# MSF Uncertainty

A propagation of uncertainty method described by Coleman et al was used to propagate the uncertainty of MSF model constants to the life estimate of the MSF routine. This process is best summarized by figure 1.

## MSF Model Equations

The MultiStage Fatigue (MSF) model separates fatigue life into three distinct regimes: incubation, microstructurally small crack (MSC), and long crack (LC). The incubation and MSC regimes were the only regimes considered in this study. $N_T = N_{INC} + N_{MSC} + N_{LC}$
(1)

Crack incubation involves nucleation plus small crack growth around of an inclusion in the order of (1/2)D (initial crack contribution to microstructurally small crack growth), where D is the inclusion particle diameter or pore size. Small crack growth (MSC) comprises crack propagation of microstructural cracks with a length of ai < a < k MS, with MS defined as a characteristic length scale of interaction with microstructural (MS) features, and k as a multiplier in the range between one and three. The physically small crack (PSC) range consists of propagation of a crack in the interval of k MS < a < O (10 MS). Depending on the microstructural inclusion morphology and texture of the matrix, the PSC regime may extend to 300–800 µm. Typically, since the math for both the MSC and PSC regime are similar, the PSC regime is folded into the MSC regime.
Crack growth in the MSC/PSC stage is governed by the range of the crack tip displacement, $\Delta CTDth$, which is proportional to the crack length, and the nth power of the applied stress amplitude, $\left( \sigma_a\right)^n$, in the high cycle fatigue (HCF) regime and to the macroscopic plastic shear strain range in the low cycle fatigue (LCF), and it is given by equation (3). Where χ is a material constant and $\Delta CTDth$ is the threshold for crack tip displacement. The crack tip displacement is a function of the remote loading (equation (8)), and $C_I$, $C_{II}$ , and $\zeta$ are material dependent parameters which capture the microstructural effects on MSC growth.

### Incubation $C_{inc} N_{inc}^{\alpha} = \beta$
(2) $\beta = \frac{\Delta \gamma_{max}^{p*}}{2} = Y [\epsilon_a - \epsilon_{th}]^q$ $\frac{l}{D}$ < $\eta_{lim}$
(3) $\beta = \frac{\Delta \gamma_{max}^{p*}}{2} = Y (1+\zeta z) [\epsilon_a - \epsilon_{th}]^q$ $\frac{l}{D}$ > $\eta_{lim}$
(4) $\frac{l}{D} = \eta_{lim} \frac{\epsilon_a - \epsilon_{th}}{\epsilon_{per} - \epsilon_{th}}$ $\frac{l}{D}$ $\eta_{lim}$
(5) $\frac{l}{D} = 1 - (1-\eta_{lim})\left(\frac{\epsilon_{per}}{\epsilon_a}\right)^r$ $\frac{l}{D}$ $\eta_{lim}$
(6)

### Microstructurally Small Crack $\left(\frac{da}{dN}\right)_{MSC}$ = $\chi \left( \Delta CTD - \Delta CTD_{th}\right)$ $a_i$ = $0.625 D$
(7) $\Delta CTD$ = $C_{II} \left(\frac{GS}{GS_0}\right)^\omega \left(\frac{GO}{GO_0}\right)^\xi \left(\frac{U \Delta \sigma}{S_{ut}}\right)^\zeta a_i +C_{I} \left(\frac{GS}{GS_0}\right)^{\omega^\prime} \left(\frac{GO}{GO_0}\right)^{\xi^\prime}\left(\frac{\Delta \gamma_{max}^{p}}{2}\right)^2$