Poissons Ratio

  • Describe and define Poisson’s ratio.

Consider a rod under an axial tensile load P such that the material is within the elastic limit. The normal stress on x plane is \(\sigma_{xx} = \dfrac{P}{A}\) and the associated longitudinal strain in the x direction can be found out from \(\epsilon_x = \dfrac{\sigma_{xx}}{E}\). As the material elongates in the x direction due to the load P, it also contracts in the other two mutually perpendicular directions, i.e., y and z directions. Hence, despite the absence of normal stresses in y and z directions, strains do exist in those directions and they are called lateral strains.
The ratio between the lateral strain and the axial/longitudinal strain for a given material is always a constant within the elastic limit and this constant is referred to as Poisson’s ratio.
It is denoted by \(nu\). Since the axial and lateral strains are opposite in sign, a negative sign is introduced in the definition to make \(\nu\) positive. \[\nu = - \dfrac{\mathrm{lateral\ strain}}{\mathrm{longitudinal\ strain}}\]
Poisson’s ratio can be as low as 0.1 for concrete and as high as 0.5 for rubber. In general, it varies from 0.25 to 0.35 and for steel it is about 0.3.
Young’s modulus = E = \(\dfrac{\mathrm{tensile\ stress}}{\mathrm{tensile\ strain}}\)
Young’s modulus = E = \(\dfrac{\mathrm{compressive\ stress}}{\mathrm{compressive\ strain}}\)
Shear modulus = G = \(\dfrac{\mathrm{shear\ stress}}{\mathrm{shear\ strain}}\)
Bulk modulus = K = \(\dfrac{\mathrm{Volumetric\ stress}}{\mathrm{volumetric\ strain}}\)
K= \(\dfrac{\mathrm{pressure}}{\mathrm{volumetric\ strain}}\) = \(\dfrac{p}{(\delta V/ V)}\)

Solved Example:

41-3-01

The value of Poisson’s ratio for steel is between:

Correct Answer: C

Solved Example:

41-3-02

Poisson’s ratio is defined as the ratio of:

Correct Answer: C

Solved Example:

41-3-03

A bar of 30 mm diameter is subjected to a pull of 60 kN. The measured extension on gauge length of 200 mm is 0.1 mm and change in diameter is 0.004 mm. Calculate Poisson’s ratio.

Correct Answer: A

Solved Example:

41-3-04

Poisson's ratio of a material is 0.5. Percentage change in its length is 0.04%. What is the change in percentage of diameter?

Correct Answer: C

Solved Example:

41-3-05

Which of the following statements is NOT true?

Correct Answer: B

Solved Example:

41-3-06

Which of the following describes the concept of Poisson's ratio most accurately?

Correct Answer: C

Solved Example:

41-3-07

What is the unit of the modulus of elasticity?

Correct Answer: C

Solved Example:

41-3-08

Within elastic limit, the volumetric strain is proportional to the hydrostatic stress. What is the constant that relates these two quantities called?

Correct Answer: D

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41-3-09

What is another term for modulus of rigidity?

Correct Answer: A

Solved Example:

41-3-10

The ratio of lateral strain to the linear strain within elastic limit is known as:

Correct Answer: D

Solved Example:

41-3-11

Young’s modulus is defined as the ratio of:

Correct Answer: C

Solved Example:

41-3-12

The materials having same elastic properties in all directions are called:

Correct Answer: C

Solved Example:

41-3-13

The value of modulus of elasticity for mild steel is of the order of:

Correct Answer: B

Solved Example:

41-3-14

A metallic rod of 500 mm length and 50 mm diameter, when subjected to a tensile force of 100 kN at the ends, experiences, an increase in its length by 0.5 mm and a reduction in its diameter by 0.015 mm. The Poisson's ratio?

Solution:

Lateral strain $=\dfrac {0.015}{50}=0.003$

Longitudinal strain $=\dfrac {0.5}{500}=0.001$

Poisson's ratio $\nu =\dfrac {0.0003}{0.001}=0.3$

Correct Answer: B