YOUNG'S MODULE: CALCULUS, APPLICATIONS, EXAMPLES, EXERCISES - PHYSICAL - 2025

The Young's modulus or elastic modulus is the constant relating the tensile or compression with the respective increase or decrease in length having the object under these forces.

External forces applied to objects can not only change their state of motion, but are also capable of changing their shape or even breaking or fracturing them.

Figure 1. The cat's movements are full of elasticity and grace. Source: Pixabay.

Young's modulus is used to study the changes produced in a material when a tensile or compressive force is applied externally. It is very useful in subjects such as engineering or architecture.

The model owes its name to the British scientist Thomas Young (1773-1829), who was the one who carried out material studies proposing a measure of the stiffness of different materials.

What is Young's model?

Young's model is a measure of stiffness. In materials with low stiffness (red) there is more deformation under an extension or compression load. Tigraan / CC BY-SA (https://creativecommons.org/licenses/by-sa/4.0)

How much can an object be deformed? This is something that engineers often want to know. The answer will depend on the properties of the material and the dimensions it has.

For example, you can compare two bars made of aluminum with different dimensions. Each has a different cross-sectional area and length, and both are subjected to the same tensile force.

The expected behavior will be the following:

- The greater the thickness (cross section) of the bar, the less stretch.

- The longer the initial length, the greater the final stretch.

This makes sense, because after all, experience shows that trying to deform a rubber band is not the same as trying to do it with a steel rod.

A parameter called the modulus of elasticity of the material is an indication of its elastic response.

How is it calculated?

Being a doctor, Young wanted to know the role of the elasticity of the arteries in the good performance of blood circulation. From his experiences he concluded the following empirical relationship:

It is possible to graphically represent the behavior of a material under the application of stress, as shown in the following figure.

Figure 2. Graph of stress versus strain for a material. Source: self made.

From origin to point A

In the first section, which goes from the origin to point A, the graph is a straight line. Hooke's Law is valid there:

F = kx

Where F is the magnitude of the force that returns the material to its original state, x is the deformation experienced by it and k is a constant that depends on the object subjected to the stress.

The deformations considered here are small and the behavior is perfectly elastic.

From A to B

From A to B the material also behaves elastically, but the relationship between stress and strain is no longer linear.

From B to C

Between points B and C, the material undergoes permanent deformation, being unable to return to its original state.

From C

If the material continues to stretch from point C, it eventually breaks.

Mathematically, Young's observations can be summarized as follows:

Stress ∝ Strain

Where the constant of proportionality is precisely the modulus of elasticity of the material:

Stress = Modulus of Elasticity x Deformation

There are many ways to deform materials. The three most common types of stress to which an object is subjected are:

- Tension or stretching.

- Compression.

- Cut or shear.

One stress that materials are commonly subjected to, for example in civil construction or automotive parts, is traction.

Formulas

When an object of length L is stretched or tensioned, it is being subjected to a traction that causes a variation in its length. A diagram of this situation is represented in figure 3.

This requires that a force of magnitude F be applied per unit area to its ends to cause stretching, such that its new length becomes L + DL.

The effort made to deform the object will be just this force per unit area, while the strain experienced is ΔL / L.

Figure 3. An object subjected to traction or stretching, experiences elongation. Source: self made.

Denoting Young's modulus as Y, and according to the above:

The answer lies in the fact that the strain indicates the relative strain with respect to the original length. It is not the same as a 1 m bar stretches or shrinks by 1 cm, as a structure 100 meters long is equally deformed by 1 cm.

For the proper functioning of parts and structures, there is a tolerance regarding the allowable relative deformations.

Equation to calculate deformation

If the above equation is analyzed as follows:

- The greater the cross-sectional area, the less deformation.

- The longer the length, the greater the deformation.

- The higher the Young's modulus, the lower the deformation.

Units of stress correspond to newtons / square meter (N / m ²). They are also the units of pressure, which in the International System bear the name of Pascal. The strain ΔL / L, on the other hand, is dimensionless because it is the quotient between two lengths.

The units of the English system are lb / in ² and are also used very frequently. The conversion factor to go from one to the other is: 14.7 lb / in ² = 1.01325 x 10 ⁵ Pa

This leads to the Young's modulus also having units of pressure. Finally, the above equation can be expressed to solve for Y:

In materials science, the elastic response of these to various efforts is important to select the most suitable for each application, whether it is manufacturing an airplane wing or an automotive bearing. The characteristics of the material to be used are decisive in the response expected of it.

To choose the best material, it is necessary to know the stresses to which a certain piece is going to be subjected; and consequently select the material that has the properties most in line with the design.

For example, the wing of an airplane must be strong, light, and capable of flexing. The materials used in the construction of buildings must resist seismic movements to a large extent, but they must also have some flexibility.

Engineers who design airplane wings, and also those who choose construction materials, must make use of stress-strain graphs like the one shown in Figure 2.

Measurements to determine the most relevant elastic properties of a material can be carried out in specialized laboratories. Thus, there are standardized tests to which the samples are subjected, to which various stresses are applied, and the resulting deformations are then measured.

Examples

As already mentioned above, Y does not depend on the size or shape of the object, but on the characteristics of the material.

Another very important note: for the equation given above to be applicable, the material must be isotropic, that is, its properties must remain unchanged throughout.

Not all materials are isotropic: there are those whose elastic response depends on certain directional parameters.

The deformation analyzed in the previous segments is just one of the many that a material can be subjected to. For example, in terms of compressive stress, it is the opposite of tensile stress.

The equations given apply to both cases, and the values ​​of Y are almost always the same (isotropic materials).

A notable exception is concrete or cement, which resists compression better than traction. Therefore, it must be reinforced when resistance to stretching is required. Steel is the material indicated for this, as it resists stretching or traction very well.

Examples of structures subjected to stress include building columns and arches, classic building elements in many ancient and modern civilizations.

Figure 4. The Pont Julien, a Roman construction from 3 BC in southern France.

Solved exercises

Exercise 1

A 2.0 m long steel wire in a musical instrument has a radius of 0.03 mm. When the cable is under a tension of 90 N: how much does its length change? Data: Young's modulus of steel is 200 x 10 ⁹ N / m ²

Solution

It is required to calculate the cross-sectional area A = πR ² = π. (0.03 x 10 ^-3 m) ² = 2.83 x 10 ^-9 m ²

Stress is stress per unit area:

Since the string is under tension, this means that it lengthens.

The new length is L = L _o + DL, where L _o is the initial length:

L = 2.32 m

Exercise 2

A marble column, whose cross-sectional area is 2.0 m ^2, supports a mass of 25,000 kg. Find:

a) The effort in the spine.

b) Strain.

c) How much shorter is the column if its height is 12 m?

Solution

a) The effort in the column is due to the weight of the 25000 kg:

P = mg = 25000 kg x 9.8 m / s ² = 245,000 N

Therefore the effort is:

b) Strain is ΔL / L:

c) ΔL is the variation of the length, given by:

ΔL = 2.45 x 10 ^-6 x 12 m = 2.94 x10 ^-5 m = 0.0294 mm.

The marble column is not expected to shrink significantly. Note that although Young's modulus is lower in marble than in steel, and that the column also supports a much greater force, its length almost does not vary.

On the other hand, in the rope of the previous example the variation is much more appreciable, although the steel has a much higher Young's modulus.

Its large cross-sectional area intervenes in the column, and therefore it is much less deformable.

About Thomas Young

1822 portrait of Thomas Young. Thomas Lawrence / Public domain

The modulus of elasticity is named after Thomas Young (1773-1829), a versatile British scientist who made great contributions to science in many areas.

As a physicist, Young not only studied the wave nature of light, revealed by the famous double-slit experiment, but he was also a physician, linguist, and even helped decipher some of the Egyptian hieroglyphs on the famous Rosetta stone.

He was a member of the Royal Society, the Royal Swedish Academy of Sciences, the American Academy of Arts and Sciences or the French Academy of Sciences, among other noble scientific institutions.

However, it should be noted that the concept of the model was previously developed by Leonhar Euler (1707-1873), and that scientists such as Giordano Riccati (1709-1790) had already carried out an experiment that would have put Young's model into practice..

References

1. Bauer, W. 2011. Physics for Engineering and Sciences. Volume 1. Mac Graw Hill. 422-527.
2. Giancoli, D. 2006. Physics: Principles with Applications. Sixth Edition. Prentice Hall. 238–249.