Introduction to Materials: Stretching and Squashing

Welcome! In this chapter of Unit 1: Materials, we are going to look at what happens to materials when we apply forces to them. Whether you are jumping on a trampoline, stretching a hair tie, or building a bridge, the way materials deform (change shape) is vital. We will explore why some things snap back to their original shape while others stay bent forever, and how much energy is hidden inside a stretched object.

Note: For the definitions of stress, strain, and the Young modulus, please refer to the dedicated chapters on those topics.

1. Elastic vs. Plastic Behaviour

When you apply a force to a material, it deforms. There are two main ways this can happen:

Elastic Deformation

In elastic deformation, the material returns to its original length and shape once the force (load) is removed. Think of a standard rubber band: you pull it, it gets longer, you let go, and it goes back to exactly how it was before. This happens because the atoms in the material are pulled slightly apart but stay in their original positions relative to each other.

Plastic Deformation

In plastic deformation, the material undergoes a permanent change in shape. Even after the force is removed, the material does not return to its original dimensions. Think of a piece of modeling clay or a metal paperclip that you have bent too far. At a microscopic level, the layers of atoms have slid past one another and settled into new positions.

Quick Review:

  • Elastic: Temporary change. Like a spring.
  • Plastic: Permanent change. Like squashed plasticine.

2. Important Points on Force-Extension Graphs

To understand how a material behaves, we often plot a force-extension graph. This shows the force \(F\) applied on the y-axis and the extension \(\Delta x\) on the x-axis. As we stretch the material, it goes through several key stages:

The Limit of Proportionality

At the start of the graph, the line is usually straight. This means the material follows Hooke's Law (\(F = k\Delta x\)). The limit of proportionality is the exact point where the graph stops being a straight line and starts to curve. Beyond this point, force is no longer directly proportional to extension.

The Elastic Limit

The elastic limit is the maximum force that can be applied to a material without causing permanent (plastic) deformation.

  • If you release the force before this limit, the material returns to its original size.
  • If you exceed this limit, the material will be permanently stretched.

The Yield Point

The yield point is where the material begins to extend rapidly with very little increase in force. Essentially, the material "gives way" and starts to behave like a plastic. Some materials have a very clear yield point where the internal structure starts to collapse or slide.

Don't worry if this seems tricky: In many exam questions, the limit of proportionality and the elastic limit are very close together. Just remember that the limit of proportionality is about the shape of the graph (straight line), while the elastic limit is about returning to the original shape.

3. Elastic Strain Energy

When you stretch a material, you are doing work on it. This work is stored in the material as elastic strain energy (\(\Delta E_{el}\)). When you let go, this energy is released (like a catapult firing a stone).

Calculating Energy for Linear Graphs (Hooke's Law)

If a material is obeying Hooke's Law (the graph is a straight line), the energy stored is equal to the area under the force-extension graph. Since the area is a triangle, we use the formula:

\(\Delta E_{el} = \frac{1}{2} \times \text{base} \times \text{height}\)

\(\Delta E_{el} = \frac{1}{2} F \Delta x\)

Where:

  • \(\Delta E_{el}\) is the elastic strain energy in Joules (\(J\)).
  • \(F\) is the force applied in Newtons (\(N\)).
  • \(\Delta x\) is the extension in metres (\(m\)).

Since \(F = k\Delta x\), we can also write this as:
\(\Delta E_{el} = \frac{1}{2} k (\Delta x)^2\)

Calculating Energy for Non-Linear Graphs

Not all materials produce a straight line (for example, rubber). For these materials, you cannot use the simple \(\frac{1}{2} F \Delta x\) formula. Instead, you must find the area under the curve.
In an exam, you can do this by counting the squares underneath the line on the graph paper and multiplying by the "value" of one square (force width \(\times\) extension height).

Did you know? When a material undergoes plastic deformation, not all the energy used to stretch it is recovered. Some of that energy is converted into heat within the material!

4. Force-Compression Graphs

Everything we have discussed about stretching (tensile force) also applies to squashing (compressive force).

  • Force-compression graphs look very similar to force-extension graphs.
  • The area under a force-compression graph still represents the work done or the energy stored.
  • Materials can have an elastic limit in compression too — think of a soda can being crushed until it can't pop back out.

5. Summary and Key Takeaways

  • Elastic behaviour: Returns to original shape after the load is removed.
  • Plastic behaviour: Permanent deformation; does not return to original shape.
  • Limit of Proportionality: The point where the straight-line relationship ends.
  • Elastic Limit: The point beyond which deformation becomes permanent.
  • Elastic Strain Energy (\(\Delta E_{el}\)): The energy stored in a deformed object, found by calculating the area under a force-extension graph.
  • Formula (for linear parts): \(\Delta E_{el} = \frac{1}{2} F \Delta x\).

Common Mistake to Avoid: Always check your units! Extensions are often given in millimetres (\(mm\)) or centimetres (\(cm\)). You must convert them to metres (\(m\)) before using the energy formula to get an answer in Joules.