A strain energy density field method to predict the life of metallic notched components under multiaxial fatigue loading

A method based on strain energy density field (SEDF) has been proposed to predict the angle of fatigue critical points and multi-axial life of the notched parts under multi-axial cyclic loading. The point bearing the maximum strain energy density was taken as the fatigue crack initiation point (fatigue critical point). The influence of stress-strain gradient around the notches were considered via combining SEDF and the critical distance theory (TCD) to calculate the multi-axial notch life. The predictive results indicate that the maximum deviation of the fatigue critical point (FCP) was 7.0°. The most of the predictive life was within the 3-time error band.


Introduction
Engineering metallic structures inevitably have some notches, such as bolt holes, keyways, shaft keys, tool retreat slots, etc.2][3][4] The state around the notch root is multi-axial even under uniaxial loading. 5For smooth specimen under multiaxial fatigue loading, Brown and Miller 6 raised the critical plane method to calculate the angle of initiation crack, and constructed the fatigue damage parameter based on the stress or strain components on the critical plane to predict the fatigue life.The fatigue damage parameters based on critical plane are collected in reference [7].The critical plane is based on the fatigue damage physical significance, and it is usually the plane of fatigue crack initiation 8 , that is, the plane where the fatigue damage reaches the maximum.
Due to the existence of the stress gradient at notch root, the maximum stress or strain at notch root are considered as the parameter to predict fatigue life generally leading to a conservative result. 9-10Yao 11 put forward the stress field intensity (SFI) method to calculate the influence of stress gradient at the notch root, as shown in Equation (1).
where, FI V is the stress field intensity of notched components.: is the fatigue damage region.V is the volume of the fatigue damage region : . ( ) M v is a weight function, representing the effect of these points around the fatigue damage region.Afterwards, Shang 12 proposed the stress-strain field intensity method to predict the low-cycle fatigue life of notched components.The stress-strain field intensity method clarifies the fatigue damage process, and a large number of experiments show that the prediction results of this method are in good agreement with experimental results.Tanaka 13 and Taylor 14 successively proposed the theory of critical distance (TCD) to consider the effect of stress gradient at the notch root.The TCD takes the critical stress within the critical distance l0 from the notch root as the fatigue damage parameter, as shown in Equation (2).And TCD is divided into point method (PM), line method (LM), area method (AM) and volume method (VM), as shown in Fig. 1 .l0 can be determined by the EL Haddad [15][16] empirical formula in Equation (3).
Where, th K ' is the crack propagation threshold range, and 1 f ' is the tension-compression fatigue limit range. .PM(Point method), LM(Line method) and AM(Area method) in TCD Under uniaxial loading and multiaxial proportional fatigue loading, the direction of principal stress does not change in the fatigue loading cycle [17][18] , so the location of the fatigue critical point is consistent with the point of maximum principal stress under static loading, which is tangent to the principal stress. 191] Therefore for notched parts under non-proportional cyclic loading, determining FCP is the basis for the prediction of fatigue crack initiation direction.For sharp notches, such as V-notches, there is serious plastic deformation at the notch root.The stress amplitude near the root is small, the tip of V-notches is regarded as the FCP. 4,228] The TCD is used to consider the stress gradient at the notch root and the critical plane method is used to consider the influence of the multi-axial loading.Susmel 22,3 combined multi-axial fatigue damage parameters based on critical plane method and PM in TCD to predict the multi-axial fatigue notch life.Although this method has good prediction results for some materials, it cannot locate the fatigue crack initiation point and the direction of fatigue crack initiation.

Materials and Experimental
The alloy GH4169 was used in the present investigation.GH4169 is a nickel-based superalloy which is widely used in aero engine and shows excellent mechanical properties under high temperature and applied stress.The mechanical parameters of GH4169 are listed in Table 1. 2 The drawings of the smooth parts and these notched parts are listed in Figure 2. The microstructure of three kinds of notches were observed with VHX-1000 three-dimensional video microscope(3-DVM).The micrographs of notches are shown in Figure 3 .

Fatigue tests results
The fatigue tests were conducted by MTS809 multi-axial testing machine.An optical camera was installed on the testing machine through holders to shot the notches every second during the multi-axial fatigue tests.Real-time fatigue crack of the notched specimens was recorded by the camera.The test failure of notch parts was defined as that the length of fatigue crack was up to l0, that was 0.14mm.The strain life curve of GH4169 alloy under symmetrical cycle loading was tested, and the results are shown in Fig.    2, where φ is the non-proportional phase.Moreover, the recorded nominal stresses during the tests are also listed in Table 2.The angle of FCP was measured for notched parts, and the results are shown in Fig. 5.The angels of FCP are defined as θ1 and θ2 in Fig. 6.The angle of TCP for all notched specimens are listed in Table 3.

Prediction of fatigue critical point (Crack initiation point)
In the process of fatigue cycle, the internal energy of the specimens has been increasing before the fatigue crack initiation due to the cyclic loading.Therefore, at the nodes area, the higher the strain energy density, the greater the fatigue damage for metallic notched parts. 29The fatigue crack usually initiate from the points bearing the maximum strain energy density at the notch root. 30or the point in the triaxial stress state, its strain energy density is shown in Equation ( 4).In the range of linear elasticity, the strain energy density is shown in Equation ( 5).
For two-dimensional notched parts the multi-axial non-proportional tension-shear loading was subjected, as shown in Fig. 7.It can be found that all points on the edge of the notch were under uniaxial stress state, and the principal stress direction was the tangential direction of the notch edge curve. 2 Assume that the multi-axial loading is as follows: sin sin( ) Where, a V is the normal stress amplitude, a W is the shear stress amplitude, M is the phase angle, and Z is the loading angular frequency.The distribution function of principal stress and principal strain near the notch is as Equation (7).And the distribution function of strain energy density on the notch edge is shown in Equation ( 8).
= ( , , ) ( , ) The total strain energy density of point T within a fatigue loading cycle is shown in Equation ( 9).The maximum strain energy density and its corresponding value are shown in Equation (10).
In general, it is not easy to solve the distribution function of the principal stress on the edge of the notch, and in most cases the distribution function of the principal stress can only be obtained by function fitting.For complex loading or boundary cases, only the approximate numerical solution of principal stress on notch edge is given by Finite Element Method(FEM).The non-proportional loading of Equation ( 6) is uniformly discretized into n loading cases in a loading cycle, denoted as C1-Cn, as shown in Fig. 8.For each loading case, Ci corresponds to the instantaneous stress state (σi,τi).The commercial software Patran and Nastran was used for FEM calculations.One section of the specimens was restrained by six degrees of freedom, and others was subjected to tensile and torsional loading.The FEM grid nodes were arranged on the edge line of the notch, and the area around it was meshed, as shown in Fig. 9.The edge line of the notch was uniformly divided into m nodes.
According to the stress state (σi,τi) and the finite element model of the notched specimen, the principal stress σij and principal strain εij of the node j on the edge of the notch can be calculated by FEM under loading case Ci, then the strain energy density of the node j under the loading case Ci is: In the whole non-proportional fatigue loading cycle, the total strain energy density of node j is as follows: The strain energy density of all nodes under the whole fatigue loading cycle, the maximum strain energy density max H X and its corresponding node d is shown in Equation (13).Node d is the fatigue critical point, that is, the fatigue crack initiation point.

Prediction of multiaxial fatigue life
The fitted strain energy density field is the sum of the strain energy density calculated after discretization.The more the loading cases, the greater the strain energy density field.Therefore, the loading discretization factor f l is introduced, and is defined as the ratio of the strain energy density after discretization to the strain energy density corresponding to the original stress amplitude: where, f l is the loading discretization factor, which is related to the waveform of fatigue loading and the number of loading discretization cases.Thus, f l is a dimensionless parameter.
For example, if the sinusoidal fatigue loading is discretized into 8 loading cases, the sinusoidal fatigue loading waveform of uniaxial tension and compression can be used instead of solving the load discretization factor.The sinusoidal waveforms are divided into 8 loading conditions, as shown in Fig. 8.The corresponding loadings of each loading case are: 0 . The sum of strain energy density H X calculated after discretization and the strain energy

, n k k z
) loading cases, then the loading discretization factor f / 2 l n .For the triangular fatigue waveform uniformly discretized into 8 loading cases, the loading discretization factor f l is equal to 3.
The strain energy density field ( , ) x y H X near the fatigue critical point A was fitted and the direction of fatigue crack initiation 0 T is given in Eq.( 18), as shown in Fig. 9. Combined with the PM, LM in TCD and SEDF, the modified point method and modified line method based on the strain energy density field can be given:

Comparison of experimental results with predicted results
The three non-proportional loadings in Fig. 8 were uniformly discretized into 8 loading cases, and then the stress distribution of the specimen under 8 different loading cases was calculated by finite element method.The numbers of nodes on the edges of the three notches are 165, 180 and 354, respectively.According to the above method, the sum of the strain energy density of the node in all loading cases was calculated, and the node corresponding to the maximum strain energy density was the fatigue critical point which are summarized in Fig. 10.The multi-axial fatigue lives of three notched components are predicted, and the results are shown in Fig. 11.It can be concluded that the predicted life of ESDF-PM and ESDF-LM is within the 4-time error band, and the prediction result of multiaxial fatigue life of ESDF-LM is always greater than that of ESDF-PM.
In addition, the prediction accuracy of ESDF-PM and ESDF-LM is higher than the M-TCD 22 .

Figure 1
Figure1.PM(Point method), LM(Line method) and AM(Area method) in TCD Under uniaxial loading and multiaxial proportional fatigue loading, the direction of principal stress does not change in the fatigue loading cycle[17][18] , so the location of the fatigue critical point is consistent with the point of maximum principal stress under static loading, which is tangent to the principal stress.19For notched parts under multi-axial non-proportional cyclic loading, the points bear the maximum principal stress change every moment because the principal stress axis rotates during the fatigue loading cycle.[20][21]Therefore for notched parts under non-proportional cyclic loading, determining FCP is the basis for the prediction of fatigue crack initiation direction.For sharp notches, such as V-notches, there is serious plastic deformation at the notch root.The stress amplitude near the root is small, the tip of V-notches is regarded as the FCP.4,22For blunt notches, such as circular notches, there is no widely accepted method to determine the FCP.[23][24][25][26]Most of the multi-axial fatigue life prediction methods are combined by the critical plane method and TCD.[27][28]The TCD is used to consider the stress gradient at the notch root and the critical plane method is used to consider the influence of the multi-axial loading.Susmel22,3 combined multi-axial fatigue damage parameters based on critical plane method and PM in TCD to predict the multi-axial fatigue notch life.Although this method has good prediction results for some materials, it cannot locate the fatigue crack initiation point and the direction of fatigue crack initiation.

Figure 2 .Table 1 . 3 .
Geometry and dimensions of three kinds of notched thin-walled tubular specimens (all the dimensions are in mm).Smooth specimen, b) D=1mm, Circular notch, c) D=2mm, Circular notch, d) waist hole notch Mechanical parameter of GH4169 2 μ D=1mm, Circular notch b) D=2mm, Circular notch c) waist hole notch Figure Micrographs of three kinds of notches 4 .

Figure 4 .
Figure 4. Strain life curve of GH4169 under symmetrical cycle loading For these notched components, five kinds of different tension-torsion fatigue loading paths were proposed.The results of multi-axial fatigue tests of notched specimens with different loading paths are summarized in Table2, where φ is the non-proportional phase.Moreover, the recorded nominal stresses during the tests are also listed in Table2.The angle of FCP was measured for notched parts, and the results are shown in Fig.5.The angels of FCP are defined as θ1 and θ2 in Fig.6.The angle of TCP for all notched specimens are listed in Table3.

X
corresponding to the original stress amplitude are shown in Equation(15).The loading discretization factor f l can be calculated as 4. For sinusoidal fatigue loading, it is discrete into n (

Table 3 .
Angles of FCP Note: "--" means no crack initiate on this side.