3.7.7Forces and Motion
Stress-Strain Graphs
5 min read 6 parts Lesson 62 of 66 Unit quiz Virtual lab
Section 1 of 6Axes and Quantities
Converting force and extension into stress and strain removes the effect of the sample’s dimensions.
Stress
- Stress σ = F/A is the force applied per unit cross-sectional area, in pascals.
Strain
- Strain ε = ΔL/L is the extension divided by the original length, so it has no unit.
Young modulus
- In the linear region E = σ/ε, so it equals the gradient of the graph and has the unit pascal.
Material property
- Unlike force-extension graphs, stress-strain graphs describe a material and do not depend on the size of the sample.
Area
- The area under the graph is the energy stored per unit volume in the material.
Section 2 of 6Key Features of a Ductile Metal
A copper or mild steel wire shows the most complete pattern.
1
The graph begins as a straight line through the origin, and the gradient of this line is the Young modulus.
2
At the limit of proportionality the line starts to curve, and shortly after this the elastic limit is reached.
3
At the yield point the material suddenly stretches a lot with little extra stress, and plastic deformation begins.
4
The stress rises to a maximum called the ultimate tensile stress, after which the wire begins to neck.
5
The wire narrows at the neck, so the stress based on the original area falls until the wire fractures at the breaking point.
Section 3 of 6Comparing Material Types
Comparing Material Types
Ductile
- A ductile material such as copper undergoes large plastic deformation and can be drawn into wires.
Brittle
- A brittle material such as glass or ceramic has a straight graph that ends abruptly at fracture, with almost no plastic region.
Polymeric
- Rubber and similar polymers show a curved graph with large strains and hysteresis between loading and unloading.
Strong and stiff
- Strength is the ultimate tensile stress, while stiffness is the Young modulus, and the two do not necessarily go together.
| Property | What the graph shows |
|---|---|
| Stiff | High gradient (large Young modulus) |
| Strong | High ultimate tensile stress |
| Tough | Large area under the whole curve |
| Ductile | Long plastic region before fracture |
Section 4 of 6Using the Graph
Finding E
- Take the gradient from two widely spaced points on the linear region, using values in Pa and not in MPa or GPa mixed together.
Safe design
- Engineers keep stresses well within the elastic region so that structures do not deform permanently.
Unloading
- Beyond the elastic limit the unloading line is parallel to the loading line, leaving a permanent strain.
Section 5 of 6Worked Example
Question
- A steel wire (Young modulus 2.0 × 10¹¹ Pa) is 2.0 m long with cross-section 1.0 × 10⁻⁶ m².
- A 150 N load is applied within the linear region.
- Find the stress, strain and extension.
- Stress = F/A = 150 / (1.0 × 10⁻⁶) = 1.5 × 10⁸ Pa.
- Strain = stress / E = 1.5 × 10⁸ / 2.0 × 10¹¹ = 7.5 × 10⁻⁴.
- Extension = strain × length = 7.5 × 10⁻⁴ × 2.0 = 1.5 × 10⁻³ m.
Answer Stress = 1.5 × 10⁸ Pa, strain = 7.5 × 10⁻⁴, extension = 1.5 mm.
Section 6 of 6Exam Checklist & Tips
3.7.7 – Key Points for the Exam
- Stress Force per unit cross-sectional area, in Pa.
- Strain Extension per unit original length, with no unit.
- Yield point The stress at which a ductile material begins to deform plastically with a large increase in strain.
- Ultimate tensile stress The maximum stress a material can withstand before it starts to neck and break.
Common Mistakes
- Using the force-extension gradient as the Young modulus instead of the stress-strain gradient.
- Quoting strain with a unit such as metres; strain is a ratio.
- Mixing up strength (ultimate tensile stress) with stiffness (Young modulus).
- Saying that brittle materials stretch a lot before breaking; they fracture with very little strain.