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. 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.

## What is the 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 modulus of elasticity. A lower modulus of elasticity means the material is more flexible and less stiff.

## Unit of Modulus of Elasticity

In Equation 1 above, Stress has a unit of N/m2 or Pascal and Strain is unit-less as it is a ratio of two lengths. Accordingly, the modulus of elasticity 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 modulus of elasticity is measured from the straight portion of the curve. The modulus of elasticity 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 modulus of elasticity 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) provides 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. Kindly review the article by clicking here to learn the major nomenclatures used for the 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 modulus of elasticity. They are:

- 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:** Sheat 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 modulus of elasticity 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.

All these three elastic moduli are interrelated between them by the following set of equations:

### 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

## 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 |

**Young’s Modulus of Common Materials**

## Factors Affecting Modulus of Elasticity

The parameters that affect the modulus of elasticity of material are:

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

## Applications of Modulus of Elasticity

The main applications of elastic modulus are:

- Modulus of Elasticity values helps in choosing the correct material for engineering design.
- Comparing different materials becomes easier if the modulus of elasticity is known.
- Knowing the modulus of elasticity 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 modulus of elasticity values to find out the load-carrying capability of their complex structures.