**9. Remaining life expectancy of dipper components**

**Figure 10.** ANSYS contours for J-integral computations.

126 Lagrangian Mechanics

**Figure 11.** SIFs for the bottom-plate crack tip.

SIFs are computed, using plane stress conditions for all the contours, and an average value of five contours (contour 2–6) is used for further fatigue analysis. Crack lengths are increased from smaller to larger crack sizes at selected locations, and SIFs are computed for each crack size. The results are used to generate the crack-growth curves and for life expectancy of dipper components. The SIF variation curves for the bottom-plate crack tip are obtained through a least square regression and curve fitting process, Eq. (33), and are plotted in **Figure 11**. The crack is in a high stress region, and, thus, the SIF is very high. Further, the SIFs show a steep increase with crack size. It is expected that the cracks at this location will propagate rapidly:

3 2 *SIF E a E a a E* = -- -+ + + 4.98 04 8.75 02 5.14 2.96 01 (33)

Fatigue life is modeled by integrating the Paris' law [25] in Eq. (34). The equation has three input parameters (c, m, and ΔK). The two material constants are computed following standard laboratory procedures. For common materials, the values for these variables can also be found in literature [31, 32]. The "c" values are between 3 and 4, and some common values for "m" are available in literature [33]. For this research, the material constants are taken from research conducted by Yin et al. [21, 22]. They estimated the crack growth for the shovel boom cracks and measured the material constants ("c" and "m") following the ASTM standard E1820:

$$N\_f = \int\_{a\_i}^{a\_f} \frac{da}{m \left(\Delta K\right)^\varepsilon} \tag{34}$$

It is also assumed that the steel properties for the dipper and teeth are similar to that for the boom. With these parameters, Eq. (34) becomes Eq. (35). As the computations become complex, it is numerically solved using Gauss-Legendre quadrature in MATLAB. The outputs from Eq. (35) include number of cycles (*Nf* ) for a crack to propagate from an initial length (ai ) to a final length (af ). The number of cycles is converted to number of days assuming that one digging cycle of shovel is equivalent to one fatigue cycle:

$$N\_f = \int\_{a\_i}^{a\_f} \frac{da}{5.89^{-12} \left(\Lambda \mathcal{K}\right)^{3.27}}\tag{35}$$

Following the Palmgren-Miner's rule [34] for equivalent damage, the total number of fatigue cycles per day is equal to the digging cycles of the shovel per day. The total number of cycles for a shovel per day is counted using the cycle time and the operational efficiency. The shovel digging cycle is assumed to be 3 s for this research. However, a typical complete excavation cycle time for P&H 4100XPC is about 30 s. The 3 s cycle time is consistent with the numerical simulation results. Using this cycle time and assuming a 95% shovel operational efficiency, the total number of digging cycles for shovel is calculated as 2730 cycles per day. This assumption is very close to field observations [22] where researchers counted 2880 cycles per day for a cable shovel working continuously over a period of 2 weeks. For this research, a middle-ground value of 2800 cycles per day is assumed to convert the cycles to days.

The remaining useful life for the cracked components can be estimated, with knowledge of critical crack lengths for dipper material. The critical crack length is the length of the crack at which the material at the crack tip starts behaving like a plastic material, and the crack propagation becomes very rapid. It is represented as the boundary between the second and third zones for a fatigue crack. The critical lengths for metals are generally measured using laboratory fatigue toughness tests following the standard procedures. A critical length limit may also be implemented based on field operating conditions or using the crack-growth curves.

The crack-growth curve for the dipper bottom plate is shown in **Figure 12**. It is observed that the crack-propagation rates become very high after a certain crack length. A critical length1 of 100 mm is set for this crack. As illustrated in this figure, the estimated life for a 50 mm bottomplate crack is 38 days. However, once the crack grows to 75 mm, the remaining life is only 16 days for the same crack.

**Figure 12.** Critical crack length and remaining life expectancy for an initial length of 50 mm.
