# Question 3: Explanation and Solutions Let's address each part step by step: --- ## **3.1 Strain Hardening for Required Flow Stress** ### **Given:** - Average flow stress = 78% of final flow stress ### **Required:** - Value of strain hardening for the required flow stress ### **Explanation:** The average flow stress (\(\bar{\sigma}\)) during deformation is given by: \[ \bar{\sigma} = \frac{\sigma_0 + \sigma_f}{2} \] where: - \(\sigma_0\) = initial flow stress - \(\sigma_f\) = final flow stress But here, the average flow stress is said to be 78% of the final flow stress: \[ \bar{\sigma} = 0.78 \sigma_f \] Set the two equations equal: \[ 0.78 \sigma_f = \frac{\sigma_0 + \sigma_f}{2} \] Multiply both sides by 2: \[ 1.56 \sigma_f = \sigma_0 + \sigma_f \] Solve for \(\sigma_0\): \[ \sigma_0 = 1.56 \sigma_f - \sigma_f = 0.56 \sigma_f \] **Strain hardening** is the increase in flow stress due to deformation: \[ \text{Strain hardening} = \sigma_f - \sigma_0 = \sigma_f - 0.56 \sigma_f = 0.44 \sigma_f \] ### **Final Answer:** > **The strain hardening for the required flow stress is \(0.44\) (or 44%) of the final flow stress.** --- ## **3.2 Determining the Strain Rate** ### **Given:** - Instantaneous height of workpiece, \(h = 130 \text{ mm} = 0.13 \text{ m}\) - Deformation speed, \(v = 15 \text{ m/s}\) ### **Formula:** \[ \text{Strain rate},\ \dot{\varepsilon} = \frac{v}{h} \] ### **Calculation:** \[ \dot{\varepsilon} = \frac{15}{0.13} \approx 115.38\ \text{s}^{-1} \] ### **Final Answer:** > **The strain rate is approximately \(115.38\ \text{s}^{-1}\).** --- ## **3.3 Effect of Strain Rate on Flow Stress at Elevated Temperature** ### **Formula:** At elevated temperatures, the flow stress (\(\sigma\)) is often related to strain rate (\(\dot{\varepsilon}\)) by: \[ \sigma = K \cdot \dot{\varepsilon}^m \] Where: - \(K\) = strength coefficient (material constant) - \(m\) = strain rate sensitivity exponent (usually between 0 and 1) ### **Interpretation:** - **If \(m > 0\):** As strain rate increases, flow stress increases. - At **elevated temperatures**, materials often show higher strain rate sensitivity (\(m\) increases), meaning the flow stress becomes more dependent on the rate at which the material is deformed. - This behavior is important in hot working processes. ### **Final Answer:** > **At elevated temperatures, as the strain rate increases, the flow stress also increases according to the power law \(\sigma = K \cdot \dot{\varepsilon}^m\), where \(m\) indicates the sensitivity of the material to strain rate.** --- If you need further clarification or example problems, let me know!