Axial-torsional high-cycle fatigue of both coarse-grained and nanostructured metals: A 3D cohesive finite element model with uncertainty characteristics

Engineering Fracture Mechanics 195 (2018) 30–43

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Axial-torsional high-cycle fatigue of both coarse-grained and nanostructured metals: A 3D cohesive finite element model with uncertainty characteristics ⁎

T

⁎

Q.Q. Suna, X. Guoa,b,c, , G.J. Wengd, G. Chene, , T. Yanga a

School of Mechanical Engineering, Tianjin University, Tianjin 300072, China Tianjin Key Laboratory of Nonlinear Dynamics and Control, Tianjin 300072, China c State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an 710049, China d Department of Mechanical and Aerospace Engineering, Rutgers University, New Brunswick, NJ 08903, USA e School of Chemical Engineering and Technology, Tianjin University, Tianjin, China b

A R T IC LE I N F O

ABS TRA CT

Keywords: Fatigue life Cohesive element Damage evolution Nanograined layer Scatter of fatigue results

In this study the combined axial–torsional fatigue life and damage evolution of both coarsegrained (CG) and nanostructured metals are modeled by a 3D cohesive finite element method with uncertainty characteristics. To account for the random nature of metal fatigue, we combine the Monte Carlo simulation with the three-parameter Weibull statistical distribution function. For both CG and nanostructured metals, we find that the axial load levels have greater effects than random fields on the amplitude of specimen rotation. Compared with the CG metals, the nanostructured metals are found to exhibit an improved fatigue resistance, for the reason that their damage process initiates from the subsurface beneath the nanograined layer and then extends to the exterior surface. Good agreements between the numerical results and experimental data are also observed. It shows the applicability of the 3D cohesive finite element method for the analysis of damage evolution and prediction of fatigue life in these two classes of metals.

1. Introduction The fatigue behavior of structural components is known to be strongly influenced by uncertainties. Nominally identical specimens under the same load can have extensive scatter in their fatigue life. The scatter of fatigue results can be attributed to the internal defects of the specimens, unavoidable wear, and machining errors, etc. In order to effectively simulate the scatter of fatigue results, the uncertainty factors should be taken into account in a quantitative manner. In most cases, probabilistic models can be invoked to consider multiple sources of randomness. Monte Carlo simulation (MCS), a widely-accepted probabilistic method, directly takes into consideration the statistical distribution and can be used in the situation with a large number of random variables. Based on the probability density function, it is able to simulate the randomness of material parameters and establish statistical models. For instance, Beaurepaire and Schuëller performed the MCS to study the variability of material parameters of aluminum alloy [1] and Bahloul et al. conducted the MCS to evaluate the effects of geometric parameters on reliability in fatigue [2]. Based on the weakest-link theory, Weibull proposed an empirical distribution regarded as Weibull distribution [3], which was

⁎ Corresponding authors at: School of Mechanical Engineering, Tianjin University, Tianjin 300072, China (X. Guo); School of Chemical Engineering and Technology, Tianjin University, Tianjin, China (G. Chen). E-mail addresses: [email protected] (X. Guo), [email protected] (G. Chen).

https://doi.org/10.1016/j.engfracmech.2018.03.025 Received 12 December 2017; Received in revised form 16 March 2018; Accepted 22 March 2018 0013-7944/ © 2018 Elsevier Ltd. All rights reserved.

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widely used to study the strength of fiber [4], ceramics [5], concrete [6], cast alloy [7], and the fatigue life of metals [8]. Its cumulative probability function is expressed as follows: V

(

σ − σu m ⎤ σ0 ⎦

P (σ ) = 1−exp ⎡− V ⎣ 0 P (σ ) = 0 for σ ⩽ σu

)

for σ > σu

and (1)

where σ0 is the scale parameter and m is the Weibull modulus, and σu is the threshold stress below which no specimen is expected to fail. V and V0 are the sample volume and the reference volume, respectively. The mean and variance of the Weibull distribution can be determined by σ0 , σu , and m. The Weibull distribution can be introduced into MCS to obtain the scatter of fatigue results and/or the description of material parameters. For instance, Yi et al. have employed it to study the effects of pore population upon the fatigue scatter, and showed that the simulated results were in a good agreement with the experiments [9]. The influence of the critical stress scatter on the regularity of crack arrest front was similarly investigated [10]. These studies have demonstrated that, although MCS needs intensive simulations due to the existence of numerous finite elements, it is an effective approach to the study of fatigue scatter and the randomness of material parameters. The two-parameter Weibull distribution is identical to the three-parameter one with σu = 0. But if the two-parameter distribution is used, there is a probability that the fracture stress of metals is smaller than the yield strength. This is not a desirable feature as the specimen should deform plastically beyond the yield strength [11]. To alleviate this shortcoming, the three-parameter Weibull distribution will be employed and σu is taken as the yield strength σy in this paper. In addition to the scatter of fatigue results, it is important to effectively simulate the fatigue damage evolution. At present, extensive experiments are being conducted [12–14], and various numerical methods are being proposed to simulate the fatigue damage evolution. The virtual crack closure technique (VCCT) is also employed to simulate the fatigue crack growth [15,16]. This technique requires the crack to be preset in the specimen, which restricts VCCT from simulating the initiation of the fatigue crack. Additionally, the extended finite element method (XFEM) is also recognized in the simulation of fatigue crack propagation [17,18]. This method requires the additional degree of freedom to represent the discontinuities and singularities on the crack surface so that there is no need to update the mesh. However, there are some limitations in XFEM applications. For instance, it is generally used in the crack tip and at present it is not able to deal well with the problems of multiple cracks [19,20]. The cohesive finite element method (CFEM) is a versatile one. It does not have the limitations of most of the traditional fracture approaches [21–30]. For instance, Dai et al. successfully used it to study the influence of microstructural characteristics on the fracture behavior of 316LN [30]. Not only is it widely useful to the study of fracture process, it is also effective to the simulation of fatigue damage evolution [31–33]. Along this line, Yamaguchi et al. have used it to predict the fatigue damage progress in fiber-metal laminates and provided good consistency with experimental data [31]. In addition, Chen et al. also employed it to analyze the progressive failure of fiber-metal laminates [32]. However, none of these studies has considered the scatter of fatigue results. This limitation can be removed, as mentioned above, through the introduction of MCS. In this context, we note that, combined with 2D CFEM, the MCS has been used to model fatigue crack initiation and propagation [1]. Based on such a combination, He et al. has considered the uncertainty parameters to simulate the fatigue crack path and fatigue life under mixed mode [34]. Bahloul et al. also integrated the MCS with 2D FEM to predict the fatigue life of the cracked structures after repair [2]. It is fair to say that, due to its simplicity, the 2D model has been employed by most researchers so far. But 3D model is a more precise approach to simulate the complex damage initiation and evolution processes of real structural components. For this reason, we will combine the 3D CFEM with the MCS to investigate the fatigue damage evolution of the metals. The metallic specimen to be considered is depicted in Fig. 1. It shows the geometry of the fatigue specimen in our laboratory. The combined axial-torsional loads with both stress ratios −1 have been applied with an in-phase sinusoidal waveform under the frequency of 1 Hz. 304 stainless steel (SS) is chosen as the material of the CG specimens. Its wide application in engineering is due to its good corrosion and oxidation resistance. However, its application is limited by its low mechanical strength. The nanograined layer (NGL) was created by surface mechanical attrition treatment (SMAT). As a novel severe plastic deformation process, it can realize self-nanocrystallization on the metal surface [35]. For instance, Bahl et al. fabricated nanocrystalline surface on 316L SS by SMAT and found the increased corrosion fatigue strength [36]. Gradient structure generated by the SMAT in Cu and 304SS exhibited good uniform elongation and high yield strength simultaneously [37,38]. In addition, the SMATed specimens were found to have other superior properties [39–41]. Due to the improved mechanical properties of the SMATed metals, the SMAT has been used as our

Fig. 1. The geometry of the experimental specimen. 31

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Fig. 2. Schematic representation of the traction-separation relation, (a) Mode-I, (b) Mode-II(III).

material (304SS) to produce the NGL. Based on the background described above, the purpose of this work is to model the fatigue life and damage evolution of both the CG and SMATed specimens under the combined axial-torsional loads. The modeling is based on the combination of the 3D CFEM with MCS. In Section 2, we briefly outline the CFEM and Chaboche hardening model [42]. In Section 3, the 3D CFE discretization for the CG and SMATed specimens will be given. In Sections 4 and 5, the effects of random fields and axial loads on the torsional rotation history and fatigue life of the CG and SMATed specimens will be presented, respectively. The main conclusions will be drawn in Section 6. 2. Computational framework 2.1. The cohesive finite element method To characterize the mixed-mode fracture behavior, the mode-I, mode-II, and mode-III traction-separation relations, as shown in Fig. 2, have been developed and described independently, and can be linked by [43]:

GI G II G + + III = 1 G IC G IIC G IIIC

(2)

Here, G IC , G IIC , and G IIIC are respectively the Mode I, II, and III critical fracture energies. G I , G II , and G III reflect the corresponding work done by the tractions and their conjugate separation in the normal, first, and second shear directions as shown in Fig. 2, and they are given by [44]:

GI =

∫0

un

∫0

G II =

G III =

Tn (u n )du n

(3.1)

Tt (ut )dut

(3.2)

ut

∫0

us

Ts (us)dus

(3.3)

where Tn , Ts , and Tt represent the normal and the two shear tractions. u n , ut , and us are the normal and two shear displacements. The calculations can be proceeded by computing Eq. (3) for triple modes until Eq. (2) is satisfied; after that, the element is regarded to be no longer capable of bearing a load, and thus it fails. Indeed, the behavior of the cohesive elements has to account for irreversibility of damage. The cohesive stiffness degrades due to damage and unloading occurs linearly at constant stiffness so that traction vanishes when the separation is zero [1]. The cohesive stiffness in the subsequent reloading is same with that in the last unloading. The cohesive damage D is defined as the scalar stiffness degradation (SDEG), which is the function of the effective separation, u m , i.e.,

um = D=

〈u n 〉2 + us2 + ut2

(4.1)

u mf (u mmax−u m0 ) u mmax (u mf−u m0 )

(4.2)

u mmax

where 〈〉 denotes Macaulay brackets such that 〈x〉 = max(0, x). is the maximum effective separation during loading process. u m0 and u mf correspond to the characteristic separation where the damage initiates and critical separation where the cohesive element fails, respectively. Under other types of cohesive law (e.g., trapezoidal one in [23] or trilinear one), the cohesive damage D can also be linked with the separation so that the associated cohesive law can be implemented. The traction-separation relation includes several parameters. Tn0 , Tt0 , and Ts0 are the cohesive strength in the normal and two shear directions. The critical fracture energies can be obtained from 2 GiC = K iC (1−υ2)/E (i = I, II)

(5.1) 32

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Table 1 Properties of the cohesive elements in combined axial-torsional model.

CG NGL

σy (MPa)

Tn0 (MPa)

Ts0 (MPa)

Tt0 (MPa)

G IC (J m−2)

G IIC (J m−2)

G IIIC (J m−2)

280 1278

562 2555

320 1456

320 1456

25,593 5896

23,443 5400

14,588 3360

2 G IIIC = KIIIC (1 + υ)/ E

(5.2)

in terms of its mode-I, mode-II, and mode-III fracture toughness KIC , KIIC , and KIIIC . Young’s modulus E and Poisson’s ratio υ of the CG CG = 75 MPa m [46]. Because the NGL of and the NGL are taken as the same, namely, 200 GPa and 0.33 [45]. For the CG 304 SS, KIC NGL is taken as 36 MPa m . KIIC and KIIIC can be determined by the strain energy density factor the 304 SS is strong but less ductile, KIC criterion [47]. We model the cohesive strength as random fields generated by Weibull distribution, and take the failure crack opening displacement u nf as constant. Tn0 , Ts0 , and Tt0 denote the mean of the cohesive strength in the normal and the two shear directions, respectively. Properties of the cohesive elements are listed in Table 1. The initial tensile stiffness k n0 together with shear stiffness k t0 and k s0 should be large enough to restrict the mesh dependence, but if they are too large, numerical ill-conditioning can occur [48–50]. In this paper, the initial stiffness can be determined as 2 × 1015 N/m . In addition, the cohesive strength can be calibrated by comparing the prediction with the experimental results. 2.2. Chaboche hardening model In order to adequately describe the elastoplastic behavior under cyclic load, it is necessary to take the combined hardening of metals into consideration. The Chaboche hardening model has been found to well predict both monotonic and cyclic behavior of metals [42,51,52]. In this paper, the Chaboche hardening model is employed for the specimens under cyclic axial-torsional loads. It describes the cyclic plastic behavior by dividing backstress into several components, i.e., 3

α=

∑

αi

(6.1)

i=1

dαi =

2 Ci dε p−γi αi dp 3

(6.2)

dε p

where α is the backstress and i represents the number of the backstress. is the increment of plastic strain tensor and dp the magnitude of the plastic strain increment. Ci is the initial kinematic hardening modulus, and γi determines the rate at which the kinematic hardening modulus decreases with increasing plastic deformation [44]. The Chaboche model parameters of CG 304 SS can be obtained [51], while the parameters of SMATed 304 SS can be calibrated by the approach in [44]. All of the parameters are listed in Table 2. 3. Finite element discretization and damage interpolation technique The combined axial-torsional fatigue experiments of the hollow thin-walled tube with the thickness of 1 mm have been conducted. As shown in Fig. 3a, the fatigue cracks of the specimen are mainly circumferential, longitudinal, or oblique. In order to simulate the damage evolution process, the 3D CFE geometric model is necessary. The present model takes the calculation burden into account so that we select half of the experimental CG/SMATed specimen without fixture part and transitional arcs in the axial direction. The CFE model consists of conventional bulk elements and cohesive elements. Fig. 3b depicts the insertion results. On one hand, we divide each hexahedron into four wedge elements (C3D6 in [44]). On the other hand, cohesive elements are inserted along the boundary of bulk elements. Specifically, as illustrated in Fig. 4, 3D 8-noded cohesive elements (COH3D8) are inserted between the adjacent wedge elements in the axial and circumferential directions, as marked in green, while 3D 6-noded cohesive elements (COH3D6) are inserted between the adjacent wedges in the radial direction, as marked in gray. According to the above mesh discretization approach, the propagation of multiple cracks can be simulated by this 3D CFE geometric model. Furthermore, the geometric modeling approach of the SMATed specimen (see Fig. 5) is the same. The number of the discretized elements of both models is listed in Table 3. More general and accurate approach for mesh discretization in 3D involves 18 propagation directions for each crack tip so that the computational load is too intensive. Our current approach maintains a balance between results accuracy and computational load. Table 2 Parameters of the Chaboche hardening model.

CG NGL

C1 (MPa)

γ1

C2 (MPa)

γ2

C3 (MPa)

γ3

115,000 110,300

3700 7348

65,000 60,000

600 820

1310 6150

2.75 2.75

33

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Fig. 3. Cohesive elements insertion, (a) The experimental fatigue failure morphology, (b) Cohesive elements insertion approach.

Fig. 4. CFE model for the CG specimen.

The damage evolution can be simulated cycle by cycle, which leads to time-consuming computation for high-cycle fatigue analysis. Therefore, extrapolations can be formulated by cycle jump to reduce the computational load [1,53,54]. The damage variable D N can be extrapolated from the current cycle N to the next N+ΔN cycles [44], as follows:

DN + ΔN ≈ DN +

dDN ΔN dN

(7)

Whether the cycle jump ΔN is acceptable can be judged by whether a satisfactory convergence is obtained [55]. Munoz et al. [56] found that the convergence was reached when ΔN was smaller than 500 and that the divergence occurred when ΔN was 2000. Similar studies were also conducted by Guo et al. [57]. In this paper, several typical levels of ΔN are selected to compare the convergence of fatigue life so that the credible ΔN can be obtained. Fig. 6 depicts the stress amplitude Δσ versus the simulated fatigue life under different ΔN. Better convergence occurs when ΔN is smaller than 500; divergence occurs when ΔN exceeds 500. The extrapolation scheme needs compromise between the solution accuracy and computational efficiency. Therefore, ΔN is chosen as 200 at stress amplitude levels larger than 300 MPa and 500 at stress amplitude levels smaller than 300 MPa. 34

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Fig. 5. CFE model for the SMATed specimen. Table 3 The number of the different elements for the CG and SMATed 304SS.

CG SMATed

C3D6

COH3D6

COH3D8

Total elements

7344 14,688

3672 11,016

10,914 21,828

21,930 47,532

Fig. 6. Effects of ΔN on the fatigue life.

4. Simulated results for the CG specimens 4.1. Effects of cohesive strength on the fatigue life Due to the existence of unavoidable defects in the specimen, the cohesive strength will change spatially, which leads to its uneven distribution [58]. In this paper, Weibull distribution is used to simulate the cohesive strength. For CG specimens, both the stress ratio is -1 and three levels of axial stress, namely, 9304 N (the stress amplitude Δσ is 344 MPa), 8588 N (Δσ = 318 MPa), and 5725 N (Δσ = 212 MPa), with the same torque 25 Nm (the shear stress amplitude is 237 MPa), are applied. In Eq. (1), we take the Weibull modulus, m, as 50 [59,60]. Five random fields of the cohesive strength can be generated. As illustrated in Fig. 7, different random fields have the same mean cohesive strength of 562 MPa and the same standard deviation of 11 MPa. Some distinctions exist in the different random fields. Specifically, different random fields indicate various weakened zones, corresponding to various distributions of the fatigue sources, and can be used to simulate the scatter of the fatigue life. There is the similar phenomenon for SMATed specimens. Fatigue life is sensitive to the cohesive strength. Therefore, the accurate cohesive strength has great significance to predict the fatigue life and it needs to be calibrated during using the CFEM. For instance, Ghovanlou et al. assumed cohesive strength to be 4 times of fatigue limit and found a good agreement with experimental results [53]. Guo et al. investigated 3D fatigue crack propagation and torsional fatigue life of the nanostructured metals and found that 3.25 σy was suitable for cohesive strength [57]. 35

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Fig. 7. Weibull random fields: (a) No. 1, (b) No. 2, (c) No. 3, (d) No. 4, and (e) No. 5.

Furthermore, Yu et al. studied the hydrogen embrittlement and cohesive strength was taken as 2 σy [61]. Recently, García-Collado et al. discussed plasticity-induced crack closure and 1.4 σy was achieved for the cohesive strength [62]. In order to determine a relationship between the cohesive strength and the yield strength, we also study the influence of cohesive strength on fatigue life. The above combined axial-torsional loads are applied to the CG specimens. The mean cohesive strength Tn0 is taken as 1.95 σy , 2.0 σy , and 2.07 σy , and the corresponding Ts0 = Tt0 are assumed as Tn0/ 3 . As shown in Fig. 8, the simulated fatigue life is smaller than the experimental results when Tn0 is 1.95 σy . But the larger results are obtained when Tn0 is 2.07 σy . Therefore, as listed in Table 1, Tn0 is calibrated as 2.0 σy , at which the simulated results are consistent with the experimental results. Furthermore, when Tn0 is taken as 2.0 σy for the CG elements in the nanostructured specimens, the cohesive strength of the NGL is calibrated as twice of its yield strength. 4.2. Effects of random fields on the torsional rotation history and fatigue life This paper puts more emphasis on the damage evolution process of the specimen. The torsional rotation history and damage dissipation diagram can be shown under the same horizontal axis which represents the period of the last damage evolution process. By fixing the torque 25 N m and the axial load 8588 N, we explore the torsional rotation history under three random fields. The rotation of metal with isotropic hardening increases linearly with tiny fluctuations and changes abruptly only in the failure stage [57], which is not consistent with the experimental observation. On the other hand, with the Chaboche hardening model, the

Fig. 8. Effects of the cohesive strength on the fatigue life for CG specimens. 36

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Fig. 9. The torsional rotation history and damage dissipation associated with No. 3 random field.

simulated torsional rotation history and damage dissipation diagram associated with No. 3 random field in Fig. 9 shows a good agreement with the experimental results. Note that the simulated rotation has a “step jump” in every 7 s. “7 s”, an artificial numerical parameter, equals the cycle time in every analysis step, and is taken mainly from the perspective of reducing the computational load. Every “step jump” of rotation corresponds to the “step jump” of damage dissipation. Therefore, it can be ascribed to damage accumulation in the elements. Fig. 10 illustrates the damage evolution process associated with No. 3 random field. The cohesive damage, SDEG, ranges from 0–1. The SDEG of the elements in Fig. 10 is larger than 0.75. Equivalently, the comparatively severe damage has occurred in those elements. In addition, six 3D damage distribution diagrams in Fig. 10 successively correspond to six “step jump” of the rotation and damage dissipation in Fig. 9. As shown in Fig. 10, the number of damaged elements increases at each “step jump”. Therefore, this validates the “step jump” of rotation in our numerical model and also demonstrates that it can effectively simulate the damage evolution process of the specimen. The damaged elements initiate from the exterior surface of the specimen and their distributions depend on the random field of cohesive strength. In these cases, the shear stress has larger influence on the damage of the specimens than the axial stress because the damaged elements are mainly longitudinal. With the cycle number increasing, the number of damaged elements along the circumferential, longitudinal, or oblique directions gradually increases, and some of the damage elements occur in the interior of the specimen, which results in the final failure. Fig. 11 depicts the torsional rotation history associated with three random fields. The rotation amplitudes are almost the same, while its mean value deviates. The similar phenomenon also occurs in the experimental results. The different random fields have slight effects on the amplitudes of the rotation. The differences between the mean of the rotation lie in a fact that the damage evolution processes vary with random fields. Therefore, the fatigue life is different under different random fields. The damage dissipation increases fast in the final failure, as shown in Fig. 12.

Fig. 10. The fatigue damage progress of the CG specimen at each “step jump”. 37

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Fig. 11. The torsional rotation history associated with three random fields.

Fig. 12. The damage dissipation associated with three random fields.

4.3. Effects of axial loads on the torsional rotation history and fatigue life By fixing both the stress ratios as −1 and the torque at 25 N m, we explore the torsional rotation history associated with No. 2 random field under three axial loads, namely, 5725 N, 8588 N, and 9304 N. Fig. 13 illustrates that the amplitude of the torsional

Fig. 13. The torsional rotation history associated with No. 3 random field under three axial loads. 38

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Fig. 14. The damage dissipation associated with No. 3 random field under three axial loads.

rotation under higher axial stress is larger, especially in the final failure stage. In this case, higher axial stress makes the specimen easier to damage, which leads to the larger amplitude of the rotation. On the contrary, the lower axial stress results in the smaller amplitude of the rotation. The damage dissipation under the above three axial loads is shown in Fig. 14. The higher axial stress leads to the faster accumulation of the damage dissipation, larger damage dissipation in the final failure, and more severe damage, which leads to the shorter fatigue life. The opposite observations can be obtained under the lower axial stress at which the fatigue life is longer. These phenomena are in agreement with the experiments that the fatigue life changes with the axial stress.

5. Simulated results for the SMATed specimens 5.1. Effects of random fields on the torsional rotation history and fatigue life The subsequent torsional rotation history and damage dissipation diagrams have the same horizontal axis which also represents the period of the last damage evolution process. The same torque 28 N m (the shear stress amplitude is 266 MPa) and axial stress 8588 N (the stress amplitude Δσ is 318 MPa) are exerted on the SMATed specimens with four random fields. Effects of the random fields on the torsional rotation history and fatigue life are investigated. Fig. 15 displays the torsional rotation history and damage dissipation associated with No. 5 random field. The simulated results have a good agreement with the experiments. The cycle time in every analysis step is also taken as 7 s. It can also be found from Fig. 15 that the damage dissipation has the same “step jump” phenomenon and the moment of “step jump” is identical to that in the rotation. A sudden increase in damage dissipation implies that the damage is further accumulated so that the cohesive strength keeps decreasing, which leads to the “step jump” in the rotation. The SDEG is also used to evaluate the damage of the SMATed specimens. The severe damaged elements (i.e., SDEG is larger than 0.8) associated with No. 5 random field are shown in Fig. 16. Each sub-figure in Fig. 16 corresponds to the “step jump” of the torsional rotation and damage dissipation in Fig. 15. Therefore, this indicates that our model can effectively keep track of the damage evolution process of the SMATed specimen. As shown in Fig. 16a-e, damaged elements occur only in the CG layer and not in the NGL. But as the cycle number increases, the elements in the NGL get damaged in Fig. 16f. Fig. 16g illustrates that the damage gradually extends to the exterior of the specimen along the circumferential, longitudinal, or oblique directions, so that the specimen eventually

Fig. 15. The torsional rotation history and damage dissipation associated with No. 5 random field. 39

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Fig. 16. The damaged elements of the SMATed specimen associated with No. 5 random field.

fails. Equivalently, the damage initiates from the subsurface and extends to the exterior surface. Due to the SMAT, the strength of the exterior surface is improved, which enhances its fatigue resistance and suppresses the initiation of the micrcracks on the exterior surface. On the other side, the similar phenomena have also been found in the experimental study. For instance, Yang et al. studied cyclic deformation of the Cu specimens with gradient NGL and found the fatigue cracks initiated in the subsurface and then propagated to the exterior surface [63]. Shortly after, Wang et al. observed that compared with CG Ti, the surface nanocrystallized Ti exhibits an improved fatigue life and the initiation locations of the fatigue cracks transform from the free surface to the subsurface [64]. Severely-damaged zones (the orange and red areas) in Fig. 16 have a larger possibility that the main crack initiates and their expansion represents the main crack propagation. However, it is still a great challenge to simulate the arbitrary propagation paths of 3D fatigue crack, especially the unstable failure shown in Fig. 3a. The direct comparison of the simulation with the experimental fractographs will be a target for our further study. As illustrated in Fig. 17, the simulated fatigue life of the SMATed specimens is in a reasonable agreement with the experiments. Fig. 18 shows the torsional rotation history associated with four random fields. The amplitudes of the rotation are almost identical. However, the mean differs modestly from each other. The simulated results agree well with the experiments.

Fig. 17. Simulated fatigue life of the SMATed specimens. 40

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Fig. 18. The torsional rotation history of the SMATed specimens in four random fields.

5.2. Effects of axial loads on the torsional rotation history and fatigue life By fixing both the stress ratios as −1 and the torque 28 N m, we explore the torsional rotation history associated with No. 1 random field under three axial loads, namely, 9304 N (Δσ = 344 MPa), 8588 N (Δσ = 318 MPa), and 7157 N (Δσ = 265 MPa). As observed in Fig. 19, the amplitude of the torsional rotation under higher axial stress is larger. In this case, higher axial stress makes the specimen easier to damage, which leads to the larger amplitude of the rotation. On the other hand, the lower axial stress results in the smaller amplitude. The damage dissipation under the above three axial loads is shown in Fig. 20. In the final failure stage, the higher the axial stress, the larger the damage dissipation, which indicates that the specimens have larger damage and shorter life. 6. Conclusions The scatter of fatigue results including the fatigue life and the intrinsic damage evolution process occur when the fatigue issue is involved. In this paper, the high-cycle fatigue behavior of 304 SS after the SMAT has been investigated. The 3D CFEM together with the MCS has been applied to model the fatigue life and damage evolution of the specimens under combined axial-torsional loads. In order to describe the complicated plastic behavior of metals, it is reasonable to use the Chaboche hardening model. The main conclusions can be drawn as follows: 1. For both the CG and SMATed specimens, by modeling spatially-varying fracture properties as random fields, different fatigue sources and thus the scatter of the fatigue life can be acquired. In addition, the axial load level significantly influences the amplitude of the torsional rotation. The simulated results of torsional rotation history and fatigue life for both the CG and SMATed specimens agree well with the experiments. Therefore, the proposed 3D CFE model together with the MCS can be validated in investigating fatigue damage evolution. 2. Accurate cohesive strength has great significance to predict the fatigue life. Therefore, it needs to be calibrated in the simulation of

Fig. 19. The torsional rotation history associated with No. 1 random field under three axial loads. 41

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Fig. 20. The damage dissipation associated with No. 1 random field under three axial loads.

the fatigue failure. It has been found that with the well-calibrated cohesive strength, the simulated fatigue life is consistent with the experiments. 3. For the SMATed specimens, by observing the 3D damage evolution process, we have found that the damage initiates from the subsurface and extends to the exterior surface. Therefore, the fatigue resistance and the fatigue life of the nanostructured metals generated by SMAT can be improved. Acknowledgments This research is supported by National Natural Science Foundation of China (Project no. 11372214) and the opening project of State Key Laboratory of Explosion Science and Technology (Beijing Institute of Technology) (Project no. KFJJ17-10M). Weng thanks the support of NSF Mechanics of Materials Program under CMMI-1162431. Chen acknowledges the financial support from the National Natural Science Foundation of China (Project no. 51471116). References [1] Beaurepaire P, Schuëller GI. Modeling of the variability of fatigue crack growth using cohesive zone elements. Eng Fract Mech 2011;78(12):2399–413. [2] Bahloul A, Ahmed AB, Mhala MM, Bouraoui CH. Probabilistic approach for predicting fatigue life improvement of cracked structure repaired by high interference fit bushing. Int J Adv Manuf Tech 2017;91:2161–73. [3] Weibull W. A statistical theory of the strength of materials. In: Proceedings of The Royal Swedish Institute for Engineering Research; 1939. [4] Peters PWM, Hemptenmacher J. Oxidation of the carbon protective coating in SCS-6 fibre reinforced titanium alloys. Compos Part A 2002;33(10):1373–9. [5] Studart AR, Filser F, Kocher P, Gauckler LJ. In vitro lifetime of dental ceramics under cyclic loading in water. Biomaterials 2007;28(17):2695–705. [6] Saucedo L, Yu RC, Medeiros A, Zhang XX, Ruiz G. A probabilistic fatigue model based on the initial distribution to consider frequency effect in plain and fiber reinforced concrete. Int J Fatigue 2013;48(1):308–18. [7] Eisaabadi GB, Davami P, Kim SK, Tiryakioğlu M. The effect of melt quality and filtering on the Weibull distributions of tensile properties in Al–7%Si–Mg alloy castings. Mater Sci Eng A 2013;579:64–70. [8] Ramsamooj DV, Shugar TA. Reliability analysis of fatigue life of the connectors—the US Mobile Offshore Base. Mar Struct 2002;15(3):233–50. [9] Yi JZ, Gao YX, Lee PD, Flower HM, Lindley TC. Scatter in fatigue life due to effects of porosity in cast A356–T6 aluminum-silicon alloys. Metall Mater Trans A 2003;34(9):1879–90. [10] Berdin C. 3D modeling of cleavage crack arrest with a stress criterion. Eng Fract Mech 2012;90:161–71. [11] Tiryakioğlu M, Campbell J. Weibull analysis of mechanical data for castings: a guide to the interpretation of probability plots. Metall Mater Trans A 2010;41(12):3121–9. [12] Xiong Y. Microstructure damage evolution associated with cyclic deformation for extruded AZ31B magnesium alloy. Mater Sci Eng A 2016;675:171–80. [13] Zhao M, Fan X, Wang TJ. Fatigue damage of closed-cell aluminum alloy foam: Modeling and mechanisms. Int J Fatigue 2016;87:257–65. [14] Fatemi A, Molaei R, Sharifimehr S, Shamsaei N, Phan N. Torsional fatigue behavior of wrought and additive manufactured Ti-6Al-4V by powder bed fusion including surface finish effect. Int J Fatigue 2017;99:187–201. [15] Collini L, Pirondi A. Fatigue crack growth analysis in porous ductile cast iron microstructure. Int J Fatigue 2014;62(2):258–65. [16] Liu W, Yao X, Ma Y, Chen X, Guo G, Ma L. Prediction on fatigue life of U-notched PMMA plate. Fatigue Fract Eng M 2016;40(2):300–12. [17] Bergara A, Dorado JI, Martín-Meizoso A, Martínez-Esnaola JM. Fatigue crack propagation in complex stress fields: Experiments and numerical simulations using the Extended Finite Element Method (XFEM). Int J Fatigue 2017;103:112–21. [18] Shu Y, Li Y, Duan M, Yang F. An X-FEM approach for simulation of 3-D multiple fatigue cracks and application to double surface crack problems. Int J Mech Sci 2017;130:331–49. [19] Song JH, Wang H, Belytschko T. A comparative study on finite element methods for dynamic fracture. Comput Mech 2008;42(2):239–50. [20] Bendezu M, Romanel C, Roehl D. Finite element analysis of blast-induced fracture propagation in hard rocks. Comput Struct 2017;182:1–13. [21] Guo X, Leung AYT, Chen AY, Ruan HH, Lu J. Investigation of non-local cracking in layered stainless steel with nanostructured interface. Scr Mater 2010;63:403–6. [22] Chen YL, Liu B, He XQ, Huang Y, Hwang KC. Failure analysis and the optimal toughness design of carbon nanotube-reinforced composites. Compos Sci Technol 2010;70:1360–7. [23] Shao Y, Zhao HP, Feng XQ, Gao HJ. Discontinuous crack-bridging model for fracture toughness analysis of nacre. J. Mech Phys Solids 2012;60:1400–19. [24] Guo X, Weng GJ, Soh AK. Ductility enhancement of layered stainless steel with nanograined interface layers. Comput Mater Sci 2012;55:350–5. [25] Saucedo L, Yu RC, Ruiz G. Fully-developed FPZ length in quasi-brittle materials. Int J Fract 2012;178:97–112.

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