Relation Between Elastic Constants

Dhristi JEE 2022-24

Learn Relation between Elastic Constants

  • We know that depending upon the type of stress being applied and the resulting strain, the modulus of elasticity had been classified into the following three types.

  1. Young’s Modulus (E): The ratio of longitudinal stress to longitudinal strain.

  2. Bulk Modulus (K): The ratio of volumetric stress to volume strain.

  3. Shear Modulus (G): The ratio of shear stress to shear strain.

  • All three elastic constants can be interrelated by deriving a relation between them known as the Elastic constant formula. But young’s modulus (E) and the Poisson ratio(𝝂) are known as the independent elastic constants and they can be obtained by performing the experiments.

  • The bulk modulus and the shear modulus are dependent constants and they are related to Young’s modulus and the Poisson ratio.

  • The relation between Young’s modulus and shear modulus is

⇒ E = 2G(1 + v)N/m2

  • The relation between Young’s modulus and Bulk modulus is:

⇒ E = 3K(1 - 2v) N/m2


Elastic Constant Formula

The relation between different elastic constants is achieved by a small derivation. For the derivation of the relation between elastic constants, we will use the relation between Young’s modulus and the bulk modulus and also the relation between Young’s modulus and the shear modulus.


Derivation of Relation Between Elastic Constants

Consider the relation between Young’s modulus and the shear modulus,


⇒ E = 2G(1 + v)N/m2 ……….(1)


Where,

E - Young’s modulus

G - Shear modulus

v - Poisson ratio

From equation (1) the value of the Poisson ratio is:


⇒ \[v = \frac{E }{2G} - 1 \]……….(2)


We know that the relation between Young’s modulus and the Bulk modulus is


⇒ E = 3K(1 - 2v) N/m2 …………..(3)


Where,

E - Young’s modulus

K - Bulk modulus

v - Poisson ratio

Substituting the value of Poisson ration from equation (2) in (3) and simplify,


⇒\[E = 3K(1 - 2\frac{E} {2G}−1)\]


⇒\[E = 3K(1 -\frac{E}{G}-2) \]


\[E = 3K(3 -\frac{E}{G})\]


Equation (4) is known as the Elastic constant formula and it gives the Relation between elastic constants.


On further simplification,


⇒ \[ E = 9K -\frac{3KE}{G}\]


Taking LCM of G and on cross multiplication,


⇒ EG + 3KE = 9KG


⇒ E(G + 3K) = 9KG


On rearranging the above expression,


⇒ \[ E = \frac{9KG}{G+3K}\] N/m2 ………..(4)


Where,

E - Young’s modulus

G - Shear modulus

K - Bulk modulus

Equation (4) is known as the Elastic constant formula and it gives the Relation between elastic constants.


Did You Know?

The relationship between different elastic constants is also given by the expression,


⇒\[\frac{1}{K} -\frac{3}{G} = \frac{9}{E}\]


Where,

E - Young’s modulus

G - Shear modulus

K - Bulk modulus

These are the different ways of writing the relationship between elastic constants, depending upon the need for the solution we should utilize the formulas.


Elastic Constants:

Elastic constants are the constants that describe the mechanical response of a material when it is elastic.  The elastic constant represents the elastic behaviour of objects.


Different Elastic Constants are as Follows:

  1. Young’s modulus

  2. Bulk modulus

  3. Rigidity modulus

  4. Poisson’s ratio


  1. Young’s Modulus

Young's modulus is based on the elastic constant which is defined as the proportionality constant between stress and strain.

  1. Bulk Modulus

Bulk Modulus of Elastic Constants is one of the measures for mechanical properties of solids. It is explained as having the ability of a material to resist deformation in terms of change in volume at the time of subject compression under pressure. 

  1. Rigidity Modulus

The modulus of rigidity that is also known as shear modulus is defined as the measure of elastic shear stiffness of a material. This property depends on the material of the member which means the more elastic the member, the higher the modulus of rigidity.

  1. Poisson’s Ratio

Poisson's ratio is defined as the ratio of the change in the width per unit width of a material in order to the change in its length per unit length which will be given as a result of strain.


Relationship between Elastic Constants

Young’s modulus, bulk modulus and Rigidity modulus of an elastic solid together can be explained as Elastic constants. In addition to this when a deforming force is acting on a solid that will result in the change in its original dimension. In such cases, we can use the relation between elastic constants to understand the magnitude of deformation.

FAQs on Relation Between Elastic Constants

1. What is the poisson ratio?

Poisson ratio is defined as the ratio of lateral strain (transverse strain) to longitudinal strain. It is denoted by 𝝂. It is an independent elastic constant and unitless scalar entity.

2. Why are some materials elastic?

The elasticity of any material is decided, depending upon the amount of transverse deformation and ability to get back to its original size and shape. Some materials have a great ability to get back to their original shape and size even after applying some force or stretching it. For example rubber.

3. What are the elastic constants?

The elastic constants are the parameters explaining the relationship between the type of stress applied and the corresponding strain.

4. What is the relationship between EG and K?

Modulus of Elasticity (E) is defined as the ratio between the normal stress and normal strain, Modulus of Rigidity (G) is termed as shear modulus or modulus of rigidity and last but not least Bulk Modulus (K) is referred to as the ratio between the normal stress and volumetric strain of material in N/m2.

5. What do you understand by an elastic constant?

It states that when a material is loaded within the elastic limit, the stress is proportional to the strain produced by the stress. This constant is known as the elastic constant.

6. What is elastic constant K?

The elastic constant K in physics is known as the rate or spring constant. This means when it is shown on a graph the line should show a direct variation as the reason is when it holds its behaviour is described as linear.

7. What is the bulk modulus of elasticity?

Bulk modulus is known for its measure of the decrease in volume with an increase in pressure. Furthermore; it is explained in detail as The “modulus of elasticity” of a liquid which varies widely depending on the specific gravity and temperature of the liquid. For example; Typical values are less than 30,000 psi to greater than 300,000 psi, depending upon the liquid.

8. Why are elastic constants structure insensitive properties?

Since these forces cannot be changed without changing the basic nature of the material, it follows that Young's Modulus is one of the most structure-insensitive of the mechanical properties.

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