Relation Between Elastic Constants - A Complete Guide

Relation Between Elastic Constants - A Complete Guide

Team Careers360Updated on 02 Jul 2025, 04:44 PM IST

Key points of this article:-

  • Elastic constant
  • Relation between young’s modulus and bulk modulus
  • Relation between modulus of elasticity and modulus of rigidity
  • Relation between young's modulus, bulk modulus and Poisson’s ratio
  • Relation between young's modulus and shear modulus
  • Relation between young’s modulus bulk modulus and modulus of rigidity.
  • What is elastic constants
  • Definition of elastic constant

Now, discuss the elastic constant and their relationship

What is elastic constant:- We can define elastic constant as Elastic constants are constants which tell us the elastic behaviour of materials.

This is the elastic constant definition

There are three elastic constants (or elastic coefficients)

Young’s modulus of elasticity (Y or E) = longitudinal stress / longitudinal strain

Bulk modulus of elasticity (K) = Normal stress/ volume strain

Modulus of rigidity/bulk modulus of rigidity (η or G) = tangential stress/ shearing stress

Poission’s ratio :- The ratio between decreasement per unit length and per unit force and the increasement per unit length and per unit force is called poissions’s ratio.

σ = 1.1.β/1.1.α

Where β = the decreasement in length

α = the increasement in length

Relation between young’s modulus, bulk modulus and Poisson's ratio:-

Let us consider a cube whose side is unit length to derive the relation between Y, K and σ
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I.e., OA = OB = OC = 1

So, the volume of the cube

V = 1 (without normal stress)

Also read -

When a normal stress is applied on a every face of cube so the increasement in unit length is α and decreasement is β due to unit force.

So, the side of the cube

OA = 1 + F.1.α - F.1.? - F.1.?

= 1 + F.α - 2F?

= 1 + F ( α - 2? )

So, OA = OB = OC = 1 + F ( α - 2? )

Then the volume of the cube

V’ = OA.OB.OC

= [1 + F ( α - 2? )]³

V’ = [ 1 + 3F ( α - 2? )]

So, change in the volume of the cube

ΔV = V’ - V

= 1 + 3F(α - 2? ) - 1

ΔV = 3F(α - 2? )

And the volume strain of the cube = change in volume/initial volume

= ΔV/V = 3F(α - 2? )/1

= 3F(α - 2? )

And the bulk modulus of elasticity of the cube (K) = normal stress/normal strain

= Force/area/ΔV/V = F/1/3F(α - 2? )

K = 1/3(α - 2? ) = 1/3α(1 - 2?/α)

= 1/α/3(1 - 2σ) ( ∵ σ = β/α)

K = Y / 3 ( 1 - 2σ)

∵ Y = long. stress/long. Strain

Y = F/A/Δl/l = F/1/F.1.α/1 = 1/α


Y = 3K( 1 - 2σ)

……(A)

Relation between E and K is

E = 3K( 1 - 2σ)

This is the relation between elastic constants- relation between young modulus and bulk modulus and poisson ratio/ relation between bulk modulus and young’s modulus.

Relation between relation between young's modulus and modulus of rigidity

For derive the relation between elastic constants - Young’s modulus (Y), modulus of rigidity (η) and the poission’s ratio. Considered a face of cube ABCD side of this face is L. Fixed the side CD and applied the tangential stress (T) on the side AD due to to this A and B point is displaced linearly (l) distance and angularly ? angle and obtained a new position A’ and B’.

So, tangential stress on the side AB

T = force/area = F/L²


  Relation between relation between young's modulus and modulus of rigidity

And shearing strain

tan? = l/L

But ? is very small tan? ∝ ?

? = l/L

So, the modulus of rigidity of face ABCD

η = T/?

η = F/l²/l/L

η = F/lL …..(i)

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Due to tangential stress T diagonal BD of ABCD face changes to B’D. To determine the change of diagonal BD to B’D a perpendicular line is drawn which cut at N point. In this position, NB’ diagonal represent the increment in the B’D diagonal.

In ΔBNB’

Cos45° = NB’/BB’

NB’ = lcos45°

NB’ = l/√2 ….(ii)

This eqn shows that the increment in the diagonal

Due to tangential stress T increments in the BD diagonal and decrement in AC diagonal. Due to unit force increment and decrement in unit length is represented by

α and β.

Due to tangential stress T increasement in BD diagonal = T.BD.α

= F/L².L√2.α

= F/L.√2.α

Due to tangential stress T decreasement in AC diagonal = T.AC.?

= F/L².L√2.?

= F/L.√2.?

So, the total change in the diagonals of ABCD face = F/L.√2.α + F/L.√2.?

= F√2/L (α + ? ) ….(iii)

From equation (ii) and (iii)

l/√2 = F√2/L (α + ? )

F/lL = 1/2(α + ? )

F/lL = 1/2α(α + ?/α) (∵ σ = ?/α)

F/lL = 1/2α(1 + σ)

From eqn (i)

η = 1/α/2(1 + σ ) ( ∵ Y = 1/α)

η = Y/2(1 + σ )

Y = 2η(1 + σ)

…….(B)

This is the relation between young modulus and shear of modulus ( or modulus of rigidity ).

And the relation between young’s modulus modulus of rigidity and Poisson’s ratio/ relation between young's modulus and poisson's ratio.

Relation between E and G is

E = 2G(1 + σ)

This is the relation between elastic constants- relation between young’s modulus and rigidity modulus

Relation between Bulk modulus, Modulus of rigidity and Poisson’s ratio:-

From eqn (A) and (B)

2η(1 + σ) = 3K(1 - 2σ)

2η + 2ησ = 3K - 6Kσ

6Kσ + 2ησ = 3K - 2η

σ( 6K + 2η) = 3K - 2η

σ = 3K - 2η/6K + 2η

………(C)

Also read :

This is the relation between bulk modulus of elasticity, poission’s ratio and modulus of

elasticity/ relation between elastic constants.

Relation between Young’s modulus bulk modulus and modulus of rigidity:-

From eqn (A)

Y = 3K ( 1 - 2σ )

Y/3K = 1 - 2σ

σ = 1/2 - Y/6K …….(a)

From eqn (B)

Y = 2η(1 + σ)

Y/2η = (1 + σ)

σ = Y/2η -1 ………(b)

From eqn (1) and (2)

1/2 - Y/6K = Y/2η -1

Y/2η + Y/6K = 1/2 + 1

Y[ 1/2η + 1/6K] = 3/2

Y/2 [1/η + 1/3K] = 3/2

3/η + 1/K = 9/Y

……(D)

or, Y = 9Kη/3K + η

This is the relation between elastic constants Young’s modulus, bulk modulus and modulus of rigidity.

Relation between E, G and K Young’s modulus, Bulk modulus and Modulus of rigidity is

E = 9KG/3K + G .

These are the elastic constant formula and elastic constants relation.

Also check-

Frequently Asked Questions (FAQs)

Q: What is the significance of the third-order elastic constants?
A:
Third-order elastic constants describe the nonlinear elastic behavior of materials under large deformations. They become important when the stress-strain relationship deviates from linearity. These constants are crucial for understanding phenomena like acoustic nonlinearity, shock wave
Q: How do elastic constants change near a material's melting point?
A:
As a material approaches its melting point, its elastic constants generally decrease. This is due to the weakening of interatomic bonds and increased atomic mobility. The rate of decrease often accelerates near the melting point, and some elastic constants (like the shear modulus) may approach zero as the material transitions to a liquid state.
Q: What is the relationship between elastic constants and thermal expansion coefficients?
A:
While not directly related, elastic constants and thermal expansion coefficients are both influenced by interatomic forces. Generally, materials with stronger interatomic bonds (and thus higher elastic constants) tend to have lower thermal expansion coefficients. However, this relationship is not strict, and exceptions exist. The Grüneisen parameter provides a more formal link between these properties.
Q: How do elastic constants influence the buckling behavior of columns?
A:
Elastic constants, particularly Young's modulus (E), play a crucial role in column buckling. The critical buckling load for a column is directly proportional to EI, where I is the moment of inertia of the column's cross-section. A higher Young's modulus results in a higher critical buckling load, meaning the column can support more weight before buckling occurs.
Q: How do elastic constants relate to the concept of strain energy?
A:
Elastic constants are directly related to strain energy, which is the energy stored in a material due to elastic deformation. The strain energy density (U) for different types of deformation is expressed in terms of elastic constants and strains. For example, for uniaxial stress, U = (1/2)Eε², where E is Young's modulus and ε is the strain. This relationship shows how stiffer materials (higher E) store more energy for a given strain.
Q: What is the Pugh's ratio and what does it tell us about a material?
A:
Pugh's ratio is defined as the ratio of shear modulus to bulk modulus (G/K). It provides information about a material's ductility or brittleness. Materials with a high Pugh's ratio (typically > 0.57) tend to be more brittle, while those with a low Pugh's ratio are generally more ductile. This ratio is useful in predicting material behavior in applications where toughness is important.
Q: How do elastic constants influence the propagation of elastic waves in solids?
A:
Elastic constants directly affect the speed and behavior of elastic waves in solids. The speed of longitudinal waves is related to both the bulk modulus and shear modulus, while the speed of transverse waves depends only on the shear modulus. The ratio of these speeds can be used to determine Poisson's ratio. Additionally, the reflection and transmission of waves at interfaces depend on the elastic constants of the materials involved.
Q: What is the relationship between elastic constants and a material's fracture toughness?
A:
While elastic constants don't directly determine fracture toughness, they play a role in it. Materials with higher elastic moduli tend to have higher stress concentrations at crack tips, which can lead to lower fracture toughness. However, the relationship is complex and also involves factors like plastic deformation and atomic bonding. Generally, a balance of high stiffness and ability to deform plastically leads to good fracture toughness.
Q: How do elastic constants relate to the concept of resilience in materials?
A:
Resilience is a material's ability to absorb energy when deformed elastically and release that energy upon unloading. It's directly related to the elastic constants, particularly Young's modulus. The modulus of resilience, which quantifies this property, is given by the area under the stress-strain curve up to the yield point. For linear elastic materials, it's expressed as σ²y / (2E), where σy is the yield stress and E is Young's modulus.
Q: What is the significance of the Poisson function in anisotropic materials?
A:
In anisotropic materials, the Poisson function replaces the single Poisson's ratio used for isotropic materials. It describes how strain in one direction relates to strains in perpendicular directions and can vary depending on the direction of applied stress. Understanding the Poisson function is crucial for accurately predicting the behavior of anisotropic materials under complex stress states.