The stress along the length of a rod with rectangular cross section) is `1%` of the Young's modulus of its material. What is the approximate percentage of change of its volume? (Poisson's ratio of the material of the rod is 0.3)

Options

A. `3%`
B. `1%`
C. `0.7%`
D. `0.4%`

Answer

To solve the problem, we need to determine the approximate percentage change in the volume of a rod subjected to a longitudinal stress that is 1% of the Young's modulus of its material. Given that the Poisson's ratio of the material is 0.3, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Stress and Strain**:
- The stress (σ) is given as 1% of the Young's modulus (E). Thus, we can write:
\[ \sigma = 0.01 E \]

2. **Relating Stress to Longitudinal Strain**:
- From the definition of Young's modulus:
\[ E = \frac{\sigma}{\epsilon_L} \]
where \( \epsilon_L \) is the longitudinal strain. Rearranging gives:
\[ \epsilon_L = \frac{\sigma}{E} = \frac{0.01 E}{E} = 0.01 \]
- Therefore, the longitudinal strain \( \epsilon_L \) is 0.01 (or 1%).

3. **Using Poisson's Ratio**:
- Poisson's ratio (μ) relates the lateral strain (ε_D) to the longitudinal strain:
\[ \mu = -\frac{\epsilon_D}{\epsilon_L} \]
- Given \( \mu = 0.3 \), we can express the lateral strain as:
\[ \epsilon_D = -\mu \cdot \epsilon_L = -0.3 \cdot 0.01 = -0.003 \]

4. **Calculating the Change in Dimensions**:
- The negative sign indicates a decrease in dimensions. Thus, the lateral strains in both width (d) and thickness (t) are:
\[ \epsilon_D = -0.003 \]
- This means:
\[ \frac{\Delta d}{d} = -0.003 \quad \text{and} \quad \frac{\Delta t}{t} = -0.003 \]

5. **Volume Change Calculation**:
- The volume \( V \) of the rod can be expressed as:
\[ V = L \cdot d \cdot t \]
- The relative change in volume \( \frac{\Delta V}{V} \) can be approximated as:
\[ \frac{\Delta V}{V} = \frac{\Delta L}{L} + \frac{\Delta d}{d} + \frac{\Delta t}{t} \]
- Substituting the values we have:
\[ \frac{\Delta V}{V} = 0.01 - 0.003 - 0.003 \]
- Simplifying this gives:
\[ \frac{\Delta V}{V} = 0.01 - 0.006 = 0.004 \]

6. **Converting to Percentage**:
- To express the change in volume as a percentage:
\[ \text{Percentage change in volume} = 0.004 \times 100 = 0.4\% \]

### Final Answer:
The approximate percentage change of the volume of the rod is **0.4%**.