Failure of a structure due to repeated cyclic loading that causes local plastic deformation ![[Pasted image 20240519023924.png|300]] $\sigma_{a} =$ stress amplitude $\Delta\sigma = \sigma _\text{max} - \sigma _\text{min} =$ stress range $\sigma_{m} =$ mean stress ### Defect-Free Approach Estimate number of cycles until crack is initiated, then assume failure is immediate #### S-N Curves Empirically determined plot of stress amplitude vs number of cycles to failure ![[Pasted image 20240519024026.png|300]] Endurance limit: some metals have a value $S=\sigma_{a}$ such that $N_{f}\to \infty$ Pseudo-endurance limit: $\sigma_{a}$ corresponding to $N_{f}=10^{7}$ #### Strain-Life Uses strain amplitude instead of stress amplitude to better model plastic deformations where failure occurs after only a few cycles $ \epsilon_{a} = \underbrace{ \frac{\sigma_{f}'}{E}(2N_{f})^{b} }_{ \text{Basquin} } + \underbrace{ \epsilon_{f}' \cdot(2N_{f})^{c} }_{ \text{Coffin-Manson }} $ High cycle fatigue — Basquins' Relation ($N_{f} > 10^{4} \text{ cycles}$) Low-cycle fatigue — Coffin-Manson relation ($N _{f}< 10^{4} \text{ cycles}$) $\sigma_{f}', \epsilon_{f}', b, c =$ material parameters ### Defect-Tolerant Approach Assume a crack of initial size $a_{i}$ exists at the most highly stressed location $a_{i}$ is set to either the largest measured crack or the minimum crack size that can be reliably measured #### Critical crack size Crack length that allows fracture to occur at maximum stress $ a_{c} =\frac{1}{\pi}\left( \frac{K_{Ic}}{Q\sigma _\text{max}} \right)^{2} $ $a_{c} =$ critical crack size ### Fatigue Crack Growth $ \Delta K_{I} = Q(\sigma _\text{max} - \sigma _\text{min})\sqrt{\pi a } $ $\Delta K_{I} =$ stress intensity range $\frac{da}{dN} =$ crack growth rate ![[Pasted image 20240519032836.png|375]] ![[Pasted image 20240519032311.png|300]] - Low $\Delta K_{I}$: Cracks don't grow beneath a threshold $\Delta K_{Ith}$ - Moderate $\Delta K_{I}$: Power law fit between $\frac{da}{dN}$ and $\Delta K_{I}$ - High $\Delta K_{I}$: Fracture occurs in a few cycles #### Paris's Law $ \frac{da}{dN} = \begin{cases} 0 & \text{if } \Delta K_{I}< \Delta K_{Ith} \\ C(\Delta K_{I})^{m} & \text{if } \Delta K_{I} > \Delta K_{Ith} \end{cases} $ $C, m =$ experimentally determined material constants So in the moderate $\Delta K_{I}$ region we have $ \frac{da}{dN} =C[Q(\sigma _\text{max} - \sigma _\text{min})\sqrt{ \pi a }]^{m} $ Integrating gives us If $m \neq 2$: $ N_{f} = \frac{2}{(m-2)C(Q\Delta\sigma\pi)^{m}}[a_{i}^{(2-m)/2}-a_{f}^{(2-m)/2}] $ If $m=2$: $ N_{f}=\frac{1}{C} \frac{1}{(Q\Delta\sigma \sqrt{ \pi })^{2}}\left[ \ln\left( \frac{a_{f}}{a_{i}} \right) \right] $