Question

A circular bar of 30 mm diameter is subjected to a pull of 60 kn. The measured elongation on a gauge length of 200 mm is 0.09 mm and the change in diameter is 0.0039 mm. Calculate poisson's ratio, young's modulus, modulus of rigidity and bulk modulus. What change in volume would a 10 cm cube of this material experience, if it is subjected to a hydrostatic stress of 50 kn/m2 .

Answer 1

Answer:

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

Explanation:

hope it's helpful

Answer 2

Step-by-step Explanation

Given: D = 30 mm, ΔD = 0.0039 mm, L = 200 mm, ΔL = 0.09 mm,

F = 60 kN, p = 50 kN/m², V = 10 cm

To Find: Poisson's ratio, values of three moduli, and change in volume of the cube

Solution:

• Calculating Poisson's ratio σ

The ratio of the lateral strain and longitudinal strain is called Poisson's ratio. Mathematical expression to find Poisson's ratio σ is;

$\sigma = - \frac{\Delta D / D}{\Delta L / L}$

Substituting the given values, we get

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

• Calculating the three moduli of elasticity

Young's modulus can be calculated as;

$Y = \frac{4FL}{\pi D^{2} \Delta L }$ . . . . . (1)

Substituting the given values in (1), we will get;

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

Poisson's ratio and three moduli, i.e., Young's Modulus (Y), Bulk Modulus (B), and Shear Modulus (G), are related as -

$Y = 3B(1-2\sigma)$ and $Y = 2G(1+ \sigma)$

Substituting the values of Young's Modulus and Poisson's ratio in the above two relations, we get B = 39.849 × 10⁹ Nm⁻² and G = 132.628 × 10⁹ Nm⁻²

• Calculating the change in volume (ΔV) for the cube of the material

For the material with bulk modulus B = 39.849 × 10⁹ Nm⁻², the cube of volume V = 10 cm is subjected to a stress of p = 50 kN/m². Therefore, a change in the volume ΔV can be calculated as;

$B = \frac{-p}{\Delta V / V} \Rightarrow \Delta V = \frac{-pV}{B}$

Substituting the required values and calculating, we will get ΔV = 1.25 × 10⁻⁹ m³

Hence, Poisson's ratio σ is -0.2889, Young's modulus is Y = 188.624 × 10⁹ Nm⁻², the Bulk modulus is B = 39.849 × 10⁹ Nm⁻², the Shear modulus is G = 132.628 × 10⁹ Nm⁻², and change in the volume of the cube is ΔV = 1.25 × 10⁻⁹ m³