To calculate Poisson's ratio and the three elastic moduli (Young's modulus, Bulk modulus, and Shear modulus), we can follow a systematic approach based on the given data.

Given dataTo calculate Poisson's ratio and the three elastic moduli (Young's modulus, Bulk modulus, and Shear modulus), we can follow a systematic approach based on the given data.

Given

• Diameter of bar (d): 30 mm
• Force applied (F): 60 kN = 60,000 N
• Gauge length (L₀): 200 mm
• Extension (ΔL): 0.09 mm
• Change in diameter (Δd): 0.0039 mm

Step 1: Calculate Poisson's ratio (ν)

Poisson's ratio is defined as the negative ratio of the lateral strain to the longitudinal strain.

$\nu = -\frac{\text{Lateral Strain}}{\text{Longitudinal Strain}}$

• Lateral strain = $\frac{\Delta d}{d} = \frac{0.0039}{30}$
• Longitudinal strain = $\frac{\Delta L}{L₀} = \frac{0.09}{200}$

Now, let's calculate Poisson's ratio:

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

Step 2: Calculate Young's modulus (E)

Young's modulus is calculated using the formula:

$E = \frac{\sigma}{\epsilon}$

Where:

• Stress (σ) = $\frac{F}{A} = \frac{60,000}{\pi \times \left(\frac{30}{2}\right)^2}$ (N/mm² or MPa)
• Strain (ε) = $\frac{\Delta L}{L₀} = \frac{0.09}{200}$

Let’s calculate the values.

Step 3: Calculate Shear modulus (G)

Shear modulus can be calculated using the relation between Young's modulus and Poisson's ratio:

$G = \frac{E}{2(1 + \nu)}$

Step 4: Calculate Bulk modulus (K)

Bulk modulus is calculated using the following relation:

$K = \frac{E}{3(1 - 2\nu)}$

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Let me calculate these values for you.### Results:

1. Poisson's ratio (ν): -0.289 (approximately)
2. Young's modulus (E): 188,628 MPa
3. Shear modulus (G): 132,629 MPa
4. Bulk modulus (K): 39,851 MPa

These values describe the material's response under the given load, where the negative Poisson's ratio suggests an unusual behavior, possibly indicating that the material expands in both directions under stress (though this could be a computational result of sign convention).

Would you like a more detailed explanation, or do you have any questions about these results?

Related Questions:

1. How does Poisson's ratio affect material properties in different scenarios?
2. What is the significance of the Bulk modulus in material science?
3. Can materials have a negative Poisson's ratio in practical applications?
4. How does shear modulus relate to the rigidity of a material?
5. What factors can influence the accuracy of modulus measurements?

Tip:

For accurate material property calculations, ensure that all measurements (like strain and deformation) are precise and account for experimental errors.