Elastic constants: Relation between elastic constants [with Pdf]

What is elastic constants?

Elastic constants indicate the resistance of the material for the strain (Longitudinal strain, shear strain, volumetric strain) due to the application of force.

Types of elastic constants:

The four main elastic constants are:

  1. Modulus of elasticity or Young’s modulus
  2. Shear modulus or Modulus of rigidity
  3. Bulk modulus
  4. Poisson’s ratio

Let’s discuss one by one,

Modulus of elasticity (E):

Modulus of elasticity or young’s modulus is the ratio of longitudinal stress to the longitudinal strain.

The modulus of elasticity indicates the resistance of the material to the axial or longitudinal strain.
The letter ‘E’ is the symbol used to denote Young’s modulus.

E = `\frac{\text{longitudinal stress}}{\text{longitudinal strain}}`

The SI unit of modulus of elasticity is N/m² and is also known as pascal while the FPS unit of modulus of elasticity is lb/ft².


Shear modulus (G):

Shear modulus or modulus of rigidity is the ratio of shear stress to the shear strain.
The modulus of rigidity state the resistance of the material to the shear strain.
The shear modulus is also denoted by the letter ‘C’, ‘G’, and ‘N’.

G =`\frac{\text{shear stress}}{\text{shear strain}}`

The SI unit of the shear modulus is N/m² or also known as pascal while the FPS unit is lb/ft².


Bulk modulus (K)

Bulk modulus indicates the resistance of the material to the volumetric stress. It is the ratio of volumetric stress to the volumetric strain and it is denoted by the letter ‘K’.

K =`\frac{\text{volumetric stress}}{\text{volumetric strain}}`

The SI unit of the bulk modulus is N/m² or also known as pascal while the FPS unit is lb/ft².


Poisson’s Ratio (μ):

The ratio of the lateral strain to the longitudinal strain is known as Poisson’s ratio and it is denoted by the Greek letter μ.

μ = `\frac{\text{lateral strain}}{\text{longitudinal strain}}`

The strain is unitless quantity therefore Poisson’s ratio is also unitless quantity.

Poisson’s ratio of some of the common materials are,

  1. Rubber = 0.5
  2. Cement concrete = 0.15
  3. Metals = 0.25 to 0.33

Relation between elastic constants:

The following equations state the relations between different elastic constants.

i] The relation between Young’s modulus, Shear modulus, and Poisson’s ratio is given by,

E = 2G(1+μ)

ii] The relation between Young’s modulus, Bulk modulus, and Poisson’s ratio is given by,

E = 3K(1−2μ)

iii] The relation between Young’s modulus, Shear modulus, and the Bulk modulus is given by,

E = `\frac{9KG}{G+3K}`

By using these relations it becomes easy to calculate the value of required elastic constants from another constant of the material.

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