We were discussing “Determination of volumetric strain for a rectangular bar subjected with an axial load in the direction of length of the rectangular bar, “determination of volumetric strain for a rectangular bar subjected with three forces mutually perpendicular with each other” and “Thermal stress and strain” in our previous posts.

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

*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.

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.

Ԑ

_{v}= Change in volume /original volume
Ԑ

_{v}= 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)
d

^{2}. 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 **

*Let us determine the final dimensions of the*

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)
[d

^{2}.L – 2.Δd.L.d + ΔL. d^{2}]###
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) [(d

^{2}.L – 2.Δd.L.d + ΔL. d^{2})-(d^{2}.L)]
ΔV= (Π/4) [ΔL. d

^{2}– 2.Δd.L.d]
Volumetric strain also known as bulk
strain will be determined as following

Ԑ

_{v}= Change in volume /original volume
Ԑ

_{v}= dV/V
Ԑ

_{v}= [(Π/4) (ΔL. d^{2}– 2.Δd.L.d)]/ [(Π/4) d^{2}.L]
Ԑ

_{v}= [(ΔL. d^{2}– 2.Δd.L.d)/ (d^{2}.L]

*Ԑ*_{v}= (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*
**
***Ԑ*_{v}= (ΔL/L) – 2(Δd/d)

*Volumetric strain = linear strain – 2 x lateral strain*

*Ԑ*_{v}= (ΔL/L) – 2(Δd/d)
Where, (ΔL/L) is the linear strain or
longitudinal strain

(Δd/d) is the lateral strain

Do you have any suggestions? Please write in comment
box

###
**Reference:**

Strength of material, By R. K. Bansal

Image Courtesy: Google

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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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