(a) Estimate the activation energy for creep (i.e., \(Q_{c}\) in Equation \(\left.8.25\right)\) for the S-590 alloy having the steady-state creep behavior shown in Figure 8.32. Use data taken at a stress level of 300 MPa (43,500 psi) and temperatures of \(650^{\circ} \mathrm{C}\) and \(730^{\circ} \mathrm{C}\). Assume that the stress exponent \(n\) is independent of temperature. (b) Estimate \(\dot{\epsilon}_{s}\) at \(600^{\circ} \mathrm{C}(873 \mathrm{~K})\) and \(300 \mathrm{MPa}\).

$$Q_{c}=-R T\ln{\left(\frac{\dot{\epsilon}_{s}}{A \sigma^n}\right)}$$ #Step 3: Use the information provided in the data points# The given data points are at a stress level of 300 MPa and temperatures of \(650^{\circ} C\) and \(730^{\circ} C\). Convert these temperatures to the absolute (Kelvin) scale: - \(T_1 = 650^{\circ} C + 273.15 K = 923.15 K\) - \(T_2 = 730^{\circ} C + 273.15 K = 1003.15 K\) From Figure 8.32, find the corresponding steady-state creep rates (\(\dot{\epsilon}_{s1}\) and \(\dot{\epsilon}_{s2}\)) at the given stress level and temperatures. Let's assume the stress exponent, \(n\), is the same at both temperatures. #Step 4: Apply the equation to find the activation energy for creep# Subtract the equations at two different temperatures:

$$Q_{c} = \frac{\Delta Q_{c}}{R (\frac{1}{T_1}-\frac{1}{T_2})}= \frac{T_1T_2R(\ln{\left(\frac{\dot{\epsilon}_{s2}}{\dot{\epsilon}_{s1}}\right)})}{T_2-T_1}$$ Plug in the known values and calculate the expected activation energy for creep, \(Q_{c}\). #Part (b) - Estimate the steady-state creep rate at 600°C and 300 MPa# #Step 1: Translate the given temperature into Kelvin# The given temperature is \(600^{\circ} C\). Convert it into Kelvin: - \(T = 600^{\circ} C + 273.15 K = 873.15 K\) #Step 2: Rearrange Equation 8.25 to find the steady-state creep rate# Rearrange the equation by keeping the steady-state creep rate the subject of the formula:

$$\dot{\epsilon}_{s} = A(300 \ \text{MPa})^{n} \exp{\left(-\frac{Q_{c}}{R * (873.15 K)}\right)}$$ By using Figure 8.32, estimate the material constant \(A\) at the given stress and temperature. Calculate the steady-state creep rate, \(\dot{\epsilon}_{s}\).