# Modulus of Resilience

## What is Modulus of Resilience

One way to describe a material is by its ability to absorb energy when deformed elastically. For instance, if you took a plastic ruler and bent it without breaking it, you’ve just stored some elastic energy in it, and once you remove the bending force, it releases that energy and snaps back to its original shape. This ability of a material to return to its original shape is what we call resilience in physics.

### Definitions:

**Resilience** is the ability of a material to absorb energy when it is deformed elastically, due to an applied force, and to release that energy once that force is removed.

**Proof resilience** is defined as the maximum strain energy that can be absorbed by a material under stress, up to the elastic limit.

**Modulus of resilience** denoted by (U_{r}) or (μ), is defined as the maximum strain energy that can be absorbed, up to the elastic limit, per unit volume.

## Modulus of Resilience Test

Various testing methods can be employed to determine the modulus of resilience of a material. The most common testing method is the tensile test. In this test, a sample of the material is clamped at both ends and stretched until it fractures.

During the test, the load and corresponding elongation of the sample are measured, and the stress and strain values are calculated. The modulus of resilience can be determined from the area under the stress-strain curve up to the yield point, which represents the elastic region of the material’s deformation. The modulus of resilience is equal to the integral of the stress-strain curve up to the yield point divided by the sample volume.

In addition to the tensile test, other testing methods, such as compression testing, flexural testing, impact testing, and hardness testing, can also be used to measure the modulus of resilience. The choice of testing method depends on the type of material and the intended application, as well as the available equipment and resources.

## Formula & Units

Modulus of resilience can be found as:

$U}_{r}=\frac{{\mathit{\sigma}}_{y}^{2}}{2E}=\frac{{\mathit{\sigma}}_{y}{\mathit{\epsilon}}_{y}}{2$Where:

- U
_{r}is the modulus of resilience - σ
_{y}is the yield stress - E is the young’s modulus
- ε
_{y}is the yield strain

The SI unit of modulus of resilience is pascal (Pa), which is equal to 1 Newton per square meter (N/m^{2}).

The US customary unit of modulus of resilience is pounds per square inch (psi).

## Modulus of Resilience Example 1.1

Let’s say we want to find the modulus of resilience for a steel material with a yield strength (σ_{y}) of 350 MPa (10^{6} N/m^{2}) and a young’s modulus (E) of 200 GPa (10^{9} N/m^{2}).

We can find the modulus of resilience for the steel material as:

$U}_{r}=\frac{{\mathit{\sigma}}_{y}^{2}}{2E}=\frac{(350\times {10}^{6}{)}^{2}}{2(200\times {10}^{9})}=3.06\times {10}^{5}\phantom{\rule{0.22em}{0ex}}\text{Pa$Modulus of Resilience Example 1.2

Suppose we needed to find the modulus of resilience for an aluminum material with a yield strength (σ_{y}) of 110 MPa (10^{6} N/m^{2}) and a yield strain (ε_{y}) of 0.0016.

We can find the modulus of resilience for the aluminum material as:

$U}_{r}=\frac{{\mathit{\sigma}}_{y}{\mathit{\epsilon}}_{y}}{2}=\frac{(110\times {10}^{6})(0.0016)}{2}=8.8\times {10}^{4}\phantom{\rule{0.22em}{0ex}}\text{Pa$## Factors Affecting Modulus of Resilience

The modulus of resilience is influenced by various factors related to the material’s composition, structure, and processing. Some of the key factors affecting the modulus of resilience include:

- Elastic modulus: The elastic modulus, which is a measure of a material’s stiffness, has a significant impact on the modulus of resilience. Higher elastic moduli generally lead to higher moduli of resilience.
- Yield strength: The yield strength, which is the stress at which a material starts to experience plastic deformation, can also affect its modulus of resilience. Materials with higher yield strengths tend to have higher moduli of resilience, as they can absorb more energy before undergoing permanent deformation.
- Temperature: The temperature at which a material is tested can also affect its modulus of resilience. Generally, most materials exhibit lower moduli of resilience at higher temperatures due to increased thermal energy and more rapid dislocation motion.
- Strain rate: The rate at which a material is loaded and unloaded can also impact its modulus of resilience. Materials subjected to high strain rates, such as in impact or explosion scenarios, tend to exhibit lower moduli of resilience than when loaded at lower rates.

## Modulus of Resilience Applications

The modulus of resilience is an essential mechanical property of materials that plays a significant role in their practical applications. Some of the applications of the modulus of resilience are:

- Impact-Resistant Applications: Materials that can absorb large amounts of energy before undergoing plastic deformation are ideal for applications where impact resistance is critical. For instance, the modulus of resilience is a key factor in selecting materials for manufacturing safety equipment, such as helmets, body armor, and protective padding.
- Designing Structural Components: The modulus of resilience is an essential parameter in designing structural components that can withstand dynamic loading conditions, such as bridges and buildings. By selecting materials with high moduli of resilience, engineers can design structures that can resist sudden impacts and vibrations while maintaining their shape and structural integrity.
- Developing High-Performance Materials: The modulus of resilience is a critical property for developing high-performance materials for various applications. For instance, it is used in the development of new materials for the automotive and aerospace industries, where lightweight, energy-absorbing materials are in high demand.
- Quality Control: The modulus of resilience is used as a quality control parameter in manufacturing processes. By measuring the modulus of resilience of a material, manufacturers can ensure that the material meets the required specifications and is suitable for the intended application.

## Modulus of Resilience Summary | ||
---|---|---|

Definition | The maximum strain energy that can be absorbed per unit volume, up to the yield point. | |

Symbol | U_{r} or μ | |

Formula | $U}_{r}=\frac{{\mathit{\sigma}}_{y}^{2}}{2E}=\frac{{\mathit{\sigma}}_{y}{\mathit{\epsilon}}_{y}}{2$ | |

Units | Si unit (Pa) | Us unit (psi) |

### Frequently Asked Questions

- How is the modulus of resilience measured?
- The modulus of resilience is measured by calculating the area under the stress-strain curve up to the yield point, which represents the elastic region of the material’s deformation. This calculation is done by dividing the integral of the stress-strain curve up to the yield point by the sample volume.

- What is the relation between modulus of resilience and young’s modulus?
- The relation between modulus of resilience and young’s modulus is described as: $U}_{r}=\frac{{\mathit{\sigma}}_{y}^{2}}{2E$

- What factors affect the modulus of resilience of a material?
- Several factors can affect the modulus of resilience of a material, including its elastic modulus, yield strength, and microstructure. Other factors such as temperature, strain rate, and loading conditions can also affect the modulus of resilience.

- Can the modulus of resilience be improved by changing the material composition or structure?
- Yes, the modulus of resilience can be improved by changing the material composition or structure. By altering the composition or structure of the material, it is possible to increase its yield strength or elastic modulus, which in turn can lead to an increase in the modulus of resilience.