# Young's Modulus Calculator

Created by Luciano Mino
Reviewed by Wojciech Sas, PhD candidate and Adena Benn
Last updated: Sep 28, 2022

With this Young's modulus calculator, you can obtain the modulus of elasticity of a material, given the strain produced by a known tensile/compressive stress.

We will also explain how to automatically calculate Young's modulus from a stress-strain curve with this tool or with a dedicated plotting software.

• What the modulus of elasticity is;
• How to calculate Young's modulus with the modulus of elasticity formula;
• What Young's modulus unit is;
• What material has the highest Young's modulus; and more.

## What is the modulus of elasticity?

Young's modulus, or modulus of elasticity, is a property of a material that tells us how difficult it is to stretch or compress the material in a given axis.

This tells us that the relation between the longitudinal strain and the stress that causes it is linear. Therefore, we can write it as the quotient of both terms.

💡 Read more about strain and stress in our true strain calculator and stress calculator!

However, this linear relation stops when we apply enough stress to the material. The region where the stress-strain proportionality remains constant is called the elastic region.

If we remove the stress after stretch/compression within this region, the material will return to its original length.

Because of that, we can only calculate Young's modulus within this elastic region, where we know the relationship between the tensile stress and longitudinal strain.

🙋 If you want to learn how the stretch and compression of the material in a given axis affect its other dimensions, check our Poisson's ratio calculator!

## Young's modulus equation

Before jumping to the modulus of elasticity formula, let's define the longitudinal strain $\epsilon$:

$\epsilon =\frac{L - L_{0}}{L_{0}},$

where:

• $L_{0}$ is the material's initial length; and
• $L$ is the length while being under tensile stress.

And the tensile stress $\sigma$ as:

$\sigma = \frac{F}{A},$

where:

• $F$ is the force producing the stretching/compression; and
• $A$ is the area on which the force is being applied.

Thus, Young's modulus equation results in:

$E = \frac{\sigma}{\epsilon}$

Since the strain is unitless, the modulus of elasticity will have the same units as the tensile stress (pascals or Pa in SI units).

## How do I calculate Young's modulus?

To calculate the modulus of elasticity E of material, follow these steps:

1. Measure its initial length, L₀ without any stress applied to the material.
2. Measure the cross-section area A.
3. Apply a known force F on the cross-section area and measure the material's length while this force is being applied. This will be L.
4. Calculate the strain ϵ felt by the material using the longitudinal strain formula: ϵ = (L - L₀) / L₀.
5. Calculate the tensile stress you applied using the stress formula: σ = F / A.
6. Divide the tensile stress by the longitudinal strain to obtain Young's modulus: E = σ / ϵ.

## Example using the modulus of elasticity formula

Let's say we have a thin wire of an unknown material, and we want to obtain its modulus of elasticity.

Assuming we measure the cross-section sides, obtaining an area of  A = 0.5×0.4 mm. Then we measure its length and get L₀ = 0.500 m.

Now, we apply a known force, F = 100 N for example, and measure, again, its length, resulting in L = 0.502 m.

Before computing the stress, we need to convert the area to meters:

A = 0.5×0.4 mm = 0.0005×0.0004 m

With those values, we are now ready to calculate the stress σ = 100/(0.0005×0.0004) = 5·10⁸ Pa and strain ϵ = (0.502 - 0.500) / 0.500 = 0.004.

Finally, if we divide the stress by the strain according to the Young's modulus equation, we get: E = 5·10⁸ Pa / 0.004 = 1.25·10¹¹ Pa or E = 125 GPa, which is really close to the modulus of elasticity of copper (130 GPa). Hence, our wire is most likely made out of copper!

## How to calculate Young's modulus from a stress-strain curve

Our Young's modulus calculator also allows you to calculate Young's modulus from a stress-strain graph!

To plot a stress-strain curve, we first need to know the material's original length, $L_{0}$. Then, we apply a set of known tensile stresses and write down its new length, $L$, for each stress value.

Lastly, we calculate the strain (independently for each stress value) using the strain formula and plot every stress-strain value pair using the $Y$-axis and $X$-axis, respectively.

### Analysing the stress-strain chart

As you can see from the chart above, the stress is proportional (linear) to the strain up to a specific value. This is the elastic region, and after we cross this section, the material will not return to its original state in the absence of stress.

Since the modulus of elasticity is the proportion between the tensile stress and the strain, the gradient of this linear region will be numerically equal to the material's Young's modulus.

We can then use a linear regression on the points inside this linear region to quickly obtain Young's modulus from the stress-strain graph.

Our Young's modulus calculator automatically identifies this linear region and outputs the modulus of elasticity for you. Give it a try!

## FAQ

### Is stiffness the same as Young's modulus?

No, but they are similar. Stiffness is defined as the capacity of a given object to oppose deformation by an external force and is dependent on the physical components and structure of the object. Young's modulus is an intensive property related to the material that the object is made of instead.

### Is tensile modulus the same as Young's modulus?

Yes. Tensile modulus is another name for Young's modulus, modulus of elasticity, or elastic modulus of a material. It relates the deformation produced in a material with the stress required to produce it.

### What material has the highest Young's modulus?

Diamonds have the highest Young's modulus or modulus of elasticity at about ~1,200 GPa. Diamonds are the hardest known natural substances, and they are formed under extreme pressures and temperatures inside Earth's mantle.

### Is the modulus of elasticity constant?

Yes. Since the modulus of elasticity is an intensive property of a material that relates the tensile stress applied to a material, and the longitudinal deformation it produces, its numerical value is constant. The resulting ratio between these two parameters is the material's modulus of elasticity.

Luciano Mino
Calculate from:
Single stress/strain value
Stress
Force (F)
N
Area (A)
in²
Stress (σ)
psi
Strain
Final length (L)
ft
Initial length (L₀)
ft
Strain (ε)
Result
Young modulus (E)
psi
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