# Elastic Constants and their Relationships

Different elastic constants are young’s modulus, modulus of rigidity, bulk modulus, lateral strain and Poisson’s ratio.

### 1. Young’s Modulus (E):

The ratio of direct stress (σ) to direct strain (ε) with the limit of proportionality.

E = σ/ε

The slope of line/part of stress strain curve.

Young’s modulus value of few materials:
Steel 200 GPa;
Aluminium 70 GPa;
Brass 100 GPa;
Bronze 80 GPa;
Copper 120 GPa;
Diamond 1200 GPa.

### 2. Modulus of Rigidity (C,N or G):

The ratio of shear stress (τ) to shear strain (ε).

G = τ/ε

### 3. Bulk Modulus (K):

When a body is subjected to like and equal direct stresses along three mutually perpendicular directions, the ratio of this direct stress to corresponding volumetric strain (εv) is called Bulk Modulus.

K = P/εv

Where P = applied pressure causes direct stress.
εv = volumetric strain

### 4. Lateral strain:

When a structure is subjected to axial load, besides the strain in axial direction there will be a lateral strain of opposite nature, in all directions at right angles.

### 5. Poisson’s Ratio:

When a bar is subjected to a simple tensile loading there is an increase in the length of the bar in the direction of the load, but a decrease in the lateral dimensions perpendicular to the load. The ratio of strain in the lateral direction to that in the axial direction is called Poisson’s ratio. It is denoted by the Greek letter μ. For most metals lie in the range 0.25 to 0.35. One new and unique material, so far of interest only in laboratory investigations, actually has a negative value of Poisson’s ratio; i.e., if stretched in one direction it expands in every other direction.

Note: The maximum possible value of Poisson’s ratio is 0.5, for an ideal elastic in-compressible material whose volumetric strain is zero.

Poisson’s ratio values for important materials:

Cork = 0;

Concrete 0.1 to 0.2;

Glass = 0.2 to 0.27;

Cast iron = 0.2 to 0.3;

Steel = 0.27 to 0.3;

Aluminium = 0.33;

Brass = 0.34;

Gold = 0.44;

Incompressible materials, clay, parrafin and Rubber =0.5

The relation between elastic constants (E, C, K and μ):

E = 3K (1-2μ)

E = 9KC/(3K +C)

μ = (3K – 2C)/(6K + 2C)

For an isotropic material (μ = 0.25) then E>K>C

If μ < 1/3  then E > K

If μ > 1/3 then E < K.

Vinodh Reddy is an Editor-in-chief of ME Mechanical. He holds Bachelor of Engineering (Honours) degree in Mechanical Engineering from BITS-Pilani.

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