We were discussing the “Derivation of relationshipbetween young’s modulus of elasticity (E) and bulk modulus of elasticity (K)”, “Elongation of
uniformly tapering rectangular rod” and we have also seen the “Basic principle of complementary shear stresses” and “Volumetric strain of arectangular 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),
modulus of rigidity (G) and Poisson ratio (ν) with the help of this post.

###
**Derivation
of relationship between young’s modulus of elasticity (E), modulus of rigidity
(G) and Poisson ratio (ν)**

Let us consider one cube with side length L and let
us assume that we have one face of cube which is displayed here by ABCD as
mentioned in following figure.

Let us think that cube is subjected with the action
of shear stresses and complementary shear stresses as shown in following
figure. We can easily say that the diagonal BD will feel a tensile stress and therefore
diagonal BD will be elongated. Whereas if we consider the diagonal AC, it will
feel compressive stress and therefore the diagonal AC will be shorted.

We will consider the joint effect on the diagonal
BD. We can easily conclude that there will be tensile strain in diagonal BD due
to tensile stress along the diagonal BD. There will be one more tensile strain
in the diagonal BD due to lateral strain produced due to compressive stress
along the diagonal AC.

Total Strain in the diagonal BD = (Tensile strain in
the diagonal BD due to the tensile stress along the diagonal BD) + (Tensile
strain in the diagonal BD due to lateral strain produced due to compressive
stress along the diagonal AC)

####
*Let
us consider*

*Let us consider*

Length of side of the cube = L

Shear stress applied = τ

Young’s modulus of elasticity = E

Modulus of rigidity = G

Poisson ratio = ν

Angle of shear = θ

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

We can say from above figure that top surface of the
cube is moved to C

_{1}D_{1}from CD due to the action of shear stresses and complementary shear stresses. Let us assume that CC_{1}is l and hence we can also say that DD_{1}=CC_{1}=l
Now consider the triangle BCC

_{1}
Tan θ = l/L

Where θ is angle of shear and it will be very small and
hence we can say that

Tan θ = θ, (Angle of shear will be measured in
radian here)

θ = l/L

Let us consider that we have drawn one perpendicular
line DE over the diagonal BD

_{1}from point D as displayed in above figure. Earlier we have seen that ∠BDC = 45^{0}and once shear stresses will be applied and diagonal BD turned to BD_{1}. Angle of shear θ will be very small and therefore we can say that ∠BD_{1}C_{1}will be very close to ∠BDC and therefore ∠BD_{1}C_{1}= 45^{0}.
Let us consider the small triangle
DD

_{1}E and from here we will determine D_{1}E. As we can see here that BD and BE will be approximately equal to each other and therefore we can easily say that D_{1}E will be termed as the elongation in the diagonal BD.
D

_{1}E = Sin 45^{0}x D_{1}D = 1/**√***2*
l= D

_{1}E x**√***2*
Let us determine the D

_{1}E
Total elongation in the diagonal BD i.e. D

_{1}E = (Elongation due to tensile stress along the diagonal BD) + (Elongation due to lateral strain produced due to compressive stress along the diagonal AC)
D
τ x BD) + (β τ x BD)

_{1}E = (α
D

_{1}E = τ x BD (α + β)
l= D

_{1}E x**√***2*
l= τ x BD (α + β) x

**√***2*
l= τ x L

**√***2*(α + β) x**√***2*, (Because BD = L**√***2)*
l= 2 x τ
x L (α + β)

l/L= 2 x τ
(α + β)

θ = 2τ (α + β)

1/ [2(α + β)] = τ/θ

τ/θ =1/[2(α + β)]

G=
1/ [2(α + β)],

Because modulus of rigidity = shear
stress (τ)/ Shear strain (θ)

G=
1/ [2α (1 + β/ α)]

G=
1/ [2α (1 + ν)]

###
**G= ****E/ [2****
(1 + ν)]**

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

Find this post for complete information about centrifugal pump i.e.

Find this post for complete information about centrifugal pump i.e.

*Centrifugal pump working principle*###
**Reference:**

Strength of material, By R. K. Bansal

Image Courtesy: Google

We will see another important topic i.e. Determination of
volumetric strain for a rectangular bar subjected with an axial load in the
direction of length of the rectangular bar, in the category of
strength of material, in our next post.

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