Compacted graphite iron (CGI) is an engineering material with the potential to fill the application gap between flake- and spheroidal-graphite irons thanks to its unique microstructure and competitive price. Despite its wide use and considerable past research, its complex microstructure often leads researchers to focus on models based on representative volume elements with multiple particles, frequently overlooking the impact of individual particle shapes and interactions between the neighbouring particles on crack initiation and propagation. This study focuses on the effects of graphite morphology and spacing between inclusions on the mechanical and fracture behaviours of CGI at the microscale. In this work, 2D cohesive-zone-element-based models with different graphite morphologies and spacings were developed to investigate the mechanical behaviour as well as crack initiation and propagation. ImageJ and scanning electron microscopy were used to characterise and analyse the microstructure of CGI. In simulations, both graphite particles and metallic matrix were assumed isotropic and ductile. Cohesive zone elements (CZEs) were employed in the whole domain studied. It was found that graphite morphology had a negligible effect on interface debonding but nodular inclusions can notably enhance the stiffness of the material and effectively impede the propagation of cracks within the matrix. Besides, a small distance between graphite particles accelerates the crack growth. These results can be used to design and manufacture better metal-matrix composites.
Keywords: CGI; CZE; crack growth; microstructure; graphite morphology; graphite spacing; numerical modelling
Cast iron has been one of the most broadly used materials in automotive engines since 1948, as it has good thermal and mechanical properties, and competitive price [[
In general, there are two main approaches used for modelling the mechanical behaviour of CGI: representative volume element (RVE) and unit cell [[
In the last decades, four main methods were used to simulate the fracture behaviours of materials [[
The complex microstructure of CGI is the main reason that hinders researchers from an in-depth exploration of its fracture behaviour. Although there are a variety of micromechanical models investigating compacted graphite iron, few of them focused on the fracture behaviour of CGI under tensile loading, and the authors are unaware of any studies into the effect of single inclusion on it. Cracks affect the mechanical properties, service life, and durability of CGI. Hence, their prediction is vital for the enhancement of the service life and sustainability of structures [[
The geometrical configuration of representative one- and two-inclusion models are presented in Figure 2 (grey and green parts represent graphite and matrix, respectively). The bottom faces of all nodes were fully constrained, and a uniaxial stress state was created by applying incremental axial displacement to the top face of the model. To achieve crack propagation, cohesive elements were inserted into the graphite and matrix domain. Furthermore, the layer around the inclusion (in red in Figure 2) depicts the interface between the matrix and graphite, with the cohesive thickness being zero.
As mentioned, CGI's microstructure has graphite inclusions with three main morphologies: nodular, flake, and vermicular. Hence, three single-inclusion models were developed in this study, corresponding to these basic inclusion geometries, presented by circular and ellipse domains, as shown in Figure 3. In all cases, the volume fraction was based on the results of microstructural characterisation and kept equal to 8.3% based on statistical results [[
Figure 4 shows the geometrical configuration of these cases with the graphite volume fraction fixed at 8.3%. Hence, the diameter of nodular graphite was 23 μm, the semi-major and semi-minor axes of the vermicular graphite were 22 μm and 6 μm, respectively, and those of the flake graphite were 44 μm and 3 μm. Also, the maximum and minimum distances were selected based on the statistical data [[
The mathematical formulations of CZM described in Section 2.3 were implemented into a 2D nonlinear FE model using Abaqus/Explicit software 2021.HF6. To prevent the generation of sawtooth particle boundaries that can result in stress concentration, a 3-node linear plane-strain triangle (CPE3) element was utilised to model the unit cell. After conducting a mesh convergence study, 2 µm was selected as the element size. The dynamic explicit method was used in this model. The model details are summarised in Table 1.
The implementation of PBCs requires opposite pairs of surfaces (or edges) to undergo identical deformations, thereby reducing the impact of edge effects on the boundary of a representative volume element (RVE) [[
(
(
where
PBCs require the number of nodes in the referred set to be equal to the number of appended nodes. PBCs conflict with cohesive elements with 4 nodes in each cohesive element. Hence, at the outermost layer of the model, cohesive elements were not inserted (see Figure 5). The PBCs were also used in the single-particle model.
A constitutive law assuming elastoplastic behaviour was employed for the metallic matrix and graphite particles, considering isotropy. Table 2 and Table 3 list the constitutive parameters for these two domains, respectively. Mechanical tests were used to derive the constitutive values for the metallic matrix, as described in [[
The modelling of material softening near the crack tip involved incorporating a traction–separation relationship into the cohesive elements (as illustrated in Figure 6). The cohesive element with zero thickness exhibited a linear-elastic response. Upon reaching the critical point defined by the parameters
Constitutive parameters of the cohesive model used in numerical simulations are shown in Table 4. The material parameters include damage initiation stress
At the first stage of the analysis, single-particle models were used to analyse the initiation and propagation of cracks. Figure 7 illustrates the distribution of von Mises stress for three graphite inclusions with different morphologies, utilising a model that incorporated both PBCs and CZEs. The edge effect triggered stress concentrations in the matrix area as well as near left- and right-hand tips of inclusions. This resulted in the occurrence of graphite debonding, with cracking subsequently propagating to the matrix. It is worth noting that cracks propagated not only around the graphite particle but also to the side borders, characterised by strong interaction with stress raisers in virtual neighbouring cells as a result of the application of PBCs. As a result, the crack propagation in mode I (i.e., horizontal) changed to a mixed-mode regime.
Crack evolution in each graphite inclusion was further assessed to evaluate the effect of graphite morphology on the characteristics of fracture propagation, as shown in Figure 8. The uncracked ligament of the CGI sample was calculated as follows:
(
where
Figure 8 illustrates the effect of graphite-particle morphology on uncracked ligaments. Overall, all the uncracked ligaments steadily declined with the increase in applied displacement after the crack initiation. The first crack initiated at a displacement of around 0.28 µm in the flake graphite, while the nodular graphite began at 0.35 µm. Although the cell with the flake graphite appears to develop cracks first, the vermicular graphite generated cracks at a faster rate. In fact, at a displacement of about 0.4 µm, the vermicular particle produced more transverse cracks than the flake one. Eventually, both cases converged in terms of transverse cracking. On the other hand, the nodular particle not only exhibited interface debonding later but also had fewer transverse cracks compared to the other two studied morphologies. In the case of the nodular particle, it prevented microcrack initiation and propagation, leading to enhanced ductility and toughness while the strength level remained high. The crack initiated and propagated more easily in the matrix in the case of vermicular particles. Apparently, the nodular graphite can effectively delay the appearance of matrix cracks (see Figure 9). It should be highlighted that graphite morphology had little effect on interface debonding. All interface debonding appeared in a relatively low range of external load—at around 0.28% to 0.35% strain. Hence, increasing the nodularity of graphite particles can improve material strength and stiffness and reduce the probability of crack occurrence.
The effect of boundary conditions and CZM was investigated by considering different combinations of periodic boundary conditions and cohesive zone elements. The crack evaluation in each microstructural model is analysed in this section for three modelling cases of unit cells: I—without cohesive elements and PBCs; II—with CZM; III—CZM combined with PBCs. The crack growth and the distribution of von Mises stress in CGI for different boundary conditions are presented in Figure 10 (the deformation scale factor was set to five to increase the visibility of the crack path).
Besides, the macroscopic engineering stress–strain curves of these models are depicted in Figure 11. Stages a, b, c, and d in Figure 10 represent different strain levels, as marked in the strain–stress diagram of Figure 11. The crack initiated when the cohesive elements completely degraded under tension. At stage a, the crack first initiated in Case III and then propagated in the matrix. The propagation of cracks in the unit cell with periodic boundary conditions was easier in stage b when comparing Cases II and III. This can be elucidated by examining the engineering stress–strain curves, as the macroscopic stress in Case III exhibited an earlier drop compared to Case II (see Figure 11). This drop reflected the crack formation accompanied by the local unloaded zones due to the shielding effect. The stress concentration usually appeared near opposite sides of the graphite inclusions (on the line perpendicular to the loading direction), consistent with the single-inclusion unit-cell model. The incomplete connection between the interface cracks and matrix cracks is attributed to the relatively low level of the macroscopic strain—only 1%—in these models. Case III was found to represent the debonding and fracture processes in CGI in the most realistic way, so the results below were obtained with this formulation.
The crack evolution was further analysed to understand the effect of micro-morphology and interparticle distance. The geometry of graphite inclusions was generated based on statistical data [[
The main results for the crack evolution are summarised in Figure 13. When a cohesive element fulfils the initial cracking criterion, damage evolution starts, eventually followed by complete failure. In a 2D cohesive-element model, the four nodes are numbered in such a way that nodes 1 and 4 are located at the side of the crack, while nodes 2 and 3 are located at its other side. Therefore, the crack length can be represented by the distance between the midpoints of node 1, node 2 and node 2, node 3. By examining the value of the damage field variable, it can be determined whether the element failed. If it failed, the length of the crack can be increased by the respective amount, thus providing the overall crack length of the model. All the curves in Figure 13a present a similar trend, with a stable initial part, followed by a rapid decline and a final plateau. Interface debonding was first exhibited in the two nodular graphite particles model with the minimum distance between them (N-N min), while the combination of vermicular and flake particles with the maximum distance (V-F max) exhibited the latest onset of debonding. It can be observed that the distance between particles had a minor influence on crack initiation and propagation. The comparison of cases with minimal (I) and maximal (II) distances showed that the former case resulted in accelerated crack initiation and propagation (see Figure 13a). The cells with the vermicular inclusion interacting with the flake one exhibited the best crack resistance compared to all the other combinations of particles, when the major axes of both vermicular and flake graphite particles were parallel, and aligned along the loading direction. A sharp contrast in crack length was evident between the cases N-N min (0.12 mm) and V-F max (0.26 mm).
The distance between the graphite inclusions influenced the crack initiation and propagation, as presented in previous sections. To further explore the impact of this distance on these processes, this section focuses on the analysis of nodular and flake graphite particles. These two types of graphite inclusions were selected because they represent two extreme graphite shapes. An examination of the fracture behaviours for these graphite particles at different distances can provide insights into their influence on the overall system.
The geometrical configurations of the models discussed in this section are illustrated in Figure 14. These models were used to further explore the effect of distance on crack growth. The distance between the two graphite particles was set at various distances: 37 μm; 23.66 μm; 12 μm; and 2 μm. Since PBCs were applied in all models, the choice of 37 μm (Figure 14a) created an infinite number of equidistant graphite particles, with flakes aligned with the loading direction). In the case of 23.66 μm, the distance between the particles was equal to the distance from the particles to the domain boundaries. The selection of 2 μm was based on statistical data, and the distance of 12 μm was selected as an intermediate value between 23.66 μm and 2 μm; the dimensions of the particles remained the same.
Apparently, the crack initiated quickest when the distance between the two particles was the smallest—2 μm (yellow lines in Figure 15), while it took the longest to initiate when the distance between the two graphite inclusions was the highest—37 μm (green lines). This observation demonstrates that for such morphology of the unit cell, the crack initiation process was influenced by the distance of the graphite inclusions, with closer positions leading to earlier crack initiation (see Figure 15a). Besides, the case with the distance of 2 μm exhibited the longest crack length compared to all other models. This suggests that the crack resistance of the material is enhanced in cases of larger spacings between the two neighbouring graphite inclusions.
In this part of the study, four different microstructure models with zero-thickness cohesive elements were developed to assess the influence of loading directions on crack initiation and propagation. The schematics of boundary conditions corresponding to the vertical and horizontal loading schemes are given in Figure 5. The influence of the loading direction on the deformation and damage behaviours of CGI is investigated in this section, accounting for the possibility of different loading directions in real-life applications, as demonstrated in Figure 16. In general, the initiation of cracks began predominantly at the interface of the graphite flake, where the local strain energy was higher due to edge effects. This phenomenon was especially prominent in the horizontal loading (see Figure 16a,c). However, the propagation of the cracks was much more complex for the vertical loading case (see Figure 16b). Multiple cracks initiated at the same time near the interface between the graphite particle and the matrix. After that, the cracks propagated into the graphite, subsequently merging with the interface debonding. At the end stage, the secondary crack initiated in the matrix and propagated towards the graphite particle. In the case of the maximum distance between the particles under the vertical loading (Figure 16d), the interface of the flake started debonding first, followed by the interface of the vermicular particle. After that, the crack initiated and propagated into the matrix. Besides, the flake inclusion could protect the vermicular inclusion from the interface debonding due to the shielding effect, as shown in Figure 16a,c. The formation of traction-free debonding of the flake particle (Figure 16a,c) caused the stress relief in the area in its vicinity along the direction of the loading force, hence producing a shielding effect.
To clearly present the damage behaviour as well as the evolution of crack growth in CGI, the results of numerical analysis for the uncracked ligament and the crack length are presented in Figure 17. It is evident that the horizontal loading significantly accelerated the crack initiation and propagation (see Figure 17b); however, the crack length was reduced due to the shielding effect from the flake inclusion.
In this work, the effects of the morphology of graphite particles in compacted graphite iron, the distance between them, and their orientation with regard to the tensile-loading direction were investigated. The main conclusions are as follows:
- Although graphite morphology had a minimal effect on the debonding of CGI interfaces, the presence of nodular inclusions could notably enhance the macroscopic stiffness of the material and effectively impede the propagation of cracks within the metallic matrix.
- The introduction of the periodic boundary conditions enhanced the propagation of cracks in the unit cells, especially with the particle's end situated close to the model boundaries.
- The minimal distance between the graphite particles could significantly accelerate the crack initiation and propagation when the loading is aligned with the main axis of graphite inclusions.
- The case of neighbouring vermicular and flake particles aligned along the loading direction exhibited the best crack resistance compared to all the other combinations of particles when both vermicular and flake graphite particles were orthogonal to this direction (for the same loading direction). The change in the loading direction for this case of particles to the orthogonal one could promote the crack initiation and accelerate the crack propagation rate, with the flake graphite preventing the interfacial debonding for another particle due to the shielding effect.
Graph: Figure 1 Microstructure of CGI.
Graph: Figure 2 Geometry and mesh used in numerical simulations: (a) Single inclusion; (b) two inclusions.
Graph: Figure 3 Geometry of nodular (a), vermicular (b), and flake (c) inclusions.
Graph: Figure 4 Geometrical parameters of unit cells for analysis of particle interaction: (a,c,e,g) maximum distance; (b,d,f,h) minimum distance.
Graph: Figure 5 Schematics of PBCs with CZM under vertical (a) and horizontal (b) loading regimes.
Graph: Figure 6 Traction–separation relationship (Reprinted with permission from Ref. [[
Graph: Figure 7 Evolution of von Mises stress distribution and crack growth under vertical tensile loading for different graphite morphology: (I) nodular; (II) vermicular; (III) flake.
Graph: Figure 8 Effect of graphite morphology on evolution of uncracked-ligament length with external displacement (N—nodular, V—vermicular, F—flake).
Graph: Figure 9 Main stages of crack propagation in terms of applied displacement for different graphite morphology (N—nodular, V—vermicular, F—flake).
Graph: Figure 10 Crack propagation and von Mises stress distribution for vertical tensile loading in CGI for various cases of formulation: (I) without CZM; (II) with CZM; (III) CZM-PBCs.
Graph: Figure 11 Effect of boundary conditions and CZM on macroscopic engineering stress–strain curve of CGI.
Graph: Figure 12 Crack propagation and von Mises stress distribution for vertical tensile loading in CGI for various graphite distances and particle morphologies: (I) minimum spacing; (II) maximum spacing (N—nodular, V—vermicular, F—flake).
Graph: Figure 13 Evolution of uncracked ligament (a) and crack length (b) for different morphologies of graphite inclusions in tow-particle models and distances between them.
Graph: Figure 14 Geometrical parameters of unit cells for N-F models with different distances between particles: (a) 37 μm; (b) 23.66 μm; (c) 12 μm; (d) 2 μm.
Graph: Figure 15 Effect of inter-particle spacing on evolution of uncracked ligament (a) and crack length (b) for N-F combination.
Graph: Figure 16 Crack propagation and von Mises stress distribution in CGI under horizontal (a,c) and vertical (b,d) loading regimes for two-particle V-F model with different spacings (1—crack initiation point; 2—beginning of crack propagation; 3—end of crack propagation).
Graph: Figure 17 Evolution of uncracked ligament (a) and crack length (b) for different loading directions for two-particle V-F model with different spacings (VL—vertical loading; HL—horizontal loading).
Table 1 Model details.
Software Analysis Type Domain Length Graphite (%) Mesh Size Element Type Loading Boundary Conditions Abaqus Dynamic explicit 0.1 mm 8.3 2 µm Triangular Tensile Periodic
Table 2 Constitutive parameters for graphite (room temperature) [[
Mass Density (Tonne/mm³) Young's Modulus (GPa) Poisson's Ratio 2.26 × 10−9 15.85 0.2 0.184 27.56
Table 3 Constitutive parameters for matrix (room temperature) [[
Mass Density (Tonne/mm³) Young's Modulus (GPa) Poisson's Ratio 6.8 × 10−9 150 0.25 0.209 323.95
Table 4 Constitutive parameters for cohesive element [[
Phase Damage Initiation Stress (MPa) Fracture Energy Failure Relative Displacement Density Matrix 600 0.06 2 × 10−4 8.65 × 10−12 Interface 1.75 1.75 × 10−4 2 × 10−4 5.00 × 10−12 Graphite 34 3.4 × 10−3 2 × 10−4 2.00 × 10−12
Conceptualisation, X.L., K.P.B. and V.V.S.; methodology, X.L., K.P.B. and V.V.S.; software, X.L.; validation, X.L.; formal analysis, X.L., K.P.B. and V.V.S.; investigation, X.L.; resources, X.L.; data curation, X.L.; writing—original draft preparation, X.L.; writing—review and editing, K.P.B. and V.V.S.; supervision, K.P.B. and V.V.S. All authors have read and agreed to the published version of the manuscript.
Data are contained within the article.
The authors declare no conflict of interest.
The authors are grateful to Evangelia Palkanoglou for the data on graphite in CGI.
By Xingling Luo; Konstantinos P. Baxevanakis and Vadim V. Silberschmidt
Reported by Author; Author; Author