Elastic Constants Notes | EduRev

Strength of Materials (SOM)

Mechanical Engineering : Elastic Constants Notes | EduRev

The document Elastic Constants Notes | EduRev is a part of the Mechanical Engineering Course Strength of Materials (SOM).
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Elastic constants

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

Various elastic constants are:

  1. Modulus of elasticity (E)
  2. Poisson’s ratio (μ or 1/m)
  3. Modulus of rigidity (G or N)
  4. Bulk modulus (K)

Materials on the basis of elastic properties

  • Homogeneous Material
    When a material exhibits Same elastic properties at any point in a given directions than the material is known as homogenous material.

Elastic Constants Notes | EduRevHomogeneous material 

  • Isotropic Material
    When a material exhibits Same elastic properties at any direction at a given point than the material is known as Isotropic Material.

Elastic Constants Notes | EduRevNon-Homogenous and isotropic material

  • Anisotropic Material
    When a material exhibits different elastic properties at every direction at a every point than the material is known as Isotropic Material.
  • Orthotropic Material
    When a material exhibits Same elastic properties at only orthogonal direction at a given point than the material is known as Orthotropic Material.
    For a homogeneous and isotropic material, the number of independent elastic constants are two.

Elastic Constants Notes | EduRev

Modules 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 increases in the direction of applied load and a stress, σ is induced in the bar.
The ratio of stress to longitudinal strain, within elastic limits, is called the modulus of elasticity (E):

Thus, modules of Elasticity E = σ/ε

Poisson's Ratio

It is the ratio of lateral strain to the longitudinal strain.
It is an unitless quantity which is generally denoted as μ or 1/m.

Poisson's Ratio = - lateral strain/longitudinal strain

Elastic Constants Notes | EduRev

Elastic Constants Notes | EduRev

Volumetric Strain

1. Volumetric Strain Due to Three Mutually Perpendicular Stress
Figure shows a parallelepiped subjected to three tensile load P1, P2 and P3 in the three mutually perpendicular direction.
Then,
δV / V = ε1 + ε2 + ε3

Elastic Constants Notes | EduRevParallelepiped subjected to Three Mutually Perpendicular StressSince 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,

Elastic Constants Notes | EduRev
Adding the three expressions of Equations we get, 

Elastic Constants Notes | EduRev

2. Hydrostatic static state of stress
In case of hydrostatic state of stress, the applied stress in all direction are equal and tensile in nature.
i.e. σ1 = σ2 = σ3 = σ

Elastic Constants Notes | EduRev
since Eϵ and σ in the above expression are positive numbers, must also be positive.
1 - 2μ ≥ 0
2μ ≤ 1
μ ≤ 0.5
Thus, maximum value poison’s ratio is 0.5

3. 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.

Elastic Constants Notes | EduRev

Elastic Constants Notes | EduRevVolumetric strain

Shear Modules or Modulus of Rigidity

The shear modules or modulus of rigidity expresses the relation between shear stress and shear strain.

Elastic Constants Notes | EduRev
where G = modulus of rigidity
ɸ = Shear strain (in radians) (also sometimes denoted by the symbol γ)

Bulk Modules

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’

Thus K = Direct stress / Volumetric strain = σ / εv

Relation Between Different Elastic Properties

E = 2G(1 + μ)
E = 3K (1 - 2μ)
E = 9KG/3K + G

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
μ ≤ 1/2
For any Engineering Material
0 ≤ µ ≤ 1/2

Elastic Constants Notes | EduRev

Elastic Constants Notes | EduRev
Always
G ≤ E
For metals

Elastic Constants Notes | EduRev

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