Question

In an experiment, a bar of 30 mm diameter is subjected to a pull of 60 kN. The measured

extension on gauge length of 200 mm is 0.09 mm and the change in diameter is 0.0039

mm. Calculate the Poisson’s ratio and the values of the three moduli.​

Answer 1

Data: d=30 mm, L=200 mm, P =60 kN, δL=0.09 mm, δd = 0.0039 mm Calculate: μ and E

A=πd24=π×3024=706.858mm2

E=PLAδL=60×103×200706.858×0.09=188628.08N/mm2

E=1.89N/mm2

μμμ= Lateral Strain  Linear Strain =(δdd)(δLL)=(0.003930)(0.09200)=0.29=0.29

Answer 2

Step-by-step Explanation

Given: original diameter of the bar $D = 30 \ mm$

change in diameter $\Delta D = 0.0039 \ mm$

the original length of the bar $l = 200 \ mm$

change in the length $\Delta l = 0.09 \ mm$

Force exerted $F = 60 \ kN$

To Find: Poisson's ratio and values of three moduli

Solution:

• Calculating Poisson's ratio

Poisson's ratio is the ratio of the lateral strain and longitudinal strain. It is  calculated as;

$\text{Poisson's ratio} \ (\sigma) = \frac{- \Delta D / D}{\Delta l / l}$

Substituting the given values in the above expression, we get

$\sigma = \frac{- \Delta D / D}{\Delta l / l} = \frac{- 0.0039 / 30}{0.09 / 200} = - 0.2889$

• Calculating the three moduli of elasticity

Young's modulus can be determined as;

$Y = \frac{4 F l}{\pi D^{2} \Delta l} \ N/m^{2}$

Substituting the given values and calculating Y, we will get;

$Y = \frac{4 \times 60 \times 10^{3} \times 200 \times 10^{-3} }{\pi \times (30 \times 10^{-3} )^{2} \times 0.09 \times 10^{-3} } = 188.624 \times 10^{9} N/m^{2}$

The relation between Poisson's ratio and three moduli, i.e., Young's Modulus (Y), Bulk Modulus (B), and Shear Modulus (G) is;

$Y = 3B (1-2 \sigma) \Rightarrow B = \frac{Y}{3 (1-2 \sigma)} N/m^{2}$

$Y = 2G (1 + \sigma) \Rightarrow G = \frac{Y}{2 (1 + \sigma)} N/m^{2}$

Therefore,

$B = \frac{188.624 \times 10^{9}}{3 (1-2 (-0.2889))} = 39.849 \times 10^{9} \ N/m^{2}$

and, $G = \frac{188.624 \times 10^{9}}{2 (1- 0.2889))} = 132.628 \times 10^{9} \ N/m^{2}$

Hence, Poisson's ratio is $-0.2889$, while Young's modulus $Y = 188.624 \times 10^{9} N/m^{2}$, the Bulk modulus is $B = 39.849 \times 10^{9} \ N/m^{2}$, and the Shear modulus is $G = 132.628 \times 10^{9} \ N/m^{2}$