the deformation is elastic if the elastic limit is 0.05 m.

Using Hooke’s Law:

$$F = k \times e = 150 \times 0.04 = 6 \text{ N}$$

Since the extension (0.04 m) is less than the elastic limit (0.05 m), the deformation is elastic, and the spring will return to its original length when the force is removed.