# The Effect of Material Static Mechanical Properties on the Fatigue Crack Initiation Life of Rail Fastening Clips

Authors: Y. Liu; Qiutong Li; Xiujie Jiang; Huan Liu; Xianpu Yuan; Zhehao Zhu
Journal: Advances in Civil Engineering
 (2021)

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## Abstract

This paper aimed to study the effect of material static mechanical properties on the fatigue crack initiation life ofω-shaped rail fastening clips, in which the Vossloh 300-1 fastener system was taken as an example. The static mechanical properties of 38Si7 steel (the material of the clip) were first investigated through a series of uniaxial tensile tests. According to the experimental outcomes, a classic assembly system was simulated afterwards using the finite element analysis (FEA) method. On the basis of the Brown–Miller criterion, an in-depth numerical study regarding the critical plane was realized, which allowed fatigue crack initiation to be successfully reproduced by FEA. Finally, a detailed parametric study was performed with the relevant sensitivity analysis. The results showed that the initiation and progression of fatigue cracks in the fastening clip occur in the plane of the maximum shear strain. The fatigue crack initiation life of the fastening clip was extremely sensitive to the elastic modulus, especially more sensitive to the tensile strength. From an engineering viewpoint, the fatigue resistance of the fastening clip could be boosted by (i) increasing the tensile strength of the material to at least 1450 MPa and (ii) rendering the elastic modulus smaller than 160 GPa.


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## Body

## 1. Introduction

With the development of the modern economy, high-speed railways have become an increasingly important part of the current society. Under long-term traffic alternative loading, several catastrophic sequences (such as corrosion, cracks, and fractures) might be attributed to the failure of rail fastening clips. For instance,ω-clip fractures appear frequently in some high-speed railway lines [1–3], leading to malfunctions of the total railway and running comfort. This engineering drawback explains the particular attention given to experimental/theoretical studies focusing on this aspect.Considerable research has been devoted to examining this engaging engineering topic over the past 20 years. Unfortunately, diverse viewpoints are frequently presented in the published literature, and no clear consensus has been attained. Many researchers [1–7] argued that the failure of the rail clip was largely due to the resonance induced by rail corrugation; conversely, other researchers [8–10] stated that the rail clip was prone to failure under heavy axis loads. To the authors’ knowledge, fatigue failure of a rail clip may occur due to the combination of multiaxial loads and complex geometric structures. In addition, many scholars [1, 3, 6, 7, 11, 12] performed fatigue experiments and numerical simulations of rail fastening clips and found that the failure crack was initiated at the position of the maximum von Mises stress. In terms of studying the factors that influence the fatigue lives of rail fastening clips, Zhu et al. [2] established a 5-part structure model including a short rail, a bolt gasket, a clip, an insulating spacer, and a gauge apron to study the mechanical behaviour of the rail clip induced by vehicle dynamic loads. Qi et al. [4] established a 5-part structure model of the ω-shaped clip as well and found that the fatigue resistance of the fastening clip could be improved by increasing the fastener stiffness. Gao et al. [5, 6] introduced a one-part clip model with simplified boundary conditions according to the assembly test in the laboratory in order to analyse the failure mechanism of the W1-type fastening clip in high-speed railways. Mohammadzadeh et al. [13, 14] modelled only one half of the SKL14 clip to assess the fracture reliability of crack growth in the spring clip. Xin et al. [11], Xiao et al. [15], Yang [16], and Hasap et al. [17] found that the fatigue life of e-shaped fastening clips decreased dramatically with the increase in clip displacement during installation. Wu et al. [18], Liu [19], and Liu [20] found that the fatigue life of W1 fastening clips was negatively correlated with the amplitude of the alternating load. Wang [21] reported that the maximum stress of the fastening clip gradually increased with the increase in the friction coefficient between the fastening clip and gauge apron, thus facilitating crack initiation.The literature review in the previous section demonstrates that the variation pattern of the fastening clip fatigue life with the installation process, cycle load condition, and system stiffness is already well understood. In addition to external factors, material static mechanical properties, surface conditions, size effects, etc., are all capable of considerably affecting the fatigue life of fastening clips. However, the influences of all these parameters have seldom been studied, and many uncertainties remain in this area, thus hindering the development of the mechanism behind. Manouchehrynia and Abdullah [22] conducted a reliability assessment to predict the fatigue of an automobile coil spring during campus, urban, and highway road surfaces and found that the Morrow model could predict a safe region of a lift data point for the three road surfaces. Abdullah et al. [23] used strain-life approaches of Coffin–Manson and Morrow and Smith–Watson–Topper (SWT) models to predict the fatigue life of the leaf spring under highway, uphill, and downhill data and found that the leaf spring had the highest fatigue life of 1,544 cycles/block under highway data. Zhang et al. [24] found that the fatigue resistance of a nickel-based superalloy was related to the heat treatment procedures applied. Wang et al. [25] investigated the effect of the modulus of the asphalt mixture at the bottom layer on the mechanical response of the bridge pavement and found that the application of a high-modulus asphalt mixture as the bottom asphalt layer decreased the stress level of the pavement structure. Zhao and Ding [26] analysed the microstructure and mechanical properties of as-deposited, quenched and tempered (QT), and stress-relief annealed (SR) specimens of 24CrNiMo alloy steel; they found that the QT specimens had the smallest wear and thermal fatigue crack lengths, excellent resistance to friction and wear performance, and prevention of crack growth, with ideal comprehensive properties. Therefore, studying the effect of material static mechanical properties on rail fastening clips is of great significance since these findings are helpful for suppliers to provide theoretical guidance to implement proper heat treatments in the manufacturing process.To fill these gaps in knowledge, based on previous research [27–30], this paper takes the ω-shaped fastening clip of the Vossloh 300-1 fastener system, which is commonly used for high-speed railways, as the research subject. A series of uniaxial tensile tests were first performed on the fastening clip material 38Si7 at three different stages. Then, based on the measured performance data of the fastening clip material, a finite element (FE) model of the complete fastener system was constructed. The critical plane method was subsequently employed to identify the critical plane at the critical point, and the fatigue crack initiation life of the fastening clip was predicted following the Brown–Miller criterion. Finally, the effects of the material static mechanical properties on the fatigue crack initiation life of the fastening clip were analysed, and a sensitivity analysis of the material static properties was performed. The findings of this study would be very helpful in (i) finding ways to effectively improve the fatigue resistance of fastening clips from the perspective of material performance and (ii) providing a theoretical framework for the advanced selection of fastening clip materials for use in engineering applications.


## 2. Materials and Methods

Among the studies of multiaxial fatigue, critical plane approaches are the mainstream [31–50]. Zhu et al. [49] evaluated and compared four typical critical plane criteria (Fatemi–Socie, Wang–Brown, modified Smith–Watson–Topper, and proposed modified generalized strain energy) for multiaxial fatigue analysis of ductile and brittle materials under four multiaxial loading conditions and found that criteria with additional material constants yielded robust life predictions for different materials. Liao et al. [50] combined the critical plane and critical distance approaches to analyse multiaxial fatigue lifetime predictions for notched components and found that using this combined approach provided better accuracy for fatigue life prediction than others. Critical plane approaches treat multiaxial fatigue life predictions as equivalent uniaxial fatigue issues on the critical plane, which connect theory with the physical observation of fatigue cracking behaviour. The stress/strain states of the maximum damage plane are selected as the key parameters of multiaxial fatigue analysis. Thus, the critical plane must be the fatigue damage plane with multiaxial loadings, and the first step is of vital importance to determine the critical plane based on multiaxial loading conditions. The definition of the critical plane can include the maximum damage plane and the maximum shear stress plane, and the definitions of the damage parameters on the critical plane are mostly different combinations of factors including shear stress, normal stress, mean value, amplitude, and maximum of hydrostatic pressure. Figure 1 depicts the process for predicting the fatigue life of a structure by critical plane methods. A brief introduction of the lifetime prediction of the ω-shaped clip is attached as follows:(1)
Through tensile tests of 38Si7 specimens, the static mechanical properties are obtained, which are utilized for (i) performing elastoplastic finite element analysis (FEA) of the investigated structure and (ii) roughly estimating the fatigue properties of the material based on a selected empirical estimation method.(2)
An elastoplastic FE calculation of the investigated structure is performed, and the results are utilized for (i) determining the dangerous regions according to the stress and strain distributions and (ii) outputting the stress-strain components within the dangerous regions.(3)
The critical plane at the dangerous point is determined via the damage parameters which are obtained by performing three-dimensional (3-D) coordinate transformation according to the 12 stress-strain components.(4)
The fatigue lifetime of theω-shaped clip is predicted by substituting the fatigue properties of the 38Si7 material and the damage parameters induced by the multiaxial loading conditions into the selected criterion.Figure 1
The prediction process for the critical plane method.Since the research objective is to analyse the effects of material static mechanical properties on the fatigue crack initiation life of a fastening clip, multivariate sensitivity analysis is then utilized for the entire prediction process. Note that the fatigue crack initiation life of theω-shaped clip (hereafter called “fatigue life”) is emphasized in this paper, and further discussion of fatigue crack propagation is beyond the scope of the paper.


### 2.1. Tensile Tests

To gain insight into the static mechanical properties of a clip, a series of uniaxial tensile tests were performed on the fastening clip material 38Si7. Because the thermoforming of clips might affect the test results [46], the 38Si7 material was tested under the following three different conditions: (i) the virgin state in the bar form prior to thermoforming, (ii) the damaged state after a long service life, and (iii) the newly formed state (see Figure 2). The preformed bar was a round bar from which two 25 mm samples (#1 and #2) could be directly sampled. For the formed fastening clips, the two middle legs were removed from the clips by cutting and polishing to meet the experimental requirements (#3 through #6).Figure 2
Material specimen of anω-shaped fastening clip at three different stages: (a) virgin state in the bar form; (b) damaged state after a long service life; (c) newly formed state.
(a)(b)(c)The tensile tests were conducted using a universal tester SHT4605 (see Figure3), with a capacity of 600 kN and a maximum speed of 140 mm/min. The standard ASTM E8 M-16 was adopted as the employed experimental programme. The speeds before and after yielding were set to be 1.0 mm/min and 24 mm/min, respectively. All tests were carried out using the displacement-controlled procedure at a controlled ambient temperature so as to eradicate the consideration of any unwanted thermocoupling phenomena. The specimens were marked and clamped at 30 mm on the top and bottom of the specimen.Figure 3
A tensile testing site for the clip material 38Si7.The test results were interpreted with respect to the engineering stress-strain curves. The elastic modulus, which was used to quantify the relationship between the tensile stress and axial strain in the linear elastic region, was determined by the slope of the straight curve at the beginning. The yield strength was taken as the stress attaining 0.2% plastic deformation, whereas the maximum stress on the engineering stress-strain curve was identified as the tensile strength.


### 2.2. Finite Element Analyses

Taking the Vossloh 300-1 fastener system as the research object, all the components are marked in Figure4. Among them, the contact relationships between the clip and other parts and the material properties of all the parts are of great importance to the stress-strain distributions of the rail clip under normal working conditions, even to the fatigue life prediction of the rail clip. However, many researchers [2, 4–6, 13, 14] established simplified models to save time and costs, in which the crucial importance of the contact relationships in the multiaxial fatigue analysis was ignored. Among the literature studies mentioned in the introduction, the simulation models are too oversimplified to represent the complex situation that includes the in situ stress and strain states inside the fastening clips. In this paper, FEA was performed on an elaborately assembled fastener system by considering all the components and contact pairs, as shown in Figure 5(a). The elastoplastic characteristics of the fastening clip material 38Si7 were determined through the aforementioned tensile tests. The material properties, mesh types, and mesh quantities of the other components are listed in Table 1. In this model, point-to-surface nonlinear contact was created between each toe end of the fastening clips and the corresponding insulating spacer, between each heel end of the fastening clips and the corresponding gauge apron, and between each middle leg of the fastening clips and the corresponding bolt gasket, as shown in Figure 5(b). The interactions of sliding surfaces were modelled using the basic Coulomb friction model, in which all the coefficients of friction were 0.3 [51]. The nodes for other interactions where the surfaces were close to one another were tied. The longitudinal displacement of the middle ring of the fastening clip (y-direction in Figure 5(b)) was restricted. Due to the complexity of the complete assembly model, the elements of the bolt, rail, and sleeper were simplified and considered part of a rigid body, resulting in faster run times at the model development stage, which could be useful for model verification purposes.Figure 4
The Vossloh 300-1 fastener system (from the official website of Vossloh LLC).Figure 5
A FE model of the Vossloh 300-1 fastener system: (a) the configuration of the fastener system; (b) locations of the point-to-surface nonlinear contacts.
(a)(b)Table 1
Material properties, mesh types, and mesh quantities of all components except the clip.
ComponentElastic modulus (MPa)Poisson’s ratioDensity (kg/m3)Mesh typeMesh quantityBolt2060000.307850R3D4658Bolt gasket85000.401150C3D8R288Insulating spacer85000.401150C3D8R512Rail2000000.307850R3D41978Rail pad24.50.30929C3D203992Iron pad1730000.307850C3D8I8168Resilient sleeper bearing19.50.49870C3D207580Gauge apron85000.401150C3D10M52533Sleeper340000.182500R3D41306A vertical load of 17 kN (Finstallation = 17 kN, minus z-direction in Figure 5(b)) was first applied to the upper surface of the bolt, representing a standard installation torque of 250 N·m [30]. Then, a cyclic load was applied to the rail head surface, which was simplified from the field-tested wheel-rail force of a high-speed railway (see Figure 6) to the roughly equivalent sinusoidal loading curve, as shown in Figure 7.Figure 6
A scene from the field tests for the wheel-rail forces of a high-speed railway.Figure 7
Measured wheel-rail forces and corresponding equivalent sinusoidal loads in the FE calculations. (a) Case 1. (b) Case 2.
(a)(b)The amplitude, mean value, and frequency of the sinusoidal loading curve were calculated with the real-time record of the wheel-rail force curve, following the norm TB/T 2489-2016. In addition, another adverse case was considered where the installation load was 19 kN, simulating the overload installation torque of 300 N·m, and the sinusoidal load was amplified (from the magenta solid curve to the green dashed curve in Figure7), simulating the overload wheel-rail force, normally due to factors such as track irregularities and cornering. Table 2 shows the relevant parameters of the equivalent sinusoidal loads.Table 2
Parameters related to installation loads and cycle loads in the FEA.
CaseInstallation loadFinstallation (kN)AmplitudeA (Fcycle)Mean valueAvg (Fcycle) (kN)Frequencyf (Hz)Case 117100−404Case 219130


### 2.3. Critical Plane Determination in 3-D Coordinates

As discussed earlier, it is of great importance to identify the critical plane at the critical point, that is, the plane of the crack at the critical position. Based on the classic theory of elasticity for 3-D space problems [52], 3-D principal strains and three maximum shear strains can first be determined at the critical point in the global coordinate system. Then, via the coordinate transformation matrix of direction cosines, the specific plane of the maximum principal strain/shear strain can be identified.Assuming that there is one principal strainεi at critical point D on plane S and taking any plane ABC parallel to S near point D, a small tetrahedron is formed in the global coordinate system that passes through critical point D (see Figure 8).Figure 8
Principal strainεi and its direction ON at critical point D.When the tetrahedron is infinitely small and approaches pointD, the strain on plane ABC is the strain on plane S where point D is located, namely, the principal strain. Assuming that the outer normal of plane ABC is ON, the projection of ON on the x-y-plane forms an angle θ with the x-axis, and the angle between ON and the x-y-plane is ϕ (see Figure 8). The direction cosines of plane ABC are expressed as follows:(1)l=cosθcosϕ,m=sinθcosϕ,n=sinϕl2+m2+n2=1.Each component of the global strain on planeABC in the global coordinate system is therefore represented as follows:(2)lεx+mγyx2+nγzx2=lεi,lγxy2+mεy+nγzy2=mεi,lγxz2+mγyz2+nεz=nεi.By solving two simultaneous equations (1) and (2), the direction and the plane of each principal strain εi can be identified, and a coordinate system based on the directions of the 3-D principal strains ε1, ε2, and ε3 can be defined. By rotating the new coordinate system 45°, the direction and plane of the local maximum shear strain can also be identified.The direction cosines of the new 3-D system (X′Y′Z′) based on the original coordinate system (XYZ) are expressed as follows:(3)lx′x=cosθcosϕ,lx′y=sinθcosϕ,lx′z=sinϕ,ly′x=−sinθ,ly′y=cosθ,ly′z=0,lz′x=−cosθsinϕ,lz′y=−sinθsinϕ,lz′z=cosϕ.Hence, the strain components in the new coordinate system are represented by the direction cosine matrix as follows:(4)εX′εY′εZ′γX′Y′2γX′Z′2γY′Z′2=lX′X2lX′Y2lX′Z22lX′XlX′Y2lX′XlX′Z2lX′YlX′ZlY′X2lY′Y2lY′Z22lY′XlY′Y2lY′XlY′Z2lY′YlY′ZlZ′X2lZ′Y2lZ′Z22lZ′XlZ′Y2lZ′XlZ′Z2lZ′YlZ′ZlX′XlY′XlX′YlY′YlX′ZlY′ZlX′XlY′Y+lX′YlY′XlX′ZlY′X+lX′XlY′ZlX′ZlY′X+lX′XlY′ZlX′XlZ′XlX′YlZ′YlX′ZlZ′ZlX′XlZ′Y+lX′YlZ′XlXZ′lZ′X+lX′XlZ′ZlX′ZlZ′X+lX′XlZ′ZlY′XlZ′XlY′YlZ′YlY′ZlZ′ZlY′XlZ′Y+lY′YlZ′XlY′ZlZ′X+lYX′lZ′ZlY′ZlZ′X+lY′XlZ′ZεXεYεZγXY2γXZ2γYZ2.


### 2.4. Brown–Miller Criterion and Seeger’s Method

The strain-based critical plane method was first proposed by Brown, Miller, and Findley [40]. They considered that a combination of shear strain and normal strain acting on the maximum shear strain plane was responsible for crack initiation and growth. The authors reported that fatigue crack initiation depended on the maximum shear strain γmax and normal strain range ∆εn in the maximum shear strain plane. However, their criterion failed to consider the effect of mean stress on the fatigue life. In order to overcome this shortcoming, the Morrow mean-stress-correction method was then employed to correct the mean stress σm, and they proposed the following formula with two model parameters:(5)γmax+Δεn=C1σf′−σmE2Nb+C2εf′2Nc,C1=1+νe+1−νe2,C2=1+νp+1−νp2,where σf′ and b represent the fatigue strength coefficient and index of the material, respectively; εf′ and c are the fatigue ductility coefficient and index of the material, respectively; νe and νp represent the elastic value and the plastic value of Poisson’s ratio, respectively; E is the elastic modulus; and N is the number of cycles to failure. The left side denotes the shear strain and normal strain amplitudes in the plane of maximum shear strain at the critical point. The right side denotes the elastic strain component and the plastic component related to the effect of complex multiaxial loading.In addition to the damage parameters obtained from FEA, the fatigue properties can generally be obtained by fitting strain-control experimental data or roughly estimated based on empirical estimation methods such as four-point correlation [53], universal slopes [53], and Seeger’s method [54]. To determine the fatigue properties of 38Si7, Seeger’s method, which is normally used for estimating the fatigue properties of plain carbon steel, middle-low alloy steel, aluminium alloy, and titanium alloy, was employed. The estimation formula is as follows:(6)σf′=1.5σb,εf′=0.59a,b=−0.087,c=−0.58,where a is a model coefficient, related to the tensile strength σb and elastic modulus E of the investigated material: a = 1 when σb/E < 0.003, whereas a = 1.375–125 σb/E when σb/E ≥ 0.003.Note that the material properties required from the fatigue life prediction model are the static mechanical properties including Poisson’s ratio and the elastic modulus and the fatigue properties which are estimated from the elastic modulus and the tensile strength based on Seeger’s method. Therefore, the material static mechanical properties required are mainly Poisson’s ratio, elastic modulus, and tensile strength. Since Poisson’s ratio of metallic material is nearly unchanged with an ignorable variation, this paper focused on the effects of the elastic modulus and tensile strength of the material on the fatigue crack initiation life of a fastening clip.


### 2.5. Multivariate Sensitivity Analysis

To evaluate the sensitivity of the static mechanical properties of 38Si7 to the fatigue crack initiation life of a fastening clip, the sensitivity functions and sensitivity factors proposed by Zhang and Zhu [55] were employed. Based on univariate sensitivity analysis, Zhang and Zhu proposed sensitivity functions and sensitivity factors for multivariate sensitivity analysis. The authors successfully identified the variable that most affected the horizontal deformation of the high sidewall of an underground power plant in the Laxiwa underground project. Cheng [56] also used this method to analyse the sensitivity of the material static mechanical properties of W6Mo5Cr4V2 steel to the fatigue crack initiation life of a drive shaft. He found that the fatigue crack initiation life of the drive shaft was extremely sensitive to the elastic modulus, yield strength, and tensile strength. In the multivariate sensitivity analysis, the first step is to find the functional relationships between the material static mechanical properties and the fatigue crack initiation life. This step consists of building a system model Nf = f (x1, x2, …, xn) based on the above fatigue crack initiation life prediction method, calculating the effect of each material static property (x1, x2, …, xn) on the fatigue crack initiation life Nf of the fastening clip, and thereby obtaining the fatigue crack initiation life of the fastening clip as a function of each material static property.Then, a dimensionless parameterization was performed for each parameter, and the dimensionless sensitivity functionS(xi), i.e., the ratio of the relative error of the fatigue crack initiation life to the relative error of the material static property, was calculated as follows:(7)Sxi=ΔNf/NfΔxi/xi=ΔNfΔxixiNf≅dNfdxixiNf,i=1,2,…,n.When using the value ofxi as the reference material static property xi∗, the function S(xi∗) is the sensitivity factor of the material static property. The larger the value of S(xi∗) is, the more sensitive the fatigue crack initiation life is to the material static property.


## 2.1. Tensile Tests

To gain insight into the static mechanical properties of a clip, a series of uniaxial tensile tests were performed on the fastening clip material 38Si7. Because the thermoforming of clips might affect the test results [46], the 38Si7 material was tested under the following three different conditions: (i) the virgin state in the bar form prior to thermoforming, (ii) the damaged state after a long service life, and (iii) the newly formed state (see Figure 2). The preformed bar was a round bar from which two 25 mm samples (#1 and #2) could be directly sampled. For the formed fastening clips, the two middle legs were removed from the clips by cutting and polishing to meet the experimental requirements (#3 through #6).Figure 2
Material specimen of anω-shaped fastening clip at three different stages: (a) virgin state in the bar form; (b) damaged state after a long service life; (c) newly formed state.
(a)(b)(c)The tensile tests were conducted using a universal tester SHT4605 (see Figure3), with a capacity of 600 kN and a maximum speed of 140 mm/min. The standard ASTM E8 M-16 was adopted as the employed experimental programme. The speeds before and after yielding were set to be 1.0 mm/min and 24 mm/min, respectively. All tests were carried out using the displacement-controlled procedure at a controlled ambient temperature so as to eradicate the consideration of any unwanted thermocoupling phenomena. The specimens were marked and clamped at 30 mm on the top and bottom of the specimen.Figure 3
A tensile testing site for the clip material 38Si7.The test results were interpreted with respect to the engineering stress-strain curves. The elastic modulus, which was used to quantify the relationship between the tensile stress and axial strain in the linear elastic region, was determined by the slope of the straight curve at the beginning. The yield strength was taken as the stress attaining 0.2% plastic deformation, whereas the maximum stress on the engineering stress-strain curve was identified as the tensile strength.


## 2.2. Finite Element Analyses

Taking the Vossloh 300-1 fastener system as the research object, all the components are marked in Figure4. Among them, the contact relationships between the clip and other parts and the material properties of all the parts are of great importance to the stress-strain distributions of the rail clip under normal working conditions, even to the fatigue life prediction of the rail clip. However, many researchers [2, 4–6, 13, 14] established simplified models to save time and costs, in which the crucial importance of the contact relationships in the multiaxial fatigue analysis was ignored. Among the literature studies mentioned in the introduction, the simulation models are too oversimplified to represent the complex situation that includes the in situ stress and strain states inside the fastening clips. In this paper, FEA was performed on an elaborately assembled fastener system by considering all the components and contact pairs, as shown in Figure 5(a). The elastoplastic characteristics of the fastening clip material 38Si7 were determined through the aforementioned tensile tests. The material properties, mesh types, and mesh quantities of the other components are listed in Table 1. In this model, point-to-surface nonlinear contact was created between each toe end of the fastening clips and the corresponding insulating spacer, between each heel end of the fastening clips and the corresponding gauge apron, and between each middle leg of the fastening clips and the corresponding bolt gasket, as shown in Figure 5(b). The interactions of sliding surfaces were modelled using the basic Coulomb friction model, in which all the coefficients of friction were 0.3 [51]. The nodes for other interactions where the surfaces were close to one another were tied. The longitudinal displacement of the middle ring of the fastening clip (y-direction in Figure 5(b)) was restricted. Due to the complexity of the complete assembly model, the elements of the bolt, rail, and sleeper were simplified and considered part of a rigid body, resulting in faster run times at the model development stage, which could be useful for model verification purposes.Figure 4
The Vossloh 300-1 fastener system (from the official website of Vossloh LLC).Figure 5
A FE model of the Vossloh 300-1 fastener system: (a) the configuration of the fastener system; (b) locations of the point-to-surface nonlinear contacts.
(a)(b)Table 1
Material properties, mesh types, and mesh quantities of all components except the clip.
ComponentElastic modulus (MPa)Poisson’s ratioDensity (kg/m3)Mesh typeMesh quantityBolt2060000.307850R3D4658Bolt gasket85000.401150C3D8R288Insulating spacer85000.401150C3D8R512Rail2000000.307850R3D41978Rail pad24.50.30929C3D203992Iron pad1730000.307850C3D8I8168Resilient sleeper bearing19.50.49870C3D207580Gauge apron85000.401150C3D10M52533Sleeper340000.182500R3D41306A vertical load of 17 kN (Finstallation = 17 kN, minus z-direction in Figure 5(b)) was first applied to the upper surface of the bolt, representing a standard installation torque of 250 N·m [30]. Then, a cyclic load was applied to the rail head surface, which was simplified from the field-tested wheel-rail force of a high-speed railway (see Figure 6) to the roughly equivalent sinusoidal loading curve, as shown in Figure 7.Figure 6
A scene from the field tests for the wheel-rail forces of a high-speed railway.Figure 7
Measured wheel-rail forces and corresponding equivalent sinusoidal loads in the FE calculations. (a) Case 1. (b) Case 2.
(a)(b)The amplitude, mean value, and frequency of the sinusoidal loading curve were calculated with the real-time record of the wheel-rail force curve, following the norm TB/T 2489-2016. In addition, another adverse case was considered where the installation load was 19 kN, simulating the overload installation torque of 300 N·m, and the sinusoidal load was amplified (from the magenta solid curve to the green dashed curve in Figure7), simulating the overload wheel-rail force, normally due to factors such as track irregularities and cornering. Table 2 shows the relevant parameters of the equivalent sinusoidal loads.Table 2
Parameters related to installation loads and cycle loads in the FEA.
CaseInstallation loadFinstallation (kN)AmplitudeA (Fcycle)Mean valueAvg (Fcycle) (kN)Frequencyf (Hz)Case 117100−404Case 219130


## 2.3. Critical Plane Determination in 3-D Coordinates

As discussed earlier, it is of great importance to identify the critical plane at the critical point, that is, the plane of the crack at the critical position. Based on the classic theory of elasticity for 3-D space problems [52], 3-D principal strains and three maximum shear strains can first be determined at the critical point in the global coordinate system. Then, via the coordinate transformation matrix of direction cosines, the specific plane of the maximum principal strain/shear strain can be identified.Assuming that there is one principal strainεi at critical point D on plane S and taking any plane ABC parallel to S near point D, a small tetrahedron is formed in the global coordinate system that passes through critical point D (see Figure 8).Figure 8
Principal strainεi and its direction ON at critical point D.When the tetrahedron is infinitely small and approaches pointD, the strain on plane ABC is the strain on plane S where point D is located, namely, the principal strain. Assuming that the outer normal of plane ABC is ON, the projection of ON on the x-y-plane forms an angle θ with the x-axis, and the angle between ON and the x-y-plane is ϕ (see Figure 8). The direction cosines of plane ABC are expressed as follows:(1)l=cosθcosϕ,m=sinθcosϕ,n=sinϕl2+m2+n2=1.Each component of the global strain on planeABC in the global coordinate system is therefore represented as follows:(2)lεx+mγyx2+nγzx2=lεi,lγxy2+mεy+nγzy2=mεi,lγxz2+mγyz2+nεz=nεi.By solving two simultaneous equations (1) and (2), the direction and the plane of each principal strain εi can be identified, and a coordinate system based on the directions of the 3-D principal strains ε1, ε2, and ε3 can be defined. By rotating the new coordinate system 45°, the direction and plane of the local maximum shear strain can also be identified.The direction cosines of the new 3-D system (X′Y′Z′) based on the original coordinate system (XYZ) are expressed as follows:(3)lx′x=cosθcosϕ,lx′y=sinθcosϕ,lx′z=sinϕ,ly′x=−sinθ,ly′y=cosθ,ly′z=0,lz′x=−cosθsinϕ,lz′y=−sinθsinϕ,lz′z=cosϕ.Hence, the strain components in the new coordinate system are represented by the direction cosine matrix as follows:(4)εX′εY′εZ′γX′Y′2γX′Z′2γY′Z′2=lX′X2lX′Y2lX′Z22lX′XlX′Y2lX′XlX′Z2lX′YlX′ZlY′X2lY′Y2lY′Z22lY′XlY′Y2lY′XlY′Z2lY′YlY′ZlZ′X2lZ′Y2lZ′Z22lZ′XlZ′Y2lZ′XlZ′Z2lZ′YlZ′ZlX′XlY′XlX′YlY′YlX′ZlY′ZlX′XlY′Y+lX′YlY′XlX′ZlY′X+lX′XlY′ZlX′ZlY′X+lX′XlY′ZlX′XlZ′XlX′YlZ′YlX′ZlZ′ZlX′XlZ′Y+lX′YlZ′XlXZ′lZ′X+lX′XlZ′ZlX′ZlZ′X+lX′XlZ′ZlY′XlZ′XlY′YlZ′YlY′ZlZ′ZlY′XlZ′Y+lY′YlZ′XlY′ZlZ′X+lYX′lZ′ZlY′ZlZ′X+lY′XlZ′ZεXεYεZγXY2γXZ2γYZ2.


## 2.4. Brown–Miller Criterion and Seeger’s Method

The strain-based critical plane method was first proposed by Brown, Miller, and Findley [40]. They considered that a combination of shear strain and normal strain acting on the maximum shear strain plane was responsible for crack initiation and growth. The authors reported that fatigue crack initiation depended on the maximum shear strain γmax and normal strain range ∆εn in the maximum shear strain plane. However, their criterion failed to consider the effect of mean stress on the fatigue life. In order to overcome this shortcoming, the Morrow mean-stress-correction method was then employed to correct the mean stress σm, and they proposed the following formula with two model parameters:(5)γmax+Δεn=C1σf′−σmE2Nb+C2εf′2Nc,C1=1+νe+1−νe2,C2=1+νp+1−νp2,where σf′ and b represent the fatigue strength coefficient and index of the material, respectively; εf′ and c are the fatigue ductility coefficient and index of the material, respectively; νe and νp represent the elastic value and the plastic value of Poisson’s ratio, respectively; E is the elastic modulus; and N is the number of cycles to failure. The left side denotes the shear strain and normal strain amplitudes in the plane of maximum shear strain at the critical point. The right side denotes the elastic strain component and the plastic component related to the effect of complex multiaxial loading.In addition to the damage parameters obtained from FEA, the fatigue properties can generally be obtained by fitting strain-control experimental data or roughly estimated based on empirical estimation methods such as four-point correlation [53], universal slopes [53], and Seeger’s method [54]. To determine the fatigue properties of 38Si7, Seeger’s method, which is normally used for estimating the fatigue properties of plain carbon steel, middle-low alloy steel, aluminium alloy, and titanium alloy, was employed. The estimation formula is as follows:(6)σf′=1.5σb,εf′=0.59a,b=−0.087,c=−0.58,where a is a model coefficient, related to the tensile strength σb and elastic modulus E of the investigated material: a = 1 when σb/E < 0.003, whereas a = 1.375–125 σb/E when σb/E ≥ 0.003.Note that the material properties required from the fatigue life prediction model are the static mechanical properties including Poisson’s ratio and the elastic modulus and the fatigue properties which are estimated from the elastic modulus and the tensile strength based on Seeger’s method. Therefore, the material static mechanical properties required are mainly Poisson’s ratio, elastic modulus, and tensile strength. Since Poisson’s ratio of metallic material is nearly unchanged with an ignorable variation, this paper focused on the effects of the elastic modulus and tensile strength of the material on the fatigue crack initiation life of a fastening clip.


## 2.5. Multivariate Sensitivity Analysis

To evaluate the sensitivity of the static mechanical properties of 38Si7 to the fatigue crack initiation life of a fastening clip, the sensitivity functions and sensitivity factors proposed by Zhang and Zhu [55] were employed. Based on univariate sensitivity analysis, Zhang and Zhu proposed sensitivity functions and sensitivity factors for multivariate sensitivity analysis. The authors successfully identified the variable that most affected the horizontal deformation of the high sidewall of an underground power plant in the Laxiwa underground project. Cheng [56] also used this method to analyse the sensitivity of the material static mechanical properties of W6Mo5Cr4V2 steel to the fatigue crack initiation life of a drive shaft. He found that the fatigue crack initiation life of the drive shaft was extremely sensitive to the elastic modulus, yield strength, and tensile strength. In the multivariate sensitivity analysis, the first step is to find the functional relationships between the material static mechanical properties and the fatigue crack initiation life. This step consists of building a system model Nf = f (x1, x2, …, xn) based on the above fatigue crack initiation life prediction method, calculating the effect of each material static property (x1, x2, …, xn) on the fatigue crack initiation life Nf of the fastening clip, and thereby obtaining the fatigue crack initiation life of the fastening clip as a function of each material static property.Then, a dimensionless parameterization was performed for each parameter, and the dimensionless sensitivity functionS(xi), i.e., the ratio of the relative error of the fatigue crack initiation life to the relative error of the material static property, was calculated as follows:(7)Sxi=ΔNf/NfΔxi/xi=ΔNfΔxixiNf≅dNfdxixiNf,i=1,2,…,n.When using the value ofxi as the reference material static property xi∗, the function S(xi∗) is the sensitivity factor of the material static property. The larger the value of S(xi∗) is, the more sensitive the fatigue crack initiation life is to the material static property.


## 3. Results and Discussion

### 3.1. Test Results of the Clip Material 38Si7

The engineering stress-strain curves of the fastening clip material at different states were obtained, as shown in Figure9. The test results of each pair of samples in the same stage (in the same colour but with different symbols) show a perfect agreement with each other, indicating that the test results were repeatable with high confidence. The curves for the damaged clip samples after a long service life (samples #3 and #4) are nearly the same as those in the newly formed state (samples #5 and #6), whereas the results for the bar before clip formation (samples #1 and #2) are slightly higher. Table 3 lists the mean values of the static mechanical properties. The comparison of the test results shows that the tensile strength, yield strength, and elastic modulus of the material for the fastening clip remain nearly constant in different states. The static mechanical properties of the material were not directly affected by the thermoforming process of the fastening clip, and deformations occurred during the in-service process. Based on the tensile properties, the fatigue properties of the fastening clip material were obtained through the Seeger algorithm, as shown in Table 4. Compared with the samples of the bars before clip formation (samples #1 and #2), the samples of newly delivered clips (samples #5 and #6) had slightly lower tensile strength, yield strength, and elastic modulus values. Moreover, the damage during long service periods (samples #3 and #4) had no effect on the tensile properties.Figure 9
The uniaxial tensile test results on the clip material 38Si7 (engineering stress-strain curve).Table 3
Tensile properties of the clip material 38Si7 (mean values).
SampleTensile strengthσb (MPa)Yield strengthσs (MPa)Elastic modulusE (GPa)#1/#214251254190#3/#413491230180#5/#613501242180Table 4
Fatigue properties of the fastening clip material.
SampleFatigue strength coefficientσf′ (MPa)Fatigue strength exponentbFatigue ductility coefficientεf′Fatigue ductility exponentc#1/#22138−0.0870.2581−0.58#3/#420240.2585#5/#620250.2581


### 3.2. FEA Results

The static mechanical properties of newly formed clips (samples #5 and #6) were considered in the FEA for the following two reasons. One, the model was a fastening clip rather than a bar before clip formation. Two, another was that the static mechanical properties of the fastening clip material for the clip damaged after a long service life were nearly the same as those of the newly delivered clip.In the FE calculations, the stress and strain distributions of the fastening clips in case 1 and case 2 were successively determined, as shown in Figure10. The global maximum stress of the fastening clip is 1507 MPa for case 1 and 1556 MPa for case 2, occurring at the inner side of the small arc at the heel end (see the red circles in Figure 10).Figure 10
Stress/strain distributions of the fastening clip in FEA: (a) von Mises stress distribution for case 1; (b) maximum principal strain distribution for case 1; (c) von Mises stress distribution for case 2; (d) maximum principal strain distribution for case 2.
(a)(b)(c)(d)The maximum von Mises stress, in either case, was located at node 20888 of the FE model. Six stress components are shown in Table5. Some stress components were beyond the fatigue limit of the material. The maximum principal strain was 15901.72 με for case 1 and 19156.69 με for case 2, which was also observed at node 20888 of the fastening clip. These strain components are shown in Table 6. The global maximum stress was extremely high, far beyond the fatigue limit of the material and even beyond the measured yield strength of the material, indicating that the elastoplastic stage was achieved under the actual load during operation, and it was prone to fatigue failure. Hence, the fatigue failure developed in the fastening clip was mainly due to the local deformation concentration of the material, and the inner side of the small arc at the heel end was the dangerous region.Table 5
Stress components at the critical point (unit: MPa).
NodeCaseσxσyσzτxyτxzτyz20888Case 1379.4274.07−178.10−670.96151.10246.98Case 2398.8279.18−178.62−678.88145.72255.30Table 6
Strain components at the critical point (unit:με).
NodeCaseεxεyεzγxyγxzγyz20888Case 15896.66−175.80−4986.65−25381.006029.2610136.30Case 27250.67−343.09−6109.37−30503.106865.5612571.00


### 3.3. Fatigue Crack Initiation Life Predictions

By performing 3-D coordinate transformation, three principal strains and three local maximum shear strains at the critical point were calculated, as listed in Table7. The maximum shear strain, in either case, is nearly twice as high as the maximum principal strain. The directions and the planes of the maximum principal strain ε1 and maximum shear strain γ1 can also be identified, as shown in Figure 11. Figure 12 shows the actual fracture plane of the fastening clip. The comparison of these figures shows that, at the critical point, the actual fastening clip is particularly prone to developing cracks on the plane of the maximum shear strain, and the maximum shear strain initiates the fatigue cracks. Moreover, the tensile strain in the plane accelerates the accumulation of crack damage. Therefore, the damage could be reasonably represented as a function of both the shear strain and the strain normal to this plane. The Brown–Miller criterion was ideally suited for predicting the fatigue life of a fastening clip.Table 7
Three principal strains and three local maximum shear strains at the critical point (unit:με).
NodeCaseε1ε2ε3γ1γ2γ320888Case 115901.72−1175.59−13991.9229848.8617089.2512759.61Case 219156.69−1453.70−16904.7836132.8720753.2015379.67Figure 11
Simulated fracture planes of a fastening clip.Figure 12
The actual fracture plane of a fastening clip.By substituting the material fatigue properties and the damage parameters into the Brown–Miller criterion, the fatigue lives of the fastening clips for cases 1 and 2 were predicted, and the distribution contours are shown in Figure13. Under normal in-service conditions, the lowest fatigue life of the fastening clip was approximately 180 million times (i.e., infinite life), occurring in the heel end area. For case 2, the lowest fatigue life was only approximately 2.399 million times, which is lower than the limit of the requirement before use (5 million times).Figure 13
Fatigue crack initiation lives of theω-shaped fastening clips: (a) case 1 and (b) case 2.
(a)(b)However, the actual in-service condition of the fatigue life was even worse due to the addition of the broadband minor-amplitude excitation induced by track irregularities and rail corrugation [57–61]. The typical practice employed to improve this adverse condition is to grind the rail profiles [61] or decrease the train speeds [62]. These measures severely impede the development of high-speed railway traffic. However, in this paper, the solution proposed is to improve the material performance at the design stage of a fastening clip. Based on this idea, the effects of the material static mechanical properties on the fatigue life of a fastening clip were discussed according to these in-service conditions.


### 3.4. Effect of the Material Static Mechanical Properties

According to relevant studies [1, 2, 4, 8, 11, 12, 17, 20], the elastic modulus of the fastening clip material is between 150 GPa and 300 GPa, and its tensile strength is normally within the range of 1000–1600 MPa. Therefore, the values of the elastic modulus and tensile strength in these ranges could be used to analyse their effects on the fatigue crack initiation life of the fastening clip. The calculations were performed following the aforementioned method.Figure14 shows that the fatigue lives of the fastening clips vary with the elastic modulus of the material. The increase in the elastic modulus leads to the shortening of the fastening clip fatigue life, and the effect gradually weakens as the elastic modulus increases. The fatigue life for case 1 is beyond the allowable limit of 5 million times only when the elastic modulus is lower than 250 GPa. However, compared with the fatigue life for case 1 in each elastic modulus (magenta rectangular symbols), the fatigue life for case 2 (yellow triangle symbols) is much lower. In such a case, to guarantee that the fastening clip conforms to the requirements before use, the elastic modulus of the material must be at least lower than 160 GPa (i.e., lower than the reference value).Figure 14
Variations in the fastening clip fatigue life with elastic modulus of the material.The effects of the material tensile strength on the fastening clip fatigue life for the two cases are shown in Figure15. The fastening clip fatigue life increases with increasing material tensile strength. The fatigue life for case 1 is beyond the allowable limit of 5 million times only when the tensile strength is higher than 1150 MPa. However, the fatigue life for case 1 for each tensile strength (megenta rectangular symbols) is much higher than that for case 2 (yellow triangle symbols). In such a case, to guarantee that the fastening clip conforms to the requirements before use, the elastic modulus of the material must exceed 1450 MPa (i.e., higher than the reference value).Figure 15
Variations in the fastening clip fatigue life with the tensile strength of the material.As shown in Table8, the sensitivity factors of the elastic modulus and tensile strength of the fastening clip material were calculated based on the definition of the sensitivity factor in equation (7). The sensitivity factor of the elastic modulus in each case is approximately 10, whereas that of the tensile strength in each case is much higher, with a value of approximately 15. With the worse in-service condition of the fastening clip (from case 1 to case 2), the sensitivity factor of each material performance parameter slightly decreases, i.e., from 10.09 to 9.89 and from 16.29 to 15.05, respectively.Table 8
Sensitivity factorsS(xi∗) of the material static mechanical properties xi∗.
Material performance parameterxi∗Elastic modulus (E)Tensile strength (σb)Case 1Finstallation = 17 kNA (Fcycle = 100 kN)10.0916.29Case 2Finstallation = 19 kNA (Fcycle = 130 kN)9.8915.05The sensitivity analysis indicates that the fatigue crack initiation life of the fastening clip is extremely sensitive to the elastic modulus, especially more sensitive to the tensile strength. For example, the sensitivity factorS(xi) of the elastic modulus E is approximately 15, which means that the change in fatigue life (|ΔNf/Nf|) is 15 times the change in the material static mechanical properties (|Δxi/xi|), according to (7). That is, the fastening clip fatigue life can be increased significantly by 80% when the elastic modulus of the fastening clip material decreased by only 6%. The elastic modulus and tensile strength of the fastening clip material have marked effects on the fatigue life of the fastening clip, which is of great significance in the reverse design of the fastening clip.Since it is difficult to reduce the elastic modulus of a raw material through conventional heat treatments and alloying techniques [24, 26], increasing the tensile strength of the material can be employed as an alternative means to improve the fatigue resistance of fastening clips. However, when the tensile strength is excessively high, the plasticity and fracture toughness of the material decreases as the tensile strength increases, which makes the fastening clip more prone to brittle failure [63]. From an engineering viewpoint, it is suggested that the tensile strength of the clip material should be properly improved to at least 1450 MPa.


## 3.1. Test Results of the Clip Material 38Si7

The engineering stress-strain curves of the fastening clip material at different states were obtained, as shown in Figure9. The test results of each pair of samples in the same stage (in the same colour but with different symbols) show a perfect agreement with each other, indicating that the test results were repeatable with high confidence. The curves for the damaged clip samples after a long service life (samples #3 and #4) are nearly the same as those in the newly formed state (samples #5 and #6), whereas the results for the bar before clip formation (samples #1 and #2) are slightly higher. Table 3 lists the mean values of the static mechanical properties. The comparison of the test results shows that the tensile strength, yield strength, and elastic modulus of the material for the fastening clip remain nearly constant in different states. The static mechanical properties of the material were not directly affected by the thermoforming process of the fastening clip, and deformations occurred during the in-service process. Based on the tensile properties, the fatigue properties of the fastening clip material were obtained through the Seeger algorithm, as shown in Table 4. Compared with the samples of the bars before clip formation (samples #1 and #2), the samples of newly delivered clips (samples #5 and #6) had slightly lower tensile strength, yield strength, and elastic modulus values. Moreover, the damage during long service periods (samples #3 and #4) had no effect on the tensile properties.Figure 9
The uniaxial tensile test results on the clip material 38Si7 (engineering stress-strain curve).Table 3
Tensile properties of the clip material 38Si7 (mean values).
SampleTensile strengthσb (MPa)Yield strengthσs (MPa)Elastic modulusE (GPa)#1/#214251254190#3/#413491230180#5/#613501242180Table 4
Fatigue properties of the fastening clip material.
SampleFatigue strength coefficientσf′ (MPa)Fatigue strength exponentbFatigue ductility coefficientεf′Fatigue ductility exponentc#1/#22138−0.0870.2581−0.58#3/#420240.2585#5/#620250.2581


## 3.2. FEA Results

The static mechanical properties of newly formed clips (samples #5 and #6) were considered in the FEA for the following two reasons. One, the model was a fastening clip rather than a bar before clip formation. Two, another was that the static mechanical properties of the fastening clip material for the clip damaged after a long service life were nearly the same as those of the newly delivered clip.In the FE calculations, the stress and strain distributions of the fastening clips in case 1 and case 2 were successively determined, as shown in Figure10. The global maximum stress of the fastening clip is 1507 MPa for case 1 and 1556 MPa for case 2, occurring at the inner side of the small arc at the heel end (see the red circles in Figure 10).Figure 10
Stress/strain distributions of the fastening clip in FEA: (a) von Mises stress distribution for case 1; (b) maximum principal strain distribution for case 1; (c) von Mises stress distribution for case 2; (d) maximum principal strain distribution for case 2.
(a)(b)(c)(d)The maximum von Mises stress, in either case, was located at node 20888 of the FE model. Six stress components are shown in Table5. Some stress components were beyond the fatigue limit of the material. The maximum principal strain was 15901.72 με for case 1 and 19156.69 με for case 2, which was also observed at node 20888 of the fastening clip. These strain components are shown in Table 6. The global maximum stress was extremely high, far beyond the fatigue limit of the material and even beyond the measured yield strength of the material, indicating that the elastoplastic stage was achieved under the actual load during operation, and it was prone to fatigue failure. Hence, the fatigue failure developed in the fastening clip was mainly due to the local deformation concentration of the material, and the inner side of the small arc at the heel end was the dangerous region.Table 5
Stress components at the critical point (unit: MPa).
NodeCaseσxσyσzτxyτxzτyz20888Case 1379.4274.07−178.10−670.96151.10246.98Case 2398.8279.18−178.62−678.88145.72255.30Table 6
Strain components at the critical point (unit:με).
NodeCaseεxεyεzγxyγxzγyz20888Case 15896.66−175.80−4986.65−25381.006029.2610136.30Case 27250.67−343.09−6109.37−30503.106865.5612571.00


## 3.3. Fatigue Crack Initiation Life Predictions

By performing 3-D coordinate transformation, three principal strains and three local maximum shear strains at the critical point were calculated, as listed in Table7. The maximum shear strain, in either case, is nearly twice as high as the maximum principal strain. The directions and the planes of the maximum principal strain ε1 and maximum shear strain γ1 can also be identified, as shown in Figure 11. Figure 12 shows the actual fracture plane of the fastening clip. The comparison of these figures shows that, at the critical point, the actual fastening clip is particularly prone to developing cracks on the plane of the maximum shear strain, and the maximum shear strain initiates the fatigue cracks. Moreover, the tensile strain in the plane accelerates the accumulation of crack damage. Therefore, the damage could be reasonably represented as a function of both the shear strain and the strain normal to this plane. The Brown–Miller criterion was ideally suited for predicting the fatigue life of a fastening clip.Table 7
Three principal strains and three local maximum shear strains at the critical point (unit:με).
NodeCaseε1ε2ε3γ1γ2γ320888Case 115901.72−1175.59−13991.9229848.8617089.2512759.61Case 219156.69−1453.70−16904.7836132.8720753.2015379.67Figure 11
Simulated fracture planes of a fastening clip.Figure 12
The actual fracture plane of a fastening clip.By substituting the material fatigue properties and the damage parameters into the Brown–Miller criterion, the fatigue lives of the fastening clips for cases 1 and 2 were predicted, and the distribution contours are shown in Figure13. Under normal in-service conditions, the lowest fatigue life of the fastening clip was approximately 180 million times (i.e., infinite life), occurring in the heel end area. For case 2, the lowest fatigue life was only approximately 2.399 million times, which is lower than the limit of the requirement before use (5 million times).Figure 13
Fatigue crack initiation lives of theω-shaped fastening clips: (a) case 1 and (b) case 2.
(a)(b)However, the actual in-service condition of the fatigue life was even worse due to the addition of the broadband minor-amplitude excitation induced by track irregularities and rail corrugation [57–61]. The typical practice employed to improve this adverse condition is to grind the rail profiles [61] or decrease the train speeds [62]. These measures severely impede the development of high-speed railway traffic. However, in this paper, the solution proposed is to improve the material performance at the design stage of a fastening clip. Based on this idea, the effects of the material static mechanical properties on the fatigue life of a fastening clip were discussed according to these in-service conditions.


## 3.4. Effect of the Material Static Mechanical Properties

According to relevant studies [1, 2, 4, 8, 11, 12, 17, 20], the elastic modulus of the fastening clip material is between 150 GPa and 300 GPa, and its tensile strength is normally within the range of 1000–1600 MPa. Therefore, the values of the elastic modulus and tensile strength in these ranges could be used to analyse their effects on the fatigue crack initiation life of the fastening clip. The calculations were performed following the aforementioned method.Figure14 shows that the fatigue lives of the fastening clips vary with the elastic modulus of the material. The increase in the elastic modulus leads to the shortening of the fastening clip fatigue life, and the effect gradually weakens as the elastic modulus increases. The fatigue life for case 1 is beyond the allowable limit of 5 million times only when the elastic modulus is lower than 250 GPa. However, compared with the fatigue life for case 1 in each elastic modulus (magenta rectangular symbols), the fatigue life for case 2 (yellow triangle symbols) is much lower. In such a case, to guarantee that the fastening clip conforms to the requirements before use, the elastic modulus of the material must be at least lower than 160 GPa (i.e., lower than the reference value).Figure 14
Variations in the fastening clip fatigue life with elastic modulus of the material.The effects of the material tensile strength on the fastening clip fatigue life for the two cases are shown in Figure15. The fastening clip fatigue life increases with increasing material tensile strength. The fatigue life for case 1 is beyond the allowable limit of 5 million times only when the tensile strength is higher than 1150 MPa. However, the fatigue life for case 1 for each tensile strength (megenta rectangular symbols) is much higher than that for case 2 (yellow triangle symbols). In such a case, to guarantee that the fastening clip conforms to the requirements before use, the elastic modulus of the material must exceed 1450 MPa (i.e., higher than the reference value).Figure 15
Variations in the fastening clip fatigue life with the tensile strength of the material.As shown in Table8, the sensitivity factors of the elastic modulus and tensile strength of the fastening clip material were calculated based on the definition of the sensitivity factor in equation (7). The sensitivity factor of the elastic modulus in each case is approximately 10, whereas that of the tensile strength in each case is much higher, with a value of approximately 15. With the worse in-service condition of the fastening clip (from case 1 to case 2), the sensitivity factor of each material performance parameter slightly decreases, i.e., from 10.09 to 9.89 and from 16.29 to 15.05, respectively.Table 8
Sensitivity factorsS(xi∗) of the material static mechanical properties xi∗.
Material performance parameterxi∗Elastic modulus (E)Tensile strength (σb)Case 1Finstallation = 17 kNA (Fcycle = 100 kN)10.0916.29Case 2Finstallation = 19 kNA (Fcycle = 130 kN)9.8915.05The sensitivity analysis indicates that the fatigue crack initiation life of the fastening clip is extremely sensitive to the elastic modulus, especially more sensitive to the tensile strength. For example, the sensitivity factorS(xi) of the elastic modulus E is approximately 15, which means that the change in fatigue life (|ΔNf/Nf|) is 15 times the change in the material static mechanical properties (|Δxi/xi|), according to (7). That is, the fastening clip fatigue life can be increased significantly by 80% when the elastic modulus of the fastening clip material decreased by only 6%. The elastic modulus and tensile strength of the fastening clip material have marked effects on the fatigue life of the fastening clip, which is of great significance in the reverse design of the fastening clip.Since it is difficult to reduce the elastic modulus of a raw material through conventional heat treatments and alloying techniques [24, 26], increasing the tensile strength of the material can be employed as an alternative means to improve the fatigue resistance of fastening clips. However, when the tensile strength is excessively high, the plasticity and fracture toughness of the material decreases as the tensile strength increases, which makes the fastening clip more prone to brittle failure [63]. From an engineering viewpoint, it is suggested that the tensile strength of the clip material should be properly improved to at least 1450 MPa.


## 4. Conclusions

Based on the measured elastoplastic characteristics of the fastening clip material, i.e., 38Si7, through a series of uniaxial tensile tests, this paper performed FEA on the complete assembly system of a fastening clip to determine the location of the dangerous region. By applying the critical plane method, the critical plane was determined. Then, the fastening clip fatigue lives in two cases were predicted following the Brown–Miller criterion. The effects of the material static mechanical properties on the fatigue life of the fastening clip were analysed, as was the sensitivity of the fatigue life to those parameters. The following conclusions can be drawn:(1)
The initiation and progression of fatigue cracks in a fastening clip occur in the plane where the maximum shear strain is included. The Brown–Miller criterion is properly employed in predicting the initiation and progression of fatigue cracks in a fastening clip and predicting the fatigue life.(2)
The material static mechanical properties of a fastening clip not only affect the stress/strain components of the fastening clip but also determine the fatigue static mechanical properties of the material. The static mechanical properties of the material have marked effects on the fatigue life of a fastening clip, which is of great significance in the reverse design of a fastening clip.(3)
The fatigue crack initiation life of a fastening clip could be improved by increasing the tensile strength of the fastening clip material to at least 1450 MPa by optimizing the metal element ratios or heat treatment of the metal. If the technical conditions permit, the elastic modulus could also be appropriately reduced to no more than 160 GPa.



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*Source: 1366007-2021-09-30.xml*