For a particular material, the modulus of resilience is the maximum resilience per unit volume of the object. To learn more about the modulus of resilience, its formula, units, and how to calculate it, continue reading the article.

**In this article, we’re going to discuss:**

- What is Modulus of resilience?
- Modulus of resilience formula:
- Modulus of resilience units:
- How to calculate modulus of resilience?
- Modulus of resilience examples:
- FAQ’s:

## What is Modulus of resilience?

**The modulus of resilience is defined as the ratio of proof resilience and the volume of the specimen or object. It is also defined as the maximum amount of energy absorbed by the unit volume of an object due to straining up to an elastic limit.**

It is denoted by the symbol `U_{r}`.

Mathematically, It is expressed as,

`U_{r}=\frac{\text{Proof resilience}}{\text{Volume of object}}`

Where proof resilience is the area under the stress-strain curve up to an elastic limit.

To understand the modulus of resilience, it is better to know about resilience and proof resilience.

**Resilience:**

When the object or specimen is strained below the elastic limit, then the energy stored in that specimen is known as resilience.

We can also say that resilience is the strain energy stored in the object when the object is strained up to the elastic limit.

It is also given by the area covered under the stress-strain curve before reaching the elastic limits.

Mathematically, it is given by,

`U=\frac{\sigma ^{2}}{2E}V`

Where,

σ = Stress

E = Young’s modulus

V = Volume of specimen

**Proof resilience:**

Proof resilience is the maximum value of resilience. **OR** it is the maximum amount of strain energy absorbed by the object up to the elastic limit.

This energy doesn’t cause permanent deformation in the object. It is also given as the area covered by the street strain curve up to the elastic limit.

## Modulus of resilience formula:

The modulus of resilience is given by,

`U_{r}=\frac{\sigma _{y}^{2}}{2E}`

Where,

`\sigma _{y}` = Yield stress

E = Modulus of elasticity or Young’s modulus

## Modulus of resilience units:

**SI unit:-**

In the SI system, the unit of proof resilience is J and the unit of volume is m^{3}. Therefore the unit of modulus of resilience is given by,

`U_{r}=\frac{\text{Proof resilience}}{\text{Volume}}`=`\frac{J}{m^{3}}`

Hence the **SI unit of the modulus of resilience is J/m ^{3}.**

**FPS unit:-**

In the FPS system, the unit of proof resilience is Btu and the unit of volume is ft^{3}. Therefore the unit of modulus of resilience is given by,

`U_{r}=\frac{\text{Proof resilience}}{\text{Volume}}`=`\frac{Btu}{ft^{3}}`

Hence the **FPS unit of modulus of resilience is Btu/ft ^{3}.**

## How to calculate modulus of resilience?

**a] From yield strength and modulus of elasticity:-**

Find the yield strength (`\sigma _{y}`) and modulus of elasticity of the material. Use the below formula for calculating the modulus of resilience.

`U_{r}=\frac{\sigma _{y}^{2}}{2E}`

**b] From stress-strain curve:-**

The modulus of resilience can also be calculated from the stress-strain curve and the volume of the specimen used for finding the stress-strain curve.

To find the modulus of resilience, find the area under the curve up to the elastic limit, and find the volume of the specimen. Then use the below formula to find the modulus of resilience.

`\text{Modulus of resilience,} U_{r} = \frac{\text{Area under stress-strain curve up to elastic point}}{\text{Volume of specimen}}`

## Modulus of resilience examples:

**a] Modulus of resilience of steel**:-

For steel, the yield strength is 550 x 10^{6} N/m^{2} and the modulus of elasticity is 207 x 10^{9} N/m^{2}. Hence the modulus of resilience is given by,

`U_{r}=\frac{\sigma _{y}^{2}}{2E}`

`U_{r}=\frac{(550\times 10^{6})^{2}}{2\times (257\times 10^{9})}`

U_{r} = 5.88 x 10^{5} J/m^{3}

Hence for steel, the modulus of resilience is 5.88 x 10^{5} J/m^{3}.

**b] Modulus of resilience of aluminum:-**

For aluminum, the yield strength is 250 x 10^{6} N/m^{2}, and the modulus of elasticity is 97 x 10^{9} N/m^{2}.

Hence the modulus of resilience is given by,

`U_{r}=\frac{\sigma _{y}^{2}}{2E}`

U_{r} = `\frac{(250\times 10^{6})^{2}}{2\times (97\times 10^{9})}`

U_{r} = 3.22 x 10^{5} J/m^{3}

Hence for aluminum, the modulus of resilience is 3.22 x 10^{5} J/m^{3}.

## FAQ’s:

What is the difference between resilience and modulus of resilience?

Resilience is the strain energy stored by an object before reaching the elastic limit while the modulus of resilience is the maximum value of resilience per unit volume.

Is modulus of resilience the same as modulus of elasticity?

No, the Modulus of elasticity is the ratio of normal stress to the normal strain in the object while the modulus of resilience is the ratio of the proof resilience to the volume of an object.

Is modulus of resilience the same as toughness?

No, the modulus of resilience is only associated with the area under the stress-strain curve up to the elastic limit while the toughness is associated with the area under the complete stress-strain curve.

What means the modulus of resilience in tension?

For the material in tension, the modulus of resilience is the maximum resilience per unit volume.