Area under a Force-Extension Graph
- Work has to be done to stretch a material
- Before a material reaches its elastic limit (whilst it obeys Hooke's Law), all the work done is stored as elastic strain energy
- The work done, or the elastic strain energy is the area under the force-extension graph
Work done is the area under the force-extension graph
- This is true for whether the material obeys Hooke’s law or not
- For the region where the material obeys Hooke’s law, the work done is the area of a right-angled triangle under the graph
- For the region where the material doesn’t obey Hooke’s law, the area is the full region under the graph. To calculate this area, split the graph into separate segments and add up the individual areas of each
Worked example
The graph shows the behaviour of a sample of a metal when it is stretched until it starts to undergo plastic deformation.
What is the total work done in stretching the sample from zero to 13.5 mm extension?
Simplify the calculation by treating the curve XY as a straight line.
Worked example
A spring is extended with varying forces; the graph below shows the results.
What is the energy stored in the spring when the extension is 40 mm?
Examiner Tip
Remember to always double-check the units on the force and extension axes on the graph before using values from it for calculations. The force can sometimes be in kN and the extension in mm!
