In this article, we shall learn about **Modulus of elasticity** its types, formulas, units, symbols and uses. We have provided a PDF for the same.

## Introduction

**Modulus of elasticity** is a material property derived from the stress-strain curve of a specific material. Also known as Elastic modulus or Elasticity Modulus, this is a measurement of a material’s elasticity.

The modulus of elasticity gives a quantified value for a material’s resistance to elastic deformation. It is a very useful parameter in engineering design, mechanics, and strength of materials. In this article, we will learn about the basics of the modulus of elasticity, its formula, unit, measurement, types, symbols, and applications.

## Definition of Modulus of Elasticity

**Hooke’s law** states that the stress of a material is directly proportional to its strain up to the proportional limit. So, mathematically we can write,

**Stress (σ) α Strain (ε) or, σ = E ε**

This constant E is known as Modulus of Elasticity (E)=σ/ε Equation 1

So, the **Modulus of elasticity can be defined **as the ratio of normal stress to longitudinal strain within the proportional limit. Alternately, it can be said that the amount of stress required to create unit strain in any material is equal to its modulus of elasticity.

As Stress (σ)=Force (F)/Area (A) and Strain (ε)=Change in length(δL)/Original length (L),

We can write, Modulus of elasticity, E=(F*L) / (A * δL)

A stiffer material has a higher value of elastic modulus. A lower elastic modulus means the material is more flexible and less stiff.

## Unit of Modulus of Elasticity

In Equation 1 above, Stress has a unit of N/m^{2} or Pascal and Strain is unit-less as it is a ratio of two lengths. Accordingly, It has the same unit as that of stress. Hence, the **unit of modulus of elasticity is Pascal**. However, as the value of elastic modulus is usually high, it is denoted by MPa (Megapascals) or GPa (Gigapascals).

1 MPa =10^{6} Pa and 1 GPa =10^{9} Pa

## Measuring Modulus of Elasticity

**Modulus of elasticity** is measured by testing the specimen on **Universal Testing Machine**. The specimen is loaded into the UTM machine. The machine slowly keeps on increasing the load on the specimen till it breaks. The stress and strain are plotted and the value is measured from the straight portion of the curve. It is basically the slope/gradient of two stress points within the elastic region. The test method followed for tensile testing is governed by ASTM D638 or ISO 527-1

The elastic modulus for different materials is established by testing the specimens in the universal testing machine. But for engineering purposes, we get established values from various codes and standards. For example, the ASME codes (ASME BPVC code) provide the modulus of elasticity values for most of the materials. So, for engineering design activities, we simply follow the relevant codes and take the elastic modulus value for the specific materials.

## Stress-Strain curve

We have already published a detailed article on the Stress-Strain curve.

The **modulus of elasticity** is a material property and the value of elastic modulus is constant for the same material at a constant temperature. However, the values of elasticity modulus change with respect to temperature. With an increase in temperature, the modulus of elasticity usually decreases.

## Types of Modulus of Elasticity

Depending on the kind of stress generated in an object, there are 3 main types of elastic modulus which are as follows

- Young’s modulus
- Shear Modulus
- Bulk Modulus

**Young’s Modulus**: This is the most frequently referred term for elastic modulus. Young’s modulus is defined as the ratio of tensile stress to the tensile strain and specifies the tendency of a material to become longer or shorter. Young’s modulus is generated under tensile or compressive force. Further details of Young’s modulus are explained here.

**Shear Modulus**: Shear modulus is defined as the ratio of shear stress to shear strain and shows the tendency of a substance to change from a rectangular shape to a parallelogram.

**Bulk Modulus**: Bulk modulus is the ratio of volumetric stress to the volumetric strain and shows the tendency of volumetric change of the material.

## Symbols of Modulus of Elasticity

Each type of elastic modulus is usually defined by different symbols in the engineering application. The common symbol for Young’s modulus is E, the popular symbol for Shear modulus is G, and the widely used symbol for Bulk modulus is K.

## Relationship Between Elastic Constants

Elastic constants refer to a set of parameters that describe how a material responds to an applied stress system. They help calculate the theoretical engineering strain of a material, which is the change in its size or shape under stress. Elastic constants enable engineers to establish a connection between the amount of stress applied and the resulting strain. In the case of a uniform and isotropic material, the elastic constants are limited to four.

The different types of elastic constant are as follows

- Young’s modulus (E)
- Shear modulus or modulus of rigidity (G)
- Bulk modulus (K)
- Poisson’s Ratio (µ)

#### E=2G(1+μ)

E=3K(1-2μ)

E=9KG/(G+3K)

Where

- E= Young’s modulus
- G=Shear modulus
- K=Bulk Modulus
- μ =Poisson Ratio

## Difference between Young’s Modulus and Shear Modulus

Parameters | Young’s Modulus | Shear Modulus |
---|---|---|

Definition | Young’s modulus describes the ratio of stress and strain in the direction of the applied force. | Shear modulus describes the ratio of stress and strain perpendicular to the applied force. |

Type of deformation | Young’s modulus is used to measure the longitudinal deformation of a material. | Shear modulus is used to measure the transverse deformation of a material. |

Formula | Young’s modulus is calculated as stress divided by strain (E = σ / ε). | Shear modulus is calculated as stress divided by shear strain (G = τ / γ). |

Materials | Young’s modulus is used for materials that undergo tension or compression, such as metals and ceramics. | Shear modulus is used for materials that undergo shearing, such as plastics and rubber. |

Measurement | Young’s modulus can be measured using tension and compression tests. | Shear modulus can be measured using torsion tests. |

## Difference between Young’s Modulus and Bulk Modulus

Parameters | Young’s Modulus | Bulk Modulus |
---|---|---|

Definition | Young’s modulus describes the ratio of stress and strain in the direction of the applied force. | Bulk modulus describes the ratio of stress and strain in response to uniform hydrostatic pressure. |

Type of deformation | Young’s modulus is used to measure the longitudinal deformation of a material. | Bulk modulus is used to measure the volumetric deformation of a material. |

Formula | Young’s modulus is calculated as stress divided by strain (E = σ / ε). | Bulk modulus is calculated as stress divided by volumetric strain (K = -VΔP / ΔV). |

Materials | Young’s modulus is used for materials that undergo tension or compression, such as metals and ceramics. | Bulk modulus is used for materials that undergo compression or expansion, such as liquids and gases. |

Measurement | Young’s modulus can be measured using tension and compression tests. | Bulk modulus can be measured using hydrostatic pressure tests. |

## Modulus of Elasticity for Materials

The following table provides Young’s modulus values for some common materials

Material | Young’s Modulus (X 10^{6 }PSI) | Young’s Modulus (GPa) |
---|---|---|

Aluminum | 10 | 69 |

Brass | 15.4 | 106 |

Bone | 2.03 | 14 |

Copper | 17 | 117 |

Diamond | 152-175 | 1050-1210 |

Glass | 6.92-12.1 | 50 to 90 |

Gold | 10.8 | 74 |

Rubber | 0.00145 to 0.0145 | 0.01 to 0.1 |

Molybdenum | 48 | 329 |

Nickel | 29 | 200 |

Steel | 29 | 200 |

Wrought Iron | 28 | 193 |

Zinc | 15.7 | 108 |

## Factors Affecting Modulus of Elasticity

The parameters that affect these values of material are as follows

- Temperature
- Presence of Impurity in the material like secondary phase particles, non-metallic inclusions, alloying elements, etc.

## Understanding youngs modulus

## Applications of Modulus of Elasticity

- These values helps in choosing the correct material for engineering design.
- Comparing different materials becomes easier if the modulus of elasticity is known.
- Knowing the values gives the user an idea about the stiffness of the material. He can easily understand if it is easier to deform any material or not.
- Civil engineers use the these values to find out the load-carrying capability of their complex structures.

## FAQ

### What is the modulus of elasticity of steel

The **modulus of elasticity of steel** typically ranges from 29,000 to 30,000 ksi. It measures the material’s ability to deform elastically under load, with a higher modulus indicating greater resistance to deformation. Steel’s high modulus of elasticity makes it a popular choice for construction and engineering applications that require strength and stability.

### what are the units for modulus of elasticity

The **units for modulus of elasticity** are typically expressed in pounds per square inch (psi) or newtons per square meter (N/m^{2}). It is a measure of a material’s ability to deform elastically under load, with a higher modulus indicating greater resistance to deformation.

### What is the modulus of elasticity for aluminium

The **modulus of elasticity for aluminum** typically ranges from 10,000 to 12,000 ksi. It measures the material’s ability to deform elastically under load, with a higher modulus indicating greater resistance to deformation. Aluminum’s value is higher than that of concrete, but it’s often used in combination with other materials due to its low strength-to-weight ratio.

### What is the modulus of elasticity for concrete

The **modulus of elasticity for concrete** typically ranges from 10,000 to 40,000 psi. It’s a measure of the material’s ability to deform elastically under load, with a higher modulus indicating greater resistance to deformation. Concrete’s value is lower than that of steel or aluminum, but it remains a useful construction material due to its other advantageous properties.

### What is the Modulus of elasticity of copper

The modulus of elasticity of copper is approximately 117 GPa (gigapascals) or 17 x 10^6 psi (pounds per square inch) at room temperature.

## Conclusion

In conclusion, the** modulus of elasticity** is a fundamental mechanical property that plays a significant role in various fields, including engineering, construction, materials science, and many others. This property is used to describe the ability of a material to resist deformation under an applied load and return to its original shape when the load is removed.

We have explored the three different types, namely Young’s modulus, shear modulus, and bulk modulus, along with their formulas, units, and symbols. We have also discussed the practical applications of these concepts, ranging from the design of structures to the development of advanced materials.

It is important to note that accurate determination of the elasticity values is crucial for designing safe and efficient structures and predicting material behavior under various conditions. Hence, understanding this property and its measurement methods are essential for researchers, engineers, and scientists working in the relevant fields. As such, the study of them continues to be an active area of research, with ongoing efforts aimed at improving its accuracy and expanding its applications.