VOLUMETRIC STRAIN OF A RECTANGULAR BODY SUBJECTED TO AN AXIAL FORCE

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.

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

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

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)

Do you have any suggestions? Please write in comment box 

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

1. Metal cutting tools i.e. Metal cutting tool definition and classification

2. Single point cutting tool nomenclature

3. Difference between rake angle and clearance angle of cutting tool

4. Main characteristics of cutting tool materials

5. Lever and its types with examples

6. Guidelines for solving friction problems in mechanics

7. Rolling resistance or rolling friction

8. Wedge friction and self-locking in engineering mechanics

9. Concept of force system in engineering mechanics

10. Difference between fan and blowers

11. Basics of centrifugal compressor

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.

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