New fatigue limit assessment approach of defective material under fully reversed tension and torsion loading

– This paper presents a new approach to predict the high cycle fatigue limit of defective material submitted to multiaxial loading. Defects are simpliﬁed to a half spherical void at the surface of a specimen. Finite Element (FE) method is used to determine stress distribution around defect for diﬀerent sizes and loading levels. Papadopoulos high cycle fatigue criterion is used to calculate equivalent stress around defect. Based on stress analysis, a deﬁnition of aﬀected area is proposed, in which the Papadopoulos criterion is violated. The evolution of the aﬀected area, versus the amplitude of loading and defect size leads to determine fatigue limit for defective material. Results are in good agreement with experimental investigations and show that the aﬀected area is a good parameter to predict the inﬂuence of a defect on multiaxial fatigue behaviour.


Introduction
The fatigue limit of components containing inherent defects is of great importance for industrial applications.Their lifespan is always depending on morphology of the defect and the loading path.Due to the production process various types of casting defects can affect mechanical parts such as: inclusion, shrinkage, and gas pore or dross defect [1].The designer needs to create balance by taking into account the real effect of the size of defect on the fatigue limit of the component to correlate the fatigue limit with defect size.The aim of this paper is to propose a new approach that allows to predict the HCF limit of a mechanical part containing defect.
Several methods are explored and different authors have proposed fatigue limit calculation corresponding to different sizes of defect.Murakami [2] proposed an interesting approach based on experimental results.A geometrical parameter, √ area, defined as the square root of the projected area of defect on the plane perpendicular to the direction of the maximal principal stress, is used to identify the defect size.This approach suggests a practical relationship between the fatigue limit, and this proposed parameter.This model is limited to the uniaxial fatigue cases and gives conservative results especially in the case of torsion loading.
a Corresponding author: hassine.wannes@gmail.comEndo [3] has extended Murakami's criterion to proportional biaxial loading, where the ratio between torsion and tension fatigue limits is supposed to be constant whatever the size of the defect.However fatigue limits under combined loading are deduced without a fatigue test.Murakami [4] considers the defect as a crack where the size is characterized by the √ area parameter.He considers that crack initiation stage is negligible.Fatigue life is therefore assumed to be controlled by the crack propagation law.
Other authors consider the defect as equivalent to a notch [5].The fatigue limit will be linked to defect size by taking into account the type of defect by means of the elastic stress concentration factor usually named K t .The defect is modelled as a half spherical hole at the surface of specimen with K t = 2.Such an approach provides good results for macroscopic notches but it is limited to uniaxial loading.
Many authors tend to determine the fatigue limit under different assumptions such as the critical distances method proposed by Taylor [6] which is applied on cracks or notches.He used the linear elastic fracture mechanics under tensile stress modified by an introduction of material parameter constant.
Nadot et al. [7] studied FGS 52 containing natural defects in surface and on the bulk, the authors conclude that for a given size, surface defects are much more harmful Article published by EDP Sciences than those located in the bulk.They conclude that the severity of the defect affects the fatigue limit.By their experimental results Billaudeau et al. [8] show that for both, tension and torsion loading, the use of a fatigue criterion made for notches or cracks does not enable to predict an accurate defect fatigue behaviour.They conclude that for a given size, the shape of the defect modifies the fatigue limit under tension and torsion loading when the stress concentration factor is lower than 2(K t < 2).For K t > 2 the fatigue limit is independent of the defect geometry.Finally they show that the ratio between torsion and tension fatigue limits, which is supposed to be constant, depends on defect geometry and balances from 0.72 to 1.
Recently, some authors [9][10][11] observe that fatigue crack initiates at the tip of the surface defect within the maximum shear plane and crack was observed to propagate in the plane which is perpendicular to the direction of the maximal principal stress.Such observations were confirmed by FE analysis describing the stress distribution around the defect.This FE analysis shows the influence of the gradient of the maximal hydro-static stress, which is taken into account on the HCF criterion for defective material.This approach has the advantage to be applicable in different load ratios.This methodology, called Defect Stress Gradient (DSG), provides good results in cases of spherical pore defects subjected to fully reversed loading for R σ = −1.However for R σ = 0.1 results are not as good as those corresponding to R σ = −1 [12].In the case of elliptical defects DSG criterion underestimates the fatigue limit [13] and confirms that the only and simple parameter size = √ area is not able to describe the ellipsoidal defect influence.
In the present paper we propose a new approach to correlate fatigue limit to defect size where triaxiality of stress gradient around defect is taken into account.Based on FE analysis, an affected area is defined, which is the area close to the defect from which the Papadoupolos fatigue criterion is violated.
Papadoupolos equivalent stress is interpolated on the plane perpendicular to the maximum principal stress.This interpolation will be used to calculate the affected area.The evolution of the affected area versus the loading amplitude and defect size is used to determine the fatigue limit for each defect size.
The proposed approach is applied for two different materials submitted to fully reversed tension and torsion loading.Results are in good agreement with experimental investigations and results are improved comparing to those given by other approaches presented in the literature.The proposed approach shows that the affected area is a good parameter to characterize the influence of a defect on the fatigue behaviour.It is important to note that the model has the advantage to be simple and does not need many parameters to identify.In addition, it can be applied to loading with mean stress and it can be extended to multiaxial loadings since an analytical solution could be determined.

Material and experimental database
This study is carried out on the 1045 steel used in aircraft and automotive industry.The main mechanical properties are: Young modulus E = 195 GPa, Poisson's ratio ν = 0.3, the 0.2% offset yield stress R p0.2% = 359 MPa, the cyclic yield limit R pc0.2% = 250 MPa, the ultimate tensile strength R m = 594 MPa and the fracture lengthening A = 31%.An experimental database containing multiaxial fatigue results for 1045 steel with induced defects has been used [13].
Massive standard cylindrical specimens, with 5 mm and 12 mm diameters are used respectively for tension and torsion fatigue tests.Artificial spherical surface defects with controlled size have been machined, at the middle length of fatigue sample, by using electric discharge machining (EDM) process [8].Due to residual stress induced by the process a tempering is done after every introduction (500 • C/h under vacuum) to minimize the residual stress effect.
The tension and torsion fatigue tests were carried out for fully reversed loading (load ratio R σ = −1) at respectively 100 and 45 Hz.The samples are subjected to a cycle number greater than 10 6 cycles.Figure 1 shows fatigue limit evolution with √ area for fully reversed tension and torsion loading for spherical pore on surface.These experimental results in Figure 1 show that the fatigue limit of a specimen containing a defect is sensitive to the defect size and to the loading path.

Papadoupolos criterion
In this study the last formulation of Papadoupolos HCF criterion is used [14].It can be defined as follows: in a material volume V , a plane Δ is defined by its normal, n, this vector makes an angle θ with z-axis of the O xyz frame as shown in Figure 2.
The projection of this vector on the xy plane makes an angle ϕ with the x-axis.
Papadoupolos et al. [13] introduce a so called resolved shear stress τ (t), which is the projection of the shear vector The amplitude of the resolved shear stress τ a acting on Δ along ζ for t ∈ P is: P is the cycle loading period.Finally, for a given plane Δ, and for a fixed couple (θ, ϕ), the generalized shear stress amplitude T a is defined as: The Papadoupolos criterion is then defined as:  The quantity σ H,max is the maximum value of the hydrostatic stress within a loading cycle.
As long as this relationship is satisfied, there is no initiation cracking risk.Both α and β parameters are function of the material fatigue properties such as: where τ D−1 and σ D−1 are the fully reversed torsion and tension fatigue limits, respectively.
4 The 3D finite element simulation of stress gradient around the defect

Finite element modelling
It is worth noticing that defects are generally the preferential sites for crack initiation.It is in great interest to characterize the stress distribution around the defect.Due to the symmetry of the problem (geometry and loading), only one fourth parallelepiped is considered for the FE simulations as shown in Figure 3. Symmetry and boundary conditions are applied as shown in Figure 4.The geometry is meshed by means of four-node linear tetrahedral   solid elements C3D4.Meshing is refined and optimized in the critical zone around the defect (Fig. 5a).Since plasticity occurs around the defect all simulations are conducted with the elastic-plastic Prandtl-Reuss behaviour type model based on Von Mises energetic criterion with a linear isotropic hardening law.The cyclic stabilized experimental stress-strain curve used to identify the model is shown on Figure 5b.

Stress distribution around defect
Based on FE calculation, the highest loaded plane (HLP) is the plane which is perpendicular to the maximum stress direction (Fig. 6).In addition the experimental works provide that crack initiation occurs always at the tip of the defect in the shear plane [8].The macroscopic crack that leads to the specimen failure grows in the HLP.Consequently we will be interested in the distribution of stress on this plane (Fig. 7).

Tension results
Numerical calculations of σ eqpap in HLP show that: σ eqpap is almost constant along the defect radius (AB) with a scattering that does not exceed 3% (Fig. 8).-In HLP the defect gradient of σ eqpap in any direction starting from the defect centre is quite similar (Fig. 9).In HLP the distribution of Papadoupolos equivalent stress is independent of the chosen direction.It depends only on the ratio r/R and the applied load.Consequently in HPL σ eqpap can be interpolated by the following equation: where: σ D−1 : Defect free fatigue limit under fully reversed tension loading.τ D−1 : Defect free fatigue limit under fully reversed torsion loading.σ a : Amplitude of applied loading.R: Defect radius.r: Distance from defect centre to the considered point.

Torsion results
Numerical calculation of σ eqpap in HLP show that: σ eqpap is almost constant along the defect radius (AB) with a scattering that does not exceed 9% (Fig. 10).-In HLP close to defect the defect gradient of σ eqpap in any direction starting from the defect centre is quite similar (Fig. 11).In HLP close to defect the distribution of Papadoupolos equivalent stress is independent of the chosen direction.It depends only on the ratio r/R and the applied load.
Consequently in HPL σ eqpap can be interpolated by the following equation:  where: τ a : Amplitude of applied loading.
Equations ( 6) and ( 7), used to interpolate σ eqpap on HLP are validated by a numerical simulation.They lead to a good results with a difference less than 5%.
5 New HCF limit assessment approch of defective material

Methodology
From the results of numerical simulations in HLP it is shown that the variation of σ eqpap is independent of the chosen direction.σ eqpap distribution in HLP shows iso-value circular arches centred around defect (Fig. 12).
In HLP there is an area, close to the defect, from which the Papadoupolos criterion is violated.Consequently the affected area for a given defect and an applied load is defined as the area where the Papadoupolos equivalent stress is greater than or equal to β.The affected area depends on applied loading and defect size.For a given defect size, under fatigue limit loading, it corresponds an affected area limit.If the affected area is less than affected area limit, there is no failure before 10 7 cycles.
At fatigue limit loading the affected area is equal to the affected area limit.
Using Equations ( 6) and ( 7) we can determine the affected area as defined above in different cases corresponding to different loading modes.

Fully reversed tension loading
The affected area is determined using R and r * (Fig. 13).From Equation (6) r * is deduced as follow: when β is equal to τ D−1 , then: and From Equations ( 10) and ( 11) the affected area is assessed as follow: where

Fully reversed torsion loading
Using Equation ( 7) the affected area in case of reversed torsion can be determinate by: where From the experimental data base (Table 1), the affected area limit is determined by Equations ( 12) or (14), at the fatigue limit loading values in each case.Now we denote: in case of reversed fully tension loading in case of fully reversed torsion loading.Figure 14 shows k w1 and k w2 versus defect size for the two loading modes.The main deduction is that k w1 and k w2 are almost constant and independent of the defect size.

Proposed approach
Based on experimental results in Table 1 for fully reversed tension and torsion loading the product of fatigue limit of defective material by the square root of affected area limit is almost constant and independent of defect size.
The proposed approach is summarized on the following steps: (i) From an experimental fatigue limit of medium defect size using Equations ( 16) or (17) k w1 or k w2 are deduced.(ii) On the same graph the following curves are plotted: -Affected area versus loading amplitude for several defect sizes Equations ( 2) or ( 14) -Affected area limit versus loading amplitude: (iii) Finally the crossing points allow to determine the fatigue limit for each defect size.The necessary steps for the proposed approach are summarized in the flowchart (Fig. 15).With a simple Matlab program fatigue limit value is determined for each defect size in both cases: fully reversed tension or torsion loading and for different materials.

Alternative tension loading
In alternative tension loading case the affected area is calculated using Equation (12).From an experimental fatigue limit of a medium defect size the parameter k w1 is determined.In this study, a defect size √ area = 400 μm corresponding to a fatigue strength σ D = 152 MPa is used.
Using the proposed approach we deduce that k w1 = 24 100 MPa.μm.
In a same graph (Fig. 16) the following curves are plotted: -Affected area versus loading amplitude for many defect sizes Equation ( 12).Using all crossing points, for each defect size Kitagawa diagram can be deduced (Fig. 19) in case of fully reversed torsion loading for 1045 steel.
Obtained results are in good agreement with experimental investigations.
For small defects, the approach is less accurate and over estimates fatigue limit ( √ area ≤ 200 μm).
In order to study the evolution of the fatigue limit with defect size, artificial surface defects are produced using the spark erosion principle method.Defects are simplified to spherical void at surface of specimen.A data base for this validation is taken from published results [12].
To apply the proposed approach we need: -Defect free fatigue limit under fully reversed tension σ D−1 .-An experimental fatigue limit of defective material with medium defect size.
For cast aluminium defect free the fatigue limit under fully reversed tension is 91 MPa.For this material, a defect size √ area = 600 μm corresponds to a fatigue strength σ D = 85 MPa.The use of assumption 1 gives: k w1 = 60 000 MPa.μm.In a same graphic (Fig. 20) the following curves are plotted: -Affected area versus loading amplitude for many defect sizes Equation (12).-Affected area limit versus loading amplitude Equation (18).
From the graphic Figure 20 using the proposed methodology explained above we can deduce easily Kitagawa diagram Figure 21.
The proposed approach provides good results for casting aluminium, containing artificial defects, subjected A definition of affected area is proposed, from which the Papadopoulos fatigue criterion is violated.The evolution of the affected area, with the amplitude of loading and defect size, allows to determine fatigue limit for different defect sizes.
In this study, the proposed approach is applied to 1045 steel and casting aluminium AS7G06-T6 with introduced defects.The results show a good agreement with the experimental observations.The triaxiality of stress gradient around defect is taken into account by using the affected area in HLP.
The representation of the equivalent stress gradient in HLP by an explicit form, improves fatigue limit prediction comparing to those using the average gradient of hydrostatic pressure or equivalent stress.The proposed approach can be used as an interesting method for engineering design, leading to a more secured HCF behaviour of defective material.This study has to be extended for combined loading with different defect morphology.

Fig. 4 .
Fig. 4. Load and boundary conditions applied in the FE model: (a) tension loading, (b) torsion loading.

Fig. 22 .
Fig. 22.Comparison between the DSG model and the proposed approach.