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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…

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Jetzt kostenlos anmeldenWhen 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.

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.

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.

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.

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.

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.

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.

A spring with a mass of 100g and a constant of 17 Newtons per metre is pulled 5cm from its resting position and then released. Calculate the velocity of the spring after its release.

We can calculate the potential energy stored in the spring using the formula below.

\[U = \frac{1}{2}k \cdot \Delta x\]

Here, k is the spring constant in Newtons per metre, while Δ is the elongation measured in metres. The potential energy will be equal to the kinetic energy when the spring is released, which is given by the following formula:

\[E_k = \frac{1}{2} m \cdot v^2\]

If we equate both and solve for v, we can obtain the velocity of the mass that is moving after the spring has been released.

\[v = \sqrt{\frac{k \cdot x}{m}} = \sqrt{\frac{17 N/m \cdot 0.05 m}{0.1 kg}} \cdot 2.915 m/s \approx 2.92 m/s\]

We can also calculate the velocity of the spring using the elastic strain energy (see the figure below). In this case, the elastic strain energy is equal to the force applied to the spring and the spring’s deflection. The stored energy is released and converted into kinetic energy when the spring is set in motion.

Energy related to elasticity in materials is a complex matter as it is related to the materials’ structure and how this behaves under mechanical stress conditions, such as when a force stretches or compresses them.

A rubber band, for instance, if stretched, deforms. At the macroscopic level, mechanical energy is stored for as long as the band is stretched. At the microscopical level, the disordered polymer chains of which the rubber band consists become ordered and stretched out. This phenomenon decreases the entropy of the rubber band while also producing heat on its structure.

- Elastic strain energy is the area below the force vs elongation curve.
- To calculate the elastic strain energy, we need to calculate half of the rectangular area with the height being equal to the applied force and the base being equal to the elongation.
- The elastic strain energy can only be calculated in the elastic zone. Above this zone, work done on the material deforms it irreversibly.
- Materials under the elastic zone, compressing or extending linearly proportional to the force acting on them, follow Hooke’s law.

To do this, we multiply the force by how much the material was deformed and divide it by 2. However, this is only valid for the elastic zone.

This means that as long the force vs the elongation of any material follows a straight line with the slope ‘m’, we can say that those materials follow Hooke’s law.

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