4.10 Elastic Strain Energy

• For a material which obeys Hooke's law, the elastic strain energy, E_el can be determined by finding the area under the force-extension graph
  • Since this area will be a triangle with sides F (force) and x (extension) the equation is:

• Where:
  • E_el = elastic strain energy (or work done) (J)
  • F = average force (N)
  • Δx = extension (m)

• Since Hooke's Law states that F = kΔx, the elastic strain energy can also be written as:  

• Where:
  • k = spring constant (N m^–1)

• 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 is 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

Linear Graphs

• For the region where the material obeys Hooke's law, the work done is the area of a right-angled triangle under the graph

Non-linear Graphs

• 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
• For the remaining part, count the squares left over
  • Before adding squares to the total they must be converted using the values on the axes
  • For example, if each division on the y-axis = 0.1 N and each division on the x-axis = 0.2 m, then each square = 0.1 × 0.2 N m = 0.02 N m

Step 1: Choose which aspect of the graph will give the answer

• • This is a force-extension graph, 
  • Since work done = force × distance, find the area under the graph

Step 2: Divide the area under the line into geometric shapes, using as much of the space as possible

Step 3: For each shape calculate the area and find the total

Total of geometric shapes = 0.025 + 0.05 + 0.75 + 1.5 +0.015 = 2.34

Step 4: For the remaining area count squares then convert

• • Total of counted squares

7 + 13 = 20

• • Value of each small square

0.1 N × 0.01 m = 0.001 N m

• • Multiply to find value of area of counted squares

20 × 0.001 = 0.02

Step 5: Add the totals together to find the area

2.34 + 0.02 = 2.36

Step 6: Write the final answer to the same significant figures as the data from the graph and include units

• • Work done stretching the material = 2.4 J