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,*

*Initial volume of the cube,*

V = Length x width x height = L

^{3}
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*

*Final Volume of the cube*

V

_{f}= L^{3 }x [1 + σ. (α – 2β)]^{ 3}
Now we will ignore the product of small quantities in order to easy understanding

V

_{f}= L^{3 }x [1 + σ. (α – 2β)]^{ 3}
V

_{f}= L^{3}+ 3 σ. L^{3}(α – 2β)Change in volume of the cube, when three mutually perpendicular tensile stresses of similar intensity are acting over the cube.

ΔV = L

^{3}+ 3 σ. L^{3}(α – 2β) - L^{3}
ΔV = 3 σ. L

^{3}(α – 2β)####
*Let us see here volumetric strain*

*Let us see here volumetric strain*

Volumetric strain in the specified cube here will be determined as displayed here

Volumetric strain = ΔV/V

Ԑ

_{V}= 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**)*

*E = 3K (1-2*

*ν*

*)*

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

You must find out and read the following quite useful engineering articles and these are as mentioned below.

###
**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.

great

ReplyDeleteWatch Online Pinoy Lambingan Tv Shows For All Episode In HD Quality Pinoy Tv, Pinoy Teleserye, Pinoy Tambayan Provide On Our PinoyTv

ReplyDeleteYour Site is very nice, and it's very helping us this post is unique and interesting, thank you for sharing this awesome information. and visit our blog site also.

ReplyDeleteSatta KingBe it any subject, once I hopped into my freshman year in college, I had to deal with a hefty coursework report that turned out to be crucial. Jotting down lectures really didn’t help me standardize the piece in a top format. Luckily, with cheap Nursing Essay Writing Service based, top elites served the proper facts and instruments needed on my coursework sheet, which in the end did the impossible. I topped my batch in the end.

ReplyDeleteVideovity has a team of extremely talented individuals, state of art technology, and a wide range of animation services. We are dedicated to producing the best typography animation to meet the needs of your business in the best way possible. At videovity, we value your input, listen to your concerns and give our all to make sure that you are satisfied with the end result. To achieve that goal, we first analyze the requirements of your company/project and then figure out what would best suit it. We consider your budget and accommodate you to our best abilities. You can rest assured, that our expertise and experience will not disappoint you.

ReplyDelete