The life curve is usually described by the following equation, which is called Goodman diagram:
${\mathrm{\sigma}}_{a}={\mathrm{\sigma}}_{a0}\left(1-\frac{{\mathrm{\sigma}}_{m}}{{\mathrm{\sigma}}_{m0}}\right)$(9)where ${\mathrm{\sigma}}_{a}$ is the stress amplitude, ${\mathrm{\sigma}}_{a0}$ is the stress amplitude when the mean is 0, ${\mathrm{\sigma}}_{m}$ is the mean stress and ${\mathrm{\sigma}}_{m0}$ is the mean stress when the amplitude is 0.

The stress cycles of each level can be represented by $({\mathrm{\sigma}}_{ai},{\mathrm{\sigma}}_{mi})$ or $({\mathrm{\sigma}}_{maxi},{R}_{i})$. Generally, $({\mathrm{\sigma}}_{maxi},{R}_{i})$ is accepted in engineering and ${\mathrm{\sigma}}_{maxi}$ can be shorted by ${\mathrm{\sigma}}_{i}$. According to Goodman diagram, $({\mathrm{\sigma}}_{i}^{\ast},{R}_{i})$, which corresponds to $({\mathrm{\sigma}}_{i},{R}_{i})$ and called symmetrical cyclic stress, can be calculated by the following equation:
${{\mathrm{\sigma}}_{i}}^{\ast}=\frac{(1-{R}_{i}){\mathrm{\sigma}}_{m0}{\mathrm{\sigma}}_{i}}{{\mathrm{\sigma}}_{m0}(1-{{R}_{i}}^{\ast})+{S}_{i}({R}^{\ast}-{R}_{i})}$(10)Then *S–N* curve and eq. (10) are used to calculate the fatigue life which corresponds to $({\mathrm{\sigma}}_{i},{R}_{i})$, the fatigue life is
${N}_{i}={\left(\frac{{S}_{\mathrm{\infty}}A}{{{\mathrm{\sigma}}_{i}}^{\ast}-{S}_{\mathrm{\infty}}}\right)}^{\frac{1}{\mathrm{\alpha}}}$(11)If ${S}_{i}$ is known, the corresponding fatigue life can be calculated by
${N}_{i}={\left(\frac{A{S}_{\mathrm{\infty}}}{{S}_{i}-{S}_{\mathrm{\infty}}}\right)}^{\frac{1}{a}}$(12)Substituting parameters into eq. (12) with ${S}_{\mathrm{\infty}}=1.18\times $ ${10}^{8}\mathrm{P}\mathrm{a}$, $A=17.9056$, $\mathrm{\alpha}=0.1945$
${N}_{i}={\left(\frac{2.1128608\times {10}^{9}}{{S}_{i}-1.18\times {10}^{8}}\right)}^{5.141388}$(13)The corresponding cyclic life by eq. (13) is shown in .

Then by using the Miner’s rule, we substitute parameters into eq. (12) ${S}_{\mathrm{\infty}}=1.18\times {10}^{8}\mathrm{P}\mathrm{a}$, $A=17.9056$, $\mathrm{\alpha}=0.1945$, and instead of ${S}_{i}$ by ${S}_{a}^{\ast}$, the ${D}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ is
$\begin{array}{c}{D}_{\text{total}}={\displaystyle \sum _{i=1}^{k}\frac{{n}_{i}}{{N}_{i}}}\\ ={\displaystyle \sum _{i=1}^{k}\frac{{n}_{i}}{{\left[\frac{2.1128608\times {10}^{9}}{{S}_{ai}^{*}-1.18\times {10}^{8}}\right]}^{5.141388}}}\end{array}$(14)Cyclic life ${N}_{i}$ and equivalent damage ${D}_{i}$ are shown in , and ${D}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}=0.018471$.

Table 5: Fatigue limit and damage based on the rotor speeds of aero-engines.

According to the calculation, the cumulative fatigue damage is 0.018471 after 2,000 flight hours. It indicates that although the main failure mechanisms of the turbine disc are low cycle fatigue, the total damage of the turbine disk is small which is caused by the centrifugal forces in 2,000 flight hours. It should be noted that, in the operation of the turbine disk, it also endures high thermal stresses. But because of the lack of experimental data, in this paper, only the centrifugal forces are used to estimate the life of the turbine disk. So we will do further study about the life of the turbine disk in the future.

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