## Main Difference – Modulus of Elasticity vs. Modulus of Rigidity

Modulus of elasticity and modulus of rigidity are two numbers that are used by materials engineers to describe how a material gets deformed. The **main difference** between modulus of elasticity and modulus of rigidity is that the **modulus of elasticity describes how a material gets deformed when a force is applied at right angles to a surface of an object**, causing the material to elongate or shorten while the **modulus of rigidity describes how a material gets deformed when a force is applied parallel to a surface of an object**, causing one of the surfaces to become shifted with respect to another surface of the same object.

## What is the Modulus of Elasticity

Modulus of elasticity (Young modulus) is a number that describes the ratio of stress to strain in an object that is being deformed by a force that is perpendicular to a surface of an object. The **stress** of a material is the deforming force per unit area. For instance, the figure below shows an object that becomes elongated as a result of a tensile force on it. In this case, the stress () is given by:

Since the deforming force acts at right angles to the face of the object, the stress is often called **normal stress**.

The **strain** is the fractional change in length of the object. Suppose the object had a length before the deforming force acted on it, and if the object gets elongated by a length under the deforming force, then the strain () is given by:

The modulus of elasticity () is then given by:

## What is the Modulus of Rigidity

Modulus of rigidity (shear modulus) is a number that gives the **shear stress** acting on a material per unit area. Here, the deforming force acts *parallel* to a face of the object, causing one face to become displaced with respect to another face. This is depicted below:

So, **shear stress (****)** is given as:

This equation has the same form as the equation for normal stress, the difference is in the way the force acts.

The **shear strain (****)** is defined as the ratio of relative displacement between the surfaces to the separation between the surfaces. Here,

Once again the **shear modulus (****)** is the ratio between shear stress and shear strain:

## Relationship between Modulus of Elasticity and Modulus of Rigidity

Modulus of elasticity () and the modulus of rigidity () are related by the following equation:

Here, represents a number called **Poisson’s ratio** given to the particular material. When the material is being stretched in one direction, it gets shortened in a perpendicular direction. In the direction that the material becomes elongated, the **axial strain (****) **is defined as the fractional increase in the length. In the direction that the material shortens, the **transverse strain (****)** gives the fractional *reduction* in length. The diagram below illustrates these changes in shape:

In this diagram, the axial strain is:

The transverse strain is:

Note that since the object *shortens* in the directiopn perpendicular to the force, . Poisson’s ratio () is defined as:

The minus sign has been introduced to ensure that takes a positive value.

## Difference Between Modulus of Elasticity and Modulus of Rigidity

### Direction of Force

**Modulus of elasticity** is used to calculate the deformation of an object when a deforming force acts at right angles to a surface of the object.

**Modulus of rigidity** is used to calculate deformations when a deforming force acts parallel to the surface of an object.

### Change in Shape

Where **modulus of elasticity** is calculated, the object under the deforming force either gets lengthened or shortened.

Where **modulus of rigidity** is calculated, one of the surfaces of the object becomes displaced with respect to another surface.

### Relative Size

For most materials, the **modulus of elasticity** is larger than the **modulus of rigidity**. The exceptions to this rule are the so-called “auxetic” materials which have *negative Poisson’s ratios*, but these materials are less common.