2. An aluminum alloy is required to withstand at least 10^7 stress cycles. The results of the tests show that when the range of stress changes is equal to 114.3 or 161.5 MPa.the part will break after 10^8 cycles or 10^5 cycles, respectively. If the fatigue behavior of this alloy can be expressed by the following equation, C=Delta sigma ^20N_(f) Where C=constant and N_(f) is the number of cycles to failure. calculate the appropriate stress __

To calculate the appropriate stress range for the aluminum alloy to withstand at least $10^{7}$ stress cycles, we can use the given equation $C=\Delta \sigma ^{20}N_{f}$.<br /><br />Given:<br />$C$ = constant<br />$N_{f}$ = number of cycles to failure<br />$\Delta \sigma$ = stress range<br /><br />We need to find the stress range $\Delta \sigma$ that will result in at least $10^{7}$ stress cycles.<br /><br />Rearranging the equation, we get:<br />$\Delta \sigma = (C/N_{f})^{1/20}$<br /><br />Substituting the given values:<br />$\Delta \sigma = (C/10^{7})^{1/20}$<br /><br />To find the value of $C$, we can use the given data points:<br />$C = (114.3)^{20} \times 10^{8}$<br />$C = (161.5)^{20} \times 10^{5}$<br /><br />Now, we can substitute these values into the equation:<br />$\Delta \sigma = (114.3)^{20} \times 10^{8})^{1/20}$<br />$\Delta \sigma = (161.5)^{20} \times 10^{5})^{1/20}$<br /><br />Calculating these values, we get:<br />$\Delta \sigma \approx 114.3$<br />$\Delta \sigma \approx 161.5$<br /><br />Therefore, the appropriate stress range for the aluminum alloy to withstand at least $10^{7}$ stress cycles is between 114.3 MPa and 161.5 MPa.