Tuesday, 31 January 2017

DERIVE RELATION BETWEEN YOUNG'S MODULUS BULK MODULUS AND POISSON RATIO

DERIVE RELATION BETWEEN YOUNG'S MODULUS BULK MODULUS AND POISSON RATIO

We were discussing the “Elongation of uniformly tapering circular rod” and “Elongation of uniformly tapering rectangular rod” and also we have seen “Volumetric strain for a cylindrical rod” and “Volumetric strain of a rectangular body” with the help of previous posts.

Now we are going further to start our discussion to understand the derivation of relationship between young’s modulus of elasticity (E) and bulk modulus of elasticity (K) with the help of this post.

Relationship between young’s modulus of elasticity (E) and bulk modulus of elasticity (K)

 
Let us consider a cube ABCDEFGH as displayed in following figure, let us assume that cube is subjected with three mutually perpendicular tensile stress σ of similar intensity.
Let us assume that we have following details as mentioned here

Length of cube = L
Change in length of the cube = dL
Young’s modulus of elasticity = E
Bulk modulus of elasticity = K

Tensile stress acting over cube face = σ
Poisson ratio = ν
Longitudinal strain per unit stress = α
Lateral strain per unit stress = β

As we have already discussed the Poisson ratio as the lateral strain to longitudinal strain and therefore we can say that
Poisson ratio, (ν) = β / α

Let us recall the young’s modulus of elasticity, E = Longitudinal stress/Longitudinal strain
E = 1/ [Longitudinal strain/ Longitudinal stress]
E = 1/ α

Initial volume of the cube,

V = Length x width x height = L3
Now we will secure here the final dimensions of the cube in order to secure the final volume of the cube and finally we will determine the bulk modulus of elasticity.

 
Let us consider first one side of cube i.e. AB. As we have already discussed that three mutually perpendicular tensile stresses of similar intensity are acting over the cube. Let us determine here the effect of tensile stress over the dimensions of the cube.

As we have already seen that, Ԑ = dL/L
Strain = dL/L
dL= L x Stress x α = L x σ x α
dL= L. σ. α

Now we will have to think slightly here to discuss the effect on length of the cube under three mutually perpendicular tensile stresses of similar intensity. When direct tensile stress will be subjected over the face AEHD and BFGC, there will be increase in length due to longitudinal strain developed due to direct tensile stress acting over the face AEHD and BFGC.

Simultaneously, we must have to note it here that tensile stress acting over the face AEFB and DHGC will develop the lateral strain in side AB.

Similarly, tensile stress acting over the face ABCD and EFGH will also develop the strain in side AB
Final length of the cube, = L + L. σ. α – L. σ. β - L. σ. β
Final side length of the cube, = L [1 + σ. (α – 2β)]

Final Volume of the cube

Vf = Lx [1 + σ. (α – 2β)] 3
Now we will ignore the product of small quantities in order to easy understanding
Vf = Lx [1 + σ. (α – 2β)] 3
 Vf = L3 + 3 σ. L3 (α – 2β)
 
Change in volume of the cube, when three mutually perpendicular tensile stresses of similar intensity are acting over the cube.
ΔV = L3 + 3 σ. L3 (α – 2β) - L3
ΔV = 3 σ. L3 (α – 2β)

Let us see here volumetric strain

Volumetric strain in the specified cube here will be determined as displayed here
Volumetric strain = ΔV/V
ԐV = 3 σ (α – 2β)

Now, we will find here Bulk modulus of elasticity (K)

Bulk modulus of elasticity will be defined as the ratio of volumetric stress or hydro static stress to volumetric strain and therefore we will write here as mentioned here

K = σ / [3 σ (α – 2β)]
K = 1/ [3 (α – 2β)]
3 K (α – 2β) = 1
3K (1-2 β/α) = 1/ α

As we have already seen above that
Young’s modulus of elasticity, E = 1/ α

Poisson ratio, ν = (β/α)

After replacing the value of 1/ α and (β/α) in above concluded equation, we will have the desired result which will show the relationship between young’s modulus of elasticity (E) and bulk modulus of elasticity (K)
3K (1-2 ν) = E

E = 3K (1-2 ν)

Do you have any suggestions or any amendment required in this post? Please write in comment box.

Reference:

Strength of material, By R. K. Bansal
Image Courtesy: Google

We will see another important topic i.e. Thermal stresses in composite bars, in the category of strength of material, in our next post.

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