Fatigue Damage Analysis of Double-Lap Bolted Joints Considering the Effects of Hole Cold Expansion and Bolt Clamping Force OPEN ACCESS

Different structural components are typically connected by fasteners such as bolts. The bolt transmits load while causes a high stress concentration at the edge of the fastener hole. In order to enhance the fatigue performance of bolted joints, application of appropriate clamping force is an effective method. According to the studies performed by Oskouei and Ibrahim [1] and Chakherlou et al. [2], fatigue performance of the double-lap bolted joint was found to be strongly dependent on the applied bolt clamping force. Two primary reasons of clamping force on fatigue life improvement of bolted joint have been demonstrated as compressive residual stress [3] and the transmission of partial load by friction force between the clamped plate [4]. However, the enhancement of clamping force alone is limited by the permissible value of force. In order to further improve the fatigue property of joints, hole cold expansion method is often combined with bolt clamping force in bolted joints. Hole cold expansion can induce a significant amount of plastic deformation around the hole by pulling an oversized tapered mandrel through the hole. After removing the tapered mandrel, compressive circumferential stresses around the hole are produced. The residual stresses reduce the maximum tensile stresses at critical locations when the component is subjected to tensile cyclic loading. Study [5] has demonstrated that the cold expansion can extend both the crack nucleation and crack propagation periods by large proportions.

Chakherlou et al. [3,6] conducted a series of fatigue tests and numerical simulations to investigate the combined effect of hole cold expansion and bolt clamping on the fatigue life and failure mode. The results showed that the effect of cold expansion in conjunction with the clamping force on fatigue behavior of joints differs from that when each factor acts separately. However, their studies failed to explain the differences of the influencing mechanisms of the combined factors. Chakherlou et al. [7,8] also adopted the finite element method combined with critical plane criteria (e.g., Smith et al. [9], Glinka et al. [10], and Fatemi and Socie [11]) to predict the fatigue lives of specimens with cold expanded holes and bolt clamping force. The predicted fatigue lives show significant deviation compared to the experimental data, which is because the critical plane methods neglect the plasticity evolution and fatigue damage evolution of hardened plastic material after cold expansion.

Due to the complicated damage mechanisms of the bolted joints pretreated by clamping force and hole cold expansion, it is very difficult to analyze quantitatively the fatigue life of joints and to reveal the combined impacts of these two pretreats. Some key aspects are needed to be considered in the analysis. The first one is the effect of residual stresses. These two pretreats both induce residual stresses around the hole. However, the hole cold expansion mainly produces compressive residual stresses along the circumferential direction, while the clamping force mainly results in compressive residual stresses along the hole axial direction. When two pretreats are carried out at the same time, the residual stress field around the hole becomes more complicated. As it is clear, the residual stresses have direct influence to the subsequent fatigue damage. The second one is the effect of plastic deformation. An accurate plasticity constitutive model is required since the material will experience significant plastic deformation during the hole cold expansion process. Moreover, the material around the hole will also undergo plasticity under high level of fatigue loadings due to the stress concentration. The third one is the continued variation of the stress and strain fields during the cycles, which is nonlinear and couples with the deterioration of the material. Therefore, in order to consider comprehensively the fatigue behavior of the bolted joint that is pretreated by hole cold expansion and clamping force, a damage-coupled plasticity constitutive model is required.

Continuum damage mechanics (CDM) [12] provides an effective method to describe the degradation of the material. Based on thermodynamics, damage-coupled elasto-plastic or visco-plastic constitutive models combined with damage evolution laws can be used to model the evolution of ductile plastic damage, fatigue, creep, and creep-fatigue interaction. The details of those methods can be found in the publications of Kachanov [13], Lemaitre and Chaboche [14], Lemaitre and Rodrigue [15], as well as Voyiadjis and Kattan [16]. Moreover, the continuum damage mechanics has been extended to describe the evolution of anisotropic damage in metal matrix composites [17,18]. Besides many theoretical models have been achieved, and CDM has also been widely used in engineering applications [19,20]. Especially, the studies performed by Basaran group have achieved abundant results on the research of fatigue and visco-plastic interaction behavior of solder joints to simulate the fatigue failure prediction [21,22] and to investigate the viscoplastic-fatigue interaction behavior of solder joints [23–25]. Two methods usually are applied to the fatigue life prediction. One is the uncoupled method, which obtains the fatigue life by integrating the damage evolution equation with ignoring the coupling effect between stress field and damage field. The other is the coupled method, which calculates the fatigue life by accumulating the damage cycle by cycle with considering the occurrence of coupling effect in each loading cycle. Generally, for the purpose of accurate applications or when the coupling between strains and damage is strong, the coupled CDM approach is very necessary. In those cases, the constitutive models need to be implemented by the numerical method, and the damage accumulation will be calculated cycle by cycle to take account of the material degradation and stress redistributions [15].

In this paper, a continuum damage mechanics based approach in conjunction with the finite element method is developed. The damage-coupled Voyiadjis plasticity constitutive model [26] is used to represent the material behavior, which is implemented by user material subroutine in abaqus. The elasticity and plasticity damage evolutions of material are described by stress-based and plastic-strain-based equations, respectively. The fatigue damage of joint is calculated using abaqus cycle by cycle. The fatigue lives of double-lap bolted joints with and without clamping force at different levels of hole cold expansion are all obtained. The characteristics of fatigue damage corresponding to the different conditions are presented to unfold the influencing mechanism of these two factors. The predicted fatigue lives and crack initiation locations are in good agreement with the experimental results available in the literature. The beneficial effects of hole cold expansion and bolt clamping force on the fatigue behavior of bolted joint are presented in this work.

The damage-coupled Voyiadjis plasticity constitutive model [26] is used to model the behavior of a damaged and hardened plastic material.

A representative volume element (RVE) [14] is introduced to describe a damaged material. The properties of the RVE are represented by the homogenized variables. The existence of a macroscopic crack of the size of the RVE is near 0.1 mm for metals. It is assumed that damage is isotropic and can be represented by the decrease in the elastic modulus [14]. The damage variable is thus defined as follows: Display Formula

where $Ẽ$ is the equivalent elastic modulus of the damaged RVE, and $E0$ is the initial elastic modulus of the material. For small strain problems, the total strain $εij$ can be additively decomposed as Display Formula

where $εije$ and $εijp$ are the elastic and plastic strain components, respectively. According to the principle of strain equivalence, the elastic strain of the damaged material is written as follows: Display Formula

where $v$ is Poisson's ratio, $σij$ is the Cauchy stress, and $δij$ is the Kronecker delta.

The von Mises yield function with the consideration of nonlinear plastic hardening flow rule is expressed as follows: Display Formula

where $sij$ is the deviatoric component of the stress, $Xij$ is the deviatoric component of the back stress, and $σy0$ is the initial yield stress. $Xij$ describes the movement of the yield surface corresponding to the kinematic hardening. The evolution law of the back stress $Xij$ proposed by Voyiadjis and Basuroychowdhury [27,28] and developed by Voyiadjis and Abu Al-Rub [26] is written as follows: Display Formula

where $Ck$, $βk$, and $γk$ are material constants that are determined from experimental tests, and $p˙$ is the accumulated plastic strain rate. In Eq. (4), $R$ is the radius of the yield surface and its evolution is defined as Display Formula

where $b$ and $Q$ are material constants, and $λ˙$ is the plastic multiplier that is determined by the plastic flow consistency condition: $f˙=f=0$. The evolution equations of the plastic strain components can be obtained as Display Formula

The damage $D$ can be considered to occur in two parts: elasticity- and plasticity-related damage, which are dependent on the cyclic stress and accumulated plastic strain, respectively [29]. Thus, one obtains Display Formula

where $S$ and $m$ are material parameters, $Rv$ is the stress triaxiality function, and $σH$ and $σeqv$ are the hydrostatic and von Mises stresses, respectively.

The evolution law of the elasticity damage has been described by Lemaitre and Chaboche [14]. For the multiaxial cyclic loading case, the damage rate equation of the nonlinear continuous damage model (NLCD) can be written as Display Formula

where $N$ is the number of cycles until failure; $α$, $χ$, $M0$, and $b2$ are material constants that are determined by fatigue tests. In Eq. (13), $AII$ and $σH,mean$ are the amplitude of the octahedral shear stress and the mean hydrostatic stress, respectively Display Formula

where $Sij,max$ and $Sij,min$ are the maximum and minimum values of the deviatoric stress components during one loading cycle, respectively, while $σH,max$ and $σH,min$ are the maximum and minimum hydrostatic stresses during one loading cycle. Also, in Eq. (13), the parameter $α$ is given by Display Formula

where $σeqv,max$ is the maximum von Mises stress during one loading cycle, $σl0$ is the fatigue limit at zero mean stress, and $b1$ is a material constant. The number of cycles until failure is given by integrating Eq. (13) from $D=0$ to $D=1$Display Formula

For comparative purposes, the Smith–Watson–Topper (SWT) critical plane criterion [9] is introduced to estimate the fatigue life. The SWT criterion is based on the product of first principal stress and principal strain range on a specified plane for tensile mode failures. The SWT life prediction employs a combined low-cycle and high-cycle fatigue equation, which is expressed as Display Formula

where $σn,max$ is the maximum normal stress on the critical plane, $Δε1$ is the principle strain range on the same plane, $σf′$ and $d$ are the high-cycle fatigue constants, and $εf′$ and $c$ are the low-cycle fatigue constants.

The material studied in this paper is Al alloy 2024-T3. Material parameters in the plasticity constitutive model, the plasticity and elasticity damage evolution laws are identified based on experimental data from uniaxial tensile tests, low-cycle and high-cycle fatigue tests. The best least square fit is adopted to determine the parameters.

First, the stress-strain data [31] of uniaxial tension is used to determine the back stress in the constitutive equations. The parameters of isotropic hardening for Al alloy 2024-T3 are obtained from the literature [32]. Three components of the back stress are used to describe the kinematic hardening behavior. The plasticity stress-strain curve of uniaxial tension is expressed as follows: Display Formula

The differential and integrated forms of the back stress can be rewritten as follows [26]: Display Formula

where $εp$ is the plastic strain. Figure 1 shows the fitting curve of stress-strain curve for Al alloy 2024-T3, and the parameters are summarized in Table 1.

Second, the material parameters in the evolution law of the plasticity damage are identified from the strain controlled low-cycle fatigue test data. The strain-life curve and cyclic stress-strain curve can be written, respectively, as follows: Display Formula

where $εf′$ and $c$ are material parameters, and $H′$ and $n′$ are material parameters. For the uniaxial case, the number of cycles until failure is obtained by integrating the evolution model of the plasticity damage Eq. (11) from $D=0$ to $Dc$Display Formula

where $Dc$ is the critical value of the damage at macrocrack initiation. The critical value of $Dc$ for ductile failure of Al alloy 2024-T3 is cited from the literature [33]. The parameters $εf′$, $c$, $H′$, and $n′$ are obtained from low-cycle fatigue tests [8,34]. The material parameters in Eqs. (11) and (24) are obtained by equating Eqs. (22) and (24) and are shown in Table 2.

Third, the high-cycle fatigue data [8] of unnotched specimens of Al alloy 2024-T3 are used to determine the parameters of the NLCD model. The fatigue limits $σl0$ and $b1$ are identified from stress-life data at different stress ratios (e.g., $R$  = −1, 0, and 0.4). The values of $M0$ and $χ$ can be determined numerically by stress-life data at stress ratio $R=−1$, and $b2$ can be determined from stress-life data at different stress ratios (e.g., $R$  = 0 and 0.4). Based on the known values of $αM0−χ$ and $χ$, $a$ can be identified numerically. The details of this method have been described by Zhang et al. [35]. The identified material parameters are shown in Table 3.

In addition, the low-cycle and high-cycle fatigue data is also used in the SWT model. The material constants of SWT are summarized in Table 4. Figure 2 shows the comparison of the SWT model and the integrated NLCD model.

The fatigue experiments of cold expanded and bolted joint specimens performed by Chakherlou et al. [6,31] are simulated in this paper. A brief overview is given to illustrate the fatigue test specimens and experimental scheme.

Figure 3(a) shows the dimensions of the bolted joint specimens. The specimens were made from 3.20 mm thick Al alloy 2024-T3. The diameter of the fastener hole is 5.90 mm. Part of these specimens were cold expanded. The details of the cold expansion process were presented in Ref. [31]. Two oversized pins with diameters of 5.988 mm and 6.177 mm were used separately on different specimens to create cold expansion degrees of 1.5% and 4.7% (Figs. 3(b) and 3(c)). The cold expansion degrees are defined as $CE=(d2−d1)/d1×100%$, where $d1$ is the diameter of the hole and $d2$ is the large diameter of the pin. A tightening torque of 4 N·m was selected to clamp some joints, while the other joints connected with no clamping force which creates clamping force equaling to 2469 N [2]. The fatigue test specimens were classified into six batches, in which different levels of cold expansion and torque clamping were combined as shown in Table 5. The first three batches (batch 1, batch 2, and batch 3) were pretreated by different levels of cold expansion without tightening torque, while the other three batches were pretreated by different levels of cold expansion with a tightening torque of 4 N·m. The fatigue tests were carried out using sinusoidal cycles at frequency of 12 Hz and loading ratio of R = 0. These tests were performed at eight loading levels from 8 to 18 kN (equivalent remote stress ranges of 139–312 MPa).

Half of the actual structural component is modeled in abaqus due to the material, geometric, and loading symmetry with respect to the X–Z plane (Fig. 4). The model of the specimen consists of six sets of elements corresponding to the plates, bolt, pin, and support. To prepare the bolted joint, two steps are used: model the hole cold expansion process and assemble the bolted joint. The three-dimensional eight-node solid elements (C3D8 in abaqus) are used to mesh the models. The damage-coupled plasticity constitutive model is used to simulate the mechanical behavior of Al alloy 2024-T3. A linear elastic material model is used for the bolt, pin, and support with the elastic modulus of 210 GPa and Poisson's ratio of 0.3. The master-slave algorithm is used to simulate all the possible contacts between the pin, support and plate, between the bolt and plates, and between the plates. For the contact pairs of pin and plate, a coefficient of friction $μ$  = 0.1 is used [31]. For the Al alloy plate pairs, the friction coefficient of 0.71 is considered according to the study of contact behavior on the bolt joints [37].

Three primary load steps are included in the simulations. In the first load step, to conduct cold expansion, the pin is pulled passing through the hole and removed from the other side by applying an incremental displacement on the upper surface of the pin in the −Z direction. In the second load step, the pin and the support are moved away, and the plates and the bolt are assembled. In the third load step, cyclic loading is applied on the middle plate to simulate the fatigue damage.

User material subroutine is used to implement the damage-coupled Voyiadjis constitutive model and simulate the fatigue damage evolution of the bolted joint. The subroutine updates the stress, solution-dependent state variables (SDV), and the Jacobian matrix at each time increment and calculates the accumulated damage at each integration point during the cycles. The solution-dependent state variables are used to save the variables of the stress update and the damage accumulation. Because significant amount of computational time is required to simulate each fatigue cycle, a cycle jumping factor $ΔN$ is introduced to reduce the number of cycles actually being calculated. It is assumed that the stresses and strains remain unchanged during each block of $ΔN$ cycles, and the damage accumulates linearly in this block. To obtain a convergent result in the numerical simulation, the cycle jumping factor $ΔN$ should meet the condition of $ΔN/NF<0.02$, which was also adopted by Zhang et al. [35]. The simplified algorithm is shown in Fig. 5. The details of the numerical simulation scheme are listed as follows:

In this paper, the six batches of bolted joints under maximum loads of 8, 10, 12, and 14 kN were simulated, respectively. Two different methods are used to predict the fatigue life of the bolted joints, as follows:

Figure 6 shows the predicted fatigue lives versus the experimental results [6] for the six batches of bolted joints. As shown in Fig. 6(a), the predicted fatigue lives using the proposed approach are in good agreement with experimental results. It is shown that 18/24 of the predicted results are located within the twice-error band, and the rest of the results are located within the three times error band. However, the SWT method (Fig. 6(b)) provides underestimated results for most of the cases.

The error index (ER) is used to assess the performance between the predicted fatigue lives and the experimental fatigue lives [38]

Table 6 shows the average absolute errors of the SWT method and the proposed method. It is shown that the error of the proposed approach shows superior performance to the SWT method. The CDM coupled with finite element analysis takes into account of the fatigue damage evolution and stress redistribution at the critical location and hence predicts longer lives compared with the SWT method.

Figure 7 shows the predicted damage fields and crack nucleation locations for different types of bolted joints under a load of 10 kN. SDV38 is a solution-dependent state variable that is used to represent the accumulated fatigue damage. For the cases of 0% and 0 N·m (Fig. 7(a)) and 0% and 4 N·m (Fig. 7(c)) which have not been pretreated with hole cold expansion, the predicted crack is shown to nucleate at the median-thickness of the hole. For the cases of 4.7% and 0 N·m (Fig. 7(b)) and 1.5% and 4 N·m (Fig. 7(d)) which have been pretreated by hole cold expansion with different levels, the crack is predicted to nucleate near the upper edge of the hole. The predicted crack nucleation locations of the aforementioned cases show good agreement with the experimental results [6] which are shown in Fig. 8. For the 4.7% and 4 N·m case, the predicted crack is shown to nucleate near the median-thickness of the hole, while the experimental crack is nucleated from the upper surface of the middle plate due to fretting damage. Since the fretting fatigue damage is a rather complicated problem related to shear stress and relative movement, which is not considered in this study, the predicted result shows certain inconformity with the experimental data.

Figure 9 shows the predicted and experimental fatigue lives versus cold expansion level for the bolted joints under a load of 10 kN. For the 0 N·m case, the predicted fatigue life is shown to increase as the cold expansion level increases. For the 4 N·m case, the predicted fatigue life is shown to increase as the cold expansion level increases from 0% to 1.5%, and decreases from 1.5% to 4%. The predicted fatigue lives show good agreement with the experimental results. Figure 10 shows the comparison of experimental and predicted fatigue lives versus tightening torque for the bolted joints under a load of 10 kN. The predicted fatigue lives for 0% and 1.5% cases are both shown to increase as the increasing of tightening torque, which agrees well with the experimental results. With the increase in tightening torque, the load transmitted by the frictional force increases and the load transmitted by the bolt decreases, thus the fatigue life is predicted to increase from 0 N·m to 6 N·m. It can also be seen from Figs. 9(a) and 10(a) that the isolated application of the hole cold expansion or clamping force can increase the fatigue life of the bolted joint. However, when the cold expansion and clamping force are combined (Figs. 9(b) and 10(b)), the influencing trend to the fatigue life is shown in a different way, especially under certain clamping force the increase of hole cold expansion level increases at first but decreases afterwards for the fatigue life. The different effects of the isolate and combined applications of the hole cold expansion and clamping force are related to the different compressive stress fields and fatigue damage evolutions.

Regarding the fact that the nucleation of fatigue cracks is related to the cyclic stresses and strains of the material, the longitudinal stress $σX$ at the critical location of different bolted joints under a load of 10 kN, which is the dominant stress component, is shown with two cycles in Fig. 11. The cyclic stress and strain can be used to partly describe the effect of the hole cold expansion and bolt clamping force. As shown in Fig. 11(a), the maximum stress of the cases pretreated by hole cold expansion with 1.5% and 4.7% shows significant decrease at the beginning of the loading cycles, which is explained by the compressive residual stress around the hole caused by the hole cold expansion. As the remote stress increases to the maximum load, the maximum stresses corresponding to the cases with hole cold expansion are shown to be smaller than that of the case without hole cold expansion. That is because the plastic deformation occurs at the critical location of the case without hole cold expansion, which slows down the increase of the stress. However, with the increase of cold expansion level, the maximum stress shows an additional slight decrease and the effect of hole cold expansion benefit is not very obvious when the remote stress is relatively large. While for the bolted joints clamped with 4 N·m (Fig. 11(b)), the stress amplitudes are shown to be significantly reduced. Besides, one can also see that the maximum stress is shown to decrease as the cold expansion level increases from 0% to 1.5%, which is related to the compressive residual stress caused by cold expansion. However, the maximum stress is shown to increase as the hole cold expansion level increases from 1.5% to 4.7%. This is explained because the clamping force will release the compressive residual stress of the hardened material. This nonmonotonic correlation of cyclic stress with the level of cold expansion combined with clamping force shows excellent agreement with the trends of the predicted fatigue lives as shown in Fig. 9(b).

Since the cyclic stress of the first two cycles cannot describe comprehensively the fatigue damage evolution of the different bolted joints, the cyclic stress-strain curves at the crack nucleation location of the six batches of bolted joints under load of 10 kN are shown in Fig. 12. It is worth noting that the SWT method considers the cyclic stresses and strains as constant during the fatigue cycles. However, the proposed approach can illustrate the variation of stress and strain with the loading cycles by counting the coupling effect between fatigue damage and material degradation. The cyclic stress can indicate the damage evolution process of the material from the original state to the fatigue failure. For the 0% and 0 N·m case (Fig. 12(a)), significant plastic strains occur during the first cycle, then the maximum stress $σX$ is shown to decrease as the number of loading cycles increases, which can be attributed to the stress redistribution due to the accumulated damage. It is also shown that cyclic plastic strain occurs at the end of the simulation due to fatigue damage and softening of the material. For the cases of 1.5% and 0 N·m (Fig. 12(b)) and 4.7% and 0 N·m (Fig. 12(c)), the maximum stresses are also shown to decrease as the number of loading cycles increases. Opposite to the 0% and 0 N·m case, the maximum strain and strain amplitude of the cases of 1.5% and 0 N·m and 4.7% and 0 N·m are shown to change little during the fatigue cycling. It is explained that the damage of the cold expanded hole is localized within a small region, and that the adjacent material with smaller damage will restrict the strain amplitude due to the significant compressive residual stress. According to the principle of strain equivalence [15], the strain constitutive equation of a damaged material is derived from the same formalism as for a nondamaged material. Therefore, the increasing strain and strain amplitude in damage evolution of 0% and 0 N·m case will yield larger damage accumulation rate than the constant strain and strain amplitude of the cases of 1.5% and 0 N·m and 4.7% and 0 N·m. It also means that the compressive residual stress of hole cold expansion not only decreases the maximum stress at critical locations, but also decreases the damage accumulation rate, which are the primary reason of fatigue life improvement. For the three batches of bolted joints clamped with 4 N·m (Figs. 12(d)–12(f)), the stress amplitudes are shown to be significantly reduced. It is also found that the effects of hole cold expansion on stress-strain curves of clamped joints are similar to the joints without clamping force. One can see that the primary effects of hole cold expansion and clamping force on fatigue life improvement and the associated fatigue damage evolution are successfully captured in the proposed approach, thus the proposed approach predicts excellent reliable results.

Based on the continuum damage mechanics, the effects of hole cold expansion and bolt clamping force on fatigue damage of bolted joint are investigated. The damage-coupled Voyiadjis plasticity constitutive model and elasticity and plasticity damage evolution laws are used to represent the material behavior, which is implemented by user material subroutine in abaqus. The simulations are performed in three main steps: the process of the hole cold expansion, the assembly of the bolted joint, and the fatigue damage accumulation cycles by cycles. Some key findings are summarized: