Let's break down the problem and find the **Poisson's ratio (ν)** step by step. ### **Given Data** - **Young's Modulus, \( E \) = 200 GPa = 200,000 MPa** - **Thermal Expansion Coefficient, \( \alpha = 11.7 \times 10^{-6} /^\circ C \)** - **Measured Strains: \( \delta_y = 0.001 \delta_x \)** - **Hydraulic Diameter \( Dh = 19 \) mm** #### **Step 1: Understanding Strain Measurements** - \( \delta_x \) = axial (longitudinal) deformation - \( \delta_y \) = lateral deformation The test shows: \[ \delta_y = 0.001 \, \delta_x \] #### **Step 2: Poisson's Ratio Definition** \[ \nu = -\frac{\text{Lateral Strain}}{\text{Axial Strain}} \] Lateral strain (\( \epsilon_{lat} \)) is the change in diameter per original diameter (\( \Delta d / d \)), and axial strain (\( \epsilon_{ax} \)) is the change in length per original length (\( \Delta L / L \)). Given the ratio: \[ \epsilon_{lat} = \frac{\delta_y}{Dh} \] \[ \epsilon_{ax} = \frac{\delta_x}{L} \] But since we are given a direct proportionality between \( \delta_y \) and \( \delta_x \): \[ \frac{\epsilon_{lat}}{\epsilon_{ax}} = \frac{\delta_y / Dh}{\delta_x / L} = \frac{\delta_y}{\delta_x} \cdot \frac{L}{Dh} \] We know for **Poisson's ratio**: \[ \nu = -\frac{\epsilon_{lat}}{\epsilon_{ax}} \] However, in most tensile tests, the lateral strain is negative (shrinks) as the length increases, so: \[ \nu = -\frac{\delta_y}{\delta_x} \] But since the problem gives \(\delta_y = 0.001 \delta_x\), we take the magnitude and apply the negative sign: \[ \nu = -0.001 \] But typically, Poisson's ratio is reported as a positive value (just the magnitude): \[ \nu = 0.001 \] **But this value is extremely low for metals (typical ν ≈ 0.3).** ### **Check:** If the measured \(\delta_y\) is the contraction (negative), and \(\delta_x\) is extension (positive), then: \[ \nu = -\frac{\delta_y}{\delta_x} = -0.001 \] But by convention, Poisson's ratio is reported as a positive number: \[ \boxed{\nu = 0.001} \] --- ## **Final Answer:** ### **a) What is the Poisson's ratio?** \[ \boxed{\nu = 0.001} \] --- ### **Note:** - This is an unusually low value for Poisson’s ratio for most engineering metals (normally about 0.3). Double-check if the data is correct or if \(\delta_y\) is not the total contraction but just a partial measurement. If you have further data or clarification, you may want to revisit this calculation. - If you need to calculate something else (like stress, strain, etc.), let me know!