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Monday, 23 January 2017

VOLUMETRIC STRAIN FOR A CYLINDRICAL ROD

Today we will see here volumetric strain for a cylindrical rod with the help of this post

Before going ahead we must recall the basic concept of volumetric strain which is explained here briefly. 

When an object will be subjected with a system of forces, object will undergo through some changes in its dimensions and hence, volume of that object will also be changed.

Volumetric strain will be defined as the ratio of change in volume of the object to its original volume. Volumetric strain is also termed as bulk strain.

Ԑv= Change in volume /original volume
Ԑv= dV/V

Let us consider one cylindrical rod or bar as displayed in following figure and let us assume that cylindrical rod is subjected with a tensile loading i.e. P as displayed in following figure.
L = Length of the cylindrical rod
d = Diameter of the cylindrical rod
Volume of the cylindrical rod, V = (Π/4) d2. L

ΔL= Change in length of the cylindrical rod
Δd= Change in diameter of the cylindrical rod
ΔV= Change in volume of the cylindrical rod

Let us determine the final dimensions of the cylindrical rod

Final length of the cylindrical rod = L+ ΔL

As we know very well that when tensile load P will act over the cylindrical rod, there will be increment in length of the cylindrical rod and decrease in diameter of the cylindrical rod.
Final diameter of the cylindrical rod = d - Δd

Final volume of the cylindrical rod = (Π/4) (d - Δd) 2. (L+ ΔL)

Let us ignore the product of small quantities and we will have  
Final volume of the cylindrical rod = (Π/4) [d2.L – 2.Δd.L.d + ΔL. d2]

Let us determine the change in volume of the cylindrical rod

Change in volume of the cylindrical rod = Final volume – initial volume
Change in volume of the cylindrical rod, ΔV = (Π/4) [(d2.L – 2.Δd.L.d + ΔL. d2)-(d2.L)]
ΔV= (Π/4) [ΔL. d2– 2.Δd.L.d]  
Volumetric strain also known as bulk strain will be determined as following
Ԑv= Change in volume /original volume
Ԑv= dV/V

Ԑv= [(Π/4) (ΔL. d2– 2.Δd.L.d)]/ [(Π/4) d2.L]
Ԑv= [(ΔL. d2– 2.Δd.L.d)/ (d2.L]  
Ԑv= (ΔL/L) – 2(Δd/d)
We can also say that volumetric strain for a cylindrical rod will be written as mentioned here

Volumetric strain = linear strain – 2 x lateral strain
Ԑv= (ΔL/L) – 2(Δd/d)

Where, (ΔL/L) is the linear strain or longitudinal strain
(Δd/d) is the lateral strain


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Reference:

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

We will see another important topic i.e. Total elongation of the bar due to its own weight in the category of strength of material.

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