Crystal plasticity based modelling and high cycle fatigue life prediction for bi-lamellar Ti-6Al-4V

Microstructural defects and inhomogeneity of titanium alloys fabricated by additive manufacturing technology make their fatigue performance much more complicated, especially reflected in the dispersion of fatigue life. This work employs crystal plasticity finite element method (CPFEM) to predict high cycle fatigue (HCF) life of bi-lamellar Ti-6Al-4V alloy. We first propose a modified VT technique to build representative volume element (RVE) models highlighting lamellar microstructure and micro-defects. Subsequently, fatigue indicator parameter (FIP) is adopted to analyse fatigue deformation under cyclic loading. Finally, HCF life determined by critical fatigue indicator parameter is compared with experimental data collected from published literatures. The results demonstrate that our approach is able to reflect the dispersion of fatigue life and to predict HCF life of bi-lamellar Ti-6Al-4V in a satisfactory manner.


Introduction
Titanium alloy Ti-6Al-4V, as a dual-phase alloy, is widely used in aerospace and marine fields as one of the most popular advanced engineering materials, due to its high strength, excellent ductility, and fatigue resistance [1][2][3].It has been revealed that the scatter of fatigue life is governed by microstructural features [4,5], such as defect morphology and size, grain size, inclusions, microstructural distribution, etc. Dual-phase Ti-6Al-4V alloy can be categorized into equiaxed structure, bimodal structure, and fully lamellar structure from the aspect of microstructure.Statistical analysis on experimental data of titanium alloys with the aforementioned structures demonstrated that bimodal Ti-6Al-4V exhibits a better fatigue performance compared with other microstructural features [6].In addition, the complex evolution of fatigue deformation emerged with various distributions of lamellar structure, resulting in different performance of fatigue resistance [7].However, with high cycle fatigue (HCF) tests on the bimodal structure and full lamellar [8], experimental results reveal that the full lamellar structure has better fatigue performance than the bimodal structure.Based on the above study, the distribution of lamellar structure can be considered as one of the main factors that affect fatigue performance.
Crystal plasticity finite element method (CPFEM) is an effective way [9][10][11] to predict mechanical properties based on microstructure characteristics, demonstrated by more applications in fatigue analysis.It shall be pointed out that pores and inclusions can be simplified as geometrical defects within the framework of the CP model.For instance, the effect of  phase fraction and average grain size of bimodal structure has been studied, illustrating that fatigue performance increases with the decrease of average grain size and  phase fraction [12].Also, a quantitative relation is formulated to depict fatigue performance with correlated defect orientation and microstructure by using crystal plasticity modeling for equiaxed Ti-6Al-4V, indicating the heterogeneous features of microstructure can be demonstrated by a linear relation between defect orientation and fatigue accumulation [13].Fatigue indicator parameter (FIP) embedded in CPFEM is generally used for life prediction analysis [14][15][16][17].For example, a Fatemi-Socie (FS) FIP is applied to capture fatigue behavior of AlSi10Mg alloy, and high cycle fatigue strength and the scatter for specimens containing different defects are predicted accurately [14].Besides, the accumulative plastic slip as FIP also successfully quantifies the variation in the deformation within the microstructure between the equiaxed and lamellar structure Ti-6Al-4V [15].The accumulative plastic slip and the accumulated energy dissipation are proved to be effective FIPs to determine low cycle fatigue life [16].
Despite the aforementioned effort, research on fatigue life with the variation of lamellar structure and defects for dual-phase Ti-6Al-4V is rather surprising, restricting its application of life prediction.In this regard, it is meaningful to develop a life prediction method with microstructural sensitivity about bi-lamellar Ti-6Al-4V alloy.
In the present work, based on the FIPs containing accumulated plastic slip and accumulative energy dissipation, we are able to predict high cycle fatigue life for bimodal Ti-6Al-4V, as well as lamellar structure analysis.Under cyclic loading, the evolution of FIPs is studied, and comparison of the two FIPs is investigated for prediction performance.

Fatigue indicator parameter
In the framework of CPFE, the crystal plasticity constitutive theory used here is formulated by Huang [18].It is further modified to accommodate the dual-phase alloys incorporating both hexagonal closed packed (HCP)  phase and body-centered cubic (BCC)  phase.Modeling parameters of the dual-phase Ti-6Al-4V are cited from references [19,20] and these parameters have been proved to be effective in our previous work [7,13].
To investigate the failure behavior, we introduce two fatigue indicator parameters that reflect fatigue damage for failure.In the process of fatigue loading, alloys undergo irreversible slip resulting in fatigue damage accumulation.Based on the CPFE modeling, plastic slip is reflected on the plastic velocity gradient, therefore, accumulated plastic slip is used to predict fatigue life in the present work and is expressed as: where   is the plastic velocity gradient,  ̇ is the accumulated plastic slip rate.In order to make comparison about the ability to capture fatigue damage of FIPs and the influence of resolved shear stress, the energy dissipation is also considered to be the indicator parameter to predict fatigue life.And the related equation is given as follows: where   and ̇ are resolved shear stress and shear strain rate, respectively, on the -th slip system.

RVE model formulation
The crystal plasticity finite element model used in this paper is established from Voronoi tessellation MATLAB.As for model settings, the average grain size is set to be 15 μm and the model is homogenized to reduce stress intensity.Microstructural effect is reflected through the different distribution of lamellar structure designated in RVE models.we build a RVE model of 240 μm *240 μm model with an elliptical defect.It contains 256 grains, of which p  phase and ( )  + phase account for 66% and 34%, respectively, here we simplify the simulation by setting  phase and p  phase to the same parameters.In our CPFE simulation, all grain orientations are set randomly and the used material parameters are cited from references [19,20].
It shall be noted that the y direction is fixed at the bottom boundary of the RVE model.Uniaxial cyclic fatigue loading is applied along the upper boundary.Stress ratio R is set to 0.06 through the cyclic loading process and there are three maximum stress levels of applied loads, namely 750MPa, 800MPa, and 850MPa.The schematic of the loading curve at 850MPa is displayed in Figure 1(b).
The lamellar structure distribution of three groups including 47 models is generated randomly, meanwhile, an elliptical defect is set in the middle of all models with the same size, as shown in Figure 1(a).Each model is assumed to be an individual case for fatigue life evaluation, and the variation of the lamellar structure distribution will lead to the scatter of fatigue life.Figures 1(c

Determination of critical fatigue indicator parameter
Under cyclic loading conditions, different FIPs are applied to describe fatigue damage, so as to evaluate the effectiveness of both FIPs to life prediction.It shall be pointed out that in the framework of CPFEM, the premise of the fatigue life prediction is to obtain the critical value of FIP by using a simple linear method [21].Figure 3(a) and Figure 3(c) give the evolution of accumulated plastic slip and accumulative energy dissipation under fatigue loadings over cycles, respectively.It is noteworthy that the displayed points in Figure 3(a) and Figure 3(c) are the maximum values extracted from 20 HCF cycles at the unloading conditions.As mentioned above, the   and   obviously have a linear relation of the number of cycles basically after 4 cycles (after the grey dotted line).Therefore, the critical value of   and   under cyclic loading can be calculated as follows ,, where   is the number of cycles to fatigue failure,  , and  , are the stable cyclic   and   as the maximum value in RVE models, respectively,  , and  , are the corresponding critical values of   and   .
Three loading conditions are used in the present work, however, the critical fatigue indicator parameter is a constant independent of the loading conditions.As shown in Figure 3(b) and Figure 3(d), the average value of  , and  , at three stress levels is 2,316 and

Experimental data
The experimental data of high cycle fatigue are cited from reference [22].Figure 4 shows the applied load of high cycle fatigue test and the corresponding number of cycles to failure, where the numbers of fatigue data at the three stress levels are 21, 17, and 9 for a total of 47 data.It is noteworthy that the dispersion of fatigue life is appreciable at all applied load of stress levels.Group A (750MPa) has the highest mean life cycles (313,839) while Group C (850MPa) has the lowest (92,844), indicating the average fatigue life is negatively correlated with the stress levels of applied load.
)-1(d) depict the different distributions of lamellar structure in which light blue grains represent the  phase and banded structure is lamellae composed of both  and  phases.

Figure 2 .
Figure 2. Accumulated plastic slip over the increase of stress levels of applied load

8 1.6 10 
MJ/ 3 m , respectively.It shall be pointed out that we generate random numbers in each group to select corresponding model values to get this constant.The error band (blue area around the dotted line in Figure3(b), 3(d)) for   and   with the range from -8.0% to 9.5% and -20.4% to 20.1%, respectively.And the maximum error band of both FIPs within the error band of ±2.Therefore, both critical FIPs are considered as independent constant with loading conditions which can reflect fatigue damage under cyclic loading.

Figure 3 .
Figure 3. FIP variation: (a) P with the number of cycles; (b) P with different stress levels; (c)W with the number of cycles; (d) W with different stress levels

Figure 4 .
Figure 4. Experimental fatigue life under different stress levels of applied load 3.4.HCF life prediction Through the above presented critical fatigue indicator parameter, Figure 5 shows the comparison of both FIPs about the predicted fatigue life and experimental data for bi-lamellar Ti-6Al-4V at different loading conditions, where nearly all data fall within the ±3 error bands and almost 95% of them lies within ±2 error bands.To further characterize the prediction ability of FIP, R-squared is adopted to show how the predicted life fits the experimental data.The relation equation is given as, ( ) ( ) 2 2 2R1ii ii

Figure 5 .
Figure 5.Comparison of predicted fatigue life and experimental data: (a) P-based fatigue life prediction, (b) W-based fatigue life prediction