Classification of Elastic Modulus

CLASSIFICATION OF ELASTIC MODULUS

Depending on the three types of strain, there are three types of elastic modulus. viz.

(i) Youngs modulus (Y) (or) modulus corresponding to longitudinal (or) tensile strain.

(ii) Bulk modulus (K) (or) modulus corresponding to the volume strain.

(iii) Rigidity modulus (n) (or) modulus corresponding to the shearing strain.

(i) Youngs modulus (Y)

Definition:

It is defined as the ratio between the longitudinal stress to the longitudinal strain, within the elastic limits.

 (i.e.) Youngs Modulus (Y) =  Longitudinal stress / Longitudinal strain  Nm^‒2 (or) pascals

Explanation:

Let us consider a wire of length ‘L’ with an area of cross section 'A'. Let one end of the wire be fixed and the other end is loaded (or) stretched as shown in Fig 1.2.

Let ‘l’ be change in length due to the action of force, then

The longitudinal stress = F/A

and the longitudinal strain = l/L

Youngs modulus Y = [ F/A ] / [ l/L ]

Y = FL / Al  Nm^‒2 (or) pascals

(II) BULK MODULUS (K)

Definition:

It is defined as the ratio between the volume stress (or) bulk stress to the volume strain (or) bulk strain within the elastic limits

(i.e.) Bulk Modulus (K) = Bulk Stress / Bulk Strain Nm^‒2 (or) pascals.

Explanation:

Let us consider a body of volume 'V' with an area of cross section 'A'. Let three equal forces act on the body in mutually perpendicular directions as shown in Fig 1.3. Let 'v' be the change in volume, due to the action of forces, then

The volume stress (or) bulk stress = F/A

The volume strain (or) bulk Strain = v/V

∴ Bulk modulus (K) = [F/A] / [v/ V]

K = FV / vA

K = PV / v Nm^‒2 (or) Pascals

Where P is the pressure = F/A

Note: The reciprocal of bulk modulus of a material is known as compressibility of that material.

(III) RIGIDITY MODULUS (N)

Definition:

It is defined as the ratio between the tangential stress to the shearing strain, within the elastic limits.

(i.e) Rigidity modulus (n) = Tangential stress / Shearing strain Nm^‒2 (or) pascal

Explanation:

Let us consider a solid cube ABCDEFGH. Whose lower face CDHG is fixed as shown in Fig 1.4. A tangential force 'F' is applied over the upper face ABEF. The result is that the cube gets deformed into a rhombus shape A'B'CDE'F'GH. (i.e) The lines joining the two faces are shifted to an angle ϕ. If ‘l’ is the original length and ‘l’ is the relative displacement of the upper face of the cube with respect to the lower fixed face, then

We can write tangential Stress = F/A

The shearing strain (ϕ) can be defined as the ratio of the relative displacement between the two layers in the direction of the stress, to the distance measured perpendicular to the layers.

We know, Rigidity modulus (n) = Tangential stress / Shearing strain

n = F / Αϕ

 (or)

Rigidity Modulus (n) = F / Αϕ Nm^‒2 (or) pascal