# Elastic Constants Study Notes for GATE, ESE & Mechanical Engineering Exams

By Akhil Gupta|Updated : April 11th, 2022
This article has complete details on elastic constants like Poisson's ratio, Types of strain, volumetric strain, Bulk modulus, Modulus of rigidity, and its special cases. You can practice a maximum number of previous years' questions based on the notes of Elastic Constant vital in Mechanical Engineering, especially if you are preparing for the GATE 2023 exam. Elastic Constant notes for Mechanical Engineering will help you understand the concept of the SOM needed for GATE ME, ESE, and other Mechanical engineering exams. Every year, around 7 to 8 marks questions come from the Subjet of Matters in the exam.

These topic-wise study notes are helpful for the preparation of various upcoming exams like GATE MEIESBARCISROSSC-JE Mechanical/ State Engineering Services examinations and other important forthcoming competitive exams.

Elastic Constant is one of the important topics of "Strength of Materials" for GATE, ESE, and other Mechanical Engineering exams.

## Elastic constants

Elastic constants are those factors that determine the deformations produced by a given stress system acting on a material.

Various elastic constants are :

(i) Modulus of elasticity (E)

(ii) Poisson’s ratio (μ or 1/m)

(iii) Modulus of rigidity (G or N)

(iv) Bulk modulus (K)

## Materials on the Basis of Elastic Properties

(i)Homogeneous Material

When a material exhibits the Same elastic properties at any point in a given direction then the material is known as a homogenous material. Fig. Homogeneous material

(ii)Isotropic Material

When a material exhibits the Same elastic properties at any direction at a given point then the material is known as Isotropic Material. Fig. Non-Homogenous and isotropic material

(iii)Anisotropic Material

When a material exhibits different elastic properties at every direction at every point then the material is known as Isotropic Material. Fig. Anisotropic material

(v) Orthotropic Material

When a material exhibits the Same elastic properties at the only orthogonal direction at a given point then the material is known as Orthotropic Material.

For a homogeneous and isotropic material, the number of independent elastic constants is two.

 Material No. of independent elastic constants Isotropic 2 Orthotropic 9 Anisotropic 21

## Modulus of Elasticity

When an axial load, P is applied along the longitudinal axis of a bar due to which length of the bar will be increased in the direction of applied load and stress, σ is induced in the bar.

The ratio of stress to longitudinal strain, within elastic limits, is called the modulus of elasticity (E): ## Poisson's Ratio

It is the ratio of lateral strain to longitudinal strain.

It is a unitless quantity which is generally denoted as μ or 1/m.  ## Volumetric Strain

Volumetric Strain Due to Three Mutually Perpendicular Stress

The figure shows a parallelepiped subjected to three tensile loads P1, P2, and P3 in the three mutually perpendicular directions.

Then,  Fig. Parallelepiped subjected to Three Mutually Perpendicular Stress

Since any axial load produces a strain in its own direction and an opposite kind of strain in every direction at right angles to this direction.

we have, Adding the three expressions of Equations we get. The hydrostatic static state of stress-

In case of the hydrostatic state of stress, the applied stress in all direction is equal and tensile in nature.

i.e. σ1 = σ2 = σ3 = σ since Eϵ and σ in the above expression are positive numbers,  must also be positive. Thus, maximum value poison’s ratio is 0.5

Volumetric Strain Due to Single Direct Stress

Figure shows a rectangular bar of length L, width b and thickness t subjected to single direct load (P) acting along its longitudinal axis. Let this stress σ generated to be tensile in nature.  Fig. Volumetric strain ## Shear Modulus OR Modulus of Rigidity

The shear modulus or modulus of rigidity expresses the relation between shear stress and shear strain. where G = modulus of rigidity

ɸ = Shear strain (in radians) (also sometimes denoted by the symbol γ)

BULK MODULUS

When a body is subjected to three mutually perpendicular like stresses of equal intensity (σ).

Then the ratio of direct stress (σ) to the corresponding volumetric strain (ϵv) is defined as the bulk modulus K for the material of the body.

Which is generally denoted as ‘K’ ## Relation Between Different Elastic Properties Value of any Elastic constant should be ≥ 0

E, K, G > 0

µ ≥ 0 [µcork = 0]

If K should be positive,

Then 1 – 2µ ≥ 0   Always

G ≤ E

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