The Microstructural Evolution of Nickel Single Crystal under Cyclic Deformation and Hyper-Gravity Conditions: A Molecular Dynamics Study

Abstract

Turbine blades are subjected to cyclic deformation and intensive hyper-gravity force during high-speed rotation. Therefore, understanding the dynamic mechanical behavior is important to improve the performance of the blade. In this work, [001](010), [110](−110), and [11−2](111) pre-existing crack models of nickel single crystals under increasing cyclic tensile deformations were studied by using molecular dynamics simulations. In addition, a novel hyper-gravity loading method is proposed to simulate the rotation of the blade. Four hyper-gravity intensities, i.e., 1 × 10¹² g, 3 × 10¹² g, 6 × 10¹² g, and 8 × 10¹² g, and different temperatures were applied during the cyclic deformation. The fatigue life decreased rapidly with the elevated hyper-gravity strength, although the plastic mechanism is consistent with the zero-gravity condition. The stress intensity factor for the first dislocation nucleation indicates that the critical stress strongly depends on the temperatures and hyper-gravity intensities. Moreover, the crack length in relation to hyper-gravity intensity is discussed and shows anisotropy along the direction of hyper-gravity. A temperature-induced brittle-to-ductile transition is observed in the [001](010) crack model. The present work enhances our understanding of the fatigue mechanism under hyper-gravity conditions from an atomistic viewpoint.

1. Introduction

Gas turbines have been widely used in aviation, ship power, and other fields. As the core component of a gas turbine, superalloy turbine blades work in a high-speed rotation and high-temperature conditions. Research on its fatigue mechanism plays an essential role in understanding fatigue behavior and improving its performance [1,2,3]. However, the dynamic and static mechanical properties of the materials are very different. Except for the tensile-compress stress, centrifugal force, which is a unique hyper-gravity environment, is created by high-speed centrifugal rotation. Therefore, characterizing the material fatigue performance under hyper-gravity conditions is important to extend the service life of turbine blades.

Fatigue fracture is the main factor for the failure of almost all mechanical structures under cyclic loading, which includes two stages of crack propagation and crack growth. The essence of crack propagation is the rupture of bonds between atoms, and the driving force behind it is the stress field caused by the loading. Forsyth et al. [4] reported a dual-stage fatigue crack growth, i.e., the microstructure-sensitive stage I and microstructure-independent stage II. The crack in stage I is driven by the maximum shear stress and aligned perpendicular to the maximum tensile direction before becoming the striation-forming crack. Laird et al. [5,6] indicated that crack propagation is a process of alternate blunting followed by a resharpening of the crack tip. The crack blunting occurs under tension due to significant slip emission. While the dislocations remain as residual products on compression, this causes crack extension along that particular plane.

Molecular dynamics simulation has proved to be a powerful method for fracture properties study [7,8,9,10,11]. Potirniche et al. [12,13,14] used the molecular dynamics method to simulate the fatigue damage of nickel and copper single crystals, and the results showed that nanocrystals experienced a large amount of plastic deformation. There are big differences in the plastic deformation and fatigue crack growth rate of each crystal orientation. Farkas et al. [15] investigated the mechanisms of fatigue behavior in nanocrystalline metals and showed that the main atomic mechanism of fatigue crack propagation was the formation of nanovoids ahead of the main crack and found that the stress intensity amplitude was consistent with experimental studies and a Paris law. Ladinek et al. [16] discussed the influence of fatigue loading order on the crack propagation of iron, indicating the cracks extend during the loading phase and the initial phase of the unloading process. Wu et al. [17] studied the microstructure evolution and stress distribution characteristics of pre-cracked single-crystal nickel at different temperatures. The simulation results indicated that the crack growth process and stress distribution characteristics of single-crystal nickel are closely related to the temperature change in the microstructure evolution before the crack growth is induced. Yang et al. [18] revealed that the fatigue cracks in the Ni-Fe alloy caused by the coalescence of nanovoids are before the main crack tips. For polycrystal, Zhang et al. [19] studied the strain rate effect on the crack propagation mechanisms of nickel. Zhou et al. [20] extracted the da/dN metrics from the fatigue crack growth in polycrystal coherent twins. Recently, we applied hyper-gravity loading to study the fatigue behavior of nickel under cyclic shear deformation [21]. The critical stress is strongly dependent on the hyper-gravity intensities and temperatures. The fatigue life decreased rapidly with the elevated hyper-gravity strength. Moreover, the crack length shows anisotropy along the direction of hyper-gravity.

In this work, a method to characterize the fatigue properties of the material under hyper-gravity conditions is proposed at the atomic scale to simulate the working condition of turbine blades. Molecular dynamics simulations are performed to investigate the fatigue mechanical properties and microstructure evolution of pre-cracked nickel single crystal under hyper-gravity conditions.

2. Materials and Methods

In this work, a pre-existing central crack in the nickel single-crystal model was adopted to study fatigue failure under cyclic loading (see Figure 1a). Three crystal orientations of the initial cracks, i.e., [001](010), [110](−110) and [11−2](111), were constructed. The cyclic deformation was applied along the y-directions. The dimensions of the specimens and their resulting number of atoms are presented in

Table 1. Model parameters and numbers of atoms in the three center crack specimens.

Crack Direction	L (nm)	H (nm)	W (nm)	Hcrack (nm)	Wcrack (nm)	Number of Atoms
[001](010)	64.4	10.6	28.2	0.7	2.8	1,754,250
[110](−110)	65.2	10.9	28.2	1.0	2.8	1,841,281
[11−2](111)	64.4	11.0	27.8	0.7	2.8	1,800,996

. The initial crack length is about 10% of the specimen width. To prevent the escaping of atoms caused by hyper-gravity, we fixed ten layers of atoms at the end of the hyper-gravity direction (y-direction). In addition, a periodic boundary condition was applied to all directions. The timestep used throughout the study is 1 fs to ensure time integration stability.

All models were first equilibrated at each temperature using the isothermal and isobaric ensemble (NPT) for 200 ps. After that, a strain-controlled cyclic tensile deformation was applied parallel to the initial crack plane in the constant number of atoms, volume, and temperature ensemble (NVT). We adopted the increasing strain amplitude cyclic loading to accelerate the fatigue process. Each specimen was subjected to one-way tensile (loading) for 10 ps (${\epsilon }_{yy}=0.01$), and then the tensile strain was unloaded for 0.008. The constant strain rate was 1 × 10⁹ s⁻¹ for both loading and unloading processes. After that, the cyclic loading was repeated, as shown in Figure 1b. Molecular dynamics simulations were performed using the LAMMPS code [22]. The force calculations between atoms were achieved by embedded-atom model potentials [23], which accurately reproduce the basic properties of nickel, such as the elastic constants, the phonon-dispersion curves, the vacancy formation, and the stacking fault energies. Furthermore, we carried out a structural analysis with the visualization and analysis software OVITO [24]. The common neighbor analysis algorithm (CNA) [25] was used to identify the local structures of atoms. Dislocations were visualized by using the DXA algorithm [26,27].

To generate the hyper-gravity environment at the atomic scale, we introduced a constant force that pointed along the hyper-gravity direction to each atom. Recently, such a hyper-gravity loading scheme has been reported by molecular dynamics and phase-field simulations [21,28]. In addition, the effect of rotation that introduces too many variables and makes the analysis difficult is ignored. The gradient tensile and compressive regions coexisted because of the fixed atoms at the periodic boundary. The region away from the hyper-gravity was subjected to compressive stress; on the contrary, the region close to the hyper-gravity was subjected to tensile stress. The crack was placed in the tensile region to simulate the working condition of the blade. We gradually increased the hyper-gravity from zero to each set value. A uniform zero pressure was applied at the initial state, and then a gradient pressure arose over time. After that, the crack models were equilibrated for 100 ps to stabilize the gradient pressure.

3. Results and Discussion

3.1. Crack Model Analysis under Zero-Gravity Conditions

Figure 2 shows the fatigue crack propagation process of single-crystal nickel with a pre-existing [001](010) central crack under cyclic strain, increasing loading at a temperature of 300 K and zero-gravity conditions. Atoms are colored based on their local structures. Red, green, and white atoms represent hexagonal close-packed (HCP), face-centered cubic (FCC), and unknown structure types. Dislocations are visualized using the DXA algorithm. The green and violet lines represent 1/6 <112> Shockley dislocation and 1/6 <110> stair-rod dislocation. The growth of the fatigue crack was not observed, and stress concentration reached σ_yy = 14.6 GPa before the 19th cycle at the crack tip. The mechanical response indicates an elastic behavior. Starting at the loading period of cycle 20, the crack began to grow and passivated at the tip. The critical stress intensity factor at the time of crack initiation is calculated as 1.06 MPa m^1/2. The stress intensity factor ($K_I$) is a parameter that describes the degree of stress concentration near the crack tip; it reflects the effect of the average stress and also contains the effect of the crack length. The driving force can be calculated by the Griffith fracture formula $K_I=\sigma \sqrt{\pi a}$, where $a$ represents the length of the crack and $\sigma$ is the average stress in the loading direction of the crack tip.

The plastic deformation is formed from the passivation area of the crack corner at the cycle 20 loading stage. 1/6 <112> Shockley partial dislocations appear first and then move along the [1−10] and [−110] crystal directions, which relax the stress concentration and decrease the fatigue crack growth. At the cycle 20 unloading stage, the dislocations moved along [1−10] and [110] in two different directions, resulting in the occurrence of short-range slip bands in different directions, i.e., [1−10](111), [1−10](11−1) and [110](1−11) directions. Then, the slip band at the crack tip disappeared and gradually transferred to the periphery of the crack, resulting in the continuous expansion of the plastic zone at the crack tip.

After that, a void formation was visible in the area near the crack tip before reaching the peak of the 24th cycle. The void volume continually increased at the unloading stage. Then, multiple voids were formed in front of the existing voids, as well as the main crack, which coalesced during the further crack opening. These voids increased in size in the first stages of the subsequent unloading phase before partially closing during the later stages of unloading. When the next loading cycle was applied, these voids coalesced with the main crack, leading to the rapid growth of the crack until fracture.

3.2. Crack Model Analysis under Hyper-Gravity Conditions

In this section, we performed cyclic tensile deformation with hyper-gravity intensities of 1 × 10¹² g, 3 × 10¹² g, 6 × 10¹² g, and 8 × 10¹² g along the y-direction, respectively. The hyper-gravity force is a constant force independent of the centrifugal position. The cyclic tensile deformation is applied when the gradient stress stabilized after 100 ps. Figure 3 shows the hyper-gravity-induced tension to compression gradient stress distribution at the different hyper-gravity intensities. The tensile and compression stress reached the maximum and minimum at the two terminals of the hyper-gravity direction (y-direction), respectively. To link the hyper-gravity force and centrifugal force, we can simply make these two forces equal. The centrifugal force is calculated using the angular velocity formula. Therefore, the centrifugal force is dependent on radial distance and angular velocity. The radial distance is set to equal to 0.5 m first. The 3 × 10¹² g hyper-gravity corresponds to the angular velocity of 7.7 × 10⁵ (rad/s). Note that this hyper-gravity is beyond the maximum that experimentalists can reach (~10¹² g, where g is the gravitational acceleration of Earth). However, it can amplify the hyper-gravity effect within the time and size constraint of the molecular dynamics method. The gradient pressure could exist, for instance, in our recent study of shock-induced plastic deformation [29,30].

Figure 4 shows the microstructure evolution of a [001](010) crack at 3 × 10¹² g. The crack began to expand in the unloading stage of cycle 13, which is earlier than zero-gravity conditions. We observed the passivation effect at the crack tip due to the stress concentration. Short dislocations were generated in the passivation region and emitted from the crack tip. In cycle 14, slip bands are formed from the passivation zone and the stress concentration is intensified at the crack tip. The direction of the slip band is identical to the one at zero-gravity conditions, indicating that the addition of gravity does not change the plastic mechanism.

Moreover, we performed the cyclic deformation under hyper-gravity of 1×10¹² g, 6 × 10¹² g, and 8 × 10¹² g. In particular, the 8 × 10¹² g condition is the maximum intensity that can be loaded for the current model and temperature. Beyond that, fractures happen because of the gradient tensile stress introduced by hyper-gravity, even before the cyclic deformation. Similar to the results at 3 × 10¹² g, the crack propagated along the [1−10] direction at 1 × 10¹² g and 6 × 10¹² g. Partial dislocations and cross slip were generated and effectively released the stress concentration at the crack tip. For the 8 × 10¹² g condition, the maximum tensile stress reached 24 GPa. Figure 5 indicates that voids were formed at the bottom of the y-direction rather than the crack tip. These voids coalesced together, leading to rapid growth until fracture.

In addition, the critical stress at the crack tip reached 14.6 GPa at the 19th cycle for zero-gravity conditions. However, at 1 × 10¹² g, 3 × 10¹² g, 6 × 10¹² g, and 8 × 10¹² g, the crack tip stress reached 15.0 GPa, 14.2 GPa, 14.7 GPa, and 14.5 GPa. The critical stress was intact with hyper-gravity intensities. Moreover, we calculated the dislocation density as a function of the cycle number at different hyper-gravity intensities, as shown in Figure 6. The increased hyper-gravity intensity increases the strength of the pre-tensile stress, which resulted in an earlier emission of dislocation. The dislocation densities started to arise at the unloading period of the 19th, 17th, 13th, and 9th cycles under zero-gravity, 1 × 10¹² g, 3 × 10¹² g, 6 × 10¹² g, and 8 × 10¹² g, respectively. Before that, the dislocation density was maintained at zero, indicating an elastic deformation. The dislocation density reached the maximum within one cycle and then decreased thereafter.

The crack length is calculated by tracking the position of the crack tip to characterize the crack growth under different hyper-gravity environments. Figure 7 shows the variation in crack length with time in nickel single crystals. As discussed in the previous section, the greater the added gravity value, the earlier the crack propagation time, and when the added gravity value is 1 × 10¹² g, 3 × 10¹² g, 6 × 10¹² g, and 8 × 10¹² g, the crack starts to expand at cycle 16, cycle 12, cycle 8 and cycle 1, respectively. The crack growth rate remains constant with the variation in hyper-gravity intensity. The 8 × 10¹² g condition is not discussed because the fracture occurred at the interface of the fixed layer and hyper-gravity loaded region.

3.3. Temperature Effect

Temperature plays an important role in the crack propagation process. In this section, we simulated the crack propagation at temperatures of 1 K, 600 K, and 900 K. Figure 8 presents a detailed observation of the stress field around the crack tip and crack propagation states under zero gravity at 1 K. The stress concentration occurs at the crack tip, which eventually expands and causes a brittle fracture along the direction of [100] at cycle 22. There is no passivation effect at the crack tip after crack initiation and propagation, which is consistent with the results of Wu et al. [17]. Under hyper conditions, the stress on the bottom of the tip is significantly greater than that on the top of the tip. Therefore, the crack propagates from the bottom of the crack, and eventually, a brittle fracture occurs. For 3 × 10¹² g, 6 × 10¹² g, and 8 × 10¹² g, cracks began to grow in cycle 15, cycle 9, and cycle 6, respectively. It shows that the greater the addition of high gravity, the greater the stress gradient, the earlier the crack tip reaches the critical stress, and the earlier the crack propagation occurs.

With the elevated temperature, the thermal motion of the atom is intensified, and the crack tip starts to emit dislocation at the cycle 18 unloading phase at 600 K. At the loading phase of cycle 19, the slip system is identified as [−110](11−1) and [−1−10](1−1−1) directions and the slip band emitted from the four crack corners. As the cyclic loading increases, the slip band gradually leaves the crack tip and moves around the crack. Due to the enhanced atom mobility at high temperatures, the activation of dislocations and slip bands become easier, which releases the stress concentration. At a hyper−gravity of 1 × 10¹² g, 3 × 10¹² g, and 6 × 10¹² g, the crack propagations undergo brittle−to−plastic deformation from cycle 16, cycle 12, and cycle 9, respectively. The dislocation emission direction of the crack tip is at a 45-degree angle to the crack, which is similar to the result of adding hyper-gravity at 300 K. After that, two slip bands symmetrical to the Y-axis are emitted from the passivation area, and the shift systems are [1−10](111) and [110](1−1−1). As the cyclic loading continues, the slip band grows along the direction [1−10] and [110] towards regions with large tensile stress regions. At the temperature of 900 K, the intensified thermal motion will cause the model to fracture before strain is added due to excessive boundary stress. Therefore, only two different hyper-gravity intensities, i.e., 1 × 10¹² g and 3 × 10¹² g, were discussed. The crack propagation processes share the same plastic mechanism with the results at 300 K. The dislocation starts to emit at cycle 17, cycle 15, and cycle 12 at zero-gravity, 1 × 10¹² g and 3 × 10¹² g, respectively. The crack length is shown in Figure 9 and the growth rate remains constant with the variation in hyper-gravity intensity.

In addition, Figure 10 shows the critical stress intensity factors at different hyper-gravity intensities and temperatures. Consistent with results of 300 K, the critical stress intensity factor decreases with the increase in hyper-gravity for each temperature. Moreover, the critical stress intensity factor decreases with elevated temperature for each hyper-gravity condition. The stress intensity factor decreases with the increase in hyper-gravity intensity, indicating a reducing barrier of plastic deformation.

3.4. [110](−110) and [11−2](111) Crack Models under Hyper-Gravity Conditions

Crystal orientation can significantly affect the mechanical response in various loading conditions. Therefore, two specimens of initial crack models of [110](−110) and [11−2](111) were constructed and deformed at the temperature of 300 K and different hyper-gravity intensities. Figure 11 shows the propagation behavior of the [110](110) crack under cyclic loading and the evolution of the surrounding microstructure at 300 K. No passivation area is observed at the tip of the crack before the first 10 cycles. After that, the dislocation is emitted from the four corners of the crack first. These dislocation lines are symmetric to the X-axis [001] and Y-axis [−110], and they expanded along the directions of [1−11] and [1−1−1]. The growth of dislocation restricts the propagation of fatigue cracks. At cycle 11, the dislocation line slips along the (−11−1) and (1−1−1) plane, one dislocation line becomes two parallel dislocation lines, and the initial position of the dislocation line always moves at the tip of the crack. Then, a void is formed at the intersection of the two dislocations. In cycle 18, multiple voids are formed where the crack tip dislocation density reaches the maximum. After that, the voids connect with the main crack, causing the crack to expand rapidly in the later stage.

For hyper-gravity conditions, the crack is emitted from the left side of the tip in cycle 7, cycle 4, and cycle 1 at 1 × 10¹² g, 3 × 10¹² g, and 6 × 10¹² g, respectively. The stress intensity factor corresponding to zero-gravity, 1 × 10¹² g, 3 × 10¹² g, and 6 × 10¹² g is 0.79 MPa m^1/2, 0.84 MPa m^1/2, 0.85 MPa m^1/2, and 0.86 MPa m^1/2. The greater the gravity, the greater the crack driving force required for cracks. Figure 12 shows the microstructure diagram of crack propagation at 3 × 10¹² g. The crack tips with higher tensile stress first emit two dislocations to the lower stress region, which restricts the crack tip propagation. The crack length of the [110](−110) crack under different values of gravity is calculated with the number of cyclic loading at 300 K. The greater the gravity, the earlier the crack tip reaches the critical stress, and the earlier the crack emits dislocations. At the later stage of crack propagation, the rapid expansion of the crack due to the connection of the holes, the greater the gravity, and the greater the crack propagation rate in the later stage.

For the [11−2](111) center crack model, the crack starts to grow at cycle 12 (see Figure 13). The plastic deformation was first observed at the tip of the crack at zero-gravity. In the early stage of the crack growth, two slip bands appeared along [−11−1](−11−1) and [−111](−111) directions. As the cyclic loading increases, the slip band [−111](−111) disappears after 13 cycles, while the direction of [−11−1](−11−1) exists. In cycle 14, the [−111](−111) slip band is emitted from the other tip of the crack. The slip bands in these two directions are the main slip system of the model.

At the hyper-gravity of 1 × 10¹² g, 3 × 10¹² g, and 6 × 10¹² g, the cracks start to emit the slip bands from the crack tip at cycle 9, cycle 7, and cycle 3, respectively. The two tips of the high-stress area are first emitted from the slip zone; this is due to the gradient stress created by hyper-gravity. Figure 14 shows the microstructure evolution of [11−2](111) crack at 300 K and 1 × 10¹² g. In cycle 9, plastic deformation was observed at the crack tip, and the direction of the slip bands is identical to the zero-gravity condition, indicating that the addition of gravity does not change the direction of slip but advances the time of crack propagation.

4. Conclusions

In this work, we performed molecular dynamics simulations to investigate the fatigue crack growth behavior under different temperatures and hyper-gravity conditions. The present work is worthy of shedding light on the mechanism of fatigue behavior under hyper-gravity conditions from an atomistic viewpoint. The main results and conclusions are summarized below:

(1)

A gradient tension–compression stress distribution is created under hyper-gravity conditions. The dynamical mechanical properties present different behaviors with statical simulation. The fatigue crack life is decreasing with increased hyper-gravity intensity.

(2)

The critical stress is strongly dependent on the hyper-gravity intensities and temperatures, which decrease with the increase in hyper-gravity and elevated temperature, indicating a reducing barrier of plastic deformation.

(3)

A brittle-to-ductile transition occurs between temperatures of 1 and 300 K in the [001](010) crack model.