3D cohesive modeling of nanostructured metallic alloys with a Weibull random field in torsional fatigue

International Journal of Mechanical Sciences 101-102 (2015) 227–240

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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci

3D cohesive modeling of nanostructured metallic alloys with a Weibull random field in torsional fatigue X. Guo a,b,n, T. Yang a, G.J. Weng c a

School of Mechanical Engineering, Tianjin University, Tianjin 300072, China Tianjin Key Laboratory of Nonlinear Dynamics and Chaos Control, Tianjin 300072, China c Department of Mechanical and Aerospace Engineering, Rutgers University, New Brunswick, NJ 08903, USA b

art ic l e i nf o

a b s t r a c t

Article history: Received 25 March 2015 Received in revised form 16 July 2015 Accepted 4 August 2015 Available online 14 August 2015

The cohesive finite element method together with Monte Carlo simulation for nanostructured metallic alloys with random fracture properties is developed to study the 3D fatigue crack propagation and torsional fatigue life. Three-parameter Weibull distribution is used to characterize the spatially random cohesive strength and fracture energy. The proposed model also considers the effects of thickness and different treatment of the nanograined layer (NGL) on the fatigue life. It is shown that the model can predict realistic crack patterns and reasonable fatigue life. The simulated fatigue cracks are mainly circumferential or oblique at an angle and they are in good agreement with the experimentally observed fracture patterns. Both different random fields and loads have significant effects on the crack initiation, crack pattern, and fatigue life. It is found that this layer plays a very important role in improving the fatigue life. As the layer thickness increases, the torsional fatigue life of the nanostructured metal also increases. The increase is particularly pronounced at high stress levels. We find that the major source of this increase is due to the increased probability for fatigue cracks to initiate from the interior surface of the tubular specimen and then propagate toward the exterior surface. This process has a profound beneficial effect on the fatigue life. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Fatigue life Cohesive element Crack pattern Weibull random field Surface mechanical attrition treatment

1. Introduction The great majority of failures, including fatigue, wear, and corrosion in engineering materials, are very sensitive to microstructure and properties of the material surface. In most cases failures originate from the exterior layers of the work piece. Therefore optimization of the surface microstructure and properties can effectively improve the failure properties and service life of the materials [1]. In recent years various techniques have been developed to produce hard bulk nanocrystalline materials, but difficulties still exist in synthesizing 3D bulk nanocrystalline specimens [2]. As an alternate route, surface mechanical attrition treatment (SMAT) has been developed to generate nanograined layer (NGL). This process uses a large number of smooth spherical steel balls in a reflecting chamber, vibrated by a generator, to impact the specimen surface. The balls could have different sizes to produce different results for the NGL [3]. The SMAT has been successfully applied to produce NGL in a variety of pure metals and alloys [4–6]. After the SMAT, the thickness of the NGL can be more

n Corresponding author at: School of Mechanical Engineering, Tianjin University, Tianjin 300072, China. Tel.: þ 86 22 2740 4934; fax: þ 86 22 8740 1979. E-mail address: [email protected] (X. Guo).

http://dx.doi.org/10.1016/j.ijmecsci.2015.08.006 0020-7403/& 2015 Elsevier Ltd. All rights reserved.

than 50 mm and the overall failure properties can be tremendously enhanced [7]. Chen et al. [8] performed uniaxial tensile tests on the SMATed 316 L austenitic stainless steel (SS) specimens and found that they showed high yield strength. Chen et al. [9] fabricated the layered 304 SS by combining SMAT with warm co-rolling and found that it exhibited both high strength and good ductility. Balusamy et al. [10] induced the formation of NGL on the surface of the Pb–Sn alloy with the aid of the SMAT, and found that corrosion resistance and the charge–discharge characteristics increased significantly. Liu et al. [11] produced NGL by means of the SMAT on tempered steel plate and found that the SMATed specimens exhibited excellent thermal stability. Furthermore, experimental investigations have been carried out to study the effects of the NGL generated by the SMAT on the fatigue behavior of metals. Roland et al. [12] investigated the axial fatigue behavior of 316 L SS and showed that the specimens treated by combining SMAT with a post-annealing had a longer fatigue life than those by SMAT alone. Li et al. [13] conducted axial fatigue tests on the SMATed carbon steel and found that the NGL improved the fatigue strength tremendously. In addition, Yang et al. [14] investigated the axial fatigue behaviors of SMATed Cu specimens and found that the fatigue cracks initiated from the subsurface layer and then propagated to the top surface; this also has a beneficial effect on the fatigue life.

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Nomenclature

T 0s

E DN

Tt T 0t

GI GIC GII GIIC GIII GIIIC K IC K IIC K IIIC K CG IC K NGL IC K m P S Scr T max Tn T 0n Ts

Young's modulus damage variable after N cycles in damage extrapolation technique work done by normal traction mode-I fracture energy work done by the first shear traction mode-II fracture energy work done by the second shear traction mode-III fracture energy mode-I fracture toughness mode-II fracture toughness mode-III fracture toughness mode-I fracture toughness of coarse-grained metal mode-I fracture toughness of nanograined layer initial stiffness of the cohesive element Weibull modulus cumulative distribution function of failure strain energy density factor critical strain energy density factor cohesive strength cohesive traction in the normal direction peak value of the nominal traction in normal direction cohesive traction in the first tangential direction

Although there are considerable investigations on the axial fatigue of SMATed metals, studies on torsional fatigue are sparse [15]. Most of the torsional fatigue research still focuses on raw metals. For instance, Stanzl-Tschegg and Mayer [16] conducted the ultrasonic torsional fatigue on high strength Al alloys to investigate the frequency dependence of high cycle fatigue; they found that different frequencies led to almost the same fatigue life. On the other hand, Kwon et al. [17] compared the pure torsional low-cycle fatigue properties of the virgin and artificially degraded (aged at 430 1C for 3600 h and quenched in water at room temperature) duplex SS and found that the transitional fatigue life of the degraded SS decreased to about 60% that of the virgin SS. McClaflin and Fatemi [18] investigated torsional fatigue behavior of both solid and thinwalled tubular specimens. They found that a stress gradient existed in solid round specimens while a uniform stress state existed in thin-walled tubes, and recommended the latter to be used in torsional fatigue experiments. Predki et al. [19] conducted some torsional fatigue tests and found that self-heating resulted in a lower damping capacity and that the loading frequency had a strong effect on the damping characteristics and a weak effect on the fatigue life. Ohkawa et al. [20] investigated the notch effect under cyclic torque for a coarse-grained (CG) austenitic SS, and found that the fatigue life of the notched specimen was longer than that of the smooth one. Similar study was also conducted by Gladskyi and Fatemi [21]. These torsional studies have shed some light on the nature of torsional fatigue, but their outcomes are still not very conclusive to provide a unified view. Testing facilities for torsional loading are very limited, but torque is still a very dominant loading mode for many industrial components. For instance, automotive transmission shafts are always subject to torque. Their torsional fatigue properties are of paramount importance to the safety and long-life operation of automobiles. The combination of limited testing facilities and this technological importance has indeed prompted us to undertake this numerical study. With the focus on the SMATed nanostructured alloys, our goal is to shed some insights into their crack initiation and propagation processes, and on how their superior fatigue life could be obtained.

V V0

δn δs δt

ΔN σ0 σu σy υ BEM CFEM CG MCS NGL SMAT SS XFEM

peak value of the nominal traction in the first shear direction cohesive traction in the second shear direction peak value of the nominal traction in the second shear direction sample volume reference volume crack opening displacement in the normal direction crack opening displacement in the first shear direction crack opening displacement in the second shear direction extrapolated forward cycles material parameter threshold stress yield strength Poisson's ratio boundary element method cohesive finite element method coarse-grained Monte Carlo simulation nanograined layer surface mechanical attrition treatment stainless steel extended finite element method

We shall adopt the cohesive finite element method (CFEM) in this investigation. The CFEM has proven to be an effective tool in investigating the fracture process during cyclic loading of structural materials [22]. For instance, in order to accurately simulate the response of the plastic zones and the stress fields, Wang and Siegmund [23] used the CFEM to investigate the constraint effects of fatigue crack growth in multi-layered structures. Abdul-Baqi et al. [24] simulated the fatigue damage in a solder bump, and found that the CFEM well described the main damage characteristics. Maiti and Geubelle [25] presented a cohesive model to simulate the fatigue crack propagation in polymeric structures subject to mode-I cyclic loading, and showed that the cohesive model successfully captured the fatigue response of epoxy. Munoz et al. [26] employed the CFEM to simulate the delamination in laminated composites under fatigue loading, and pointed out that relatively large cohesive elements could lead to significant oscillations in crack growth rate. Ural et al. [27] implemented cohesive analysis to discuss the crack growth process and the blocking effects under axial fatigue and obtained good agreement with experimental observations. These investigations have pointed to the reliability of CFEM. But it must be cautioned that scattering of fatigue test data is a common phenomenon, so it is desirable to introduce some probabilistic approach into the CFEM to account for such scattering. This is an inevitable outcome because of the internal defects of the specimen, unavoidable wear, processing errors, etc. In our recent experimental investigations [28], we found that fatigue test data scattered for a group of nominally identical specimens under the same loading condition. To our best knowledge, such statistical treatments have never been implemented in previous CFEM studies on fatigue. For this reason, our CFEM will be complemented with a statistical analysis to fully account for the scattering nature of torsional fatigue problems. Of course random statistical distributions have been widely used to describe the heterogeneous nature such as those occurring in microstructures and material defects. For instance, using Voronoi cells and Delaunay triangulation, Al-Ostaz et al. [29]

X. Guo et al. / International Journal of Mechanical Sciences 101-102 (2015) 227–240

presented a framework to study the local stress field of the microstructure in composites. Sfantos and Aliabadi [30] proposed a Poisson–Voronoi tessellation method to simulate the intergranular micro-fracture evolution in polycrystalline brittle materials with random distribution of grain orientations. For the nonlinear fracture problems, Monte Carlo simulation (MCS) is a very desirable approach to estimate the uncertainty and reliability of structures, but it is more difficult to introduce due to the high nonlinearity and heavy computational cost associated with generating a large number of finite element meshes [31]. So far CFEM combined with MCS has been successfully employed to study the fracture behavior. For instance, Yang and Xu [32] developed a heterogeneous cohesive crack model by MCS to predict macroscopic strength of concrete specimens based on remeshing procedure. Using the MCS, Yang et al. [33] simulated complex 2D crack propagation in quasi-brittle materials and found that the random model could predict complicated fracture processes. Shortly after, Su et al. [31] extended the above analysis to 3D cases, and showed that the 3D model could predict more realistic fracture surfaces and higher structural load-carrying capacity compared to the 2D model. The MCS not only can be used to investigate the fracture process, but can also be effectively employed to analyze the fatigue failure process. Based on the MCS, Cetin et al. [34] proposed a theoretical statistical model to describe the random defects distribution in both smooth and notched specimens under cyclic loading and successfully predicted the fatigue strength. Pashah and Arif [35] used the MCS together with the 2D FEM to investigate the fatigue crack initiation in the adhesive joint under thermal cyclic stress, with a result that agreed well with experiments. Up to now, 2D models have been adopted by most researchers, but it is known that it cannot accurately simulate the complexity in the inherently-3D phenomena (such as the influence of the thickness of the NGL and the different failure modes) in the fatigue crack propagation. For this reason, a 3D model will be adopted in this investigation. Our fully-reversed torsional fatigue experiments were conducted with a sinusoidal waveform under the frequency of 1 Hz [28]. We shall use the geometry of that specimen in this study, which is shown in Fig. 1. The effect of the diameter of the steel balls in the SMAT on fatigue properties has been compared. Specifically, for the 304 SS which has been SMATed by balls with diameters 2 and 3 mm, the torsional fatigue life improved remarkably, but the 3 mm series gave even better results. In this work we have developed a 3D CFEM model to study the crack initiation, propagation, and torsional fatigue life with the aid of MCS. In Section 2, the CFEM, together with the cohesive law and the quantization of the constitutive parameters, is elaborated. In Section 3, Weibull cumulative distribution function is introduced. In Sections 4, 3D cohesive element modeling for the CG and SMATed specimen is developed. In Section 5, the effects of the random fields and loads on the crack initiation, crack pattern, and fatigue life of the CG specimens are investigated. In Section 6, the dependences of fatigue life on the treatment in SMAT and on the

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thickness of NGL are discussed. Main conclusions are drawn in Section 7.

2. The cohesive finite element method The CFEM is developed to model the material separation process. Many variations have been proposed and successfully applied to predict crack propagation. The CFEM allows the damage initiation/evolution and fracture processes to be modeled explicitly. Moreover, it is an effective tool to model the spontaneous multiple crack initiation, branching, and coalescence, so it has been widely used to investigate brittle, quasi-brittle, and ductile fracture [36–47]. Furthermore, the CFEM has significant advantages compared to other methods including the boundary element method (BEM) and the extended finite element method (XFEM). At present, in using the BEM to model a crack, the main problem arises from the fact that two source points coincide at the same location on the cracked boundary, which leads to a mathematical degeneration of the numerical solution [48]. XFEM encounters some difficulties in modeling nucleation and propagation of multiple microcracks since its major shortcoming is to inject the discontinuity on the basis of a failure criterion and its use of level sets favors the propagation of a single crack [49]. Our focus is on the highly nonlinear problems involved in the cohesive fatigue fracture process with multiple microcracks; in this case the CFEM is more appealing. When the crack initiation site or propagation path is not known in advance, two approaches including the intrinsic and extrinsic CFEM are available. The former embeds cohesive elements along the boundaries of all volumetric elements [50]. The latter gradually inserts the cohesive elements into the model as fracture develops, based on an extrinsic fracture initiation criteria [51]. Compared to the intrinsic CFEM, it is known that the extrinsic one involves some issues in model implementation and result interpretation [52]. Therefore the intrinsic CFEM is adopted in this paper.

2.1. Failure criterion The CFEM reflects the cohesion of the material by means of the traction–separation relation, which has been used to control the separation of material on both sides of the crack. Many cohesive laws, which specify the constitutive relation between traction and separation, have been developed for different conditions [53]. Our experimental fracture morphology indicated that the specimen had mixed-mode cracks under torsional fatigue loading [28]. To characterize the fracture behavior of the mixed-mode geometries, the mode-I, mode-II, and mode-III traction–separation relations have been developed and described independently, but linked through the simple mixed-mode failure criterion [54–57] GI GII GIII þ þ ¼1 GIC GIIC GIIIC

ð1Þ

Here GIC , GIIC , and GIIIC represent the total areas under the traction–separation curve for individual mode. The mode-I and mode-II(III) traction–separation relations are shown in Fig. 2(a and b), respectively. GI , GII , and GIII denote the work done by the tractions and their conjugate relative displacement in the normal, first, and second shear directions, respectively, and given by [58]

Fig. 1. Geometry of the tubular torsional fatigue specimen (in mm).

GI ¼

Z δn 0

T n ðδn Þdδn

ð2:1Þ

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

GII ¼

GIII ¼

Z δt 0

Z δs 0

T t ðδt Þdδt

ð2:2Þ

T s ðδs Þdδs

ð2:3Þ

where δn , δt , and δs are the normal and two shear displacements, respectively. T n ,T s , and T t represent the normal and the two shear tractions, respectively. The numerical calculations proceeded by computing Eq. (2) for triple modes until the failure condition of Eq. (1) is met and the cracks advance; at this point the element is considered to be no longer capable of bearing a load, and thus it fails. Note that unloading from any point C follows path CO and subsequent reloading follows OC and then CB. Part of the work expended on causing the separation in this regime is irreversible, as illustrated by the hysteresis loop OAC which implies dissipation during the softening. Correspondingly, there is a decrease in the cohesive strength. This is reflected in the elastic reloading along OC and further softening along CB. 2.2. Quantization of constitutive parameters The cohesive law includes several parameters. It is reasonable to assume that only the cohesive strength and the fracture energy play key roles. T max is the cohesive strength, at which damage initiates. G is the fracture energy, i.e., the external work required to create and fully break a unit surface area of the cohesive element. It is obtained from GiC ¼ K 2iC ð1  υ2 Þ=Eði ¼ I; IIÞ and GIIIC ¼ K 2IIIC ð1 þ υÞ=E in terms of its mode-I, mode-II, and mode-III fracture toughness K IC , K IIC , and K IIIC . The Young's modulus E and Poisson's ratio υ of the CG and the NGL are taken as the same as those in Frontan et al. [59], namely, 200 GPa and 0.33. For the CG 304 SS, pffiffiffiffiffi K CG high IC ¼ 75 MPa m [60]. Because the NGL of the 304 SS has the pffiffiffiffiffi strength but low ductility, K NGL is taken as 36 MPa m , an IC intermediate value between the fracture toughness of typical ceramics and that of the CG 304 SS. K IIC and K IIIC can be determined by the strain energy density factor (S) criterion proposed by Sih [61] for the prediction of the load and propagation direction under monotonic/mixed mode loading. This criterion indicates that the direction of crack propagation for the mixed problems of mode-I and mode-II is governed by the critical strain energy density factor Scr . The basic assumption is that crack initiation occurs when the interior minimum of S reaches a critical value Scr . It has been widely employed to study the mixed–mode crack propagation. For instance, Badaliance [62]

used it to investigate the crack propagation speed under mixed fatigue loading. In order to assess the mixed fatigue crack growth quantitatively, Sih and Barthelemy [63] studied the influence of the strain energy density factor on the crack geometry, complex loading, and material properties. Pandey and Pater [64] employed the strain energy density factor approach to characterize the mixed mode fatigue crack growth, and to discuss the effect of crack angle on the fatigue life of the component. Biner [65] also used the same criterion to estimate the crack propagation directions in 304 SS specimens under mixed–mode (mode-I and modeII) loading. This criterion has a significant effect on the fatigue crack growth under mixed loading. Based on it, fracture toughness parameters were determined as [66] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3ð1  2υÞ K IIC ¼ K IC ; namely; K IIC ¼ 0:91K IC 2ð1  υÞ  υ2

K IIIC ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffi 1  2υ K IC ; namely; K IIIC ¼ 0:58K IC

ð3:1Þ

ð3:2Þ

For the simulation of fatigue crack propagation, the range of the cohesive strength (T max ) has been extensively discussed. It can be related to the yield strength (σ y ) of the material. Nguyen et al. [67] proposed a 2D CFEM to resolve the near-tip plastic fields in fatigue loading, and the initial T max was found to be 2:6 σ y . Jiang et al. [68] suggested that T max varied in the range of ð2  3:5Þσ y because a higher T max led to a lower crack propagation speed. Guo el al. [40] employed the CFEM to study the non-localized fracture of the 304 SS and compared the effect of T max on the ductility of the material, and found that the material has a higher ductility when T max was equal to 1.479 times of the yield strength of the NGL. Alvaro et al. [69] developed a 3D CFEM to simulate the crack initiation of hydrogen-induced fracture and found that it was reasonable to take T max as 2 σ y . Kim et al. [70] proposed another 3D CFEM to investigate the fatigue crack propagation in 4130 steel. They took T max as the ultimate strength of the substrate material, (2  3)σ y , which could predict the fatigue life under different stress ratio reasonably. In this paper T max is calibrated by comparing the prediction with the experimental results. In addition, the initial tensile stiffness K n0 and shear stiffness K t0 and K s0 should be large enough to represent the uncracked material, but care must be exercised as too large values may cause numerical ill-conditioning [33,40–44]. The initial stiffness is determined to be K ¼ 2  1015 N=m by trials and errors.

X. Guo et al. / International Journal of Mechanical Sciences 101-102 (2015) 227–240

3. Fracture properties characterized as Weibull random field Weibull distribution is widely used in the analysis of the static strength of ceramics, fibers and composite materials and fatigue life of metallic materials [71]. The Weibull statistical strength theory is based on the probability distribution function and the weakest link principle. It assumes that strength is related to flaws which are randomly distributed in the materials. Our preliminary attempts show that the mean field approach for the data of heterogeneous materials could result in unrealistic crack patterns and fatigue life due to lack of the fatigue sources. These phenomena are inconsistent with experimental observations. Similar with the approach in [31–33], the bulk material properties of each phase are assumed not to be spatially random for simplicity so that this study can focus on the effects of spatially randomly distributed fracture properties on the fatigue life. In this context the cohesive strength T max is modeled by a random field, f and the failure crack opening displacement δn is assumed to be constant. This implies that the fracture energy GiC is also a random field. The cumulative distribution function of failure of the material subject to a stress σ can be represented by     V σ  σu m Pðσ Þ ¼ 1  exp  for σ 4 σ u and Pðσ Þ ¼ 0 if σ r σ u V0 σ0 ð4Þ where σ 0 the material parameter, m the Weibull modulus, σ u the threshold stress below which the failure probability is zero, V the sample volume, and V 0 the reference volume. The variance and mean of the Weibull distribution can be determined by σ 0 , σ u , and m. If σ u is taken as zero, Eq. (4) degenerates into the twoparameter form, which has been used in the random analysis of brittle materials [71]. Random numbers can be generated and the corresponding Weibull random fields of cohesive strength can be obtained by the inverse function of Eq. (4). In the process of generating random numbers, the Weibull modulus, which describes the degree of scatter of the material strength, ranges between 50 and 60 for steels [71,72], so m ¼50 is adopted in this investigation. In the case of a conservative estimate, the threshold stress σ u can be taken to be the yield strength σ y , and the material parameter σ 0 can be taken to be the ultimate strength σ f [71]. The number of random values is equal to that of cohesive elements included in the model. Lastly, the random distribution of yield strength transforms into that of cohesive strength through the corresponding multiple relationship. The bulk material properties such as Young's modulus and Poisson's ratio are assumed to be constant for simplicity. The plastic property is defined as isotropic, and the specific material parameters for the CG specimen are taken from the experimental studies in Frontan et al. [59], while the flow stress of the NGL is estimated from experimental results of the SMATed 316 L SS in Chen et al. [8]. The material parameters are listed in Table 1.

4. Geometric modeling The hollow thin-walled tube with the thickness of 1 mm has been used in our torsional fatigue experiments [28]. On

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observation of the fractured specimens, as shown in Fig. 3(a), we find that the fatigue cracks are mainly circumferential or oblique at an angle. Therefore, a 3D CFE model is developed based on the above experimental observation in this paper, and the frontal view of the insertion process is illustrated in Fig. 3(b). The initial meshes consist of 6-noded wedge elements (C3D6 in [58]). 3D 8-noded cohesive elements (COH3D8) can be inserted between the adjacent wedge elements in the axial direction. 3D 6-noded cohesive elements (COH3D6) can be inserted between the adjacent wedges in the radial direction. Meanwhile, 3D 8-noded cohesive elements (COH3D8) can be inserted between the adjacent wedges in the circumferential direction. This CFE model can allow the crack to propagate along multiple directions. Moreover, from the perspective of saving computing resources, we only select a quarter of the test section of the specimen in the axial direction and remove fixture part and transitional arcs. As shown in Fig. 4, the CG specimen is discretized with  10 k elements. Specifically, the number of the linear wedge elements (C3D6) is  3 k, that of the 8-noded cohesive elements (COH3D8) is  5 k, and that of the 6noded cohesive elements (COH3D6) is  1 k. For the SMATed specimen, in order to investigate the influence of the thickness of the NGL on the crack initiation, crack pattern, and fatigue life, we choose the NGL with the thickness of 20 mm, 40 mm, and 60 mm. The geometrical modeling methodology is the same as that of the CG CFE model. The corresponding CFE model is shown in Fig. 5(a–c). The SMATed specimen with the NGL thickness of 20 mm is discretized with 20 k elements. Specifically, the number of the linear wedge elements (C3D6) is  4 k, that of the 8-noded cohesive elements (COH3D8)  9 k, and that of the 6noded cohesive elements (COH3D6)  6 k. The SMATed specimen with the NGL thickness of 40 mm is discretized with  27 k elements. Specifically, the number of the linear wedge elements (C3D6) is  8 k, that of the 8-noded cohesive elements (COH3D8)  12 k, and that of the 6-noded cohesive elements (COH3D6)  6 k. The SMATed specimen with the NGL thickness of 60 mm is discretized with 35 k elements. Specifically, the number of the linear wedge elements (C3D6) is  10 k, that of the 8-noded cohesive elements (COH3D8) 15 k, and that of the 6-noded cohesive elements (COH3D6)  8 k. The boundary conditions and loading process are consistent with experiments. Specifically, the bottom of the specimen is fixed, while the top is subject to the torque. Set up a reference point and then build a kinematic coupling between the reference point and the node set consisting of all nodes on the top of the specimen by creating coupling constraint in Interaction module of ABAQUS. After that, a torque can be directly applied at the reference point in Load module [58]. ABAQUS/Standard static analysis combined with direct cycle analysis is used to solve the highly nonlinear problems. It is computationally expensive and thus intractable to perform a cycle-by-cycle simulation for a high-cycle fatigue analysis [26,73]. To accelerate the high-cycle fatigue analysis, each increment extrapolates the current damaged state in the material forward over many cycles to a new damaged state after the current loading cycle is stabilized [58]. Specifically, if the damage initiation criterion is satisfied in a material point at the end of a stabilized cycle N, ABAQUS/Standard extrapolates the damage

Table 1 Constitutive parameters for the CG and SMATed 304 SS. Material

σ y ðMPaÞ

T n ðMPaÞ

T s ðMPaÞ

T t ðMPaÞ

GIC ðJ=m2 Þ

GIIC ðJ=m2 Þ

GIIIC ðJ=m2 Þ

CG NGL (2 mm) NGL (3 mm)

280 1278 1533

840 2556 3066

504 1533 1839

504 1533 1839

25059 5774 5774

20599 4815 4815

8615 1964 1964

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Fig. 3. Cohesive elements insertion approach, (a) Experimental observation of fatigue failure and (b) Cohesive elements inserting mode.

Fig. 4. CFE model for the CG specimen.

variable DN , from the current cycle forward to the next increment over a number of cycles ΔN. We can specify the cycle number over which the damage is extrapolated forward in any given increment. Munoz et al. [26] selected three different values of ΔN: 500, 1000, and 2000 cycles. Although different values of ΔN provided almost identical crack propagation speed, no convergence was achieved for the largest value 2000. Naghipour et al. [73] proposed that the cyclic increment needs to be determined properly in order to obtain the converged solution. For very small ΔN, the computational effort would be huge. On the other hand, when ΔN was too large, the predicted damage values for the next loading cycle DN þ ΔN would be rather different from the exact solution, so the ΔN was chosen as 100, 200 or maximum 500 cycles in their numerical simulations [73]. In this paper, in order to obtain the reliable ΔN, it is chosen as 100, 200, 500, 1000, 2000, and 5000 to compare the convergence of the fatigue life, as shown in Fig. 6.

When ΔN is less than 1000, the predicted fatigue life shows better convergence; when ΔN reaches 5000, the fatigue life diverges. Therefore, from the perspective of computational accuracy and efficiency, ΔN is selected as 200 at higher stress level and 1000 at lower stress level.

5. Results and discussion for the CG specimens 5.1. Effect of the cohesive strength T max on the fatigue life As mentioned in Section 2, in order to get the accurate relation between the cohesive strength T nmax and yield strength σ y of the matrix material, T nmax ¼ p 2:7 ffiffiffi σ y , 3.25 σ y , and 3.8 σ y , the corresponding T tmax ¼ T smax ¼ T nmax = 3, are selected to study the effect of T max on the fatigue life under the sinusoidal torque with the amplitude

X. Guo et al. / International Journal of Mechanical Sciences 101-102 (2015) 227–240

233

Fig. 5. CFE model for the SMATed specimen with a thickness of the NGL (a) 20 mm, (b) 40 mm, and (c) 60 mm.

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

Fig. 7. Effect of the cohesive strength T max on the fatigue life.

41 Nm (the corresponding shear stress amplitude is 390 MPa), 39 Nm (the corresponding shear stress amplitude is 371 MPa), and 36 Nm (the corresponding shear stress amplitude is 343 MPa). As shown in Fig. 7, when T nmax is 2.7 σ y , the simulation results are lower than the experimental value, when T nmax is 3.8 σ y , the simulation results are higher than the experimental results. Therefore, T nmax is calibrated as 3.25 σ y . Similarly, the cohesive strength of the NGL is calibrated as twice of yield strength of the nanocrystalline materials. When T nmax is taken as 3.25 σ y , we do a systematic mesh convergence check: (1) refining mesh in Z (height) direction, (2) refining mesh in R (radial) direction, (3) refining mesh in C (circumferential) direction, and (4) refining mesh in both C

(circumferential) and Z (height) directions, by reducing the corresponding characteristic mesh size to one half of the former one. The simulated results of the fatigue life versus shear stress amplitude in different mesh refinements are illustrated in Fig. 8. It shows that the results obtained by the mesh that we use for the CG specimens (termed as “coarse mesh”) are fairly consistent with those obtained by several kinds of refined meshes except when the shear stress amplitude is in a very high level. This observation is consistent with the finding in Yang et al. [33] and Su et al. [31] that the heterogeneous model predicted realistic fracture processes and load-carrying capacity of little mesh-dependence while the homogeneous model led to incorrect crack patterns and load– displacement curves with strong mesh-dependence.

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5.2. Effect of the random field on the crack pattern and fatigue life As mentioned above, we find that when the homogeneous material model is employed, unrealistic crack pattern occurs, as shown in Fig. 9. Due to lack of the fatigue sources, the cohesive elements open and close at the same rate during the loading process, which leads to uniformly-distributed cracks with the same width. During the simulations where the cohesive strength and corresponding fracture energy are modeled by Weibull random distribution, three groups of torque are selected, namely, 41 Nm, 39 Nm, and 36 Nm. All of the torques change with a sinusoidal waveform under the frequency of 1 Hz and the stress ratio of –1. For each torque, five random series are generated. Fig. 10(a–e) illustrates five groups of the Weibull random fields of cohesive strength with a mean 912 MPa and a standard variance 71 MPa. Different random fields show various weak areas and produce various fatigue sources, which lead to the scatter of fatigue life. When the torque is 41 Nm, effects of the different random fields on the crack pattern and fatigue life are compared. The numerical results show that the random material model predicts the more reasonable and realistic crack pattern, and that cracks are mainly circumferential or oblique at 451, 601, or 1201. As shown in Fig. 11, No. 1 random field, as the cycle number increases, small cracks in the weak area gradually initiate, and a few circumferential cracks are linked with the oblique cracks and then merge into major cracks. After the main cracks form, crack propagation in

Fig. 8. Fatigue life versus shear stress amplitude in different mesh refinements.

other areas is suppressed, so it results in crack localization and mode-II cracks slide further. Finally, the specimen fails as the major cracks pass through multiple weak areas. This simulated crack pattern agrees well with the experimental one. In Fig. 12(a– d), the same torque is applied to the material but with different random fields, namely, No. 2, No. 3, No. 4, and No. 5 random field. Different crack pattern is observed. The failure modes are mainly dominated by the mode-I cracks. In general different random fields lead to various internal “defects” in the material, and they effectively simulate the scatter of the fatigue life in the experiments. Fig. 13 illustrates the relation between the rotation and cycle number. When the specimen is in the failure stage, its rotation changes abruptly. In this case the cycle number is defined as the fatigue life. Note that due to the damage extrapolation, we can output only the rotation after each damage extrapolation (ΔN cycles) and we do not have the rotation during each damage extrapolation. This is the reason that the cyclic change of the rotation in individual cycle cannot be seen.

5.3. Effect of the load on the crack pattern and fatigue life By keeping the same random series (No. 1 random field), the effect of different load on the crack pattern and fatigue life can be studied. As shown in Fig. 14(a–c), different loads, namely, 41 Nm, 39 Nm, and 36 Nm, lead to various crack propagation patterns. At a higher torque, there are less small cracks initiated in the local region and thus the major cracks dominate the specimen. In this case, elements in the weak area are subject to larger force and local stress concentrates, the small cracks in other area do not propagate, and the specimen fails rapidly. On the other hand, when a lower torque is applied, some small cracks initiate on the exterior surface of the specimen. In this case, many elements around the weak area are mobilized to resist the load, which makes it possible to initiate more small cracks. Finally, the weaker areas become even weaker and the fracture paths form. Fig. 15 illustrates the relation between the rotation and cycle number under different loads. The overall trend is that decreased torque leads to increased cycle number. As shown in Fig. 16, the shear stress amplitude versus fatigue life obtained from this simulation is in a good agreement with the experiments. The Weibull random fields in Fig. 10(a-e) deviate slightly from the ones with the same mean but a smaller standard deviation of 19 MPa. It has only small quantitative effect but no qualitative effect on the fatigue life.

Fig. 9. Homogeneous crack patterns.

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Fig. 10. Weibull random fields: (a) No. 1 field, (b) No. 2 field, (c) No. 3 field, (d) No. 4 field, and (e) No. 5 field.

Fig. 11. Crack propagation under 41 Nm with No. 1 random field.

6. Results and discussion for SMATed specimens In the experiment [28], 304 SS has been treated by SMAT with 2-mm-diamerer and 3-mm-diameter balls and a treatment time of 15 min. The SMATed specimens exhibit longer fatigue life, especially in the 3 mm series. This difference is related to many factors, such as the flow stress, residual compressive stress, surface roughness, and thickness of the NGL. Moreover, Dai and Shaw [74] found that surface roughness had only slight effect on the fatigue life and that residual compressive stress also had less contribution as compared to NGL in enhancing the fatigue strength. In order to investigate these features, we here focus on the effects of flow stress and thickness of the NGL in both treatments on fatigue life. 6.1. Effect of flow stress on the fatigue life Based on the experimental result that the 3 mm treated series has a longer fatigue life, the flow stress of the NGL in this series is

assigned a 20% increase. In the computations the material parameters are also listed in Table 1. We use linear strain hardening in the plastic constitutive relation. As shown in Table 1, the relation that the cohesive strength of the NGL is twice of the corresponding yield strength is guaranteed for the 2-mm series and the 3-mm series to figure out the effect of the flow stress on the fatigue life. The initial thickness of the NGL is taken as 20 mm. In the SMATed specimens, we refine the surface mesh and assign the properties of nanocrystalline 304 SS to the NGL. The properties of the CG 304 SS are assigned to the matrix material. Three groups of torque are selected, namely, 45 Nm (the corresponding shear stress amplitude is 428 MPa), 43 Nm (the corresponding shear stress amplitude is 410 MPa), and 41 Nm (the corresponding shear stress amplitude is 390 MPa). All of the torques change with a sinusoidal waveform under the frequency of 1 Hz and the stress ratio of 1. For each torque, five random series are generated. We use the same random series (No. 3 random field) under the fatigue load 45 Nm to compare the crack pattern and fatigue life in the specimens

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Fig. 12. Crack patterns under 41 Nm with (a) No. 2 random field, (b) No. 3 random field, (c) No. 4 random field, and (d) No. 5 random field.

Fig. 13. Rotation versus cycle number under 41 Nm.

of 2-mm and 3-mm ball series. Numerical results show that the pattern of crack propagation in the 2-mm series is similar to that in the 3-mm series. The small cracks all initiate from the exterior surface of the specimens, and gradually propagate. The increased flow stress of the NGL has resulted in an increased fatigue life by about 24%, which is clearly demonstrated in Fig. 17. It implies that the increased flow stress in the 3-mm series exhibits higher resistance against fatigue crack initiation on the NGL. It is an outcome of the higher yield strength in the 3-mm series as compared to the 2-mm one. The simulated fatigue life in both series is also found to capture general trend of experimental results, as shown in Fig. 18. This is another remarkable effect of the NGL. Because of its contribution, the torsional fatigue life of the SMATed alloy is much superior to that made of traditional CG metals.

6.2. Effect of thickness of the NGL on the crack pattern and fatigue life Now the thickness of the NGL in the SMATed specimens is increased to 40 mm and 60 mm. Our objective is to uncover the effect of thickness of the NGL on the crack pattern and fatigue life. The same mesh size and material properties of the NGL (2-mm series) are used. We also apply the same torques as in the previous SMATed model.

For each torque, five random series are generated. Numerical results show that when the thickness of the NGL is increased to 40 mm, the fatigue life can be improved at a higher stress. Fig. 19 shows the mode-II crack initiation in No. 5 random field under the torque of 45 Nm. Cracks also initiate from the exterior surface of the specimen, and gradually propagate. This is the same as the failure pattern for the NGL thickness 20 mm. As crack propagates and crack tip opening displacement increases, the stress state around those mode-II small cracks becomes complicated, i.e., a multi-axial stress state exists. Therefore, mode-II fracture changes into mixed-mode fracture so that the circumferential major crack forms. We also discover that, when the thickness of the NGL increases to 60 mm, a new mechanism develops: cracks now initiate from the interior surface of the tubular specimen and then gradually propagate to the exterior surface in the cases of (i) 43 Nm with No. 2 random field and (ii) 45 Nm with No. 5 random field. Fig. 20 shows the crack propagation in the first case. As the cycle number increases, small cracks are found to initiate from the interior surface of the tubular specimen, and later on other small cracks are seen to initiate from the exterior surface. This feature is quite different from the failure pattern when the NGL thickness is 40 mm. The similar phenomenon has also been observed in the fatigued gradient nanograined Cu where the cracks initiated from the subsurface layer and then propagated to the top surface [14]. Moreover, lesser small cracks are found now. This indicates that the increased thickness of NGL acts as obstacles to crack initiation and propagation. Smaller cracks are now suppressed by the thicker NGL. On observation of the rotation versus cycle number in Fig. 21, we find that when the cracks initiate from the interior surface, the NGL can help retard crack propagation, and fatigue life of the specimens is prolonged. Fig. 22 illustrates the simulated fatigue life with different NGL thickness. A remarkable feature is that torsional fatigue life of the specimen is markedly higher with the increase of NGL thickness at high stress levels.

7. Conclusions A torsional fatigue model for nanostructured metallic alloys has been proposed in this study. We follow an efficient approach to insert the cohesive elements into the initial bulk elements. An

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Fig. 14. Cracks patterns in No. 1 random field under (a) 41 Nm, (b) 39 Nm, and (c) 36 Nm.

Fig. 15. Rotation versus cycle number in No. 1 random field.

Fig. 17. Rotation versus cycle number in two series.

Fig. 16. Simulated fatigue life of CG specimens.

Fig. 18. Effect of flow stress of the NGL on the fatigue life.

irreversible CFEM combined with MCS has been employed to describe the 3D fatigue fracture. The proposed model has been verified by the experimental results. The following main conclusions can be drawn

3. The NGL plays a very important role in improving the torsional fatigue life of the material. The increase of NGL thickness is an effective means to improve the fatigue life. When the thickness of the NGL reaches 60 mm from 20 or 40 mm, there is a strong probability that fatigue cracks initiate form the interior surface of the tubular specimen and then propagate to the exterior surface. This mechanism has a very beneficial effect to the improvement of torsional fatigue life for nanostructured alloys.

1. By characterizing spatially-varying fracture properties as random fields, more reasonable crack pattern and fatigue life can be predicted, and the scatter of the fatigue life in the experiment can also be captured. 2. For the CG specimen, cohesive strength T max has a significant effect on the fatigue life. For this reason, the T max value should be carefully calibrated in the simulation of fatigue failure. Both random field and load level also have significant influence on crack pattern and fatigue life.

We believe that this study has provided a new idea to use numerical simulation to study the torsional fatigue failure, and to improve the fatigue life through use of SMATed alloys. This numerical approach can also be applied to study axial fatigue, fretting fatigue, and three-point bending fatigue. The calculated

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Fig. 19. Crack propagation in a specimen with the NGL thickness 40 mm (No. 5 random field).

Fig. 20. Crack propagation in a specimen with the NGL thickness 60 mm (No. 2 random field).

Fig. 21. Rotation versus cycle number of SMATed specimens with the NGL thickness 60 mm.

Fig. 22. Simulated fatigue life with different thickness of NGL.

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results can also provide guidelines for future experimental investigation using surface treatment to enhance the fatigue life.

Acknowledgments We wish to thank anonymous reviewers for their helpful comments. This research is supported by the National Natural Science Foundation of China (Project nos. 11372214 and 11102128) and Elite Scholar Program of Tianjin University. G.J. Weng thanks the support of NSF Mechanics of Materials Program under CMMI1162431. References [1] Lu K, Lu J. Surface nanocrystallization (SNC) of metallic materials — presentation of the concept behind a new approach. J Mater Sci Technol 1999;15:193–7. [2] Ya M, Xing Y, Dai F, Lu K, Lu J. Study of residual stress in surface nanostructured AISI 316L stainless steel using two mechanical methods. Surf Coat Tech 2003;168:148–55. [3] Lu K, Lu J. Nanostructured surface layer on metallic materials induced by surface mechanical attrition treatment. Mater Sci Eng A 2004;375:38–45. [4] Tao NR, Zhang HW, Lu K, Lu J. Development of nanostructures in metallic materials with low stacking fault energies during surface mechanical attrition treatment (SMAT). Mater Trans 2002;44:1919–25. [5] Tao NR, Wang ZB, Tong WP, Sui ML, Lu J, Lu K. An investigation of surface nanocrystallization mechanism in Fe induced by surface mechanical attrition treatment. Acta Mater 2002;50:4603–16. [6] Zhang HW, Hei ZK, Liu G, Lu J, Lu K. Formation of nanostructured surface layer on AISI 304 stainless steel by means of surface mechanical attrition treatment. Acta Mater 2003;51:1871–81. [7] Lu AQ, Zhang Y, Li Y, Liu G, Zang QH, Liu CM. Effect of nanocrystalline and twin boundaries on corrosion behavior of 316L stainless steel using SMAT. Acta Metall Sin 2006;19:183–9. [8] Chen XH, Lu J, Lu L, Lu K. Tensile properties of a nanocrystalline 316L austenitic stainless steel. Scripta Mater 2005;52:1039–44. [9] Chen AY, Zhang JB, Lu J, Lu W, Song HW. Necking propagated deformation behavior of layer-structured steel prepared by co-warm rolled surface nanocrystallized 304 stainless steel. Mater Lett 2007;61:5191–3. [10] Balusamy T, Venkateswarlu M, Murthy KSN, Vijayanand S, Narayanan TSN S. Effect of surface mechanical attrition treatment (SMAT) on the surface and electrochemical characteristics of Pb–Sn alloy. Int J Electrochem Sci 2014;9:96–108. [11] Liu WN, Zhang C, Yang ZG, Xia ZX. Microstructure and thermal stability of bulk nanocrystalline alloys produced by surface mechanical attrition treatment. App Surf Sci 2014;292:556–62. [12] Roland T, Retraint D, Lu K, Lu J. Fatigue life improvement through surface nanostructuring of stainless steel by means of surface mechanical attrition treatment. Scripta Mater 2006;54:1949–54. [13] Li D, Chen HN, Xu H. The effect of nanostructured surface layer on the fatigue behaviors of a carbon steel. App Surf Sci 2009;255:3811–6. [14] Yang L, Tao NR, Lu K, Lu L. Enhanced fatigue resistance of Cu with a gradient nanograined surface layer. Scripta Mater 2013;68:801–4. [15] Shamsaei N, Fatemi A. Deformation and fatigue behaviors of case-hardened steels in torsion: experiments and predictions. Int J Fatigue 2009;31:1386–96. [16] Stanzl-Tschegg SE, Mayer H. Fatigue and fatigue crack growth of aluminium alloys at very high numbers of cycles. Int J Fatigue 2001;23:S231–7. [17] Kwon JD, Park JC, Kim JH. Low cycle fatigue evaluation of duplex stainless steel with material degradation effect under torsional load. Int J Mod Phys B 2003;17:1561–6. [18] McClaflin D, Fatemi A. Torsional deformation and fatigue of hardened steel including mean stress and stress gradient effects. Int J Fatigue 2004;26:773–84. [19] Predki W, Klonne M, Knopik A. Cyclic torsional loading of pseudoelastic NiTi shape memory alloys: damping and fatigue failure. Mater Sci Eng A 2006;417:182–9. [20] Ohkawa C, Ohkaw I. Notch effect on torsional fatigue of austenitic stainless steel: comparison with low carbon steel. Eng Fract Mech 2011;78:1577–89. [21] Gladskyi M, Fatemi A. Notched fatigue behavior including load sequence effects under axial and torsional loadings. Int J Fatigue 2013;55:43–53. [22] Turon A, Costa J, Camanho PP, Davil CG. Simulation of delamination in composites under high-cycle fatigue. Compos: Part A 2007;38:2270–82. [23] Wang B, Siegmund T. A numerical analysis of constraint effects in fatigue crack growth by use of an irreversible cohesive zone model. Int J Fatigue 2005;132:175–96. [24] Abdul-Baqi A, Schreurs PJG, Geers MGD. Fatigue damage modeling in solder interconnects using a cohesive zone approach. Int J Solids Struct 2005;42:927–42. [25] Maiti S, Geubelle PH. A cohesive model for fatigue failure of polymers. Eng Fract Mech 2005;72:691–708.

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