Force-extension and stress-strain graphs

CIE A-Level Physics· Unit 6: Deformation of Solids· 15 min read

1. Key Features of Force-Extension Graphs★★☆☆☆⏱ 5 min

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.

📘 Definition

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.

📘 Definition

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.

2. Why Stress-Strain Graphs Are Useful★★★☆☆⏱ 6 min

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.

📘 Definition

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 ( $σ = F / A$ ) and strain normalises extension by original length ( $ϵ = Δ l / l_{0}$ ). 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.

3. Graph Shapes For Different Materials★★★☆☆⏱ 7 min

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
1. 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.

4. Calculating Young Modulus From Graphs★★☆☆☆⏱ 4 min

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:

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📐 Worked Example

A wire of original length 2.0 m and cross-sectional area $1.0 \times 1 0^{- 6} m^{2}$ has a force-extension gradient of $2.5 \times 1 0^{4} Nm^{- 1}$ 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.

5. Common Pitfalls

Wrong move:

Confusing limit of proportionality with elastic limit

Why:

The two points are very close on the graph but have different definitions

Correct move:

Limit of proportionality = end of proportionality; elastic limit = end of fully elastic deformation

Wrong move:

Comparing stiffness of different samples from force-extension graphs

Why:

Force-extension gradient depends on sample size, not just material

Correct move:

Always use stress-strain graphs to compare material properties between different samples

Wrong move:

Calculating Young modulus from the gradient of the entire graph including plastic region

Why:

Young modulus is only defined for the elastic proportional region

Correct move:

Only measure the gradient of the initial straight line section

Wrong move:

Taking gradient from the origin when axes are offset

Why:

Many plotted graphs have axes that do not start at (0,0), so gradient from origin will be wrong

Correct move:

Calculate gradient using two points on the straight line section of the graph

Wrong move:

Claiming brittle materials always have lower breaking stress than ductile materials

Why:

Brittle materials often have higher breaking stress than ductile materials, just lower breaking strain

Correct move:

Brittle materials break at lower strain (lower extension), not necessarily lower stress

6. Quick Reference Cheatsheet

Feature	Force-Extension Graph	Stress-Strain Graph
Gradient (linear region)	$F / e$ = sample stiffness	$σ / ϵ = E$ = 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)

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · 12
Identify elastic limit on F-e graph
2023 · 22
Calculate Young modulus from stress-strain
2021 · 11
Compare ductile and brittle graphs

Going deeper

practical guideCIE A-Level Young modulus practicalReference for required practical on this topic

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.