# Crystal Plasticity Finite Element Modeling on High Temperature Low Cycle Fatigue of Ti2AlNb Alloy

^{1}

^{2}

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## Abstract

**:**

_{2}phase. Because of the difference in the mechanical characteristics between phases, Ti2AlNb alloy often exhibits deformation heterogeneity. Based on EBSD images of the Ti2AlNb alloy, a crystal plasticity finite element model (CPFEM) was built to study the effects of O phase and β phase (dominant phases) on stress and strain distribution. Four types of fatigue experiments, and the Chaboche model with 1.2%~1.6% total strain range were conducted to verify the CPFEM. The simulation results showed that the phase boundary was the important position of stress concentration. The main reason for the stress concentration was the inconsistency deformation of grains which resulted from the different deformation abilities of the O and β phases.

## 1. Introduction

^{4}life cycles under periodic plastic loading; this phenomenon is often called the low cycle fatigue (LCF) [1,2]. Aeroengine components are subjected to periodic variations of centrifugal forces and high temperature environments during operation, and the repeated action of high temperature alternating stress results in stress or strain concentration, and then causes local micro plastic deformation. The local micro-plastic deformation causes microcracks to form and spread further in the weak region of the alloy, causing the component to fail in the form of LCF [3]. TiAl-based alloys have received more and more attention because of their excellent high temperature properties and low density [4,5,6]. In the early of 21st century, many researchers found that adding Nb to TiAl alloys improved high temperature properties and creep resistance, thus, research on TiAlNb alloys has increased in recent years. As an important lightweight alloy, Ti

_{2}AlNb alloy is an intermetallic compound. Its long-range ordered super-lattice structure has the effect of weakening dislocation diffusion and high temperature diffusion, which gives the alloy the advantages of high strength, creep resistance, fracture toughness and oxidation resistance [6,7]. Therefore, it is important to study the high temperature LCF of Ti2AlNb alloy by experiments and simulations.

_{2}-phase and α

_{2}/β boundaries are important initiation positions for fatigue cracks, and that the lamellar O-phase plays a role in preventing fatigue crack propagation and causes the direction of the microcracks to deviate. Ding et al. [10,11] found that high temperature LCF will lead to strain-induced phase transformations and dynamic recrystallization. The large difference in lattice strain between different phases results in crack nucleated at and propagated along the phase boundary, which are detrimental to the fatigue life of the TiAlNb alloy. Kruml et al. [12] studied the LCF of TiAlNb alloy at 750 °C, and significant cyclic hardening characteristic was observed for 2%Nb alloy, but cyclic hardening was not observed for 7% Nb alloy, as the higher Nb content increased the stacking fault energy. Fatigue experiment results are lengthy, so establishing models to simulate the LCF process can improve efficiency. At present, the common modeling methods for LCF are macroscopical constitutive models, such as the Chaboche model, and microscopical models, such as the CPFEM.

_{2}AlNb alloy LCF was established by using the Chaboche constitutive model. Then, the CPFEM of O and β phases Ti

_{2}AlNb alloy LCF was established, the macroscopic stress–strain responses were obtained and then compared with the results of the experiments and Chaboche model. By analyzing the stress distribution and strain distribution of β and O phase, it was found that the phase boundary is the most important position of stress concentration, and that the main reason for stress concentration is the inconsistet deformation of grains resulting from the different deformation abilities of the O and β phases.

## 2. Experiment and Modeling Theory

#### 2.1. Experiment Materials and Procedures

_{2}AlNb alloy is cut from the combustion chamber casing blank. The microstructure of the initial Ti

_{2}AlNb alloy is characterized at room temperature, and the results are shown in Figure 1. The chemical composition of as-received Ti2AlNb alloy is shown in Table 1. According to EBSD (electron backscattered diffraction) results, alloy is composed of O phase, β phase and α

_{2}phase, accounting for 52%, 30% and 7% of the area, remainder is unresolved area, as shown in Figure 2. The different contents of Ti, Al and Nb elements and the solution elements lead to the variable phase composition of Ti2AlNb alloy. The β phase is BCC structure, the α

_{2}phase is HCP structure, and O phase is orthorhombic structure. EBSD and EDS (energy dispersive spectroscopy) analyses are performed using a ZEISS Sigma 300 scanning electron microscope, which is equipped with an electron backscatterer and an energy spectrometer. LCF tests are carried out at 550 °C. Figure 3a shows the sample sizes of tests. LCF tests are carried out on the Instron-8802 fatigue testing machine at a constant loading rate of 0.005 s

^{−1}with strain control loading. When the samples fracture or the stress of the testing machine decrease by 25% instantaneously, the failure of the sample was judged. As shown in Figure 3b, the loading waveforms of LCF tests are triangular wave and the strain ratio is −1. And the Figure 3c is the evolution of maximum tensile stress per cycle. The specimens at different strain levels show slight cyclic hardening in the front and middle stages of low-cycle fatigue. At the end of the curve, there is obvious cyclic softening. The fatigue life of four strain range is shown in Table 2. When the total strain range reaches 1.6%, the fatigue life decreases to 482 cycles, the strain range can no longer increase. Therefore, the total strain ranges Δε

_{t}of fatigue experiment are 1.2%, 1.3%, 1.4% and 1.6%, respectively.

#### 2.2. Chaboche Cyclic Deformation Constitutive Model

**D**is the elasticity tensor, p is the equivalent plastic strain, F is the yield function, α is back stress tensor and ${\sigma}^{\prime}$ is deviatoric stress tensor. The nonlinear follow-up hardening criterion decomposes the back stress tensor

**α**into three back stress components ${\alpha}_{i}$:

_{i}is initial value of follow-up hardening parameters, γ

_{i}is rate of change of following hardening parameters, p is the equivalent plastic strain. The size of yield surface Q is:

_{0}is the initial value of the yield surface size, Q

_{∞}is the maximum change value of the yield surface size, and b is the change rate of the yield surface size.

#### 2.3. Crystal Plasticity Finite Element Model

**F**can be expressed by a multiplicative decomposition of elastic deformation gradient

**F**

^{e}which describes the stretching and rotation of the crystal lattice and plastic deformation gradient

**F**

^{p}associated with slip:

**L**

^{p}can be written as the sum of slip rate on all activated slip systems:

**m**

^{α}and

**n**

^{α}are the slip direction and slip plane normal of the αth slip system, respectively. In the current work, in order to describe the crystal plasticity under cycle loading and the effect of temperature on rate-dependent deformation behavior, a thermally activated flow rule [28,29], with ${\dot{\gamma}}^{\alpha}$ as a function of τ

^{α}, B

^{α}and S

^{α}, is adopted as follows:

_{0}and θ are the reference strain rate, Boltzmann constant, Helmholtz free energy for material and absolute temperature, respectively. τ

^{α}, B

^{α}and S

^{α}represents the resolved shear stress, back stress, and isotropic slip resistance on the αth slip system. τ

_{0}is the critical slip resistance at the current temperature, and function <x> indicate that:

^{α}is defined as

^{αβ}is the hardening matrix to indicate the cross-hardening behavior between the slip systems α and β. S

_{sat}and S

_{0}are the saturated and initial slip resistance, respectively. The evolution of the back stress B

^{α}follows Armstrong–Frederick’s kinematic hardening rule [23,30,31]:

_{B}are r

_{D}material constants.

## 3. Model Parameters Identification and Validation

#### 3.1. Chaoboche Cyclic Deformation Constitutive Model

#### 3.2. Crystal Plasticity Finite Element Model

#### 3.2.1. RVEs and Boundary Conditions

_{2}phase accounted for 52%, 30% and 7%, respectively. Due to the least content of α

_{2}phase, this was ignored, so the Ti

_{2}AlNb was regarded as a two-phase material. The RVE contained 100 sets, the proportion of O phase and β phase was 2:1. Figure 6a shows that the O phase is the dominant phase, and the β phase is dispersed in the O phase. The RVEs were generated by Neper [32], the mesh scale was 100 × 100, and were divided into 100 areas. Then the RVE files were imported into ABAQUS. The boundary conditions for the RVE model are shown in Figure 6, the periodic boundary conditions (PBC) were implemented, and can be expressed as:

#### 3.2.2. Identification of Material Parameters

## 4. Results and Discussion

#### 4.1. Comparison of Simulation Ability between Cyclic Deformation Model and CPFEM

#### 4.2. Strain and Stress Distribution between the O Phase and b Phase

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Microstructure characterization and element proportion of Ti2AlNb—base alloy: (

**a**) grain morphology under OM (200×); and (

**b**) phase composition under EBSD (β is β phase, Ti

_{2}AlNb is O phase, Ti

_{3}Al is α

_{2}phase).

**Figure 3.**Fatigue experiments of Ti2AlNb alloy: (

**a**) shape and dimensions of the LCF tests specimens; (

**b**) fatigue loading waveform; and (

**c**) evolution of maximum tensile stress per cycle [23].

**Figure 5.**Simulation, (

**a**) and experimental results of uniaxial tension and LCF steady-state hysteresis loop at 1.4% strain range, (

**b**).

**Figure 7.**Crystal plasticity simulation and experimental results: (

**a**) 2% uniaxial tension; and (

**b**) steady-state hysteresis loop at 1.4% cycle deformation.

**Figure 8.**Stress–strain curves of experiment, CP simulation and Chaboche model: (

**a**) 1.2% strain range; (

**b**) 1.3% strain range; (

**c**) 1.4% strain range; and (

**d**) 1.6% strain range.

**Figure 9.**Stress and strain distribution figures of three loading points: (

**a**,

**b**) 0.7% strain; (

**c**,

**d**) 0% strain; (

**e**,

**f**) −0.7% strain. (The O and β phases are also shown in figure, as same as Figure 6a).

Ti | Al | Nb | Zr |
---|---|---|---|

55 | 21 | 23 | 1 |

**Table 2.**Fatigue life of four strain ranges [23].

Test | Strain Range/% | Life |
---|---|---|

Fatigue | 1.2 | 9660/cycle |

1.3 | 1081/cycle | |

1.4 | 742/cycle | |

1.6 | 482/cycle |

E/GPa | Q_{0}/MPa | C_{1}/MPa | γ_{1} | C_{2}/MPa | γ_{2} | C_{3}/MPa | γ_{3} |
---|---|---|---|---|---|---|---|

109.191 | 570 | 1,634,660 | 25,468 | 142,294 | 1405.3 | 36,262 | 187.15 |

Parameter | Unit | Value(O|β) | ||
---|---|---|---|---|

Elastic constants | C_{11} | GPa | 184 | 135 |

C_{12} | GPa | 86.2 | 113 | |

C_{44} | GPa | 49 | 55 | |

Flow parameters | ${\dot{\gamma}}_{0}$ | s^{−1} | 120 | 120 |

F_{0} | kJ/mol | 150 | 250 | |

τ_{0} | MPa | 200 | 200 | |

Hardening parameters | h_{B} | MPa | 1000 | 950 |

r_{D} | MPa | 8 | 10 | |

S_{sat} | MPa | 250 | 160 | |

S_{0} | MPa | 300 | 225 | |

h^{αβ} | MPa | 350 | 350 |

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**MDPI and ACS Style**

Wang, Y.; Zhang, Z.; Wang, X.; Yang, Y.; Lan, X.; Li, H.
Crystal Plasticity Finite Element Modeling on High Temperature Low Cycle Fatigue of Ti2AlNb Alloy. *Appl. Sci.* **2023**, *13*, 706.
https://doi.org/10.3390/app13020706

**AMA Style**

Wang Y, Zhang Z, Wang X, Yang Y, Lan X, Li H.
Crystal Plasticity Finite Element Modeling on High Temperature Low Cycle Fatigue of Ti2AlNb Alloy. *Applied Sciences*. 2023; 13(2):706.
https://doi.org/10.3390/app13020706

**Chicago/Turabian Style**

Wang, Yanju, Zhao Zhang, Xinhao Wang, Yanfeng Yang, Xiang Lan, and Heng Li.
2023. "Crystal Plasticity Finite Element Modeling on High Temperature Low Cycle Fatigue of Ti2AlNb Alloy" *Applied Sciences* 13, no. 2: 706.
https://doi.org/10.3390/app13020706