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.

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 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

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

Let us determine the 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
Ԑv= Change in volume /original volume
Ԑv= dV/V
Ԑv= (b. t. ΔL + L. b. Δt + L. t. Δ b)/ (L. b. t)
Ԑv= (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.
Ԑv= (ΔL/L) + 2 x Lateral strain
Ԑv= 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 Ԑv= Linear strain + 2 x [- Poisson ratio (ν) x Linear strain]
Volumetric strain Ԑv= Linear strain – 2 x ν x Linear strain
Volumetric strain Ԑv= Linear strain x (1 – 2ν)

### Volumetric strain Ԑv= (ΔL/L) x (1 – 2ν)

Do you have any suggestions? Please write in comment box

### Reference:

Strength of material, By R. K. Bansal
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