Modulus of Elasticity

The modulus of rigidity is one of the primary traits of any material. It helps in estimating the extent of force needed to alter the object’s shape. 

The rigidity modulus of elasticity tells us how efficiently a solid will resist its deformation if a force is applied. Therefore, it has a significant role in engineering designs. So, if you want to learn more about the implications of rigidity modulus of elasticity, keep reading this article.

Basics of bulk matter and elasticity

Before beginning the topic of rigidity modulus, let’s look at the basics of bulk matter.

Bulk matter

Matter as an observable scale is called bulk matter. The various states of matter include solid, liquid, and gas.

Solids are rigid substances. But, they tend to behave elastically under certain conditions. It means they can stretch or compress when force is applied. 

They have specific properties which bring about the change in their shape and size. 

Elasticity

Elasticity is one of the properties of bulk matter. As you know, by applying a force, a solid object changes its shape and size. 

But, on removing the force, if the object regains its shape, it is elastic. This property of solid matter is called elasticity.

Stress

The change in shape and size or deformation of a solid on applying a force may not be visible to the naked eye. The deformation could be minute or massive. When the deforming force is applied, a restoring force in the solid acts to regain its form. The magnitude of the deforming and restoring force is equivalent but has an opposite direction, which is referred to as stress. 

Stress corresponds to the restoring force acting per unit area. It is denoted by,

The magnitude of stress = Force/Area, SI unit N/m2 

If the solid is stretched, then the restoring force is tensile stress. 

If the solid is compressed, then the restoring force is compressive stress.

Strain

The deforming force brings a change in the length of the solid, and it is said to “stretch” the solid. The strain tells us the extent of stretching or deformation. It corresponds to the ratio of change in the length of the solid to its original length. We call it longitudinal strain as it deals with the change in length of the solid.

Longitudinal strain = (change in length(∆L))/(original length(L)) 

Since it is a ratio, it has no unit.

By now, you must have gotten well-acquainted with elasticity, stress, and strain. So, let’s take a quick look at elastic moduli and their types.

Modulus of elasticity

Modulus of Elasticity is of great importance in manufacturing and structural engineering designs. There is a point when the solids will no longer behave like elastic, and the deformation tends to become permanent. It means the solids will not regain their original form even after removing the deforming force or stress. The modulus of elasticity measures the material’s resistance to elasticity.

Modulus of Elasticity is nothing but defines a quantitative relation between stress and strain. It is obtained from the slope of the strain-strain curve and is a characteristic property of a material. 

Modulus types

The three types of Moduli of Elasticity are:

• Young’s modulus
• Bulk modulus
• Rigidity modulus of elasticity or shear modulus

In this article, we will be discussing the rigidity modulus of elasticity. 

Rigidity modulus of elasticity or shear modulus (η)

Modulus of Rigidity gives the rigidity of a solid. The more the Shear Modulus, the more is the rigidity of the object. 

Solid deformation occurs when a force is applied to one side of the solid while the other side is kept fixed, then the rigidity modulus of elasticity comes into play. 

Suppose a deforming force is applied on a solid with a length (l) and area of cross-section (A). The deforming force is applied to the parallel face of the solid, with the opposite face being kept fixed. Then the parallel face of the solid displaces by a distance of ∆x. The parallel face on which the force is applied is called the sheared face.

The restoring force caused due to deforming force per unit area of the solid is called the shearing stress or tangential stress. And the strain created is referred to as the shearing strain.

Shearing stress = Force/Area

Shearing strain = ∆x/L = tan θ

Since the angle is generally too small, tan θ is nearly equal to θ.

Shearing strain = ∆x/L = θ

The rigidity modulus of elasticity (η) is the ratio of the shearing stress and the shearing strain. It is given by, 

η = (Shearing stress)/(Shearing strain)

η = (F/A)/(∆x/L) = (F×L)/(A×∆x)

Since, ∆x/L = θ

η = F/(A× θ) 

The S.I.unit of rigidity modulus is N/m2 or Pascal.

Note: The entire solid does not move. It means the solid remains fixed at the place, but there is a slant in the length of the solid. Imagine it as a punching bag affixed to the ground.

Conclusion

It’s the modulus of rigidity that determines the rigidity of a solid. Therefore, the rigidity modulus of elasticity is a significant characteristic to know the resistance of a solid when a deforming force is applied. The rigidity modulus of elasticity is of great importance in manufacturing and structural engineering designs.