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
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
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
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.
No. of independent elastic constants
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):
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 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.
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.
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 modules 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)
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
G ≤ E
For the detailed schedule of the GATE Mechanical Engineering(ME) 2022 Champion Study Plan, click here
Sahi Prep Hai Toh Life Set Hai