When materials are affected by forces such as tension and compression, work is done to them. The work, in this case, is mechanical and must be produced by using energy. As the material compresses or expands, provided it is elastic, the work transmitted to the material in the form of deformation will not change its shape but will be stored as energy.
What is the energy in a material?
When a material is compressed or elongated, the internal structure of the material compresses or elongates, too. Depending on the elasticity of the material, this results in more or less deformation. If the material is still in the elastic zone, the structure of the material will revert to its original state.
Figure 1. For elastic materials, a force ‘F’ produces an elongation δl=a, as in the first image. If the material is under the elastic zone after the force has acted on it (the black cross in the second image), it means that the material has reverted back to its original length lo. The elongation then is δl=0. Source: Manuel R. Camacho, StudySmarter.
Under the material’s elastic zone, the work done on the material to stretch it by a certain length is related to a property known as elastic strain energy.
The energy that is given to the material under the elastic zone can be recovered when the force is removed. However, if the deformation occurs in the plastic area, the energy won’t be recovered. To determine the energy used to deform the material, you only calculate the area under the curve.
How to calculate the elastic strain energy
Because the elastic strain energy occurs in the elastic zone, the area below the curve for the elastic area can be calculated using the following formula:
\[\text{Elastic Strain Energy} = \frac{1}{2} F \cdot \Delta I\]
Here, F is the maximum force where the material reaches its elastic limit, which is measured in Newtons, while Δl is the extension reached in metres.
Figure 2. In the zone where the material has an elastic response (elastic zone), the elastic strain energy can be calculated if we know the force F and the elongation Δl. Source: Manuel R. Camacho, StudySmarter.
The main difference between this curve and the stress-strain curve is that the stress is the force per area while the strain is the deformation relative to the object’s original length.
A metal bar is deformed by an increasing force. The material reaches its elastic limit where the relationship between the force and the elongation is not linear after deforming the bar by 10mm and the force reaches 7.5N. Calculate the elastic strain energy.
As the material is being tested under elastic conditions, the relationship between the force and the deformation is a straight line. You can draw this line from the initial point where the force and the deformation are 0 to the point where F = 7.5N and s = 10 mm.
Figure 3. In this example, the force reaches 7.5N, after which the material begins its plastic phase. The material reaches an elongation of 10mm. Source: Manuel R. Camacho, StudySmarter.
The area below the curve will be equal to half the area of the rectangle, with a height of F = 7.5 and a side of s = 10.
\(\text{Area} = \frac{1}{2} \cdot 7.5 N \cdot 0.001 m = 3.75 \cdot 10^{-3} J\)
In the area where the relationship between the stress and the strain and the force and the extension is linear, the material follows Hooke’s law.
Energy stored by a spring
Springs are also elastic objects, which deform under certain circumstances and come back to their original shape if the force acting on them is not too large. In spring and mass systems, energy is stored as potential energy when they are pulled from their relaxed length. This stored energy can be converted into movement (kinetic energy).
The relaxed length is equal to the spring length when no force is acting on it.
Figure 4. A spring in a compressed, relaxed, and tensed state. Source: Manuel R. Camacho, StudySmarter.
Explanations 
Textbooks 
Exams 
Magazine 
