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Longitudinal Strain: Definition, Formula, Poisson's Ratio

You must have observed that after the application of longitudinal force the body undergoes deformation. Look at the image below where a man is stretching the rubber wire using his hands. After stretching the rubber its length increases. The ratio of change in the length of wire with respect to the original length is defined as strain. Now we know the body undergoes strain after the application of force. In this article we will be studying particularly about the longitudinal strain in which deforming force is normal to the cross section of the object. 


Table of Contents 

  • Strain 
  • Types of strain
  • Longitudinal strain
  • Poisson’s ratio
  • Practice problems
  • FAQs

Strain 

When a deforming force is applied on the body, there is change in the configuration of the body. The body is said to be strained or deformed. The ratio of the change in configuration (shape, length, or volume) to the original configuration of the body is known as strain. 

Strain,

Note: Strain is unitless and dimensionless. Generally it is represented in percentage. 

Types of Strain 

Corresponding to the three types of stress, there are three types of strains.

  • Longitudinal strain ()
  • Shear strain ()
  •  Volumetric strain ()

Longitudinal Strain

This type of strain is produced when the deforming force causes a change in the length of the body. It is defined as the ratio of the change in length to the original length of the body.

Consider a rod of length (L) having an area of cross-section (A). Weight is added at the free end of the rod that applies force (F) on the rod. Initially, the rod is hinged from the roof and the weight is not added. After the weight is added to the rod, the length of the rod gets extended.

Original length = L

Area of cross-section = A 

Applied force due to weight = F 

Final length = L + ∆L 

Change in length = Final length - Original length = 


Longitudinal strain is given by,

Poisson’s Ratio

Poisson’s ratio is related to the simultaneous strain in the transverse (lateral) direction and longitudinal (axial) direction of an object due to an external force. It is defined as the negative ratio of the transverse (or lateral) strain to the longitudinal (or axial) strain.


Consider a rectangular bar of length l and breadth b. Due to an external force F, which acts in the outward direction in two opposite faces of the bar as shown in the figure (just like the bar is stretched along its length), the length of the bar will be increased and there will be a consequent decrease in the breadth.

So, if the change in length is, , and the change in breadth is,

then the transverse strain is, , here is a negative quantity since breadth has decreased. And longitudinal strain  

So poisson’s ratio,

Practice Problems

Q. A rod having length is compressed such that its final length is . Find the percentage strain in the rod?

A.

Q. A rod is stretched axially so that its transverse strain is and longitudinal strain is . Find the poisson’s ratio of the given rod?

A.  

poisson’s ratio,

Q. A brass rod of length 1 m is fixed to a vertical wall at one end, the other end is kept free to expand. When the temperature of the rod is increased by 120 degrees, the length increases by 3 cm. What is the strain?

A. After the rod expands to the new length, there are no elastic forces developed in it. 

So,

Q. A steel rod 2 m long is suspended from the ceiling. When the mass is hung at its lower end, the increase in the length recorded is 1 cm. Determine the strain in the wire?

A.

FAQs

Q. Under what condition will the value of strain be negative?
A. If the deforming force applied on the body is compressive in nature then the final configuration will be less than its original value. So the value of strain will be negative. 

Q. Under what condition will the value of strain be positive?
A. If the stretching force is applied on the body then the final configuration will be larger than its original value so the value of strain will be positive. 

Q. Why is the strain a dimensionless quantity?
A. Strain is the ratio of change in the configuration to the original configuration. So the ratio of two quantities having the same unit has no unit. Therefore strain is unitless and dimensionless. 

Q. How longitudinal strain and transverse strain are related?
A. Longitudinal strain and transverse strain are related with each other by poisson’s ratio i.e.

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