The stabilized hysteresis loop for a material under cyclic loading is determined to have the following values:

Strain amplitude $(\frac{}{2})=0\mathrm{.}002$

Mean stress $({}_m)=-9\mathrm{.}5$ksi

Maximum stress $({}_{\text{m}\text{a}\text{x}})=30\mathrm{.}5$ksi

The strain$-$life parameters for the material are $E=30{10}^3$ksi,${}_f^{\text{'}}=133$ksi,$b=-0\mathrm{.}095,{}_f^{\text{'}}=0\mathrm{.}26,$ and $c=-0\mathrm{.}47.$ Determine the cycles to failure, $N_f,$ using each of the following three relationships that can be used to account for the mean stress effect on life:

$(1)$ Morrow: $\frac{}{2}=\frac{{}_f^{\text{'}}-{}_m}{E}(2N_f{)}^b+{}_f^{\text{'}}(2N_f{)}^c$

$(2)$ Manson$-$Halford: $\frac{}{2}=\frac{{}_f^{\text{'}}-{}_m}{E}(2N_f{)}^b+{}_f^{\text{'}}(\frac{{}_f^{\text{'}}-{}_m}{{}_f^{\text{'}}}{)}^{\frac{c}{b}}(2N_f{)}^c$

$(3)$ Smith$-$Watson$-$Topper $($SWT$)$: ${}_{\text{m}\text{a}\text{x}}\frac{}{2}=\frac{({}_f^{\text{'}}{)}^2}{E}(2N_f{)}^{2b}+{}_f^{\text{'}}{}_f^{\text{'}}(2N_f{)}^{b+c}$

Q$2.$

given below are the cyclic stress$-$strain and strain$-$life parameters for a steel:

$E=30{10}^3$ksi,$H^{\text{'}}=174\mathrm{.}6$ksi,$n^{\text{'}}=0\mathrm{.}202,{}_f^{\text{'}}=133$ksi,$b=-0\mathrm{.}095,{}_f^{\text{'}}=0\mathrm{.}26,$ and $c=-0\mathrm{.}47$

Determine the life of this material under the following strain histories. History $A$ is constant amplitude, while histories B and C have initial overloads which set up residual stresses. Use the Morrow, Manson$-$Halford, and SWT relationships for these calculations. Compare the predictions made using the three methods.