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Image-based numerical simulation of the local cyclic deformation behavior around cast pore in steel
Materials Science & Engineering A 678 (2016) 347–354
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Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
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Image-based numerical simulation of the local cyclic deformation behavior around cast pore in steel ⁎
Lihe Qiana,b, , Xiaona Cuia,b, Shuai Liua,b, Minan Chena, Penghui Maa,b, Honglan Xiec, Fucheng Zhanga,b, Jiangying Menga a b c
State Key Laboratory of Metastable Materials Science and Technology, Yanshan University, China National Engineering Research Center for Equipment and Technology of Cold Strip Rolling, Yanshan University, China Shanghai Synchrotron Radiation Facility, Shanghai Institute of Applied Physics, China
A R T I C L E I N F O
A BS T RAC T
Keywords: Cyclic deformation Low-cycle fatigue Local stress/strain behavior Image-based simulation Chaboche's model
The local cyclic stress/strain responses around an actual, irregular pore in cast Hadfield steel under fatigue loading are investigated numerically, and compared with those around a spherical and an ellipsoidal pore. The actual pore-containing model takes into account the real shape of the pore imaged via high-resolution synchrotron X-ray computed tomography and combines both isotropic hardening and Bauschinger effects by using the Chaboche's material model, which enables to realistically simulate the cyclic deformation behaviors around actual pore. The results show that the stress and strain energy density concentration factors (Kσ and KE) around either an actual irregular pore or an idealized pore increase while the strain concentration factor (Kε) decreases slightly with increasing the number of fatigue cycles. However, all the three parameters, Kσ, Kε and KE, around an actual pore are always several times larger than those around an idealized pore, whatever the number of fatigue cycles. It is suggested that the fatigue properties of cast pore-containing materials cannot be realistically evaluated with any idealized pore models. The feasibility of the methodology presented highlights the potential of its application in the micromechanical understanding of fatigue damage phenomena in cast pore-containing materials.
1. Introduction Fatigue damage is frequently encountered in some key engineering materials; it is originated from the repeated plastic strain at stress concentrations under fatigue loading. Microstructural imperfections or processing defects are stress concentrators and thus potential sites for fatigue damage [1–6]. Cast pores, as one of the critical damaging defects, are inevitably existent in cast components, and tend to cause severe fatigue damage in engineering applications [7–9]. Thus, a large volume of research has been conducted, both experimentally and theoretically, to understand the correlation between fatigue damage and cast pores. For elucidating the mechanical sources of such fatigue damage, analytical and computational models have been applied to analyze stress/strain concentrations at pores [10–14]. These models are often constructed by idealizing actual cast pores into ones with regular shape, such as sphere or ellipsoid. For example, several authors applied a sphere-equivalent method for analyzing stress concentrations around pore, by approximating an irregular pore to a sphere with its volume equivalent to the actual pore's volume [13,14]. This sphere-
⁎
equivalent method and other theoretical models available in literature are valuable, because they give a convenient estimation of stress concentrations around actual cast pores. However, those previous models, ignoring the shape irregularity of pores, are unable to provide reliable information on realistic local deformation behavior around actual pores. This is because stresses and strains at different locations around an irregular pore may interact with each other, and thus the local deformation behavior around an actual pore can be affected by the irregularity of the pore [15–19]. Indeed, several authors have studied the mechanical behavior of pore-containing materials using image-based finite-element (FE) models by taking into account the real shapes of pores, and explored the effects of pore shape on the local deformation behavior under monotonic loading [20– 23]. Furthermore, the effects of the second phase on the monotonic deformation behaviors in some heterogeneous materials have also been investigated via image-based FE models [24–26]. However, no studies are available on the local cyclic deformation behavior around actual pores under fatigue loading, although it is well known that fatigue damage results from the accumulation of repeated plastic strain.
Corresponding author at: State Key Laboratory of Metastable Materials Science and Technology, Yanshan University, Qinhuangdao 066004, China. E-mail address: [email protected] (L. Qian).
http://dx.doi.org/10.1016/j.msea.2016.10.017 Received 5 August 2016; Received in revised form 4 October 2016; Accepted 5 October 2016 Available online 06 October 2016 0921-5093/ © 2016 Elsevier B.V. All rights reserved.
Materials Science & Engineering A 678 (2016) 347–354
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2. Modeling methodology
The purpose of the present work was to investigate how an actual irregular pore affects the local stress/strain, strain energy density and their evolutions with fatigue loading in cast Hadfield steel via threedimensional (3D) image-based FE modeling, and also to make a comparison with those around idealized pores. For fatigue problems, not only the shape of the pore but also the cyclic hardening characteristic of the material has an influence on the stress and strain responses locally around the pore [4]. The actual pore-containing model was created by considering the real shape of pore. For this, a sample of cast Hadfield steel was visualized by the high-resolution synchrotron X-ray computed tomography (CT), and thus a volume of 3D images of cast pores were acquired. Note that only one irregular pore was intentionally extracted from a volume of pores to create a model so as to exclude the possible effects of other pores. Furthermore, as already identified in literature [27,28], Hadfield steel shows Bauschinger effect, i.e. kinematic hardening. Therefore, in order to realistically simulate the cyclic hardening behavior of Hadfield steel, the Chaboche's model [4,29], which incorporates both the isotropic and kinematic hardening, was adopted.
2.1. Acquisition of 3D images of pores Cast Hadfield high manganese steel, which is widely applied for making railway crossings, was taken as the subject. The actual cast crossing contains a large number of cast pores, mainly associated with the solidification of casting. The nominal chemical composition of the steel is Fe–13Mn–1.2C (wt%) [30]. 3D images of cast pores in the steel were obtained by highresolution synchrotron X-ray CT [15–18,31,32]. For CT experiments, a monochromatic X-ray beam coming from a double crystal monochromator was applied. The X-ray source to sample distance was 34 m. The CCD detector was a cooled 2048×2048 element PCO 2000 camera. The beam energy and sample-to-detector distance were chosen to be 36 keV and 200 mm, respectively. The obtained raw radiograph data was reconstructed using a filtered back-projection algorithm. The voxel size of the reconstructed 3D image is 0.74 µm. Segmentation technique was used to extract the volumes and capture the real shapes of irregular pores. 3D images of pores were rendered by stacking the slice images using a commercial visualization software. A 3D distribution of various sizes and shapes of pores in a small volume is typically shown in Fig. 1. 2.2. Construction of image-based FE model In order to simulate the local deformation behavior around a single pore and to rule out the possible effects of other pores, one irregular pore only was taken from the CT images for creating the porecontaining FE model (Fig. 2). The outer surfaces of the chosen pore were extracted by thresholding the gray value and tracing the iso-grayvalue surfaces with triangular surface meshes [19,22]. Based on the surface meshes of the pore, the 3D geometrical model and hence the 3D FE model containing the pore were created. The created model, a boneshaped plate specimen, has a nominal gauge section of 15×10×3 mm3, and it consists of ~54,000 20 node tetrahedral elements. Finer meshes were used in regions close to the pore to conform to its irregular shape [32]. Furthermore, for purpose of comparison, two idealized models, one containing a spherical pore and the other containing an ellipsoidal pore, were created. The geometries of the two idealized pores are illustrated in Fig. 3. The long-axis length of the ellipsoidal pore and the diameter of the spherical pore were assigned to be the same as the maximum length of the actual pore. Thus, the volumes of the spherical and ellipsoidal pores are 41.9×105 and 5.13×105 μm3, respectively, both being much larger than that of the actual pore, 2.31×105 μm3.
Fig. 1. (a) 3D rendering of cast pores (gray-white) in a small volume; (b) an irregular pore typically extracted from the CT image dada for construction of an actual porecontaining FE model.
Fig. 2. Construction of an actual pore-containing FE model, constraint condition and cyclic loading schedule. Note that the loading direction is perpendicular to the long axis of the pore.
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Fig. 3. The geometries of a spherical pore and an ellopsoidal pore for construction of idealized pore models.
Fig. 4. Experimental and simulated stress/strain hesteresis loops of Hadfield steel during fatigue at a strain amplitude of 0.004: (a) in the first cycle; (b) in the 20th cycle; (c) experimental and simulated peak and valley stress responses versus the number of cycles.
2.3. Fatigue test for acquiring fatigue parameters
Constraint was applied to the lower end of each model specimen. Axial symmetrical tension-compression cyclic strain was applied to the upper end, with a triangular wave shape under a constant strain amplitude of Δε/2=0.004. Note that the loading direction is perpendicular to the long axis of the actual and ellipsoidal pores. Here, 40 cycles of fatigue loading only were simulated, because more cycles of simulation would take much more computation time. The numerical simulations were performed with the general purpose ANSYS finite-element soft package.
The constructed pore-containing FE model consists of a homogeneous medium of Hadfield steel embedded with a single pore. For a better understanding of the local stress/strain behaviors around the pore during fatigue loading, it is necessary to input actual fatigue parameters into the FE model. For acquiring the parameters, symmetrical tension-compression fatigue tests were performed under strain amplitude control [27,33]. Bone-shaped plate specimens, similar to the FE model specimen (Fig. 2), were used. The specimen was taken from 349
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The Bauschinger effect and cyclic hardening observed here are in agreement with the literature [24,25]. Thus, it is necessary to take into account these two fatigue phenomena in FE model for realistically simulating the cyclic deformation behavior. For this, the steel was modeled as an elastic-plastic medium with combined nonlinear isotropic/kinematic hardening, following the Chaboche's model [29]. The nonlinear isotropic hardening is described by
Table 1 Elastic modulus, initial yield stress and cyclic hardening parameters of Hadfield steel. E (MPa)
σ0 (MPa)
Q∞ (MPa)
b
C (MPa)
γ
196900
286
62
3.4
84099
540
the subsurface region of a cast crossing, thus having fewer pores, because of the fast cooling rate near the surface. An axial extensometer with a gauge length of 10 mm was used for measuring strain. A triangular strain wave shape was applied, with strain amplitudes ranging from 0.004 to 0.008 at a constant strain rate of 6×10−3 s−1. After fatigue test, fatigue parameters necessary for fatigue simulations were deduced from the macroscopic cyclic stress/stain responses.
pl
σ y = σ 0 + Q∞ (1 − e−be )
(1)
where σ y is the yield stress, σ 0 is the initial yield stress, e pl is the accumulated equivalent plastic strain, and Q∞ and b are constants. The kinematic hardening is represented by
α̇ = 2.4. Acquisition of fatigue parameters
2 pl Ce ̇ − γαe ̇ pl 3
(2) stress, e pl
where α is the back is the equivalent plastic strain, C is the initial kinematic hardening modulus, and γ determines the rate at which the kinematic hardening modulus decreases with increasing plastic deformation. These parameters were determined by means of the nonlinear fitting to strain-controlled symmetrical tension-compression fatigue tests data using Matlab software. By taking advantage of the back stresses and the plastic strains measured from the stabilized hysteresis loops of multiple fatigue tests at different strain amplitudes, the yield stresses as well as the plastic strains accumulated with increasing the cyclic number, the associated fatigue parameters of the
The experimental stress/strain hysteresis loops at a constant strain amplitude of 0.004 are typically shown in Fig. 4(a) and (b) for cycles 1 and 20 (black curves), respectively. The variations of the peak and valley stresses with increasing number of cycles are shown in Fig. 4(c) (black curves). The results clearly show Bauschinger effect (i.e., the compressive yield stress decreases after prior tensile loading, Fig. 4(a) or (b). In addition, cyclic hardening is also observed (i.e., at a constant strain amplitude of 0.004, the stress amplitude increases with the increase of the number of cycles, Fig. 4(c).
Fig. 5. The contour patterns of von Mises equivalent stress around (a) actual pore, (b) ellipsoidal pore and (c) spherical pore in the first cycle at loading stages P1–P4 (see the upper inset for the loading schedule). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 6. The contour patterns of equivalent strain around (a) actual pore, (b) ellipsoidal pore and (c) spherical pore in the first cycle at loading stages P1–P4 (see the upper inset for the loading schedule).
concentration remains around the pore and this stress concentration becomes even larger than those at all previous loading stages (P1–P3). This even larger stress concentration occurring near the pore at zero strain (P4) is due to the even larger residual tensile stress generated around the pore after one-cycle fatigue. These results indicate that when the externally applied strain is decreased from tensile (or compressive) strain to zero strain, larger compressive (or tensile) residual stress is generated near the pore and it increases with fatigue loading. For the idealized ellipsoidal and spherical pores (Fig. 5(b) and (c)), the variation tendencies of stress concentration are similar to those around the actual pore. However, the degrees of stress concentration are much smaller around the idealized pores (red regions are almost invisible throughout the first cycle), as compared with those around the actual pore (red regions appear and the sizes of the red regions increase with loading from P1 to P4). Furthermore, the equivalent strain contour patterns locally around the three pores at different loading stages in the first cycle are shown in Fig. 6(a)–(c). Clearly, strains are also concentrated near the pores whatever the loading levels are. It is especially noted that, with the complete unloading of the applied strain from either tensile or compressive side (i.e. when the externally applied strain is returned to zero), the local strain remains largely concentrated around each pore. It is indicated that the evolution of the strain locally around a pore during fatigue loading is not synchronized with the variation of
steel were obtained, as given in Table 1. Fig. 4(a) and (b) shows the simulated far-field stress/strain hysteresis loops at cycles 1 and 20, as compared with the experimental ones. Fig. 4(c) shows the simulated macroscopic peak and valley stresses responses, in comparison with those obtained by fatigue tests. Clearly, the cyclic deformation behaviors simulated using the parameters in Table 1 coincide well with the experimental results. Accordingly, the fatigue parameters in Table 1 were used to simulate the local stress/strain responses around pore in Hadfield steel during fatigue loading, as addressed in the following. 3. Results Fig. 5(a)–(c) shows the von Mises equivalent stress contours locally around the actual, ellipsoidal and spherical pores, respectively, at different loading stages in the first cycle. For the actual pore (Fig. 5a), when tensile strain is increased to the maximum of 0.004 (P1), significant stress concentration occurs around the pore. When unloaded to zero strain (P2), stress concentration remains near the pore, but it becomes even larger than that at the maximum tension (P1). This is associated with the residual compressive stress generated near the pore when the applied macroscopic tension strain is forced back to zero. When reverse loading is applied and reaches the maximum compressive strain of −0.004 (P3), even larger stress concentration is observed near the pore than that at loading levels P1 and P2. Afterwards, when unloaded to zero strain (P4) again, stress 351
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Fig. 7. (a) Far-field y-direction stress/strain hysteresis loops (i.e. along the applied loading direction, Fig. 2), and the y-direction stress/strain hysteresis loops concentrated around (b) the spherical pore, (c) ellipsoidal pore, and (d) actual pore for several typical cycles.
three times larger than those around the two idealized pores (3.5 and 5.4 for the spherical and ellipsoidal pore, respectively). Furthermore, the strain amplitude (half width of the hysteresis loop) locally around each pore decreases with increasing number of fatigue cycles, but gradually approaches to a stabilized value after a certain number of cycles (Fig. 7(b)–(d)); all these stabilized values of the local strain amplitude remain much larger than the far-field strain amplitude value (Fig. 7(a)). In contrast, the local stress amplitude increases rapidly with increasing number of cycles. These observations are related to cyclic hardening of Hadfield steel. The concentration factors of stress, strain and strain energy density at cycle 40 are shown in Fig. 8(b). As compared with those in the first cycle 8(a), in the fortieth cycle, the values of Kσ near all the three pores increase while the Kε values decrease slightly, and thus, the corresponding KE values increase. A comparison among different pores demonstrates that the Kσ value around the actual pore, 3.9, is roughly two times larger than those around the idealized pores. And, the values of Kε and KE around the actual pore reach 8.0 and 31, respectively, which are approximately three-to-eight times larger than those around the two idealized pores.
the applied far-field strain. Fig. 7 shows the hysteresis loops of the maximum stress and strain along the y-direction (i.e. along the loading direction, Fig. 2) concentrated around the three pores at fatigue cycles 1, 10, 20, 30 and 40. The corresponding far-field hysteresis loops are also shown for comparison. As it may be expected, the hysteresis loops of the maximum stress and strain around the three pores (Fig. 7(c)–(d)) are much wider and higher than the far-field hysteresis loops (Fig. 7(a)), indicating higher concentrations of both stress and strain around each pore. To get a quantitative comparison of the local stress and strain concentrations around the three pores, the stress and strain concentration factors (Kσ and Kε), as defined as the ratio of the concentrated stress and strain locally around the pore to the far-field stress and strain at the maximum tensile load, are shown in Fig. 8 for cycles 1 and 40. Note that the “local” refers to a region covering a certain distance from the internal surface of the pore, and the stress/strain concentrations may exist exactly at the internal surface and may also exist at a distance away from the internal surface. Additionally, the strain energy density concentration factor (KE), defined as the ratio of the concentrated strain energy density locally around the pore to the far-field value at the maximum tensile load, is also shown in Fig. 8 for cycles 1 and 40. The strain energy density consumed in one cycle, which is often used to represent the damage of material during one-cycle fatigue, is determined from the area of one hysteresis loop. It is observed from Fig. 8(a) that, in the first cycle, stress concentration factor around the actual pore, 1.5, is slightly larger than those around the two idealized pores, the same small value of 1.3. In contrast, the strain concentration factor near the actual pore reaches 13, over three times larger than the ones around the two idealized pores. Furthermore, the local strain energy density factor around the actual pore (as large as 19) is also over
4. Discussion Fatigue failure of components stems primarily from the plastic strain accumulated at stress concentrations, such as at cast pores, during cyclic loading. Therefore, several models have been developed to analyze stress/strain concentrations [10–14]. These models are mostly based on idealizing irregular cast pores into regular ones. Such models are useful; however, they do not consider the irregularity of pore morphology, unable to disclose the realistic local stress/strain behavior. Recently, several authors attempted to consider the real 352
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with the increase of pore volume (Figs. 5 and 6). Furthermore, it is considered that, in the case of multiple pores with a larger porosity and a smaller spacing between neighboring pores, the stress/strain fields around neighboring pores may interact with each other and the maximum stress/strain concentrations may thus increase with the increase of porosity. Given that a large number of various irregular pores exist in a practical cast material, and on the basis of the present study, it is thus necessary to conduct further simulations that consider the entire microstructure or a representative volume of pores, for understanding the entire mechanical behavior by investigating the mutual interactions between various pores and their synergetic effect, although constructions and computations of even more complicated models with a huge number of FE meshes are a tough and timeconsuming work. 5. Conclusions The local cyclic stress/strain behaviors around an actual pore in cast Hadfield high manganese steel were investigated via image-based numerical simulation. The numerical model took into account the real shape of the pore imaged by synchrotron X-ray CT and incorporated both isotropic and kinematic hardening effects, with the parameters obtained by fitting fatigue experiments data, thus being able to realistically simulate the cyclic deformation behavior around actual pores. The simulation results based on the actual pore-containing model were compared with the results based on the idealized pore models. Under a strain-controlled fatigue condition, the evolution of local strain around either an actual pore or an idealized (spherical or ellipsoidal) pore is not synchronized with the variation of the far-field strain applied. As the far-field strain is decreased to zero, large residual stress is generated around each pore and it increases with fatigue loading. The stress concentration factor (Kσ) around either an actual pore or an idealized (spherical or ellipsoidal) pore is always smaller than the concentration factors of strain and strain energy density (Kε and KE). With increasing number of fatigue cycles, Kσ and KE increase whereas Kε decreases slightly. The three concentration factors, Kσ, Kε and KE, around the actual pore are always larger than those around the two idealized pores whatever the number of fatigue cycles. In the first cycle, although the increase in Kσ caused by actual pore is not so significant, the values of Kε and KE around the actual pore reach 13 and 19, respectively, which are over three times larger than those around the idealized pores. At a higher number of fatigue cycles, e.g., at cycle 40, the Kσ value around the actual pore, 3.9, is roughly two times larger than those around the idealized pores; the Kε and KE values reach 8.0 and 31, respectively, which are approximately three-to-eight times larger than those around the two idealized pores. Further work is necessary to investigate the interactions among various pores and their synergetic effects on the local/general stress/ strain responses of materials under fatigue loading, by creating models with a volume of pores rather than a single one.
Fig. 8. (a) Concentration factors of stress (Kσ), strain (Kε) and strain energy density (KE) around three different pores during fatigue loading (a) in the first cycle and (b) in the fortieth cycle.
shapes of pores in simulating the monotonic mechanical behavior of materials containing irregular pores [20–23]. In contrast, the present work focused on the cyclic stress/strain responses locally around an irregular pore during fatigue loading, via 3D image-based simulation. The simulation methodology not only took into account the actual 3D shape of a cast pore visualized via synchrotron X-ray CT, but also incorporated the combined effects of isotropic hardening and Bauschinger effect evaluated by fatigue experiments, thereby being able to realistically simulate the cyclic deformation behavior around an actual pore. Furthermore, in constructing the actual pore-containing model, a single pore only was extracted from a volume of pores, with the intent to exclude the possible effects of other pores. This was intended to clarify the role of the irregularity of a cast pore, through a comparison of the cyclic deformation behaviors locally around an actual pore and idealized pores. The methodology and the results presented are of importance for micromechanically understanding the effect of an irregular pore on fatigue damage phenomena in pore-containing materials. The present work was directed towards the cyclic deformation behavior locally around a pore under the strain-controlled fatigue condition. Strain-controlled fatigue, i.e. low-cycle fatigue, frequently occurs at high cyclic loads or high stress concentrations where strain is constrained. The methodology is also applicable to simulating the local stress/strain responses around pores at low-stress levels under stresscontrolled fatigue (i.e. high-cycle fatigue) conditions. The results demonstrated that the degrees of stress and strain concentrations around an actual pore are significantly larger than those around an idealized pore, regardless of the number of fatigue cycles. It appears that the volume of a single pore does not have an obvious effect on the maximum value of the local stress/strain concentrations. However, the sizes of higher-stress/strain regions increase obviously
Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant no. 51171166). The synchrotron radiation experiments were performed at the beamline BL13W1 of Shanghai Synchrotron Radiation Facility (SSRF). References [1] Y. Hong, Z. Lei, C. Sun, A. Zhao, Int. J. Fatigue 58 (2014) 144–151. [2] Q.G. Wang, D. Apelian, D.A. Lados, J. Light Met. 1 (2001) 73–84. [3] M.J. Caton, J.W. Jones, J.M. Boileau, J.E. Allison, Metall. Mater. Trans. A 30
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Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
crossmark
Image-based numerical simulation of the local cyclic deformation behavior around cast pore in steel ⁎
Lihe Qiana,b, , Xiaona Cuia,b, Shuai Liua,b, Minan Chena, Penghui Maa,b, Honglan Xiec, Fucheng Zhanga,b, Jiangying Menga a b c
State Key Laboratory of Metastable Materials Science and Technology, Yanshan University, China National Engineering Research Center for Equipment and Technology of Cold Strip Rolling, Yanshan University, China Shanghai Synchrotron Radiation Facility, Shanghai Institute of Applied Physics, China
A R T I C L E I N F O
A BS T RAC T
Keywords: Cyclic deformation Low-cycle fatigue Local stress/strain behavior Image-based simulation Chaboche's model
The local cyclic stress/strain responses around an actual, irregular pore in cast Hadfield steel under fatigue loading are investigated numerically, and compared with those around a spherical and an ellipsoidal pore. The actual pore-containing model takes into account the real shape of the pore imaged via high-resolution synchrotron X-ray computed tomography and combines both isotropic hardening and Bauschinger effects by using the Chaboche's material model, which enables to realistically simulate the cyclic deformation behaviors around actual pore. The results show that the stress and strain energy density concentration factors (Kσ and KE) around either an actual irregular pore or an idealized pore increase while the strain concentration factor (Kε) decreases slightly with increasing the number of fatigue cycles. However, all the three parameters, Kσ, Kε and KE, around an actual pore are always several times larger than those around an idealized pore, whatever the number of fatigue cycles. It is suggested that the fatigue properties of cast pore-containing materials cannot be realistically evaluated with any idealized pore models. The feasibility of the methodology presented highlights the potential of its application in the micromechanical understanding of fatigue damage phenomena in cast pore-containing materials.
1. Introduction Fatigue damage is frequently encountered in some key engineering materials; it is originated from the repeated plastic strain at stress concentrations under fatigue loading. Microstructural imperfections or processing defects are stress concentrators and thus potential sites for fatigue damage [1–6]. Cast pores, as one of the critical damaging defects, are inevitably existent in cast components, and tend to cause severe fatigue damage in engineering applications [7–9]. Thus, a large volume of research has been conducted, both experimentally and theoretically, to understand the correlation between fatigue damage and cast pores. For elucidating the mechanical sources of such fatigue damage, analytical and computational models have been applied to analyze stress/strain concentrations at pores [10–14]. These models are often constructed by idealizing actual cast pores into ones with regular shape, such as sphere or ellipsoid. For example, several authors applied a sphere-equivalent method for analyzing stress concentrations around pore, by approximating an irregular pore to a sphere with its volume equivalent to the actual pore's volume [13,14]. This sphere-
⁎
equivalent method and other theoretical models available in literature are valuable, because they give a convenient estimation of stress concentrations around actual cast pores. However, those previous models, ignoring the shape irregularity of pores, are unable to provide reliable information on realistic local deformation behavior around actual pores. This is because stresses and strains at different locations around an irregular pore may interact with each other, and thus the local deformation behavior around an actual pore can be affected by the irregularity of the pore [15–19]. Indeed, several authors have studied the mechanical behavior of pore-containing materials using image-based finite-element (FE) models by taking into account the real shapes of pores, and explored the effects of pore shape on the local deformation behavior under monotonic loading [20– 23]. Furthermore, the effects of the second phase on the monotonic deformation behaviors in some heterogeneous materials have also been investigated via image-based FE models [24–26]. However, no studies are available on the local cyclic deformation behavior around actual pores under fatigue loading, although it is well known that fatigue damage results from the accumulation of repeated plastic strain.
Corresponding author at: State Key Laboratory of Metastable Materials Science and Technology, Yanshan University, Qinhuangdao 066004, China. E-mail address: [email protected] (L. Qian).
http://dx.doi.org/10.1016/j.msea.2016.10.017 Received 5 August 2016; Received in revised form 4 October 2016; Accepted 5 October 2016 Available online 06 October 2016 0921-5093/ © 2016 Elsevier B.V. All rights reserved.
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2. Modeling methodology
The purpose of the present work was to investigate how an actual irregular pore affects the local stress/strain, strain energy density and their evolutions with fatigue loading in cast Hadfield steel via threedimensional (3D) image-based FE modeling, and also to make a comparison with those around idealized pores. For fatigue problems, not only the shape of the pore but also the cyclic hardening characteristic of the material has an influence on the stress and strain responses locally around the pore [4]. The actual pore-containing model was created by considering the real shape of pore. For this, a sample of cast Hadfield steel was visualized by the high-resolution synchrotron X-ray computed tomography (CT), and thus a volume of 3D images of cast pores were acquired. Note that only one irregular pore was intentionally extracted from a volume of pores to create a model so as to exclude the possible effects of other pores. Furthermore, as already identified in literature [27,28], Hadfield steel shows Bauschinger effect, i.e. kinematic hardening. Therefore, in order to realistically simulate the cyclic hardening behavior of Hadfield steel, the Chaboche's model [4,29], which incorporates both the isotropic and kinematic hardening, was adopted.
2.1. Acquisition of 3D images of pores Cast Hadfield high manganese steel, which is widely applied for making railway crossings, was taken as the subject. The actual cast crossing contains a large number of cast pores, mainly associated with the solidification of casting. The nominal chemical composition of the steel is Fe–13Mn–1.2C (wt%) [30]. 3D images of cast pores in the steel were obtained by highresolution synchrotron X-ray CT [15–18,31,32]. For CT experiments, a monochromatic X-ray beam coming from a double crystal monochromator was applied. The X-ray source to sample distance was 34 m. The CCD detector was a cooled 2048×2048 element PCO 2000 camera. The beam energy and sample-to-detector distance were chosen to be 36 keV and 200 mm, respectively. The obtained raw radiograph data was reconstructed using a filtered back-projection algorithm. The voxel size of the reconstructed 3D image is 0.74 µm. Segmentation technique was used to extract the volumes and capture the real shapes of irregular pores. 3D images of pores were rendered by stacking the slice images using a commercial visualization software. A 3D distribution of various sizes and shapes of pores in a small volume is typically shown in Fig. 1. 2.2. Construction of image-based FE model In order to simulate the local deformation behavior around a single pore and to rule out the possible effects of other pores, one irregular pore only was taken from the CT images for creating the porecontaining FE model (Fig. 2). The outer surfaces of the chosen pore were extracted by thresholding the gray value and tracing the iso-grayvalue surfaces with triangular surface meshes [19,22]. Based on the surface meshes of the pore, the 3D geometrical model and hence the 3D FE model containing the pore were created. The created model, a boneshaped plate specimen, has a nominal gauge section of 15×10×3 mm3, and it consists of ~54,000 20 node tetrahedral elements. Finer meshes were used in regions close to the pore to conform to its irregular shape [32]. Furthermore, for purpose of comparison, two idealized models, one containing a spherical pore and the other containing an ellipsoidal pore, were created. The geometries of the two idealized pores are illustrated in Fig. 3. The long-axis length of the ellipsoidal pore and the diameter of the spherical pore were assigned to be the same as the maximum length of the actual pore. Thus, the volumes of the spherical and ellipsoidal pores are 41.9×105 and 5.13×105 μm3, respectively, both being much larger than that of the actual pore, 2.31×105 μm3.
Fig. 1. (a) 3D rendering of cast pores (gray-white) in a small volume; (b) an irregular pore typically extracted from the CT image dada for construction of an actual porecontaining FE model.
Fig. 2. Construction of an actual pore-containing FE model, constraint condition and cyclic loading schedule. Note that the loading direction is perpendicular to the long axis of the pore.
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Fig. 3. The geometries of a spherical pore and an ellopsoidal pore for construction of idealized pore models.
Fig. 4. Experimental and simulated stress/strain hesteresis loops of Hadfield steel during fatigue at a strain amplitude of 0.004: (a) in the first cycle; (b) in the 20th cycle; (c) experimental and simulated peak and valley stress responses versus the number of cycles.
2.3. Fatigue test for acquiring fatigue parameters
Constraint was applied to the lower end of each model specimen. Axial symmetrical tension-compression cyclic strain was applied to the upper end, with a triangular wave shape under a constant strain amplitude of Δε/2=0.004. Note that the loading direction is perpendicular to the long axis of the actual and ellipsoidal pores. Here, 40 cycles of fatigue loading only were simulated, because more cycles of simulation would take much more computation time. The numerical simulations were performed with the general purpose ANSYS finite-element soft package.
The constructed pore-containing FE model consists of a homogeneous medium of Hadfield steel embedded with a single pore. For a better understanding of the local stress/strain behaviors around the pore during fatigue loading, it is necessary to input actual fatigue parameters into the FE model. For acquiring the parameters, symmetrical tension-compression fatigue tests were performed under strain amplitude control [27,33]. Bone-shaped plate specimens, similar to the FE model specimen (Fig. 2), were used. The specimen was taken from 349
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The Bauschinger effect and cyclic hardening observed here are in agreement with the literature [24,25]. Thus, it is necessary to take into account these two fatigue phenomena in FE model for realistically simulating the cyclic deformation behavior. For this, the steel was modeled as an elastic-plastic medium with combined nonlinear isotropic/kinematic hardening, following the Chaboche's model [29]. The nonlinear isotropic hardening is described by
Table 1 Elastic modulus, initial yield stress and cyclic hardening parameters of Hadfield steel. E (MPa)
σ0 (MPa)
Q∞ (MPa)
b
C (MPa)
γ
196900
286
62
3.4
84099
540
the subsurface region of a cast crossing, thus having fewer pores, because of the fast cooling rate near the surface. An axial extensometer with a gauge length of 10 mm was used for measuring strain. A triangular strain wave shape was applied, with strain amplitudes ranging from 0.004 to 0.008 at a constant strain rate of 6×10−3 s−1. After fatigue test, fatigue parameters necessary for fatigue simulations were deduced from the macroscopic cyclic stress/stain responses.
pl
σ y = σ 0 + Q∞ (1 − e−be )
(1)
where σ y is the yield stress, σ 0 is the initial yield stress, e pl is the accumulated equivalent plastic strain, and Q∞ and b are constants. The kinematic hardening is represented by
α̇ = 2.4. Acquisition of fatigue parameters
2 pl Ce ̇ − γαe ̇ pl 3
(2) stress, e pl
where α is the back is the equivalent plastic strain, C is the initial kinematic hardening modulus, and γ determines the rate at which the kinematic hardening modulus decreases with increasing plastic deformation. These parameters were determined by means of the nonlinear fitting to strain-controlled symmetrical tension-compression fatigue tests data using Matlab software. By taking advantage of the back stresses and the plastic strains measured from the stabilized hysteresis loops of multiple fatigue tests at different strain amplitudes, the yield stresses as well as the plastic strains accumulated with increasing the cyclic number, the associated fatigue parameters of the
The experimental stress/strain hysteresis loops at a constant strain amplitude of 0.004 are typically shown in Fig. 4(a) and (b) for cycles 1 and 20 (black curves), respectively. The variations of the peak and valley stresses with increasing number of cycles are shown in Fig. 4(c) (black curves). The results clearly show Bauschinger effect (i.e., the compressive yield stress decreases after prior tensile loading, Fig. 4(a) or (b). In addition, cyclic hardening is also observed (i.e., at a constant strain amplitude of 0.004, the stress amplitude increases with the increase of the number of cycles, Fig. 4(c).
Fig. 5. The contour patterns of von Mises equivalent stress around (a) actual pore, (b) ellipsoidal pore and (c) spherical pore in the first cycle at loading stages P1–P4 (see the upper inset for the loading schedule). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 6. The contour patterns of equivalent strain around (a) actual pore, (b) ellipsoidal pore and (c) spherical pore in the first cycle at loading stages P1–P4 (see the upper inset for the loading schedule).
concentration remains around the pore and this stress concentration becomes even larger than those at all previous loading stages (P1–P3). This even larger stress concentration occurring near the pore at zero strain (P4) is due to the even larger residual tensile stress generated around the pore after one-cycle fatigue. These results indicate that when the externally applied strain is decreased from tensile (or compressive) strain to zero strain, larger compressive (or tensile) residual stress is generated near the pore and it increases with fatigue loading. For the idealized ellipsoidal and spherical pores (Fig. 5(b) and (c)), the variation tendencies of stress concentration are similar to those around the actual pore. However, the degrees of stress concentration are much smaller around the idealized pores (red regions are almost invisible throughout the first cycle), as compared with those around the actual pore (red regions appear and the sizes of the red regions increase with loading from P1 to P4). Furthermore, the equivalent strain contour patterns locally around the three pores at different loading stages in the first cycle are shown in Fig. 6(a)–(c). Clearly, strains are also concentrated near the pores whatever the loading levels are. It is especially noted that, with the complete unloading of the applied strain from either tensile or compressive side (i.e. when the externally applied strain is returned to zero), the local strain remains largely concentrated around each pore. It is indicated that the evolution of the strain locally around a pore during fatigue loading is not synchronized with the variation of
steel were obtained, as given in Table 1. Fig. 4(a) and (b) shows the simulated far-field stress/strain hysteresis loops at cycles 1 and 20, as compared with the experimental ones. Fig. 4(c) shows the simulated macroscopic peak and valley stresses responses, in comparison with those obtained by fatigue tests. Clearly, the cyclic deformation behaviors simulated using the parameters in Table 1 coincide well with the experimental results. Accordingly, the fatigue parameters in Table 1 were used to simulate the local stress/strain responses around pore in Hadfield steel during fatigue loading, as addressed in the following. 3. Results Fig. 5(a)–(c) shows the von Mises equivalent stress contours locally around the actual, ellipsoidal and spherical pores, respectively, at different loading stages in the first cycle. For the actual pore (Fig. 5a), when tensile strain is increased to the maximum of 0.004 (P1), significant stress concentration occurs around the pore. When unloaded to zero strain (P2), stress concentration remains near the pore, but it becomes even larger than that at the maximum tension (P1). This is associated with the residual compressive stress generated near the pore when the applied macroscopic tension strain is forced back to zero. When reverse loading is applied and reaches the maximum compressive strain of −0.004 (P3), even larger stress concentration is observed near the pore than that at loading levels P1 and P2. Afterwards, when unloaded to zero strain (P4) again, stress 351
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Fig. 7. (a) Far-field y-direction stress/strain hysteresis loops (i.e. along the applied loading direction, Fig. 2), and the y-direction stress/strain hysteresis loops concentrated around (b) the spherical pore, (c) ellipsoidal pore, and (d) actual pore for several typical cycles.
three times larger than those around the two idealized pores (3.5 and 5.4 for the spherical and ellipsoidal pore, respectively). Furthermore, the strain amplitude (half width of the hysteresis loop) locally around each pore decreases with increasing number of fatigue cycles, but gradually approaches to a stabilized value after a certain number of cycles (Fig. 7(b)–(d)); all these stabilized values of the local strain amplitude remain much larger than the far-field strain amplitude value (Fig. 7(a)). In contrast, the local stress amplitude increases rapidly with increasing number of cycles. These observations are related to cyclic hardening of Hadfield steel. The concentration factors of stress, strain and strain energy density at cycle 40 are shown in Fig. 8(b). As compared with those in the first cycle 8(a), in the fortieth cycle, the values of Kσ near all the three pores increase while the Kε values decrease slightly, and thus, the corresponding KE values increase. A comparison among different pores demonstrates that the Kσ value around the actual pore, 3.9, is roughly two times larger than those around the idealized pores. And, the values of Kε and KE around the actual pore reach 8.0 and 31, respectively, which are approximately three-to-eight times larger than those around the two idealized pores.
the applied far-field strain. Fig. 7 shows the hysteresis loops of the maximum stress and strain along the y-direction (i.e. along the loading direction, Fig. 2) concentrated around the three pores at fatigue cycles 1, 10, 20, 30 and 40. The corresponding far-field hysteresis loops are also shown for comparison. As it may be expected, the hysteresis loops of the maximum stress and strain around the three pores (Fig. 7(c)–(d)) are much wider and higher than the far-field hysteresis loops (Fig. 7(a)), indicating higher concentrations of both stress and strain around each pore. To get a quantitative comparison of the local stress and strain concentrations around the three pores, the stress and strain concentration factors (Kσ and Kε), as defined as the ratio of the concentrated stress and strain locally around the pore to the far-field stress and strain at the maximum tensile load, are shown in Fig. 8 for cycles 1 and 40. Note that the “local” refers to a region covering a certain distance from the internal surface of the pore, and the stress/strain concentrations may exist exactly at the internal surface and may also exist at a distance away from the internal surface. Additionally, the strain energy density concentration factor (KE), defined as the ratio of the concentrated strain energy density locally around the pore to the far-field value at the maximum tensile load, is also shown in Fig. 8 for cycles 1 and 40. The strain energy density consumed in one cycle, which is often used to represent the damage of material during one-cycle fatigue, is determined from the area of one hysteresis loop. It is observed from Fig. 8(a) that, in the first cycle, stress concentration factor around the actual pore, 1.5, is slightly larger than those around the two idealized pores, the same small value of 1.3. In contrast, the strain concentration factor near the actual pore reaches 13, over three times larger than the ones around the two idealized pores. Furthermore, the local strain energy density factor around the actual pore (as large as 19) is also over
4. Discussion Fatigue failure of components stems primarily from the plastic strain accumulated at stress concentrations, such as at cast pores, during cyclic loading. Therefore, several models have been developed to analyze stress/strain concentrations [10–14]. These models are mostly based on idealizing irregular cast pores into regular ones. Such models are useful; however, they do not consider the irregularity of pore morphology, unable to disclose the realistic local stress/strain behavior. Recently, several authors attempted to consider the real 352
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with the increase of pore volume (Figs. 5 and 6). Furthermore, it is considered that, in the case of multiple pores with a larger porosity and a smaller spacing between neighboring pores, the stress/strain fields around neighboring pores may interact with each other and the maximum stress/strain concentrations may thus increase with the increase of porosity. Given that a large number of various irregular pores exist in a practical cast material, and on the basis of the present study, it is thus necessary to conduct further simulations that consider the entire microstructure or a representative volume of pores, for understanding the entire mechanical behavior by investigating the mutual interactions between various pores and their synergetic effect, although constructions and computations of even more complicated models with a huge number of FE meshes are a tough and timeconsuming work. 5. Conclusions The local cyclic stress/strain behaviors around an actual pore in cast Hadfield high manganese steel were investigated via image-based numerical simulation. The numerical model took into account the real shape of the pore imaged by synchrotron X-ray CT and incorporated both isotropic and kinematic hardening effects, with the parameters obtained by fitting fatigue experiments data, thus being able to realistically simulate the cyclic deformation behavior around actual pores. The simulation results based on the actual pore-containing model were compared with the results based on the idealized pore models. Under a strain-controlled fatigue condition, the evolution of local strain around either an actual pore or an idealized (spherical or ellipsoidal) pore is not synchronized with the variation of the far-field strain applied. As the far-field strain is decreased to zero, large residual stress is generated around each pore and it increases with fatigue loading. The stress concentration factor (Kσ) around either an actual pore or an idealized (spherical or ellipsoidal) pore is always smaller than the concentration factors of strain and strain energy density (Kε and KE). With increasing number of fatigue cycles, Kσ and KE increase whereas Kε decreases slightly. The three concentration factors, Kσ, Kε and KE, around the actual pore are always larger than those around the two idealized pores whatever the number of fatigue cycles. In the first cycle, although the increase in Kσ caused by actual pore is not so significant, the values of Kε and KE around the actual pore reach 13 and 19, respectively, which are over three times larger than those around the idealized pores. At a higher number of fatigue cycles, e.g., at cycle 40, the Kσ value around the actual pore, 3.9, is roughly two times larger than those around the idealized pores; the Kε and KE values reach 8.0 and 31, respectively, which are approximately three-to-eight times larger than those around the two idealized pores. Further work is necessary to investigate the interactions among various pores and their synergetic effects on the local/general stress/ strain responses of materials under fatigue loading, by creating models with a volume of pores rather than a single one.
Fig. 8. (a) Concentration factors of stress (Kσ), strain (Kε) and strain energy density (KE) around three different pores during fatigue loading (a) in the first cycle and (b) in the fortieth cycle.
shapes of pores in simulating the monotonic mechanical behavior of materials containing irregular pores [20–23]. In contrast, the present work focused on the cyclic stress/strain responses locally around an irregular pore during fatigue loading, via 3D image-based simulation. The simulation methodology not only took into account the actual 3D shape of a cast pore visualized via synchrotron X-ray CT, but also incorporated the combined effects of isotropic hardening and Bauschinger effect evaluated by fatigue experiments, thereby being able to realistically simulate the cyclic deformation behavior around an actual pore. Furthermore, in constructing the actual pore-containing model, a single pore only was extracted from a volume of pores, with the intent to exclude the possible effects of other pores. This was intended to clarify the role of the irregularity of a cast pore, through a comparison of the cyclic deformation behaviors locally around an actual pore and idealized pores. The methodology and the results presented are of importance for micromechanically understanding the effect of an irregular pore on fatigue damage phenomena in pore-containing materials. The present work was directed towards the cyclic deformation behavior locally around a pore under the strain-controlled fatigue condition. Strain-controlled fatigue, i.e. low-cycle fatigue, frequently occurs at high cyclic loads or high stress concentrations where strain is constrained. The methodology is also applicable to simulating the local stress/strain responses around pores at low-stress levels under stresscontrolled fatigue (i.e. high-cycle fatigue) conditions. The results demonstrated that the degrees of stress and strain concentrations around an actual pore are significantly larger than those around an idealized pore, regardless of the number of fatigue cycles. It appears that the volume of a single pore does not have an obvious effect on the maximum value of the local stress/strain concentrations. However, the sizes of higher-stress/strain regions increase obviously
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