We were discussing “Elongation
of uniformly tapering circular rod”, “Total elongation of the bar due toits own weight” and “Thermal stress and strain” in our previous posts. Today we
will see here the determination of volumetric strain for a rectangular bar
subjected with an axial load in the direction of length of the rectangular bar.

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 rectangular bar as
displayed in following figure.

L = Length of the rectangular bar

b= Width of the rectangular bar

t= Thickness or depth of the rectangular
bar

V= Volume of the rectangular bar = L x b
x t

ΔL=Change in length of the rectangular
bar

Δb=Change in width of the rectangular
bar

Δt= Change in thickness of the
rectangular bar

ΔV= Change in volume of the rectangular
bar

P= Axial load acting in its direction of length

###
*Let us determine the final dimensions of the
rectangular bar *

*Let us determine the final dimensions of the rectangular bar*

Final length of the rectangular bar =
L+ΔL

Final width of the rectangular bar = b
+Δb

Final thickness or depth of the
rectangular bar = t +Δt

Final volume of the rectangular bar = (L+ΔL)
x (b +Δb) x (t +Δt)

Let us ignore the product of small
quantities and we will have

Final volume of the rectangular bar = L.
b. t + b. t. ΔL + L. b. Δt + L. t. Δb

**change in volume**of the rectangular bar

Change in volume of the rectangular bar
= Final volume – initial volume

ΔV= (L. b. t + b. t. ΔL + L. b. Δt + L.
t. Δb) - L x b x t

ΔV= b. t. ΔL + L. b. Δt + L. t. Δb

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}= (b. t. ΔL + L. b. Δt + L. t. Δ b)/ (L. b. t)
Ԑ

_{v}= (ΔL/L) + (Δt/t) + (Δb/b)
Let us brief here first lateral strain
before going ahead; lateral strain will be the strain at perpendicular or right
angle to the direction of applied force.

Therefore we can define here lateral
strain such as lateral strain will be basically defined as the ratio of change
in breadth of the body to the original breadth of the body.

Therefore (ΔL/L) is the linear strain
and other two terms i.e. (Δt/t) and (Δb/b) are the lateral strain.

Ԑ

_{v}= (ΔL/L) + 2 x Lateral strain
Ԑ

_{v}= Linear strain + 2 x Lateral strain
Let us brief here first “Poisson ratio”
before going ahead

Poisson ratio = Lateral strain /Linear
strain

We can also say that, Lateral strain =
Poisson ratio (ν) x Linear strain

As we have already seen that, lateral
strain will be opposite in sign to linear strain and therefore above equation
will be written as following

Lateral strain = - Poisson ratio (ν) x Linear strain

### Therefore volumetric strain will be shown as following

Volumetric strain Ԑ

_{v}= Linear strain + 2 x [- Poisson ratio (ν) x Linear strain]
Volumetric strain Ԑ

_{v}= Linear strain – 2 x ν x Linear strain
Volumetric strain Ԑ

_{v}= Linear strain x (1 – 2ν)###
**Volumetric
strain Ԑ**_{v}= (ΔL/L) x (1 – 2ν)

_{v}= (ΔL/L) x (1 – 2ν)

Do
you have any suggestions? Please write in comment box

###
**Reference:**

Strength of material, By R. K. Bansal

Image Courtesy: Google

We will see another important topic i.e. What is elasticity and elastic limit?, in the category of strength of material, in
our next post.