**8. Fatigue failure modeling of dipper components**

Rope shovel excavation is cyclic in nature. The stresses on the front-end assembly vary continuously during the duty cycle [24]. This variation, combined with the material flaws, can initiate fatigue cracks in shovel components. Environmental factors (e.g., freezing temperatures and corrosive materials) affect metal toughness. Fatigue crack may grow rapidly to undesirable lengths under cyclic loading conditions. There are three modes for metal fatigue failure: (i) mode-I (crack opening), (ii) mode-II (in-plane shear or crack opening), and mode-III (out-of-plane shear or crack twist). Metal failure can also be a result of mixed-mode fatigue. Mode-I fatigue research has dominated the fatigue analysis and life-expectancy field. Three common fatigue failure analysis approaches are typically used, including the stress life, strain life, and fracture mechanics. Each approach has its own application with overlapping boundaries. The fracture mechanics approach is used to estimate a crack's propagation life. For this approach, the initial crack lengths are either assumed or known (welds, known defects, porosities, and cracks found during nondestructive testing). Fracture mechanics principles and theories are applied to estimate crack-propagation rates and, thus, crack-propagation lives.

A typical crack-growth curve has three distinct regions: crack initiation, crack propagation, and rapid increase in crack growth leading to failure. Fatigue life may, however, occur for a longer period of time during the crack-propagation phase as the majority of crack time is spent during this phase. A number of models are available to predict the crack-propagation phase (the middle region on the curve). Paris' law [25], defined by Eq. (32), is the most commonly used method to estimate the crack propagation. The slope of the linear region of the fatigue curve defines the crack-growth rate with every cycle. The material constants (C, m) can be found for different metals in literature or computed using standard tests (ASTM E647):

$$\frac{da}{dN} = C\left(\Delta K\right) \tag{32}$$

K is the stress intensity factor (SIF). According to the linear elastic fracture mechanics (LEFM) theory, computation of stress intensity factor (SIF) at the crack tip is necessary to predict the crack growth. SIFs for many simple geometries and loading situations are available in published literature [26–29]. For complex geometries and stress loading conditions, numerical methods are used to compute the reliable SIF values. The most common methods for calculating SIF are the J-integral and energy release rate. Finite element techniques evaluate the SIF using the energy release rate method. Many commercial software packages, such as ANSYS Workbench (R15), have options for calculating these parameters. In these techniques, energy release is estimated around the crack tip nodes in close loops, in the form of contours. For 2D cases, the node at the crack tip forms the first contour, while for 3D cases, all nodes forming the crack front determine the first contour. The shape, length, and depth of a crack determine the crack life at a specific location.

For fracture modeling of the dipper, representative cracks can be induced in the virtual shovel. **Figure 8** illustrates the location and geometry of an elliptical crack in the dipper's bottom plate. The crack is induced in the high stress region and is along the stress lines. This representative crack is 3 in. long and is 1 in. deep. A localized reference system defines the geometry of this crack. The crack plane lies on this XZ plane, while the width of crack is along the Y direction of this plane. The crack grows along the X and Z directions. A number of parameters control the crack failure modeling, including crack geometry, shape, orientation, and stress environment. The crack definition is explained in **Figure 9**. This crack front is divided into a number of segments, and six circular contours (also divided into segments) are generated around the crack front. All these divisions represent the node locations for the finite element model. The J-integral values are computed for every contour along the crack front. These values are used to compute the SIFs at the crack tip and for life estimations.

The crack size is the most critical aspect of fatigue crack modeling and dipper life estimation. The SIFs are highly dependent upon the crack size (length and depth). The relationship between SIF and crack length is nonlinear. Estimating the variation of SIFs with size is the most important and critical aspect for life estimation. The experimentation is performed for multiple size cracks. ANSYS R15 software is used to compute the J-integrals around the predefined crack tips. As explained in **Figure 9**, contours are generated around the crack front to represent the closed paths for J-integral. There are six contours around every modeled crack as illustrated in **Figure 10**. A very fine mesh size is generated around the crack tip, and J-integrals are computed for all these contours. The first contour is very close to the tip and may represent erroneous results. Therefore, the J-integral values for the first contour are ignored to ensure accuracy [30].

**Figure 8.** Elliptical crack at the side of the dipper bottom plate.

A typical crack-growth curve has three distinct regions: crack initiation, crack propagation, and rapid increase in crack growth leading to failure. Fatigue life may, however, occur for a longer period of time during the crack-propagation phase as the majority of crack time is spent during this phase. A number of models are available to predict the crack-propagation phase (the middle region on the curve). Paris' law [25], defined by Eq. (32), is the most commonly used method to estimate the crack propagation. The slope of the linear region of the fatigue curve defines the crack-growth rate with every cycle. The material constants (C, m) can be found for different metals in literature or computed using standard tests (ASTM E647):

( ) *da C K*

K is the stress intensity factor (SIF). According to the linear elastic fracture mechanics (LEFM) theory, computation of stress intensity factor (SIF) at the crack tip is necessary to predict the crack growth. SIFs for many simple geometries and loading situations are available in published literature [26–29]. For complex geometries and stress loading conditions, numerical methods are used to compute the reliable SIF values. The most common methods for calculating SIF are the J-integral and energy release rate. Finite element techniques evaluate the SIF using the energy release rate method. Many commercial software packages, such as ANSYS Workbench (R15), have options for calculating these parameters. In these techniques, energy release is estimated around the crack tip nodes in close loops, in the form of contours. For 2D cases, the node at the crack tip forms the first contour, while for 3D cases, all nodes forming the crack front determine the first contour. The shape, length, and depth of a crack determine

For fracture modeling of the dipper, representative cracks can be induced in the virtual shovel. **Figure 8** illustrates the location and geometry of an elliptical crack in the dipper's bottom plate. The crack is induced in the high stress region and is along the stress lines. This representative crack is 3 in. long and is 1 in. deep. A localized reference system defines the geometry of this crack. The crack plane lies on this XZ plane, while the width of crack is along the Y direction of this plane. The crack grows along the X and Z directions. A number of parameters control the crack failure modeling, including crack geometry, shape, orientation, and stress environment. The crack definition is explained in **Figure 9**. This crack front is divided into a number of segments, and six circular contours (also divided into segments) are generated around the crack front. All these divisions represent the node locations for the finite element model. The J-integral values are computed for every contour along the crack front. These values are used

The crack size is the most critical aspect of fatigue crack modeling and dipper life estimation. The SIFs are highly dependent upon the crack size (length and depth). The relationship between SIF and crack length is nonlinear. Estimating the variation of SIFs with size is the most important and critical aspect for life estimation. The experimentation is performed for multiple size cracks. ANSYS R15 software is used to compute the J-integrals around the predefined crack tips. As explained in **Figure 9**, contours are generated around the crack front to represent

= D (32)

*dN*

the crack life at a specific location.

124 Lagrangian Mechanics

to compute the SIFs at the crack tip and for life estimations.

**Figure 9.** Crack definition in ANSYS for J-integral computations.

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

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)

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