Elastic Constants

Elastic moduli are constants that describe the behaviour of materials under stress, how they deform, and how they rebound to their initial structure after the stress is removed. Among the elastic constants are the Lame constant, Poisson’s ratio, shear modulus, and Young’s modulus. Seismology relies on elastic constants and rock density to determine the velocity of seismic waves.

The formula for the elastic constant (Ec)

E=9KG  ⁄  G+3k

This property of a material is known as its Bulk modulus or K, and the shear modulus of rigidity modulus G  is the modulus of elasticity of Young’s modulus.

What is an elastic constant?

Deformation occurs when an external force is exerted on the body. A body is elastic only if it can return to its original shape and size after being subjected to an external force. Elasticity refers to the ability of some materials to return to their original position after an external force is removed. The body can only return to its original shape and size if the deformation caused by the external force is kept within a certain range. Deformation completely disappears by removing forces that fall within a certain range of force. The elastic limit of any material is the stress level at which the material cannot withstand further stress. This answers the question of what is elastic constant as well.

Hooke’s Law:

Stress is directly proportional to strain when a material is subjected to an elastic load. This means that within the elastic limit, the stress-to-strain ratio is constant.

  Stress /Strain = Constant

This constant is known as the elastic constant.

Types of Elastic Constants:

There are three types of elastic constants;

• Normal stress/ Normal strain = Young’s Modulus
• Shear stress/ Shear strain = Shear modulus or 
• Direct stress/ Volumetric strain = Bulk modulus (K)

1.Young’sModulus:

This law states that the stress applied to an object is proportional to the strain within the elastic limits of that object when tensile or compressive stress is applied to it. Young’s modulus, or the modulus of elasticity, is a constant ratio of applied stress to strain.

The normal stress-to-strain ratio is the ratio of normal stress (σ) to longitudinal strain (s) (e).

E = (σn) / (e)

2. Modulus of Rigidity:

the measure of shear stress to strain ratio (es). When referring to it, use G or C.

G= τ/es

3. Bulk Modulus or Volume Modulus of Elasticity (K):

Normal stress divided by volumetric strain can be defined as the ratio of normal stress to volumetric strain.

K serves as an identifier. Volume change without changing the material’s shape or form is called Bulk modulus.

K = Direct Stress / Volumetric strain

= σ/ev

Relation between elastic constants

Elastic constants include a solid’s Young’s modulus, Bulk modulus, and rigidity modulus. A solid’s original dimension changes when a deforming force is applied. For example, we can use the relation between elastic constants to estimate the deformation’s magnitude.

        E=9KG ⁄ G+3k

   The Bulk modulus K, on the other hand,

Shear modulus or rigidity modulus G

Elasticity modulus, or Young’s modulus, is abbreviated as E.

Derivation of the relation between elastic constants

It is possible to derive the relationship between elastic constants by combining terms individually.

Poisson’s ratio and Bulk modulus can be used to calculate the Young modulus.

 E=3K(1-2μ)

Young’s modulus can also be expressed using the rigidity modulus and Poisson’s ratio as

 E=2G(1+2μ)

We can derive a relationship between Young’s modulus, Bulk modulus k, and modulus of rigidity by combining the above two equations and solving them to eliminate Poisson’s ratio.

 E=9KG ⁄ G+3k

Is the constant elastic equivalent to the spring constant?

When spring is elastic, it will return to its original shape once the external force (the mass) has been released. The linear equation describes the relationship between the force and the displacement. As a result, because the spring constant is constant (a feature of the spring itself), it may be concluded that the relationship is linear.

How do you calculate the relationship between elastic constants?

A small derivation is needed to connect the various elastic constants. Using the relationships between Young’s modulus and bulk modulus and shear modulus, we can derive the relationship between elastic constants.

Conclusion

Body deformation happens when a force outside the body is applied to the deformed object. Elasticity is defined as the ability of some materials to return to their previous position after being subjected to a force external to themselves. Hooke’s law states that when an elastic load is applied to a material, the stress produced is proportionate to the strain experienced by the material.

How can the relationship between elastic constants be used to determine the extent of deformation? Elastic constants are a general term that refers to all of the elastic constants of an elastic solid. And the SI unit is N/m.