 Original Article
 Open access
 Published:
Closure Effect of I + II Mixedmode Crack for EA4T Axle Steel
Chinese Journal of Mechanical Engineering volume 37, Article number: 81 (2024)
Abstract
The crackclosure effect is a crucial factor that affects the crack growth rate and should be considered in simulation analysis and testing. A mixedmode I + II loading fatigue crack growth test was performed using EA4T axle steel specimens. The variation of the plasticinduced crack closure (PICC) effect and the roughnessinduced crack closure (RICC) effect during crack deflection in the mixedmode is examined in this study. The results show that the load perpendicular to the crack propagation direction hinders the slip effect caused by the load parallel to the crack propagation direction under mixedmode loading, and the crack deflection is an intuitive manifestation of the interaction between the PICC and RICC. The proportion of the R_{A} value change on the crack side caused by contact friction was reduced by the interaction between PICC and RICC. The roughness of the crack surface before and after the crack deflection is different, and the spatial torsion crack surface is formed during the crack propagation process. With the increase of the crack length, the roughness of the fracture surface increases. During the crack deflection process, the PICC value fluctuates around 0.2, and the RICC value is increased to 0.15.
1 Introduction
Fatigue damage and fracture are inevitable phenomena in the service life of engineering structures and materials [1]. When fatigue damage accumulates to a certain level, local plastic deformation occurs at the location of the highest stress region. Plastic deformation is one of the significant reasons for the occurrence of crack closure effects, and the existence of crack closure effects is conducive to restraining the opening of the crack. The crack closure effect encompasses factors such as plasticity, roughness, the presence of oxides, the small crack phenomenon, overload, and compression overload [2,3,4]. A reasonable understanding and application of the closure effect contribute to researchers in engineering structure design and lifespan assessment.
Cracks can be divided into three types based on linear elastic fracture mechanics (LEFM): opening (type I), sliding (type II), and tearing types (type III), the different operating conditions in actual service lead to fatigue cracks in either isolated or mixed forms. To provide a reasonable description or assessment of the process of crack propagation, researchers have conducted longterm and systematic theoretical and experimental studies [5,6,7,8,9], including methods related to the Elber crack closure model, the incremental (or partial) crack closure model, and the zero crack propagation load method. With the help of relevant experimental methods, the relationship between the crack growth rate determined by farfield loading conditions, crack geometry and the stress intensity factor range (ΔK) under smallscale yield conditions is established. The contribution of oxidation or roughness to crack closure is shown to be smaller than the contribution of the slope change of the loaddisplacement curve. In addition, the plastic strainrelated mechanism at the crack tip can explain most of the test results, but some test results cannot be explained according to the plasticinduced crack closure (PICC) effect, and even contradict the crack closure method, other factors may play a role, such as the degradation of material mechanical properties (hardening and softening) during creepfatigue process, and interaction between micro defects and microstructure (metal inclusions) [10, 11].
Research on the crack closure effect mainly focuses on PICC, roughnessinduced crack closure (RICC) and oxideinduced crack closure (OICC) effects. Under the condition of uniaxial loading, materials are prone to PICC. Under mixedmode loading conditions, crack deflection leads to mutual contact between fracture surfaces, resulting in the RICC effect. In humid environments, some materials are subject to OICC effects, as fatigue crack surfaces come into contact with the air, initiating oxidationreduction reactions that accumulate oxidationreduction products on the fracture surfaces. Peters and Yamaguchi et al. [12, 13] presented a noncontact fullfield optical measurement method to investigate the PICC effects. McClung [14] used an elasticplastic finite element method (FEM), which is considered PICC, to simulate fatigue crack propagation (FCG), the traditional IrwinRice estimate for crack tip plastic zone size in plane stress is found to be generally consistent with the FEM results. Christopher et al. [15, 16] established a new cracktip stress field model (Christopher–James–Patterson, CJP model) based on the traditional Westergaard’s model, the results show that the CJP model can better explain the PICC effect. Yang et al. [17] used the LevenbergMarguardt (LM) iterative method to solve for the crack tip position and apply it to the CJP crack tip field model. Li et al. [18] considered the effect of the dislocation field on the plastic zone of the crack tip and hence on the elastic field by introducing a plastic flow factor ρ, the plastic flow correction provides a more accurate crack tip field model and improves the CJP crack growth relationship. Yang et al. [19] applied the CJP model to the FCG of U71MnG rail steel standard compact tensile (CT) specimens, the results point out that the CJP model can describe the contour of the plastic zone, and can characterize the influence of crack tip compatible stress and the crack closure effect on crack propagation under different stress ratios. Paysan et al. [20] applied an elastic–plastic 3D FEM to study the evolution of plasticity during FCG. The result can reproduce the major features of fatigue crack closure, the 3D and loaddependent evolution of PICC. Gonzales et al. [21] provided an experimental methodology to determine the reversed plastic zone size at the crack tip under cyclic loading through highresolution digital image correlation (DIC) measurements.
Although modern Xray microtomography 3D technology can achieve submicron resolution, cracks cannot be regarded as simple planar discontinuities in isotropic media, there is no direct method to measure the actual development of fullthickness crack closure, so the FEM is generally used in the study of RICC or OICC. Chen et al. [22] introduced elastic wedges into elastic cracks to simulate RICC or OICC effects and applied LEFM to solve the relationship between externally applied stress and crack opening displacement. Field et al. [23] conducted a general fractographic analysis of the fatigue fracture surface to understand what correlation there may be between surface roughness and the FCG rate. Gonzalez et al. [24] reviewed and discussed the key points on the RICC that are considered to affect crack tip shielding, the dislocationbased models have indicated that the crack tip shielding effect from a single asperity is relatively minor. Pippan et al. [25] explored the RICC effect from asymmetric dislocations and measured the local crystal orientation of fatigue cracks in the nearthreshold region of Cu and Fe. Crapps et al. [26] studied the crackclosure problem under the combined action of PICC and RICC by introducing a serrated crack wake and changing the roughness and geometric shape period of the crack surface. The results show that the average crack closure degree increased with an increase in the roughness of the crack surface, whereas the geometric shape period has little effect on RICC. Pokluda et al. [27, 28] characterized the size of the crackclosure force component caused by roughness, the results are superior to the CJP model in evaluating the closure force component caused by roughness.
The FCG behavior in the mixedmode has been further investigated since the introduction of compact tensile shear (CTS) specimens and relative loading devices by Richard [29]. Previous studies have reported that the effect of the L/r_{p} ratio (L and r_{p} represent the crack roughness length scale and plastic zone size, respectively) during a single peak overload may be similar to the RICC under constant amplitude (CA) loading [30, 31]. Parry et al. [32] studied the opening behavior of the crack tip section and the residual shear deformation of the asperity in the crack wake and described the crack deflection under CA loading from the perspective of micromechanics. Singh et al. [33] discussed and analyzed the competitive relationship between RICC and the PICC process after an overload. The material exhibited different degrees of deflection in mixedmode loading conditions [34]. Under pure modeI loading, FCG is not affected by RICC. However, under mixedmode loading, it is affected by the mutual contact and interaction forces of the fracture surface. Previous work [35] showed that the crack propagation behavior can be described more effectively because the solution includes PICC and RICC effects, compared with the traditional mixed model crack growth models with higher fitting correlation coefficients. The relevant studies of OICC are not described in this paper, because no redox reaction was found in the previous experiments, and no oxide accumulation was observed on the fracture surface.
The above studies are mainly devoted to exploring or explaining the mechanism of the PICC or RICC effect while proposing new means or using FEM and advanced technologies et al. to simulate the crack propagation process, and most of the studies focus on uniaxial loading conditions. Few literatures have described the interaction between PICC and RICC during crack deflection in the mixedmode so far, so it is necessary to further study the crack closure effect in the process of crack deflection. In this study, EA4T axle steel was used to explore the variation of PICC and RICC under mixedmode I + II loading conditions and the interaction between the two crack closure effects during crack deflection propagation. The DIC technique and the replica method were employed to record the crack tip displacement field, deflection angle and crack length, the change of SIFs during crack deflection was simulated by FEM.
2 Calculation of the Crack Closure
2.1 PICC Calculation
The CJP model established by Christopher et al. [36] considers the influence of compressive stress between crack surfaces and shear stress between crack sides, as shown in Figure 1, it shows the idealized diagram of the elasticplastic interface stress at the crack tip during fatigue crack propagation (F_{A} is mode–I force; F_{B} is an additional force caused by mode–II loads; F_{T} is a force caused by positive Tstress; F_{S} is interface shear force in the elastic–plastic interval; F_{C} and F_{P} produce a shielding effect together; F_{C} is the contact force between the sides of the crack, and F_{P} is the force generated by the compatibility constraint of the plastically deformed material.). The crack tip plastic zone gradually transitions from the crack front to the crack wake with the crack propagation, a plastic inclusion is formed at the crack tip and a plastic wedge is formed between the crack sides to cause the PICC effect. The two effective crack propagation driving factors are used in the model, as shown in Eq. (1).
where K_{F} is the SIF that drives crack growth, and K_{R} is the SIF that acts to retard crack growth. K_{F,max}, K_{R,max}, K_{F,min}, K_{R,min} at K_{F} and K_{R} under the maximum and minimum stress states.
The CJP model is defined as [36]:
The elastic stress fields near the crack tip are given by:
The K_{F} and K_{R} can be obtained in Eq. (4), which represent the driving SIF and the delaying SIF respectively.
The fiveparameter can be solved from displacement fields:
where u_{x} and u_{y} are the horizontal and vertical displacements respectively, μ = E/2(1+v); k = 3−4v (plane strain) or k = (3−v)/(1+v) (plane stress); u_{x} and u_{y} are shown explicitly below with the assumption D+E = 0. After obtaining the crack tip displacement field, the equivalent stress value is solved according to the generalized Hooke’s law. The value is compared with the yield strength value of the material to distinguish which stress values of the pixels obtained by DIC technology are in the linear elastic range. The position information of this part of the points is stored in the document. Substituting the displacement values of all points in the document into Eqs. (6) and (7), a linear equation set can be formed. The linear fitting method can be used to solve the parameters in the CJP model, and then the parameters such as K_{F} and K_{R} can be obtained.
The crack closure ratio proposed by Pokluda [37] is
where K_{PICC} is the SIF value calculated after considering PICC; K_{max} is the maximum SIF value; c=b/10(π)^{0.5}B; b is the distance between the upper and lower grains of the material, B ≈ 10^{10} m; c is a dimensionless constant independent of the material, c ≈ 0.1. It should be noted that the evaluation of PICC in the Pokluda model is relatively straightforward, requiring only certain material parameters for calculation. The listing of this expression is for analysis, comparison, and validation of the results obtained in this experiment.
In related studies, the load corresponding to crack opening is denoted as P_{op}, and U is used to represent the degree of crack closure [38, 39], as shown in Eq. (9). In this study, to quantify the PICC component, the P_{op}/P_{max} value is employed to represent the crack closure component. Based on the forcedisplacement data recorded during the experiment, combined with the paraboliclinear method [40], the crack opening force at each stage of crack deflection can be determined.
where U is a dimensionless constant; P_{max} and P_{min} are the maximum and minimum load, respectively.
2.2 RICC Calculation
The asymmetry of dislocations and roughness of the fracture surface led to different shear displacements on both sides of the crack, resulting in the RICC effect. Assuming that the roughness geometry is simple, as shown in Figure 2, the dotted line indicates the crack propagation direction. The calculation expression is given in Eq. (10) [27].
where K_{RICC} is the RICC SIF; K_{max} is the maximum SIF, and R_{A} is the roughness of the fracture. η is a statistical parameter, it obeys the Weibull distribution. S_{Rc}∈(0.2,1) is a fitting parameter, and a value of 0.5 is generally acceptable [41]. d_{m} is the characteristic microstructure distance, and r_{p} is the static plastic zone size [28].
A theoretical analytical solution of the CJP model in mixedmode is presented in the previous work, as shown in Eq. (12). The crack deflection propagation behavior can be described more effectively by considering the influence of PICC and RICC [35]. The SIF value in the process of crack deflection is calculated by Eq. (12) in this study.
where ∆K_{PR} is the ∆K_{eff} value with the consideration of PICC and RICC effects, and x is the linear scaling factor. R_{A} value can be obtained by measuring the fracture area after the testing.
3 Experimental Procedure
The sample material used in this study was EA4T axle steel. The specimens were subjected to a mixedmode I + II loading FCG test, which was performed using an electronic tensiletorsion testing machine (INSTRON E10BMT). For the formal test, P_{max} = 4 kN, R = 0.1, and f = 25 Hz, and CA loading was carried out in the directions of 30°, 45°, and 60°. During the test, a replica method was adopted to record the crack propagation trajectory, so the crack tip location and angle can be defined after a certain number of cycles. The crack propagation length was measured after a specific number of cycles, and loaded at a low speed (the maximum load remained unchanged). The sampling frequency was adjusted to 1000 Hz to collect sufficient loaddisplacement data. A DIC device, i.e., Revealer 2DDIC was used to obtain the displacement field of the FCG and calculate the change in the SIF value at the crack tip during crack deflection. The resolution of DIC equipment is 0.00638 pixels/mm. An industrial microscope with a resolution of 10 μm was used to observe and record crack propagation paths in realtime.
The fatigue crack propagation process in the mixedmode can be divided into two stages. In the first stage, under the interaction of shear and tensile stresses perpendicular to the prefabricated crack surface, cracks initiated and propagated in the surface slip band under shear stress. In the second stage, crack propagation was approximately perpendicular to the principal tensile stress, as shown in Figure 3a. As the crack propagated from the prefabricated crack position to the first stage of crack propagation, the slip system formed a certain angle with the crack propagation direction, resulting in crack deflection. The mixedmode loading device and crack deflection are shown in Figure 3a, the fixture designed by Richard was employed for loading [29]. The dimensions of the specimen are shown in Figure 3b, and the field test device is shown in Figure 4. The chemical compositions and basic mechanical properties of the materials are listed in Tables 1 and 2, respectively, where Et. is element, Wt. is weight (%). E is elastic modulus (GPa), v is poisson’s ratio, σ_{s} is yield strength , σ_{b} is tensile strength (MPa), δ is elongation (%), Ψ is reduction of area (%) [42]. It is worth noting that the degradation phenomenon of material mechanical properties is not considered in this study, mechanical properties, such as yield strength, ultimate tensile strength and elongation are key factors in damage evaluation. Sun et al.’s work showed that the degradation of mechanical properties is influenced by cyclic softening/hardening characteristics of materials, and a comprehensive characterization method is proposed using a tensile plastic strain energy density parameter [43].
4 Results and Discussions
4.1 Simulation Results
The research shows that the FEM is used to simulate the crack propagation process, which can reflect the change trend and range of each parameter to a certain extent. In Singh et al’s work [33], a single crack deflection angle of θ = 45° and deflection lengths between L = 18.75 and L = 225 μm are considered, fatigue surface roughness is idealized as a simple regular zigzag geometry for RICC studies. This modelling method is convenient for researchers to explore the changing trend of the RICC effect, but the crack deflection angle is fixed, while the crack deflection angle of the material or structure in service is changed in real time. In Nittur et al's work [44], only the right half of the specimen is modelled due to symmetries of the loading and geometry and appropriate boundary conditions are imposed along the vertical symmetry line. The model consists of 16 layers of 8 noded isoparametric ‘C3D8R’ brick elements and has finer mesh towards the free surface and coarser mesh towards the midplane. The crack propagation is accomplished by releasing the constraints on the current crack front nodes at the end of the load cycle. Since the crack deflection occurs during the mixedmode crack propagation process, the submodel technique was not used in this study. In Antunes et al’s work [45], the finite element mesh is refined at the crack front to model the severe plastic deformation gradients and enlarged at remote positions to reduce the numerical effort. In this study, the model crack propagation path element was also preprocessed to reasonably simulate the crack propagation process.
In this study, a commercial finite element software Franc3D 8.0 was used to simulate FCG in mixedmode. The element type was selected as an 8node isoparametric plane stress element (CPS8 element). According to Richard distribution equation [46], the CTS specimens were subjected to force and constraint. It is worth noting that in the study of elasticplastic stress (strain), simulation results are influenced by the loading path when plastic stress/deformation is involved. For instance, if tensile loading is applied before shear loading or shear loading is applied before tensile loading, the results can differ significantly [47, 48]. The above problems are avoided by using the Richard distribution equation for loading and constraint setting.
CTS modelling was established in the finite element software of HyperWorks 21.0. The cell grid was divided into three parts based on its size: the first part was a circular area with a diameter of 14 mm centred on the crack tip. Based on the sensitivity analysis results of Wang et al. [34], the cell grid size of this part was 0.05 mm. The grid size increases from the inside to the outside, and the other two parts of the grid size are 0.1 mm and 0.2 mm, respectively, as shown in Figure 5. Material parameters were set according to the values provided in Table 2, and Paris constants were determined based on experimental fitting results, with c = 4.92×10^{9} and m = 2.1676 [35]. A 2 mm elliptical crack at the initial crack tip throughout the entire thickness of the CTS specimen was inserted. Subsequently, a combination of the extended finite element method (XFEM) was used to simulate mixedmode crack propagation. This study calculated the SIF based on the displacement extrapolation method [49], as shown in Eq. (13).
where E is Young's modulus, r is the distance from the crack tip, u_{1} and u_{2} are the opening displacement and sliding displacement of the two relative points above and below the crack.
The maximum tangential stress (MTS) [50] and strain energy density factor (S) criteria [51] were sat as the mixedmode crack growth criterion in this paper, the theoretical expressions of crack propagation direction θ of MTS criterion and S criterion are shown as follows:
MTS criterion:
S criterion:
where S is the strain energy density coefficient, S = a_{11}K_{I}^{2} + 2a_{12}K_{I}K_{II} + a_{22}K_{II}^{2}, the coefficients a_{11} = (1+v/8πE)(1+cosθ)(k–cosθ), a_{12} = (1+v/8πE)sinθ[2cosθ–(k–1)], a_{22} = (1+v/8πE)[(k+1)(1–cosθ)+(1+cosθ)(3cosθ–1)]. E and v are Young’s modulus and Poisson’s ratio, respectively, k = (3–v)/(1+v) under plane stress condition and k = 3 – 4v under plane strain condition.
As there was a small difference between the calculated values of the two criteria in previous work, only the MTS results were plotted here, as shown in Figure 6. The variation in K_{I} and K_{II} values with crack length is depicted in the figure, K_{I} and K_{II} are the SIF values of modeI and modeII loading conditions, respectively. In Figure 6a, with increased crack propagation length, the K_{I} value shows an overall upward trend, consistent with the results obtained using the Westergaard method. The curve in Figure 6a showed an alternating trend of increase and decrease during ascent, which was related to crack deflection. The a–b segment in Figure 6b is defined as the first stage and the b–c segment is defined as the second stage, the dotted line is the fitted line of the simulation data, which has no clear physical meaning for the following analysis. The K_{II} value gradually increased with increasing crack length in the first stage and decreased in the second stage. The growth trend of the K_{II} value in the first stage was similar to that of the Paris crack propagation curve. As the crack length increased, the contact area of the crack side also increased. In combination with the crack deflection diagram in Figure 3a, the variation trend of K_{II} value is explained. As shown in Figure 7, P_{I} and P_{II} are the loads corresponding to the directions perpendicular and parallel to the crack propagation, the ac in Figure 7 shows the load projection when the mixed mode's loading angle is set to 30 °, 45 ° and 60 ° respectively. The first stage region in Figure 3a was noncontact in the mixedmode loading when the crack propagated, and as the crack deflected (the stages 1 to 2), although the P_{II} value increased to the maximum load of the loading cycle, the P_{I} value hindered the slipeffect of the P_{II} value parallel to the crack propagation direction and the K_{II} value eventually stabilized as the crack continued to propagate, as shown at point c in Figure 6b. Due to the microgeometric structure of the material and the influence of inclusions and defects in this process, the crack propagation path changed slightly, but the overall propagation path direction was approximately perpendicular to the loading axis.
Before analyzing the variation of SIFs during the crack deflection process, it is essential to understand how the applied load is projected into the directions parallel and perpendicular to the crack propagation under mixedmode conditions. This is crucial for gaining a clear insight into the changes in SIFs in Figure 6. With an increase in the initial loading angle, the P_{I} value decreased, and the P_{II} value increased. The variation in crack deflection angle leads to changes in the projected loads in different directions, P_{I} and P_{II} interact with each other in this process. The presence of P_{I} hinders the slip effect parallel to the crack propagation direction caused by P_{II}, while the presence of P_{II} results in the deflection of the crack propagation direction. P_{I} and P_{II} dominated the variation trends of K_{I} and K_{II}, the final manifestation of the competitive relationship between P_{I} and P_{II} was the competitive relationship between K_{I}dominated PICC and K_{II}dominated RICC. The results indicated that the crack deflected in the first expansion stage and the crack propagation direction was perpendicular to the loading axis in the second.
The alternating effect of the loading angle α=45° in Figure 6a was not apparent compared with other loading angles, and the reason for this result was analyzed. Under the initial condition of α=45°, the values of P_{I} and P_{II} were equal, as shown in Figure 7b, and the crack deflected after a certain number of cycles. Although RICC has an effect on the PICC during the deflection process, the effect of the interaction was not significant due to the same magnitude of the load, so the peak value of K_{II} at α=45° in the first stage of Figure 6b was significantly lower than that at the other two angles. In addition, the mean value of K_{II} value under three loading angles in the range of 2–5 mm was maintained at the same level, because the crack deflect needs to undergo a certain number of cycles.
4.2 Experimental Results
The roughness values (R_{A}) on both sides of the crack in the fatigue crack propagation test were measured by the optical microscope (Atometrics ER230) after the test, and the crack roughnessinduced closure scale was analyzed and calculated. In addition, the fracture morphology was analyzed by SEM (JSM 7800F). When measuring the roughness of the fracture surface, the fracture surface area is divided into four regions ((a)–(d)) according to the Paris crack propagation curve division method, which represents the prefabricated crack, lowspeed, mediumspeed, and highspeed propagation areas, respectively, as shown in Figure 8ad.
The crack surface R_{A} was measured, as shown in Figure 9ad, the coordinate axes x and y represent the measurement range of roughness measurement. The measurement results in Figure 9ad are derived from the corresponding positions in the region outlined in Figure 8ad. The size of the measured fracture surface area is 900 × 500 μm. The result showed that the surface roughness on both sides of the crack gradually increased with crack propagation. The RICC value can be obtained by substituting the measurement results into Eq. (10). The reason for the change of roughness in Figure 9 is that the fatigue fracture surface contacts during the crack deflection process. Under the action of cyclic load, the friction of the contact surface leads to the increase of the roughness value of the fracture surface.
The calculation results of PICC and RICC obtained from the test are plotted in Figure 10. The overall level of the PICC value in Figure 10(a) was approximately 0.2, which was consistent with the theoretical calculating values of Eq. (8). However, the PICC value in the curve was not fixed at a certain value, and the PICC curve extended in a wavy manner. The final crack closure degree P_{op}/P_{max} tended to be constant at approximately 0.2, it was similar to the calculation results of Wang et al. [34] and Lu et al. [38]. The results from Wang et al. [34] also indicate an inconsistent trend in the P_{op}/P_{max} variation between the base material and the welded joint. This inconsistency is attributed to the fact that residual stresses in the weld are not eliminated through postweld heat treatment. As the crack propagates, the P_{op}/P_{max} decreases and gradually stabilizes. Since the material used in this test was EA4T base metal, the crack closure trend of the material after welding was not studied.
The abscissa in Figure 10b represents the experimentally measured fracture surface R_{A}, while the ordinate represents the RICC results corresponding to different R_{A}. The four measuring points in Figure 10b correspond to the precrack, the initial crack deflection, the middle crack deflection and the end of the crack deflection stage. A/B/C/D is used to represent the four stages for the convenience of analysis and description. The variation range of R_{A} from stage B to C was significantly narrower than the stage A to B in Figure 10b, it was the result of the interaction between the PICC and RICC. The interaction or competition between the two closure effects aggravates the contact between the crack fracture surfaces, resulting in a large change in the R_{A} value of the fracture surface. The RICC value presented in Figure 10b was illustrated by combining the simulation results shown in Figure 6b. The K_{II} value provides the driving force for RICC, and the RICC effect was correspondingly enhanced with the increase of K_{II}. After the crack propagation entered stage D, the crack deflection ended and the crack propagation direction was approximately perpendicular to the loading axis. The change after stage D was not analyzed and discussed, as this study focuses on the crack deflection process under mixedmode loading.
Figure 11 shows the recorded results of the fatigue crack growth rates (FCGR) during the experimental process: the FCGRs showed an overall upward trend, and there was also an alternating pattern of increase and decrease in the early stages of crack propagation. This trend was consistent with the results in Figure 6a, where K_{I} values act as the driving force for crack propagation and PICC. The alternating changes in K_{I} values contribute to the alternating pattern in FCGRs.
4.3 Discussions
Based on the above simulation and experiment, the authors surveyed relevant studies on crack closure under mixedmode conditions. Partial experimental results were presented to conduct a comparative analysis with the findings of this study. Zhang et al’s [52] simulation results show that the crack closure level under the plane stress state is about 0.4, and the results are consistent with the estimated level of Eq. (8). The closure ratio in plane stress conditions is 2–2.5 times that in plane strain conditions, the calculated PICC average value of this test was 0.2, the experimental results in the plane strain state have the same trend as the above results. Furthermore, it was observed that as the stress ratio increased from 0.1 to 0.3, the degree of crack closure increased. This is attributed to the increase in P_{min} in the denominator of Eq. (9). A higher stress ratio means that the loading minimum load P_{min} increased, and the denominator in Eq. (9) decreased resulting in a significant upward shift of the crack closure level.
The FCG behaviors of AlLiCuMg alloy (8090) and traditional AlCuMg alloy (2004) are compared in a study by Rodrigues et al. [39]. The crack closure level of Al2024 is in the range of 0.4–0.6, but the crack closure level of Al8090 alloy is 0.7–0.9, the reason is that the fracture analysis results of Al8090 alloy indicated a transgranular fracture mode (high roughness, high crack deflection), it is characteristic of extensive crack closure caused by the roughness wedge. Suresh et al. [53] proposed the RICC geometric model of a fracture surface and considered the contribution of the crack tip displacement of modes I and II in early research. The results show that the displacement ratio is positively correlated with the crack closure level. The crack closure level corresponding to the test data of the two materials (Rail steel and 1018 steel) is approximately 0.4, which is of the same order of magnitude as the crack closure level of the PICC under CA in Eq. (8). The material used in this experiment was EA4T steel, which exhibits consistency in crack closure behavior with the material properties used in Suresh’s experimental study.
The crack deflected in the prefabricated direction follows the same mechanism as the composite direction of the force defined in physics under mixed loading, as illustrated in Figure 7. As the deflection angle changes, the projected values onto the X and Y axes change accordingly. The plastic zone at the crack tip caused by angle deflection was a butterfly asymmetric structure. The asymmetric structure leads to crack propagation along the smaller side of the plastic zone. The crack propagation rate was slowed down because the load in the crack tip region generated the residual compressive stress. When the crack is deflected, the constraint of the crack along the propagation direction is reduced due to the smaller load projected onto modeII. At this moment, the crack propagation along the singleslip system was dominated by modeI load, resulting in a jagged crack propagation path. The contact area of the serrated crack propagation path increased as the load projected onto modeII increased and the crack deflection progressed. The crack tip was passivated under cyclic loading, and the ratio of P_{op} to P_{max} approximately represented the PICC value. The presence of RICC caused the crack side to make contact in advance, resulting in a small P_{op} value in the subsequent loading cycle. The change in P_{op} was positively correlated with the PICC value, thus the PICC value fluctuates under cyclic loading. After crack deflection was completed, the crack propagation direction was approximately perpendicular to the loading axis. The crack propagation can be approximated as modeI crack propagation, the PICC value tends to be a fixed value now, as shown in the latter half of the curve in Figure 10a. Moreover, the degree of crack closure was negatively correlated with the test loading angle, consistent with similar findings reported in Ref. [52].
As the contact area of the serrated crack propagation path increased, the contact friction between the fracture surfaces caused a larger R_{A} value in the contact area, as described in the calculation expression of the RICC value in Eq. (10). The RICC value shows a positive correlation with R_{A} value. The RICC values obtained from the test results were plotted in Figure 10b, demonstrating an approximately linear relationship with the R_{A} value, with the RICC value being around 0.15. The crack tip displacement field was obtained by the DIC technique, and then the plastic zone at the crack tip was calculated using the Tresca yield criterion, the position of the solution area is shown in Figure 12a. During the crack deflection process, the plastic zone at the crack tip exhibited an asymmetrical butterfly shape, as shown in Figure 12b. The larger side of the plastic zone at the crack tip facilitated the elastic recovery of the plastic inclusion when the load was at its minimum, which led to a decrease in the distance between the plastic wakes on both sides of the crack [54]. However, due to the serrated propagation path of the crack, the upper and lower fracture surfaces are contacted in advance during the elastic recovery stage hindering the PICC process.
The fracture surface of the specimen was analyzed using SEM at the end of the test, as shown in Figure 13. Figure 13 shows the macroscopic schematic of the mixedmode fracture surface and the local amplification diagram of the crack deflection line position, respectively. In Figure 13a, the black arrow indicates the crack propagation direction, the fracture surface was rough and had no metallic lustre. The fatigue arc and crack deflection line angle were approximately 25°, and the crack deflection angle was approximately 30°. Notably, the roughness of the crack surface before and after crack deflection was different, the roughness of the crack surface before crack deflection was smooth, as shown in Figure 13b. The crack formed a symmetrical butterflyshaped plastic zone at the crack tip under modeI loading. Owing to the mixedmode loading conditions, irreversible residual shear deformation was introduced at the crack tip, and a spatially torsional crack surface was formed during crack propagation.
During the crack propagation process, the shape of the plastic zone changed due to contact with the crack surface. The alterations in the plastic zone and fracture surface profile before and after crack deflection in the mixedmode are illustrated in Figure 14, where ① – ③ represents the prefabricated crack stage, the crack deflection stage, the stage with crack propagation direction perpendicular to the loading axis, respectively. Crack deflection occurred in the second stage, the plastic zone at the crack tip was asymmetric during the deflection process. The crack propagation line was used as the dividing line, and the area of the plastic zone at the crack tip in the M and the N regions changed alternately. The results of the interaction between PICC and RICC during crack deflection were again confirmed.
Considering the influence of fracture surface roughness on crack propagation rate, from the findings in this work, the evolution law of the fracture surface roughness is explored in the following work, and the parameters of the fracture surface roughness evolution are set onto the mixedmode crack growth path, and the crack growth process can be obtained by combining the XFEM method. It is worth mentioning that the specimen is always in a tensile state under a highstress ratio, and the crack closure effect will be affected by the stress ratio, which is also the influencing factor to be considered in future work.
5 Conclusions

(1)
The driving force of the PICC and RICC is the K_{I} and K_{II}, the K_{I} and K_{II} values in the mixedmode increased with crack propagation, and the latter then decreased to a stable value. The reason for this decrease is that the crack propagation direction gradually becomes perpendicular to the loading axis after crack deflection, the mixedmode loading can be approximated as modeI loading at this moment. The P_{I} value hindered the slip effect of the P_{II} value parallel to the crack propagation direction during the process of crack deflection.

(2)
The interaction relationship between K_{I} and K_{II} values during crack deflection slowed the crack propagation rate to a certain extent. After crack deflection, the crack propagation rate increased linearly. The asymmetric butterflyshaped plastic zone at the crack tip caused by crack deflection facilitated crack propagation along the smaller side of the plastic zone.

(3)
The value of the PICC result was fluctuation due to early contact with the crack surface caused by RICC. The serrated crack propagation path hindered the elastic recovery of the plastic inclusion at the crack tip when the load was at its minimum. The result of the interaction between PICC and RICC was that the variation in roughness caused by contact friction between the fracture surfaces was slowed.

(4)
The roughness difference of the crack surface before and after crack deflection was obvious. Owing to the mixedmode loading, irreversible residual shear deformation was introduced at the crack tip, and a spatially torsional crack surface was formed during crack propagation. The changing trend of the crack tip plastic zone area was consistent with the changing trend of the fracture surface profile roughness, the plastic zone at the crack tip was asymmetric during the deflection process.
Data availability
Date will be available from the corresponding author on reasonable request.
References
F Bjørheim, S C Siriwardane, D Pavlou. A review of fatigue damage detection and measurement techniques. International Journal of Fatigue, 2022, 154: 106556.
T Oplt, T Vojtek, R Kubíček, et al. Numerical modelling of fatigue crack closure and its implication on crack front curvature using ΔCTOD_{p}. International Journal of Fatigue, 2023, 171: 107570.
B Yang, S Wang, J Li, et al. Crack growth behavior of U71MnG rail steel under overload conditions described using a dislocation correction mode. International Journal of Fracture, 2024, 115.
D Kujawski, A K Vasudevan, R E Ricker, et al. On 50 years of fatigue crack closure dispute. Fatigue & Fracture of Engineering Materials & Structures, 2023, 46(8): 28162829.
P Zhang, L Xie, C Zhou, et al. Experimental and numerical investigation on fatigue crack growth behavior of commercial pure titanium under III mixed mode loading at negative load ratios. International Journal of Fatigue, 2020, 138: 105700.
Z S Hosseini, M Dadfarnia, B P Somerday, et al. On the theoretical modeling of fatigue crack growth. Journal of the Mechanics and Physics of Solids, 2018, 121: 341362.
S Qi, L X Cai, C Bao, et al. Analytical theory for fatigue crack propagation rates of mixedmode I–II cracks and its application. International Journal of Fatigue, 2019, 119: 150159.
Y Z Li, S P Zhu, D Liao, et al. Probabilistic modeling of fatigue crack growth and experimental verification. Engineering Failure Analysis, 2020, 118: 104862.
S Stoychev, D Kujawski. Methods for crack opening load and crack tip shielding determination: a review. Fatigue & Fracture of Engineering Materials & Structures, 2003, 26(11): 10531067.
M Skorupa. Load interaction effects during fatigue crack growth under variable amplitude loading a literature review. Part II: qualitative interpretation. Fatigue & Fracture of Engineering Materials & Structures, 1999, 22(10): 905926.
M L Z, J Long, F Z Xuan. Fatigue life and mechanistic modeling of interior microdefect induced cracking in high cycle and very high cycle regimes. Acta Materialia, 2018, 157:259275.
W H Peters, W F Ranson. Digital imaging techniques in experimental stress analysis. Optical Engineering, 1982, 21(3): 427431.
I Yamaguchi, T Furukawa, T Ueda, et al. Accelerated laser speckle strain gauge. Optical Engineering, 1986, 25(5): 671676.
R C McClung. Crack closure and plastic zone sizes in fatigue. Fatigue & Fracture of Engineering Materials & Structures, 1991, 14(4): 455468.
C J Christopher, M N James, E A Patterson, et al. Towards a new model of crack tip stress fields. International Journal of Fracture, 2007, 148: 361371.
C J Christopher, M N James, E A Patterson, et al. A quantitative evaluation of fatigue crack shielding forces using photoelasticity. Engineering Fracture Mechanics, 2008, 75(14): 41904199.
B Yang, Z Wei, F A Diaz, et al. New algorithm for optimised fitting of DIC data to crack tip plastic zone using the CJP model. Theoretical and Applied Fracture Mechanics, 2021, 113: 102950.
J Li, B Yang, S Wang, et al. Modified model of crack tip stress field considering dislocation slip accumulation and crack tip blunting. Chinese Journal of Mechanical Engineering, 2023, 36(1): 47.
B Yang, S Wang, J Li, et al. Study on fatigue crack growth in rail steel at numerical and experimental approaches. Materials, 2023, 16(8): 2981.
F Paysan, E Breitbarth. Towards three dimensional aspects of plasticityinduced crack closure: A finite element simulation. International Journal of Fatigue, 2022, 163: 107092.
G L G. Gonzales, J A O. González, F V Antunes, et al. Experimental determination of the reversed plastic zone size around fatigue crack using digital image correlation. Theoretical and Applied Fracture Mechanics, 2023, 125: 103901.
D L Chen, B Weiss, R Stickler. A model for crack closure. Engineering Fracture Mechanics, 1996, 53(4): 493509.
I Field, B Dixon, E Kandare, et al. The relationship between surface roughness and fatigue crack growth rate in AA7050T7451 subjected to periodic underloads. International Journal of Fatigue, 2023, 167: 107355.
O Gonzalez, J A P Castro, J T Meggiolaro, et al. Challenging the "ΔK_{eff} is the driving force for fatigue crack growth" hypothesis. International Journal of Fatigue. 2020, 136: 105577.
R Pippan, G Strobl, H Kreuzer, et al. Asymmetric crack wake plasticitya reason for roughness induced crack closure. Acta materialia, 2004, 52(15): 44934502.
J Crapps, S R Daniewicz. A macrostructural model for simulating the combined effects of roughness and plasticity induced fatigue crack closure. International Journal of Fatigue, 2012, 45: 1530.
J Pokluda, R Pippan. Analysis of roughnessinduced crack closure based on asymmetric crackwake plasticity and size ratio effect. Materials Science and Engineering: A, 2007, 462(12): 355358.
J Pokluda, P Šandera, J Horníková. Statistical approach to roughnessinduced shielding effects. Fatigue & Fracture of Engineering Materials & Structures, 2004, 27(2): 141157.
H A Richard, K Benitz. A loading device for the creation of mixed mode in fracture mechanics. International Journal of Fracture, 1983, 22: R55R58.
K H Khor, N Kamp, K D Singh, et al. Micromechanistic crack closure modelling of constant amplitude and variable amplitude fatigue in damage tolerant airframe aluminium alloys. Materials Forum. 2004, 28: 850855.
N Kamp, M R Parry, K D Singh, et al. Analytical and finite element modelling of roughness induced crack closure. Acta Materialia, 2004, 52(2): 343353.
M R Parry, S Syngellakis, I Sinclair. Numerical modelling of combined roughness and plasticity induced crack closure effects in fatigue. Materials Science and Engineering: A, 2000, 291(12): 224234.
K D Singh, K H Khor, I Sinclair. Roughness and plasticityinduced fatigue crack closure under single overloads: Analytical modelling. Acta Materialia, 2006, 54(17): 43934403.
Q Wang, X Liu, W Wang, et al. Mixed mode fatigue crack growth behaviour of NiCrMoV high strength steel weldments. International Journal of Fatigue, 2017, 102: 7991.
S Wang, B Yang, J Li, et al. Mixed mode crack growth behaviour considering plasticityinduced and roughnessinduced closure. Engineering Fracture Mechanics, 2023, 289: 109430.
C J Christopher, G Laboviciute, M N James, et al. Extension of the CJP model to mixed mode I and mode II. Frattura ed Integrita Strutturale, 2013, 7(25): 161166.
J Pokluda. Dislocationbased model of cracktip shielding effects. Proceedings of 1st IJFatigue and FFEMS joint workshop on characterisation of crack tip stresses, Gruppo Italiano Frattura. Forni di Sopra, Italy, Marh 7–9, 2011: 162–169.
Y Lu, F Yang, T Chen. Effect of single overload on fatigue crack growth in QSTE340TM steel and retardation model modification. Engineering Fracture Mechanics, 2019, 212: 8194.
E M Rodrigues, A Matias, L B Godefroid, et al. Fatigue crack growth resistance and crack closure behaviour in two aluminium alloys for aeronautical applications. Materials Research, 2005, 8: 287291.
Y G Xu, P J Gregson, I Sinclair. Systematic assessment and validation of compliancebased crack closure measurements in fatigue. Materials Science and Engineering: A, 2000, 284(12): 114125.
J Pokluda, P Šandera. Micromechanisms of fracture and fatigue, Springer London: Springer Science & Business Media, 2010, 125241.
X Li, J Zhang, B Yang, et al. Effect of microshot peening, conventional shot peening and their combination on fatigue property of EA4T axle steel. Journal of Materials Processing Technology, 2020, 275: 116320.
L Sun, X C Zhang, R Z Wang, et al. Evaluation of fatigue and creepfatigue damage levels on the basis of engineering damage mechanics approach. International Journal of Fatigue, 2023, 166: 107277.
P G Nittur, A M Karlsson, L A Carlsson. Implementation of a plastically dissipated energy criterion for three dimensional modeling of fatigue crack growth. International Journal of Fatigue, 2013, 54: 4755.
F V Antunes, A G Chegini, R Branco, et al. A numerical study of plasticity induced crack closure under plane strain conditions. International Journal of Fatigue, 2015, 71: 7586.
K Benitz, H A Richard. Eine Bruchmechanikprobe und Belastungsvorrichtung zur Bestimmung von Rißzähigkeiten bei überlagerter Normal‐ und Schubbeanspruchung. Materialwissenschaft und Werkstofftechnik, 1981, 12(8): 297300.
H J Hoh, Z M Xiao, J Luo. On the plastic zone size and crack tip opening displacement of a Dugdale crack interacting with a circular inclusion. Acta Mechanica, 2010, 210(3): 305314.
Y M Zhang, Z M Xiao, W G Zhang. On 3D crack problems in offshore pipeline with large plastic deformation. Theoretical and Applied Fracture Mechanics, 2013, 67: 2228.
L Achchhe, M B Vaghela. Numerical investigation of an orthotropic plate with interactions of crack, inclusions and oids under uniaxial tensile loading by XFEM. International Journal of Applied Mechanics, 2020, 12(10): 2050113.
F Erdogan, G C Sih. Closure to "Discussion of 'On the crack extension in plates under plane loading and transverse shear'". Journal of Basic Engineering, 1963, 85(4): 519525.
G C Sih, B MacDonald. Fracture mechanics applied to engineering problemsstrain energy density fracture criterion. Engineering Fracture Mechanics, 1974, 6(2): 361386.
P Zhang, L Xie, C Zhou, et al. Experimental and numerical investigation on fatigue cack growth behavior of commercial pure titanium under III mode loading at negative load ratios. International Journal of Fatigue, 2020, 138: 105700.
S Suresh, R O Ritchie. A geometric model for fatigue crack closure induced by fracture surface roughness. Metallurgical transactions A, 1982, 13: 16271631.
M N James, M N Pacey, L W Wei, et al. Characterisation of plasticityinduced closurecrack flank contact force versus plastic enclave. Engineering Fracture Mechanics, 2003, 70(17): 24732487.
Acknowledgements
Not applicable.
Funding
Supported by National Natural Science Foundation of China (Grant No. 52375159), National Railway Administration of China (Grant No. KF2023025) and the Independent Research Project of the State Key Laboratory of Traction Power (Grant No. 2022TPL_T03).
Author information
Authors and Affiliations
Contributions
BY was in charge of the whole trial; SW wrote the manuscript; SW and SZ carried out the experimental analyses and the numerical simulations; JL and SX guided the experiments, BY and SZ revised the manuscript. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing Interests
The authors declare no competing financial interests.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Wang, S., Yang, B., Zhou, S. et al. Closure Effect of I + II Mixedmode Crack for EA4T Axle Steel. Chin. J. Mech. Eng. 37, 81 (2024). https://doi.org/10.1186/s10033024010611
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1186/s10033024010611