Austempered ductile iron (ADI) is composed of an ausferritic matrix with graphite nodules and has a wide range of applications because of its high mechanical strength, fatigue resistance, and wear resistance compared to other cast irons. The amount and size of the nodules can be controlled by the chemical composition and austenitizing temperature. As the nodules have lower stiffness than the matrix and can act as stress concentrators, they influence crack propagation. However, the crack propagation mechanism in ADI is not yet fully understood. In this study, we describe a numerical investigation of crack propagation in ADIs subjected to cyclic loading. The numerical model used to calculate the stress intensity factors in the material under the given conditions is built with the aid of Abaqus commercial finite element code. The crack propagation routine, which is based on the Paris law, is implemented in Python. The results of the simulation show that the presence of a nodule generates a shear load on the crack tip. Consequently, even under uniaxial tensile loading, the presence of the nodule yields a non-zero stress intensity factor in mode II, resulting in a deviation in the crack propagation path. This is the primary factor responsible for changing the crack propagation direction towards the nodule. Modifying the parameters, for example, increasing the nodule size or decreasing the distance between the nodule and crack tip, can intensify this effect. In simulations comparing two different ADIs with the same graphite fraction area, the crack in the material with more nodules reaches another nodule in a shorter propagation time (or shorter number of cycles). This suggests that the high fatigue resistance observed in ADIs may be correlated with the number of nodules intercepted by a crack and the additional energy required to nucleate new cracks. In summary, these findings contribute to a better understanding of crack propagation in ADIs, provide insights into the relationship between the presence of nodules and the fatigue resistance of these materials, and support studies that associate the increased fatigue resistance with a higher number of graphite nodules. These results can also help justify the enhanced fatigue resistance of ADIs when compared to other cast irons.

The first ductile cast irons (DCIs) were developed in the 1940s after magnesium or cerium were added to conventional cast iron [

ADI has a lower relative material cost compared with cast and forged steels [

Another factor impacting the properties of ADI is the size and quantity of graphite nodules. Given their lower mechanical strength and stiffness compared to the matrix, ADI graphite nodules function as discontinuities and, consequently, act as stress concentrators [

Fatigue crack propagation in ADI was investigated by Greno et al. [

Several authors [

In this work, we investigate the influence of nodules on crack propagation in ADI subjected to cyclic loads using a two-dimensional finite element model. The presence of nodules induces a mixed state of stress combining normal and shear stresses. To assess crack behavior under cyclic loading, including growth rate, crack path and estimation of the fatigue lifetime of the component, we implement an iterative crack growth method based on the Paris law. This study contributes by offering insights to enhance our understanding of crack propagation in ADI, with a particular focus on highlighting the presence and influence of the stress intensity factor in mode II. This aspect has received limited investigation until now. While our two-dimensional approach may initially appear simplified and may not provide quantitatively accurate values, it does offer a clear qualitative depiction of the underlying phenomena.

To simulate crack propagation in ADI and analyze the stress intensity factors as the crack grows, a Python-based computational routine was developed in the Abaqus environment. The routine iteratively calculates crack propagation considering various parameters such as geometry, material properties, initial crack conditions (size, position and angle), boundary conditions and the size and location of nodules.

The routine reads the input data and communicates it to the Abaqus finite element commercial program, which processes the model. This involves applying the specified loads, obtaining stresses, strains and displacements and calculating the stress intensity factors. At each iteration, the crack propagates, and its size is updated accordingly. The simulation continues until a stop criterion is met, such as reaching the maximum number of cycles or the crack encountering a nodule.

Some simplifications are made in the mechanical model. Firstly, the FEA model is restricted to two dimensions (2D) and represents the graphite nodules as circumferences that approximate cylinders in 3D. Additionally, the model assumes linear elastic fracture mechanics theory and neglects the effects of plasticity. Therefore, the stress analysis at the crack tip is only used to calculate the stress intensity factors. Lastly, the interface between graphite and austenite in the ADI is considered fully bonded, and no allowance is made for a gradual variation of properties between the two materials. Despite these simplifying assumptions, the qualitative results obtained are in line with those of experimental studies and offer important insights into crack propagation in ADIs, as can be seen in

The bidimensional finite element mesh is constructed in Abaqus with six-node triangular quarter-point elements around the crack tip and eight-node quadratic elements in the other regions. Quarter-point elements are specifically used to model singularities, such as crack tips, and provide a more accurate calculation of the stress intensity factors compared with commonly used elements [

The elastic properties of the ADI used in the model were obtained by Yan et al. [

Material | Young’s modulus (GPa) | Poisson’s ratio |
---|---|---|

Ausferrite | 210 | 0.290 |

Graphite nodule | 35 | 0.126 |

The number of cycles associated with crack growth size is calculated with the Paris law and an effective (or equivalent) stress intensity factor (_{eff}) proposed by Tanaka [_{i}, based on the value of the equivalent stress intensity factor range, Δ_{eff}. Therefore, the number of cycles for each iteration, Δ_{i}, is given by

The values used for ^{−11} and 2.74, respectively).

The total number of cycles, _{i}, for all

To calculate the direction of crack propagation,

_{II}

The routine ends when one of the stopping criteria is reached. This may be a maximum number of iterations, an admissible value of _{eff}, or if the crack intercepts a nodule.

In this case study we adopted a geometry based on that used by Silva et al. [

Three subcases are studied, each with a different nodule diameter (_{0}) is 1 mm, as shown in

The results for crack propagation are shown in

The stress intensity factor range variation as the crack propagates is also examined. _{I}) and mode II (shearing-Δ_{II}) as a function of the horizontal coordinate (_{II} is at least 10 times lower than Δ_{I}. Consequently, the equivalent stress intensity factor range (_{II} appeared after several iterations, resulting in deviation of the crack away from the nodule.

The relationship between crack length (

In another subcase, of which, however, for the sake of brevity, only the conclusion is reported here, the variation in the distance

Once again, using the geometry proposed by Silva et al. [

The results of crack propagation for all the subcases studied are presented in

The equivalent stress intensity factor range along the horizontal coordinate of the crack path is illustrated in _{II} remains close to zero throughout all iterations.

Finally, the

In this case study, a representative area measuring 1 mm × 1 mm in ADI is subjected to cyclic tensile loading in the vertical direction ranging from 0 to 60 MPa. To mitigate potential edge effects caused by the applied load, a central area measuring 0.8 mm × 0.8 mm is defined within which the graphite nodules are positioned. The characteristics of the materials, as described by Gans et al. [

Graphite fraction area (%) | Nodule count (nodules/mm²) | Nodule diameter (μm) | |
---|---|---|---|

ADI I | 13 | 196 | 29.1 |

ADI II | 13 | 532 | 17.6 |

The overall element size for these simulations is set at 0.025 mm. However, mesh refinement is applied in the region of the nodules, as depicted in the detail shown in

The equivalent stress intensity factor range is determined as the crack propagates, as shown in _{eff} at certain times despite the increase in crack size. This could potentially be attributed to the increasing distance between crack tip and nodule.

When the equivalent stress intensity factor ranges for ADI I and ADI II are compared, a notable difference in the initial values of this factor becomes apparent. At the onset of the crack, the value for ADI I is approximately 12.2 MPa m^{1/2}, whereas for ADI II, it is slightly lower at around 10 MPa m^{1/2}.

Examination of the

Furthermore, the results demonstrate that cracks in the material with more nodules (ADI II) tend to propagate shorter distances before intercepting a new nodule along their path, as illustrated in

It is worth recalling that the cracks tend to propagate towards the nodules and that the magnitude of this attraction depends on nodule size and the distance to the nodule. This characteristic improves the fatigue resistance of ADIs compared with that of other cast irons.

A numerical and computational approach was implemented in conjunction with commercial finite element code to study crack propagation in ADI.

In the cases studied to demonstrate the influence of graphite nodules on the stress intensity factor range and crack propagation, the proximity of the crack tip to the nodule and the size of the nodule directly affected the value of the stress intensity factor range. Two stress intensity factors were analyzed: mode I (opening) and mode II (in-plane shear). Under tensile load, the presence of the nodule resulted in a non-zero stress intensity factor in mode II, leading to a perturbation in the crack propagation path. Consequently, the presence of nodules primarily influences the direction of crack propagation and can accelerate propagation because of the increased stress intensity factor range. However, these factors alone do not fully explain the high fatigue resistance of these materials observed in experimental tests.

In the last case study, where the nodule distribution and the mechanical properties of the material are those of an ADI structure, the crack tended to intercept a nodule after fewer cycles in materials with smaller diameter nodules. As a result, for the same area fraction of graphite, a crack propagating in a material with smaller nodules tends to intercept more nodules than a crack in an ADI with larger nodules. This suggests a potential correlation between the protective effect attributed to nodules and the number of these that are intercepted, supporting studies that associate increased fatigue resistance with a higher number of graphite nodules.

To summarize, the study yielded the following findings:

i) Influence of Graphite Nodules: The proximity of a crack tip to graphite nodules and the nodules’ size directly affect the stress intensity factor range, influencing crack propagation;

ii) Stress Intensity Factors: Two stress intensity factors, mode I (opening) and mode II (in-plane shear), were examined. Under tension, nodules introduced a non-zero mode II stress intensity factor, altering crack propagation;

iii) Crack Propagation Direction: Graphite nodules primarily influence crack propagation direction and can accelerate it by increasing the stress intensity factor range. However, these factors alone do not fully explain the high fatigue resistance observed in experiments;

iv) Nodule Distribution and Fatigue Resistance: Materials with smaller diameter nodules, but a higher nodule count, exhibited a tendency for cracks to intercept nodules more rapidly. These nodules may act as barriers, slowing down crack propagation. This observation suggests a correlation between nodule count and fatigue resistance, reinforcing existing studies linking increased fatigue resistance to more graphite nodules.

These findings contribute to a deeper understanding of how graphite nodules influence crack propagation and fatigue resistance in ADI. They suggest that the presence of nodules can significantly influence the direction and speed of crack propagation, which may have implications for the design and use of these materials in various applications. Further research in this area could explore how to optimize the distribution and size of nodules to enhance the fatigue resistance of such materials.

The authors would like to thank the Structural Mechanics Laboratory at the Federal University of Technology–Paraná (Curitiba, Brazil) for providing the computing resources for the development of this study.

The authors received no specific funding for this study.

The authors confirm contribution to the paper as follows: study conception and design: M. A. Luersen, C. H. Silva; numerical implementation: G. V. França; data collection: G. V. França, R. L. Assumpção; analysis and interpretation of results: G. V. França, M. A. Luersen, C. H. Silva; draft manuscript preparation: R. L. Assumpção, M. A. Luersen. All authors reviewed the results and approved the final version of the manuscript.

The comprehensive dataset resulting from the numerical simulations, whose compiled results are presented in this study, is available upon request from the corresponding author (

The authors declare that they have no conflicts of interest to report regarding the present study.