Yanwen Liu， Guangyuan Jin， Jinjun Gao and Yajie Liu

（1. School of Mechatronics Engineering, Harbin Institute of Technology, Harbin 150001, China；2. Heilongjiang Institute of Industrial Technology, Harbin 150001, China）

Abstract: A research on the stable fatigue crack propagation of 16MnR steel is investigated sys?tematically in this paper. First, control experiments of 16MnR with compact tension specimen is conducted to study the effect of R?ratios, specimen thickness and notch sizes. The experiments show that the fatigue crack growth (FCG) rate in stable propagation was insensitive to these factors.Then, the stress intensity factor (SIF) is computed and compared by displacement interpolation method, J integral and interaction integral method respectively. The simulation shows that optimiz?ation on the mesh density and the angle of singular element improved the computational efficiency and accuracy of SIF and the interaction integral method has an obvious advantage on stability. Fi?nally, the FCG rate is modeled by the Jiang fatigue damage criterion and the extended finite ele?ment method (XFEM) respectively. The simulation results of FCG rate are in line with experi?ments data and indicate that XFEM method is more accurate than Jiang fatigue damage method.

Key words: 16MnR；crack propagation；stress intensity factors；fatigue damage；extended finite element method

16MnR steel is most widely used in pressure vessels in China[1]. The use of 16MnR steel under these circumstances is likely to lead to crack ini?tiation considering pressure vessels undergo high pressure in terrible working conditions. Many studies concentrated on fatigue crack growth(FCG) have been published over the years. As the most popular FCG model, Paris Law de?scribes the log?linear relationship by relating the FCG rate da/dN to the stress intensity factor(SIF) range ΔK[2]. However, some influencing factors should be considered when applying the formula into practice. Therefore, Walker has put forward an improved model to predict the effect of stress?ratios on crack propagation and fatigue life[3], Kujawski gave a further modification on the positive stress?ratios while Silva studied FCG at negative stress?ratios[4?5], Corsetti studied the effect of mean stress and environment on corro?sion and Gingell researched the effects of fre?quency and microstructure on FCG[6?7]. However,the effect of different factors on FCG varies based on the experimental setup due to the lack of a unified model that can be suitable for all kinds of materials. On the other hand, it is diffi?cult to obtain the analytical solution of SIF be?cause of the complex configuration of fatigue cracks in practical steel structures[8]. Finite ele?ment analysis (FEA) is widely used to calculate the SIF for non?standard crack configurations[9].Three common estimation methods used to com?pute the SIF are the displacement interpolation method, J integral method and the interaction integral method respectively[10?12]. The basic idea of the aforementioned methods is relating the SIF with physical quantities that can be determined from FEA. However, the accuracy and efficiency of FEA is sensitive to mesh density owing to the singularity at the crack tip.

It is believed that fatigue failure will hap?pen when the fatigue damage on a critical plane accumulates to an extent[13]. Several fatigue cri?teria have been developed based on observations on cracking behavior of various materials. The Fatemi?Socie criterion defines the critical plane as the plane with the maximum shear strain while short crack criterion proposed by D?ring R is expressed in terms of the range of the cyclic J?Integral[14?15]. Jiang developed a new multiaxial fatigue criterion based on the cyclic plasticity,which was assumed to be the major cause of fa?tigue damage[16]. The three criteria made great ef?fort to reveal the multiaxial fatigue mechanism by defining different damage parameters and crit?ical plane approaches. The advantage of using the Jiang criterion was that it could predict the FCG orientation and calculate the FCG rate by considering the mean stress effect and loading se?quence effect comprehensively[17]. At the same time, the extended finite element method(XFEM) proposed by Belytschko overcame the limitations of the traditional finite element meth?od by addressing the discontinuity problem in crack propagation[18]. XFEM does not need to up?date the grid dynamically, which reduces the amount of elements and improves the computa?tional efficiency and accuracy[19]. As an emerging numerical method, XFEM opens up many excit?ing possibilities for its further development and advancement.

The aim of this paper is to systematically analyze the fatigue crack propagation of 16MnR steel based on Paris Law. The effect of different factors on FCG was investigated and described through the experiments. Three current numeric?al techniques to calculate SIF were compared to the theoretical solution. The effect of mesh dens?ity and the angle of a singular element on the ac?curacy of SIF was simulated by FEA. Both the Jiang criterion and XFEM were applied to simu?late the FCG behavior respectively, which were then verified experimentally. The advantages and disadvantages of the Jiang fatigue model and XFEM are summarized at the end.

1 Crack Propagation Experiments

Mechanical properties of 16MnR steel should be determined by a uniaxial tensile test at ambi?ent temperature before the crack propagation ex?periments. This test was conducted according to GB228–2002 in China. The mechanical con?stants reported in Tab. 1 show that the meas?ured value of yield strength is higher than the standard yield strength 345 MPa, while the measured value of tensile strength is within the standard range 510 MPa–655 MPa.

Tab. 1 Mechanical constants of 16MnR

Crack propagation experiments were conduc?ted according to the GB/T 6398–2000 “Stand?ard test method for fatigue crack growth rates of metallic material” in China. The detailed dimen?sions of a compact tension (CT) specimen is shown in Fig. 1. Wire?electrode cutting was ad?opted to process the notch. In order to observe how R?ratios, specimen thickness, notch sizes and load amplitude affect the FCG rate, four control groups were set in the experiments (Tab. 2). It is noted that crack length is the distance in the x?direction from the center of the circle to the crack tip.

The main piece of equipment used in these experiments was an Instron?8802. FCG displace?ment was measured by the extensometer fixed on the side of the CT specimen (Fig. 2). The con?stant?amplitude loading frequency was 10 Hz with the sine load wave. Data acquisition in the all experiments was automated by a computer.

Fig. 1 Standard CT specimen (all dimensions are in millimeters)

Tab. 2 Control groups in crack propagation experiments

Fig. 2 Experiments equipment Instron?8 802

The acquired data was processed in the fol?lowing section. Seven point incremental polyno?mial method was adopted to calculate the FCG rate da/dN while ?K was calculated through the given empirical formula defined as[20]

In Eq.(1), ?P is the difference between the minimum and maximum of the loading force and W is the width of the CT specimen. α=a/W, a denotes the crack length.

The FCG rate curve of 16MnR fitted by the least squares method is depicted in the log?log scale as shown in Fig. 3, which is characterized by the standard form of the Paris Law[20]

where C and m are the material constants.

The values of C and m remain unchanged with increasing specimen thickness (Fig. 3a).The results indicate that FCG is insensitive to the specimen thickness. The effect of R?ratios on stable FCG can be shown in Fig. 3b. The values of C and m show a trend of fluctuations on a small scale but overall remains constant. Stress relaxation can account for this phenomenon. The stress is readjusted when the local yield zone at the crack tip occurs, which results in a small change in the real R?ratio at the damage area.The R?ratio has a little influence on stable FCG under the circumstances. The values of R?ratio adopted in these experiments are relatively small on the other hand. R?ratios are considered to be of a little impact on stable FCG at a low R?ratio range. It is obvious that the values of C and m keep constant with increasing notch size(Fig. 3c). It is generally accepted that the notch size affects the early stage of FCG. It is demon?strated that notch size is irrelevant to the stable FCG by this experiment.

Fig. 3 FCG rate curves in experiments

2 Simulation on SIF

Due to the complexity of theoretical deriva?tion, there is a lack of a unified method for com?puting SIF under different boundary conditions.FEA can greatly reduce the computational diffi?culty and computation time. The stress near the crack tip is directly proportional to r–0.5, which results in a singularity at the crack tip[8]. Quad?ratic isoparametric finite element with 1/4 node should be adopted to reflect the singularity in FEA.

The dimensions of the singular elements can affect the efficiency and accuracy on computing SIF. The main influencing factors are the radius and angle of the singular elements (Fig. 4).

Fig. 4 Singular elements in SIF model

Simulation results on the three methods (A denotes the displacement interpolation method,B denotes the J integral method and C denotes the interaction integral method) can be seen in Fig. 5. K0represents the theoretical value. It can be seen in Fig. 5a and Fig. 5b that the displace?ment interpolation method is more sensitive to the angle of the singular elements and the J in?tegral method is more sensitive to the radius of the singular elements. However, it is obvious that the interaction integral method is insensitive to the radius and the angle of the singular elements,which shows excellent stability. Fig. 5c shows that the simulation results of the J integral method and the interaction integral method reach a steady state ultimately with the integral path moving outward.

Overall, the interaction integral method has more advantages on maintaining the stability,compared to the other two methods. To guaran?tee the efficiency and accuracy of the simulation,the size of the singular element was determined to be 2.8 mm with the displacement interpola?tion method, 0.05 mm with the J integral meth?od and 3 mm with the interaction integral meth?od while the angle of the singular element was adopted as 15°. The SIF based on the three methods are basically of the same accuracy with a maximum relative error of 0.34%, 0.46% and 0.44% respectively (Tab. 3). The J integral method and interaction integral method do not exploit their advantages to their full extent con?sidering that 16MnR belongs to isotropic materi?al and CT specimen model is a simple model. By general comparison of the efficiency and accur?acy, the interaction integral method is the optim?al method to calculate SIF.

Fig. 5 Simulation analysis with the three methods

Tab. 3 Simulation results on SIF with three methods

3 Simulation on FCG Rate

3.1 Simulation on Jiang fatigue damage model

A new fatigue criterion combining the ma?terial memory and an energy concept proposed by Yanyao Jiang takes the following form[16]

Fig. 6 Fatigue damage curves

Von Mises yield criterion and bilinear kin?ematic hardening model were adopted to build a finite element model of 16MnR in ANSYS. The CT specimen can be assumed to be in the plane strain state considering the specimen thickness.On the other hand, only half of the CT specimen was modeled, assuming symmetry. The grids near the crack tip need to be subdivided considering the singularity at crack tip (Fig. 7). It is noted that the simulation results are greatly affected by the element size. The element size that is twice or three times the grain size can contribute to a satisfactory result, which was determined as 0.125 mm in this paper. Stress field and strain field of the model are shown in Fig. 8 (Pmax=10 kN, R=0.1, W=8 mm, r=0.2 mm).

Hysteresis loop at the crack tip is shown in Fig. 9. The results are in good agreement with the elastic?plastic constitutive model. It implies that the plastic deformation and the fatigue dam?age increases with the increasing crack length. a denotes the crack length, the same below.

As shown in Fig. 10, the stress decreases with the crack length increasing. The crack tip is in the three?dimensional stress state. The stress in every direction is larger than the Mises stress,which indicates that plastic deformation is diffi?cult near the crack tip. It can also be found that σmris approximately equal to yield strength. A new material constant B can be created to re?place the material constants in the Jiang cri?terion, which includes σ0, m, σmrand D0as

where Dndenotes the new defined fatigue damage.

Fig. 7 Finite element model of 16MnR

Fig. 8 Stress field and strain field near the crack tip (the unit of stress is MPa)

Fig. 9 Hysteresis loop at the crack tip

Fig. 10 Stress distribution curves

The parameter B can be determined by com?bining the simulation results with the test data,shown as

f(B) is the objective function and the sub?script t represents the data obtained from the test. B equals 2×10?3when the error is the least.However, it can be easily found that B is irrelev?ant to the slope of the FCG rate curve.

The distribution of Dncan be seen in Fig. 11(b=0.38). It is obvious that the damage region and the value of damage increases with increas?ing crack length. Fig. 12 implies that the FCG rate remains constant generally when the crack length is short. As a result, the FCG rate curve can be obtained by simulation. Fig. 13 shows that simulation curve and the test curve(Pmax=10 kN, R=0.1, W=8 mm, r=0.2 mm).The relative error of the value of m is 1.67%.Simulation of the Jiang criterion fits the experi?ments perfectly, which can prove the accuracy of the Jiang criterion.

Fig. 11 Distribution of Dn with the change of r

Fig. 12 Relation between Dn and the crack length

Fig. 13 Simulation curves by the Jiang criterion

3.2 Simulation on XFEM

The crack is permitted to propagate through the elements in XFEM based on discontinuous shape function. As shown in Fig. 14. Low cycle fatigue (LCF) analysis is based on the relative energy release rate ?G as

where c3and c4are the material constants. In Fig. 14, Gthrepresents the threshold value of the energy release rate. The crack does not propag?ate when G

Fig. 14 da/dN curve with energy release G

The conditions of crack initiation are demon?strated in

where c1and c2are the material constants. N de?notes the number of loading cycles with constant amplitude.

The finite element model of 16MnR in Abaqus can be seen in Fig. 15a. C3D8 plane ele?ments are adopted to mesh the grids. Prefabric?ated crack surface needs to be assembled with the model. The stress distribution is shown in Fig. 15b. The results demonstrating that the stress distribution symmetrical in the x?axis and the maximum stress occurs in the crack tip matches our priors.

Simulation results by combining XFEM and LCF are shown in Fig. 16 ( c1= 0.5, c2= –0.1,c3= 2.05×10?5, c4=1.42). Fig. 16a shows the a?N curve by XFEM simulation with the maximum relative error of 2.63% (Pmax=10 kN, R=0.1,W=8 mm, r=0.2 mm). Fig. 16b shows the FCG rate curve by XFEM simulation. The relative er?ror of the value of m is 0.31%. It indicates that XFEM simulation fits perfectly with the experi?ments, which proves the accuracy of XFEM.

In summary, XFEM has advantages over the Jiang Fatigue damage model in a range of applic?ations and accuracy, while the adjustment of the material constants in XFEM is more complic?ated than the Jiang model.

Fig. 15 Model simulation by XFEM

Fig. 16 XFEM simulation about FCG rate curves

4 Conclusions

The systematic research on fatigue crack propagation according to the Paris Law is reiter?ated here.

① According to the experiments, it can be found that the FCG rate in stable propagation was insensitive to the specimen thickness, R?ra?tios and notch sizes.

② The SIF based on the aforementioned three methods are of the same accuracy gener?ally for the CT specimen model of 16MnR.However, the simulation results of the interac?tion integral method are insensitive to the angle and the size of the singular element compared to the J integral method and the displacement in?terpolation method, which is the optimal meth?od to calculate SIF by general comparison of the efficiency and accuracy.

③ The new material constant, B, is determ?ined to be 2×10?3for the improved Jiang fatigue damage model. The relative error of material constant between the simulation curve by the Ji?ang fatigue damage model and the test curve is 1.67%.

④ The parameters of XFEM simulation based on LCF are determined as c1= 0.5, c2=–0.1,c3= 2.05×10–5, c4=1.42. The relative error of ma?terial constant between the simulation curve from the XFEM model and the test curve is 0.31%.

⑤ XFEM has advantages over the Jiang fa?tigue damage model in a range of applications and accuracy while the adjustment of the materi?al constants in XFEM is more complicated than the Jiang model.