A crystal plasticity finite element (CPFE) model has been developed to study the microplasticity behavior of Ti-5553 alloy with equiaxed dual-phase microstructure under high-cycle fatigue (HCF) loading process. The microstructure and density distributions of geometrically necessary dislocations (GNDs) after HCF loading were characterized by electron backscattered diffraction (EBSD). Experimental results show that the microplasticity accumulation behavior happens in α phase during the HCF loading process. The developed CPFE model captures the essential physics of microplasticity accumulation in the primary α phase and stress concentration at the softer α/β interface. Besides, the effect of stress levels and volume fraction of primary α phase on microplasticity behavior was discussed. Results indicate that the stress concentration can be taken as fatigue indicator parameter (FIP) for high stress level HCF loading conditions, while the cumulative shear strain can be taken as FIP for low volume fraction of primary α phase conditions. This work provides a strategy not only contributing to understand the microplasticity behavior during HCF loading process but also choosing a suitable FIP for different loading conditions and microstructure states.

High-cycle fatigue (HCF) performance is one of the most important issues for the applications of titanium alloys in aero-engine and airframes [1–3]. It is reported that HCF is a pervasive problem affecting all engine sections (especially for fan and turbine blades) and a wide range of materials due to it usually resulted in significant unpredictable failure [4,5]. Furthermore, the high microstructure sensitivity of titanium alloys [6] makes the unpredictable failure more serious [7]. Microplasticity is a local plastic phenomenon of a solid body whereby it is globally subjected in the elastic domain. The microplasticity in metals mainly results from the activity of the certain slip systems when the loading stress is much less than yield strength of the metals [8–11]. Although the occurrence of the microplasticity in a single loading cycle is insignificant and has no effect on the fatigue life of metals, the accommodated strain after thousands of cycles loading cannot be ignored [3,12,13]. R.J. Morrissey et al. found that the accumulation of cyclic microplasticity in dual-phase Ti-6Al-4 V alloy under HCF was related to the bulk-dominated damage and ratchetting [13]. Crystal plasticity approach is an effective technique to study the local plasticity deformation and predict the relationship between microstructure and mechanical performance [14–16]. Recently, special crystal plasticity models have been developed to describe the slip activity behavior [17,18] and damage formation under HCF loading [12]. According to the crystal plasticity simulations, Florent Bridier et al. [17] observed the progressive occurrence and relative percentages of activated slip systems in duplex Ti-6Al-4 V titanium alloy during the HCF loading process. Besides, single slip is mostly observed at the low applied cyclic strain levels. Their work directly contributes to the HCF life prediction and the understanding of micro-damage mechanism.

As the typical near-β titanium alloy, Ti-5553 (Ti-5Al-5V-5Mo-3Cr) alloy has been widely used in aircraft components, such as landing gear assemblies and fastener stock, due to its deeper hardenability and high strength properties compared to the α + β titanium alloys (such as Ti-6Al-4V and TC21 alloys) [19–21]. HCF is a significantly important performance for Ti-5553 alloy in service conditions. Up to now, neither relationship between microstructure and HCF performance nor life prediction of the Ti-5553 alloy has been clearly understood and controlled since the discovery of the alloy due to the inherent microstructure sensitivity [22,23]. The aim of the work is to investigate the microplasticity behavior of Ti-5553 alloy with equiaxed dual-phase microstructure under HCF loading process by using crystal plasticity finite element (CPFE) simulations. The strain accumulations in primary α grains with different orientations during the cyclic loading process were studied.

2Experimental proceduresThe Ti-5553 alloy used in present work was prepared by three times vacuum arc remelting (VAR) followed by multistep cogging and forging. After forging, the Ti-5553 ingot was then cut into 10 × 10 × 10 mm3 samples for microstructure observation and Φ16 × 90 mm samples for HCF experiments, respectively. HCF tests were performed on QBG-50 fatigue machine with sinusoid waveforms at room temperature in air. They were carried out at a high frequency f =50 Hz, a stress ratio R = σmin/σmax = 0 and a stress level of 350 MPa. The orientations of the primary α phase and β phase were determined in a SEM instrument with the help of EBSD technique. The operating parameters of the EBSD procedure are as follows: accelerating voltage 20 keV, specimen tilt 70° and scan step size 0.1 μm. After EBSD detecting, HKL-Channel 5 system was used for EBSD acquisition and data manipulation. The specimens for EBSD observations were prepared by using conventional metallographic methods (mounting, grinding and polishing) followed by etching in a solution composed of 10 vol% HF, 10 vol% HNO3 and 80 vol% H2O. The initial microstructure of Ti-5553 ingot investigated in present work are shown in Fig. 1. Fig. 1(a) is the band contrast image in which the light grey represents α phase and the dark grey represents β matrix. The alloy is composed of 22% equiaxed primary α phase and 78% β matrix from statistical results as shown in Fig. 1(b). The inverse pole figure (IPF) in Fig. 1(c) exhibits the orientation distribution for all the α grains. It can be found that most of α grains own random orientation and a weak local texture exists in the microstructure.

3Crystal plasticity finite element method3.1Model developmentThe theory framework of rate-dependent crystal plasticity used in present work is based on the models developed by Asaro and Peirce [24–26]. The total deformation gradient F can be separated into two parts: elastic stretching and rigid body rotation of the lattice (described by Fe) and dislocation slip in the undeformed lattice (described by Fp) [27]. Therefore, the total deformation gradient can be written as an elastic-plastic form:

The plastic velocity gradient Lp can be decomposed into the symmetric and anti-symmetric parts [28]:

By considering the contributions of all slip systems in certain phases, the plastic velocity gradient can be given as: where γ˙i is shear strain rate, si and ni are slip direction vector and the normal vector to the slip plane of any given slip system i, respectively. N is the number of the active slip systems in certain phases.As reported by Anand and Kothari [29], the shear strain rate γ˙i on slip system i can be taken as a power law expression. The model was improved to consider the plasticity deformation during cycle loading by inducing back stress χi[17,30].

where, γ˙0i is the reference strain rate, n is the inverse strain-rate sensitivity exponent, τi is the resolved shear stress, ξi is the slip resistance of slip system i. The χi was divided into two components, both obeying the Armstrong-Frederick format [30]where C1, C2 and d2 are constants. d1 is allowed to change to express the cyclic softening behavior [30].The total slip system resistance is associated with statistically stored dislocations (SSDs) and geometrically necessary dislocations (GND). The deformation resistance rate can be expressed as [31,32]

where hij describes the rate of increase of the deformation resistance on slip system i due to shearing on slip system j[29]. If i = j, it is the self-hardening moduli. If i ≠ j, it is the latent hardening moduli. ∑jhijγ˙j describes the strain hardening caused by SSD [33]. The second item describes the strain hardening caused by GND [32]. ξ0i is the initial slip resistance on slip system i.k0 is a material constant. b is Burgers vector magnitude. G is the shear modulus. α^ is a non-dimensional parameter, taken to be 1/3 in present work [34]. λj is a parameter associated with the density of GND on slip system i[31]. It can be given by as a function of slip plane normal nj and incompatibility tensor Λ.where Λ can be expressed using the curl of plastic deformation gradient tensor FP.The self-hardening moduli and latent hardening moduli can be written as [35,36]

where h0 is the initial hardening modulus following the initial yield point, hs is the hardening modulus during easy glide within the first hardening stage, γi is the cumulative shear strain in slip system i (same for γj), and q is the latent hardening parameter. γ0 is the shear strain after which the interaction between two slip systems from same family reaches the peak strength. fij is the influence of the interaction between two slip systems on the magnitude of strength. If i = j, fij=f0. When i ≠ j, fij=fs. Meanwhile, the effects of interaction between slip systems from different families are defined by γs and fs given in Table 2.Crystal plasticity parameters of α phase and β phase for Ti-5553 alloy [35,36].

Phases | γ˙0 | n | h0MPa | hsMPa | γ0 | γs | f0 | fs | ξ0MPa | ξsMPa | |
---|---|---|---|---|---|---|---|---|---|---|---|

β phase | 0.001 | 50 | 7482 | 3.8 | 0.00091 | 0.000314 | 14.6 | 17.9 | 300 | 304 | |

α phase | Basal | 0.001 | 50 | 7482 | 3 | 0.00091 | 0.000314 | 14.6 | 15.0 | 200 | 202 |

Prismatic | 50 | 50.5 | |||||||||

Pyramidal | 100 | 101 |

The rate-dependent non-local cyclic crystal plasticity developed in present work is solved by a semi-implicit forward gradient integration method [25]. The simulations were carried out in two-dimension condition with the real size of 90 μm × 90 μm for the two-phase polycrystalline Ti-5553 alloy. The average grain size is about 8 μm and there are 150 grains in the microstructure. The polycrystalline microstructure is artificially generated by employing the Voronoi tessellation algorithm, as shown in Fig. 2. For simplification, some of smaller grains in Fig. 2(a) are chose to be identified as α phase (as shown in Fig. 2(b)) and the remaining is β phase. There are 22 α grains in Fig. 2(b), indicating a volume fraction of 15% for primary α phase. The grain orientations in Fig. 2 are randomly distributed in the microstructure model, distinguishing as different colors.

The HCF loading is imposed through a uniaxial stress uniformly applied in the X direction to the right side of the finite element model. The HCF loading history is given as

where σ0 is the stress level, taken as 350 MPa in this work. It indicates a frequency of 50 Hz and a stress ratio (σmin/σmax) of 0.The elastic parameters and crystal plasticity parameters of Ti-5553 alloy have been obtained by A.F. Gerday et al. based on macroscopic tests and nanoindentation simulations [35,36]. The elasticity in the model was taken as isotropic and the elastic moduli of α phase and β phase were list in Table 1. Twelve 110<111> slip systems of β phase and three basal 0001<112¯0>, three prismatic 101¯0<112¯0> and six pyramidal 101¯1<112¯0> slip systems of α phase have been considered in this work. The corresponding parameters of α phase and β phase for Ti-5553 alloy were mentioned in Table 2. According to the critical resolved shear stress (CRSS) of α phase and β phase reported by Gerday et al. [35,36], the α phase is softer than β phase due to the lower CRSS of α phase.

Elastic parameters of α phase and β phase for Ti-5553 alloy [35,36].

Phases | elastic moduli, E(GPa) | Poisson’s ratio, μ | |
---|---|---|---|

β phase | 85 | 0.35 | |

α phase | Basal | 125 | 0.33 |

Prismatic | |||

Pyramidal |

The CPFE calculations were conducted to simulate HCF loading conditions for up to 100 cycles and the results data were reserved at every 10 cycles. Based on the calculation results, the effect of stress level and volume fraction of primary α phase on microplasticity deformation during the HCF loading process was discussed. According to the CRSS values of α and β phases shown in Table 1 and 2, the α phase is softer than β phase. However, this view is a controversial issue in titanium alloys. Feaugas et al. also considered that α phase is a soft phase and experiences most of the plastic deformation [37,38]. However, B.Z. Jiang, C.S.Tan and K. Kapoor et al. [3,39,40] proposed that the β phase is softer than the α phase. For the near β titanium alloys with two-phase microstructure, the harder β phase maybe due to the distribution of nano-size α needles in β grains which cannot be detected by SEM observations. Therefore, we suppose that the accumulation of microplasticity during the HCF loading process mainly present in α grains, which may be related with the initiation of fatigue cracks. Similar results have been reported by Tan et al. [3]. The volume fractions of primary α phase and the stress levels selected for the simulations were 10%, 15%, 20%, 25% and 300 MPa, 350 MPa, 400 MPa, 450 MPa, respectively.

4.1Microplasticity behavior during HCF loadingIn present work, the slip systems with largest deformation and the accumulated shear strain in corresponding slip system of α grains were analyzed. As shown in Table 3, the largest deformation mainly happens in prismatic < a> slip systems {101¯0}<112¯0> and pyramidal < a> slip systems {101¯1}<112¯0>. The value of accumulated shear strain is related to the angle between c-axis and loading axis and the Schmid factors.

The deformation and slip systems at different α grains in the microstructure.

Grain number | Angle between c-axis and loading axis | Slip systems with largest deformation | Schmid factor | Accumulated shear strain in slip system |
---|---|---|---|---|

1 | 119.76° | (11¯01)[112¯0] | −0.1079 | 8.44E-4 |

2 | 89.14° | (011¯0)[2¯110] | −0.3283 | 0.01792 |

3 | 35.84° | (11¯00)[112¯0] | 0.1189 | 4.38E-4 |

4 | 60.61° | (1¯011)[12¯10] | −0.4782 | 2.88E-4 |

5 | 102.41° | (011¯0)[2¯110] | 0.4339 | 0.0227 |

6 | 113.60° | (01¯11)[2¯110] | −0.0598 | 1.27E-4 |

7 | 106.14° | (1¯011)[12¯10] | 0.3178 | 6.91E-4 |

8 | 101.22° | (011¯0)[2¯110] | 0.4575 | 0.02577 |

9 | 158.31° | (11¯00)[112¯0] | −0.0507 | 0.03597 |

10 | 147.92° | (101¯0)[12¯10] | −0.1234 | 3.86E-4 |

11 | 159.56° | (011¯0)[2¯110] | −0.0603 | 4.57E-4 |

12 | 49.44° | (011¯1)[2¯110] | 0.1021 | 0.00219 |

13 | 141.07° | (11¯01)[112¯0] | 0.0245 | 0.00148 |

14 | 153.96° | (11¯01)[112¯0] | −0.1074 | 0.00113 |

15 | 35.87° | (101¯0)[12¯10] | −0.1699 | 0.00156 |

16 | 41.48° | (11¯00)[112¯0] | −0.0250 | 9.58E-4 |

17 | 138.38° | (11¯00)[112¯0] | −0.0485 | 0.01369 |

18 | 152.94° | (11¯01)[112¯0] | −0.1888 | 1.75E-4 |

19 | 76.68° | (101¯0)[12¯10] | −0.4258 | 0.00628 |

20 | 153.38° | (11¯01)[112¯0] | −0.0910 | 0.00131 |

21 | 16.16° | (011¯0)[2¯110] | −0.0280 | 0.00123 |

22 | 128.50° | (011¯0)[2¯110] | 0.2946 | 0.08947 |

Fig. 3 presents the relationship between c-axis/loading axis angle and accumulated shear strain. For soft orientation, the c-axis is perpendicular to loading axis, while c-axis is parallel to load axis for hard orientation [41]. It can be seen from Fig. 3 that the accumulation of microplasticity in α grains with softer orientations was larger than that of in harder α grains. As shown in Fig. 3, two prismatic < a> slip systems, named (011¯0)[2¯110] and (11¯00)[112¯0], were found be mainly activated in Grain 2# with softer orientation, and the largest shear strain mainly exists in (011¯0)[2¯110] slip system. For Grain 12# with middle orientation, pyramidal < a> slip system (011¯1)[2¯110] and prismatic < a> slip system (011¯0)[2¯110] are mainly activated during the HCF loading process, and the largest shear strain exists in pyramidal < a> slip system. For Grain 21# with harder orientation, prismatic < a> slip system (011¯0)[2¯110] and pyramidal < a> slip system (011¯1)[2¯110] are mainly activated, and the largest shear strain exists in prismatic < a> slip system. However, the accumulated microplasticity is also associated with the orientation of surrounded β grains and the slip resistance of the α/β interfaces [17]. Therefore, the angle between c-axis and loading axis is not the only criterion to determine the degree of accumulated microplasticity in α grain. The surrounding β matrix and α/β interfaces also has great influence on microplasticity accumulation.

Fig. 4 shows the evolution of accumulated shear strain at α grains during the HCF loading process. As shown in Fig. 4(a), although most of the α grains have a low accumulated shear strain after HCF loading (below 0.03), a special α grain, named Grain 22#, has an extreme large accumulated shear strain. According to Fig. 4(b), prismatic < a> slip system (011¯0)[2¯110] are mainly activated during the loading process, and the accumulated shear strain gradually increased with the increase of loading cycles. Although the accumulation rate decreased continuously with the increase of loading cycles, the accumulated shear strain has the possibility to achieve a relative larger value after millions of loading cycles. Therefore, we suppose that the abnormal microplasticity accumulation in individual α grain may contribute to the formation of fatigue micro-cracks.

As we have well known, the GND in α grains will increase with the accumulation of microplasticity during the HCF loading process. Therefore, HCF loading experiments and EBSD observations were taken to confirm the microplasticity accumulation. Fig. 5 shows the distribution of GND after different HCF loading cycles. As shown in Fig. 5(d), the GND density increased with the increase of loading cycles. The average GND density at original state is 128.362 (1012/m2), while it increases to 132.586 and 246.768 after 4200 cycles and 11,200 cycles, respectively. Note that the microplasticity accumulation behavior happens in α phase during the HCF loading process.

In order to get a detailed understanding about the influence of grain orientations on fatigue behavior, the interaction between the α grains with harder orientation and softer orientation was discussed. Fig. 6 exhibits the GND distribution in the hard-soft grain pair consisted by Grain 15# and 19#. As shown in Fig. 6(a), the GND in Grain 19# with softer orientation is more lager than that of in Grain 15#. Besides, no increasing phenomenon of GND was found in α/α interface. However, in the α/β interface, there is an obviously increasing of GND, especially in the triple boundary junctions, as shown in Fig. 6(b). Besides, the interfaces between softer α grain (Grain 19#) and β grains have larger stress concentration than harder α grain/β interface and α/α interface. Therefore, the softer α grain/β interfaces are also the potential nucleation sites of fatigue cracks.

4.2Effect of stress level on microplasticity deformation of dual-phase microstructureFig. 7 shows the Mises stress distribution in the microstructure after 100 cycles HCF loading at different stress levels. As seen from Fig. 7(a) and (b), the stress concentration mainly exists in Grain5#, 22#, 19# and 8# when the stress levels were set as 300 MPa and 350 MPa, while the stress concentration around Grain 12# and 13# increased obviously when the stress level increased to 400 MPa and 450 MPa. Besides, the sites with maximum stress were found change with the stress levels. For 300 MPa and 450 MPa stress levels, the maximum stress happens at Grain5#/β interface, while it changes to Grain19#/β interface when the stress level is 350 MPa or 400 MPa. From 300 MPa to 450 MPa, the maximum stress increased from 333 MPa to 726 MPa. Note that the stress concentration may be an important fatigue indicator parameter (FIP) for high stress level HCF loading conditions.

Fig. 8 shows the distribution of cumulative shear strain after 100 cycles HCF loading at different stress levels. As shown in Fig. 8(a), the cumulative shear strain is relatively small (the maximum is about 0.087) at 300 MPa stress levels. When the stress level increases to 350 MPa, 400 MPa and 450 MPa, cumulative shear strain mainly exists in Grain 2#, 17#, 9# and 22#. Besides, there is an abnormal large cumulative shear strain at Grain 22# when the stress level is 400 MPa. Fig. 9 gives the evolution of maximum Mises stress and maximum cumulative shear strain with the increase of stress levels for different loading cycles. As shown in Fig. 9(a), the maximum Mises stress increase dramatically with the increase of stress level. However, the change of maximum cumulative shear strain experienced a tendency that first increase and then be stable with the increase of stress level. Note that maximum cumulative shear strain is not an idea FIP for high stress level HCF loading conditions.

Fig. 10 shows the accumulate process of shear strain and Mises distribution after 100 cycles loading for the microstructure with different volume fraction of primary α phase. As seen from Fig. 10(a ˜ c), two of the 15 grains (Grain 1# and Grain 6#) have extremely large cumulative shear strain, while the stress concentration mainly happens at the β/α interfaces around Grain 13#, 5# and 3#. For 15% volume fraction (see Fig. 10(d ˜ f)), only one of the 22 grains (Grain 22#) has extremely large cumulative shear strain and the stress concentration mainly happens at the β/α interfaces around Grain 5#, 19# and 8#. Increase the volume fraction of primary α phase to 20% (see Fig. 10(g ˜ i)) and 25% (see Fig. 10(j ˜ l)), we found that the largest cumulative shear strain and the difference between the grains were reduced obviously. Note that high volume fraction of primary α phase contributes to the uniform distribution of cumulative shear strain and restricts the appearance of extremely large values. Besides, the change of maximum Mises stress value is relatively small at different volume fractions of primary α phase, although the stress concentration happens at different β/α interfaces. Note that the influence of the volume fraction of primary α phase on stress concentration is not noticeable. According to above discussion, the cumulative shear strain can be taken as FIP for low volume fraction of primary α phase conditions.

5ConclusionsIn summary, a CPFE model was established to simulate HCF loading process of near-β titanium alloy with equiaxed dual-phase microstructure. The model quantitatively described the microplasticity accumulation in the primary α phase and stress concentration at the softer α/β interface during the HCF loading process. This model captures the important behavior of microplasticity deformation and slip system activation in α phase during HCF. The main conclusions are:

- (1)
The largest deformation mainly happens in prismatic < a> slip systems {101¯0}<112¯0> and pyramidal < a> slip systems {101¯1}<112¯0>. An extremely large cumulative shear strain was found in individual α grain, which may contribute to the formation of fatigue micro-cracks.

- (2)
The maximum Mises stress increase dramatically with the increase of stress level. Note that the stress concentration can be taken as a FIP for high stress level HCF loading conditions.

- (3)
The extremely large cumulative shear strain mainly appears at the conditions of low volume fraction of primary α phase. Note that the cumulative shear strain can be taken as FIP for low volume fraction of primary α phase conditions.

We certify that no conflict of interest exists in the article and will not be published elsewhere in the same form, in any language.

We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.

We certify that the article has been written by the stated authors who are ALL aware of its content and approve its submission.

This work was financially supported by the National Key Research and Development Program of China (No. 2016YFB0701303) and the Natural Science Foundation of Shaanxi Province (2018JM5174).

*et al*.

*et al*.

*et al*.