Relationship Between Elastic Constants

The ability of an object or material to resort back to its original shape after being twisted is called elasticity. The opposite is plasticity, wherein the object or material will break or remain in a deformed shape after being twisted. We will try to derive the relationship between elastic constants in this chapter through concepts like Hooke’s Law. It is defined mathematically as

F = kx

where F is the force or strain applied, k is the constant, and x is the amount of stress generated. Further, we will explore the different types of elastic constants available.

Elastic Constants explained

The relationship between the stress applied and the strain generated can be measured through the following types of constants. The derivation of the relationship between elastic constants will be conducted through these different constants and try to achieve complete knowledge of the subject.

Types of Elastic Constant

1. Young’s Modulus – As we explained with Hooke’s law, the stress applied to an object is directly proportional to the strain generated, resulting in perfect plasticity. This is the first and most basic understanding of the derivation of the relationship between elastic constants.
   This is referred to as Young’s Modulus.
2. Bulk modulus – When the same amount of force is exerted on an object from two or more sides with an equal amount of pressure, the equal and similar stress created is constant as per the Bulk Modulus. It is expressed mathematically as
   Bulk modulus = direct stress/ volumetric strain
   Wherein Bulk Modulus is considered as K.
3. Rigidity Modulus – When unequal and parallel stress, also known as shear stress,  is applied to an object, the object’s shape is changed from its basic structure into a different formation. The Rigidity modulus is the ratio between the unequal and parallel pressure applied to the unequal and parallel tension created. It is expressed as
   Modulus of Rigidity= shear stress/ shear strain 
4. Poisson’s Ratio – When a force is applied to an object, and the resulting strain causes the basic shape of the object to change perpendicular to the stress applied, the ratio of lateral strain to longitudinal strain is called Poisson’s ratio. 

Poisson’s Ratio= lateral strain/ longitudinal strain 

Relationship between the elastic constants

When we study the derivation of the relationship between elastic constants, we tend to understand the different ways in which pressure can be applied to a material and what can be the impact of the resulting strain. The physical properties of a given material to withstand elasticity and their application in different physics areas are determined by studying these constants.

To derive the relationship between elastic constants, which are Young’s modulus (denoted as Y), Rigidity Modulus (denoted as η or Eta), Bulk Modulus (denoted as K), and Poisson’s ratio (denoted by μ or Mu symbol).

Take a look at the relationship between Young’s modulus and shear modulus.

E = 2G(1 + v)N/m2………. (1)

Where,

Young’s modulus (E)

G is the shear modulus.

Poisson ratio v

The Poisson ratio is calculated as follows from equation (1):

v=E2G-1………. (2)

The relationship between Young’s modulus and the Bulk modulus is well known.

E = 3K(1 – 2v) N/m2………… (3)

Where,

Young’s modulus (E)

K denotes the bulk modulus.

Poisson ratio v

Substituting the Poisson ratio value from equation (2) in (3) and simplifying,

⇒E=3K(1−2E)(2G−1)

⇒E=3K(1−EG−2)

E=3K(3−EG)

Equation (4) describes the relationship between elastic constants and is known as the Elastic constant formula.

E=9K3KEG can be simplified further.

Taking the LCM of G and multiplying by cross multiplication, EG + 3KE = 9KG E(G + 3K) = 9KG

Rearranging the foregoing formula, E=9KG/G+3K N/m2………..

(4) Where E is the Young’s modulus.

G is the shear modulus.

K denotes the bulk modulus.

Equation (4) describes the relationship between elastic constants and is known as the Elastic constant formula.

Conclusion

Studying the elastic properties of objects is significant to understanding the behaviour of materials so we can improve or replace the ones in use currently. Plastic hair clips can hold the hair in place. However, rubber bands can handle hair volume and different positions of tying it up. Hair clips cannot withstand a lot of pressure and will break instantaneously or twist. However, the elastic capabilities of rubber bands allow us to not only twist them as per our requirement. However, it also allows us to hold more volume than its relatively basic structure. This allows for more capacity.