# Force-extension and stress-strain graphs

> CIE A-Level Physics · 9702
> Source: https://www.owlsprep.com/study/cie-9702-u6-force-extension-and-stress-strain/

This module teaches you to interpret force-extension and stress-strain graphs, identify key characteristic points, calculate material properties like Young modulus, and compare mechanical behaviour of different materials for CIE A-Level Physics.

**Prerequisites:** [Hooke's Law and elastic deformation](https://www.owlsprep.com/study/cie-9702-u6-hookes-law/); [Stress, strain and Young modulus](https://www.owlsprep.com/study/cie-9702-u6-stress-strain-young-modulus/)

## Learning objectives

- Interpret key features of force-extension and stress-strain graphs
- Distinguish between key characteristic points like proportionality limit and elastic limit
- Calculate Young modulus from both force-extension and stress-strain graphs
- Compare graph shapes for brittle, ductile and polymeric materials

## Key Features of Force-Extension Graphs

A force-extension ($F$-$e$) graph plots the applied force on a material sample against the resulting extension. All elastic materials share common key points that describe their behaviour.

**Limit of Proportionality** — The first point where the graph deviates from a straight line, so force is no longer proportional to extension. Hooke's Law does not apply beyond this point.

*Example:* For a typical metal wire, this occurs at less than 1% of the original length.

**Elastic Limit** — The maximum force beyond which the material no longer returns to its original length when force is removed, and permanent (plastic) deformation occurs.

**Worked example:** A student marks two points on an F-e graph: P where the line stops being straight, and E where the wire retains 0.1 mm permanent extension after force is removed. Identify P and E.

1. A straight line on an F-e graph means force is proportional to extension. The point where proportionality ends is:
2. P = Limit of proportionality
3. The point after which deformation becomes permanent is defined as the elastic limit, so:
4. E = Elastic limit

> **Exam tip:** In CIE exams, these two points are often tested: they are very close but not the same, so you must name them correctly.

## Why Stress-Strain Graphs Are Useful

Unlike force-extension graphs, which depend on the dimensions of the specific sample you test, stress-strain graphs are an intensive property: they are the same for any size sample of the same material. This makes them ideal for comparing different materials.

**Yield Point** — The point on a stress-strain graph where plastic deformation begins, with a rapid increase in strain for very little increase in stress.

**Worked example:** Why can't you compare the stiffness of a 1 mm diameter steel wire and a 2 mm diameter steel wire directly from their force-extension graphs?

1. Force-extension graphs depend on sample cross-sectional area: a thicker wire requires twice the force to produce the same extension as a thinner wire of the same material.
2. Stress normalises force by area ($\sigma = F/A$) and strain normalises extension by original length ($\epsilon = \Delta l / l_0$). The gradient of a stress-strain graph is:
3. $$E = \frac{\sigma}{\epsilon}$$
4. Young modulus $E$ is a property of the material only, independent of sample size, so only stress-strain graphs can be directly compared.

> **Exam tip:** Always check if the question asks for force-extension or stress-strain: mixing these up is a common source of lost marks.

## Graph Shapes For Different Materials

Different classes of materials have characteristic stress-strain shapes that describe their mechanical behaviour:

| Material Type | Key Graph Features | Example |
| --- | --- | --- |
| Brittle | Linear elastic region, no plastic deformation, breaks immediately at low strain | Glass, cast iron |
| Ductile | Linear elastic, clear yield point, large plastic region, necking before breaking | Copper, mild steel |
| Polymeric | Low initial gradient, gradient increases at large strain, very high extensions before breaking | Natural rubber |

**Worked example:** State two key differences between the stress-strain graphs of brittle glass and ductile copper.

1. 1. Glass (brittle) has no plastic region: the linear elastic region continues all the way to breaking, while copper has a large plastic region after the yield point.
2. 2. Glass breaks at a much lower strain (≈0.1%) than copper, which can extend by 20-30% before breaking.
3. Glass does not have a yield point, while ductile copper has a clear yield point where plastic deformation starts.

## Calculating Young Modulus From Graphs

Young modulus is equal to the gradient of the linear (elastic) region of a stress-strain graph. You can also calculate it from a force-extension graph using the relationship:

Where $F/e$ is the gradient of the force-extension graph, $l_0$ is original length, and $A$ is cross-sectional area.

**Worked example:** A wire of original length 2.0 m and cross-sectional area $1.0 \times 10^{-6} \text{ m}^2$ has a force-extension gradient of $2.5 \times 10^4 \text{ Nm}^{-1}$ in the linear region. Calculate Young modulus.

1. Use the formula relating gradient of F-e to E:
2. $$E = \frac{\text{gradient} \times l_0}{A}$$
3. Substitute the given values:
4. $$E = \frac{(2.5 \times 10^4) \times 2.0}{1.0 \times 10^{-6}} = 5.0 \times 10^{10} \text{ Pa}$$

> **Exam tip:** Always check units: Young modulus has units of Pa (N m⁻²), common errors leave out the area unit leading to wrong units.

## Common pitfalls

- **Wrong:** Confusing limit of proportionality with elastic limit
  - Why it fails: The two points are very close on the graph but have different definitions
  - Correct: Limit of proportionality = end of proportionality; elastic limit = end of fully elastic deformation
- **Wrong:** Comparing stiffness of different samples from force-extension graphs
  - Why it fails: Force-extension gradient depends on sample size, not just material
  - Correct: Always use stress-strain graphs to compare material properties between different samples
- **Wrong:** Calculating Young modulus from the gradient of the entire graph including plastic region
  - Why it fails: Young modulus is only defined for the elastic proportional region
  - Correct: Only measure the gradient of the initial straight line section
- **Wrong:** Taking gradient from the origin when axes are offset
  - Why it fails: Many plotted graphs have axes that do not start at (0,0), so gradient from origin will be wrong
  - Correct: Calculate gradient using two points on the straight line section of the graph
- **Wrong:** Claiming brittle materials always have lower breaking stress than ductile materials
  - Why it fails: Brittle materials often have higher breaking stress than ductile materials, just lower breaking strain
  - Correct: Brittle materials break at lower strain (lower extension), not necessarily lower stress

## Cheatsheet

| Feature | Force-Extension Graph | Stress-Strain Graph |
| --- | --- | --- |
| Gradient (linear region) | $F/e$ = sample stiffness | $\sigma/\epsilon = E$ = material Young modulus |
| Limit of proportionality | End of straight line | End of straight line |
| Elastic limit | Maximum force for elastic deformation | Maximum stress for elastic deformation |
| Yield point | Not well-defined (sample dependent) | Clear distinct point |
| Fracture point | Breaking force (sample dependent) | Breaking stress (material property) |

## What's next

Mastery of force-extension and stress-strain graphs is core to understanding deformation of solids, and this topic appears in both multiple choice and structured questions in CIE A-Level exams. These concepts are the foundation for analysing energy stored in elastic materials, selecting materials for engineering applications, and understanding how materials fail under load. You can build on this knowledge by exploring related topics below.

- [Elastic and plastic deformation](https://www.owlsprep.com/study/cie-9702-u6-elastic-and-plastic-deformation/)
- [Strain energy](https://www.owlsprep.com/study/cie-9702-u6-strain-energy/)
- [Waves](https://www.owlsprep.com/study/cie-9702-u7-overview/)

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