1 Introduction

Fatigue is the main reason of material failure, so it is very important to study the internal mechanisms of fatigue crack initiation and propagation. To understand these mechanisms, it is necessary to investigate the atomic configuration and field distribution evolution of the material subjected to a repeated load. However, when the specimen size is reduced from visible to nanoscale, experiments face many insurmountable difficulties in spatial accessibility and time scale coordination. Atomistic simulations method is a good choice and much research has been done by using this approach [1,2,3,4].

In a single crystal, crack propagation is usually induced by dislocations [5], which are emitted from the crack tip and move along the slip direction on its slip plane. Sung et al. [6] found that the partial dislocations nucleated at the crack tip and slipped along the close-packed plane until complete fracture. Zhou et al. [7] observed the emission of blunting and jog dislocations in the three-dimensional crack growth simulation. In general, a jog dislocation has a very limited contribution to crack growth, because its emission does not produce a new surface. Tang et al. [8] found that under cyclic loading, if there are simultaneously multiple active slip systems at the crack tip, it is easy to form micro-voids at the crack front. The connection between these micro-voids and the main crack is an important mechanism for fatigue crack propagation. Crystal orientation has an important effect on fatigue crack propagation. It can change the angle between the dislocation slip system at the crack tip and the crack extension line, affect the distribution of atomic stress near the crack tip, and further change the crack propagation rate and path. Potirniche et al. [9, 10] analyzed the crack propagation in 10 different orientation crystals and found that the crystal orientation had a significant effect on the plastic deformation and fatigue crack propagation rate.

In addition, loading mode also has an important effect on the fatigue crack propagation. Spielmannova et al. [] found that in a Fe single crystal under tensile loading, a crack with orientation of (001) [110] induces twins in the < 111 > {112} direction, while the crack with orientation of (\( \bar{1} \)10) [110] motivates dislocations in this direction. Wu et al. [12] found that under a variable amplitude load, both dislocation emission and permanent slip band formation would change the stress distribution and hinder a crack to grow, but under a constant amplitude load, the crack opening and closure would always be accompanied by void formation ahead of the crack.

Unlike the single crystal case, grain boundaries play a very important role in the fracture of polycrystalline or nanocrystalline metals [13, 14]. In a loading process, the dislocations emitted from the crack tip may be absorbed by the grain boundaries, while in an unloading process, the dislocation absorbed by the grain boundaries may be re-emitted and return to the crack tip [15, 16]. Ma et al. [2] carried out the fatigue crack propagation simulations on the bicrystals with grain boundaries of ∑3 < 112 > {111} and ∑5 < 310 > {001}, respectively. The results showed that if the two slip bands at both ends of the grain boundary are symmetrical, the crack propagation would be significantly inhibited, that is to say, the crack tip would be obviously blunted. When the grain boundary with a big misorientation angle is perpendicular to the crack surface, it could act as a barrier [17]. In the case of the same misorientation angle, the effect of a twist grain boundary on the crack is stronger than that of a tilt grain boundary [18]. In a grain boundary model with a tilt dislocation angle, partial dislocations are more possibly emitted from the intergranular crack tip to the interior of the grain [19]. On the other hand, a grain boundary with a twist dislocation angle has a stronger ability to absorb dislocations, so it can promote a crack to grow [20]. In addition, the grain size distribution also has an important influence. Wang et al. [21] found that a large grain size gradient is conducive to improving the cracking resistance of materials. When a crack encounters a large-size grain in front of it, it could change its propagation mode from intergranular to intragranular.

Although much progress has been made in the fracture of polycrystalline or nanocrystalline metals [13, 22,23,24], the microscopic mechanism of fatigue fracture is still vague. Some fatigue researches focused on damage accumulation and crack propagation during multiple load cycles [1,2,3, 22], but ignored the crack propagation behavior within a single load cycle. In this paper, the molecular dynamics method is used to investigate the propagation behaviors of the intracrystalline and intergranular cracks in nanocrystalline Ni under single-cycle mode I loading, in an attempt to make some progress in exploring the crack propagation mechanisms and the crack-tip blunting and sharpening during a loading and unloading cycle by comparison.

2 Models and Computational Method

A nanocrystalline Ni model with the size of 500 × 500 × 20 Å3 was created. It has 14 grains with size of 195 Å and the lattice constant \({\text{a}}_{0}\) of 3.52 Å. In order to reduce the influence of grain shape on the results, the modified Voronoi method [25, 26] was used to roughly set the grain shape to a regular hexagon. A rectangular coordinate system was established with [110] as the X axis, [001] as the Y axis, and [1\( \bar{1} \)0] as the Z axis [27, 28], as shown in Fig. 1. As the smallest angles of grains with respect to the X direction, the grain orientation angles in the middle of the models were set to 0°, while the rest were generated randomly. To avoid the occurrence of low angle grain boundaries, the differences between adjacent grains were controlled to be greater than 10°. Similar to the bicrystal cases [19, 29], the tilt grain boundaries at the crack tips were set symmetrical to eliminate the effect of the grain orientation on the crack propagation direction.

Fig. 1
figure 1

Nanocrystalline models with a unilateral crack

Full size image

The large-scale atomic/molecular massively parallel simulator (LAMMPS) [30] was used for molecular dynamics simulations with the embedded-atom method (EAM) potential function for Al and Ni [31] to describe the interaction between Ni atoms. Before loading, the models relaxed at 300 K for 100 ps in order to obtain an equilibrium atomic configuration. Subsequently, the temperature decreased to 5 K and the models continued to relax for 100 ps. After the relaxation, an intragranular or intergranular edge crack with a length of 150 Å and a width of 5 Å was introduced in the middle of the models by removing atoms. The intragranular or intergranular crack models have the total atomic numbers of 454,600 and 454,400, respectively.

Two atomic layers with a thickness of 10 Å on the upper and lower surfaces of the models were treated as rigid loading regions. The displacement load with strain rate of \(1 \times 10^{9} \text{s}^{ - 1}\) was applied in the Y direction. In the loading stage, while the atoms in the upper loading region were given an upward velocity, those in the lower loading region were given a downward velocity with the same magnitude. During unloading, however, their velocity directions were reversed. The time step was set to 0.001 ps, the Z direction had the periodic boundary condition, and the NVT ensemble was adopted. The models were stretched for 100 ps until the nominal strain reached 0.1, and then they were unloaded for 100 ps until the nominal strain returned to 0.

3 Results and Analyses

The visualization software OVITO [32] was used to post-process the simulation results and the common neighbor analysis (CNA) method [33] was used to distinguish different crystal structures. The face centered cubed (FCC) atoms were marked green, and the hexagonal close-packed (HCP) atoms were marked red. In addition, a single HCP atomic line represented a twin boundary, two adjacent HCP atomic lines represented an intrinsic stacking fault, and two HCP atomic lines sandwiched with an FCC atomic line represented an extrinsic stacking fault. The dislocation extraction algorithm (DXA) [34] was used to analyze the dislocation emission and motion. In order to distinguish it from the CNA method, the FCC atoms were marked blue, while the HCP atoms were still marked red.

3.1 Intragranular Crack

Figure 2 shows the local atomic configurations around the intragranular crack tip at different loading moments. As can be seen from Fig. 2a, before loading, all the HCP atoms lie at the grain boundary, and none is inside the grains. When the strain is 0.04, the crack tip begins to open, there are extrinsic stacking faults occurring in the oblique upward direction at an about 45° angle from the X-axis, and microcracks begin to form at the grain boundary falling on the X-axis in front of the crack, as shown in Fig. 2b. When the strain reaches 0.07, the crack opens further. Due to the guidance of the stacking faults, the crack changes its direction and extends in an oblique upward direction at an about 45° angle from the X-axis. At the same time, there is an intrinsic stacking fault near the crack tip in the direction of −45° from the X-axis. Some microcracks or microvoids gradually nucleate in the grain boundary in front of the crack, and then aggregate to a new crack. The new crack develops in the same direction as the original crack, and form a bifurcation until it reaches the triple junction, as shown in Fig. 2c. When the strain reaches its maximum of 0.1, both the two cracks would open fully. Due to the induction of the intrinsic stacking fault, the original crack changes its direction again, and propagates downward in the direction of -45° angle from the X-axis. As a result, the two cracks have a tendency to connect with each other. It means that if the loading continues, the intragranular crack propagation would change to intergranular crack propagation. This is very consistent with some previous results [14, 21]. There are intrinsic and extrinsic stacking faults occurring near both the two crack tips, as shown in Fig. 2d. In the unloading stage, when the strain decreases to 0.04, both the original crack and the new crack contract obviously, and a large number of twin boundaries, intrinsic and extrinsic faults form between the two crack tips, as shown in Fig. 2e. As the strain returns to 0, both the original crack and the new crack contract further. Although a part of the crack that grows during the loading stage is completely closed, the new crack has not closed completely yet, as shown in Fig. 2f. It can be found by comparison of Fig. 2a, f that, after going through a loading and unloading cycle, the original crack has taken a step forward with a certain length. In addition, the dislocation density in the vicinity of the crack tip always increases during the whole cycle. It is noted that some survival disordered atom clusters induce new dislocations as the cracks shrink in the unloading process.

Fig. 2
figure 2

Local atomic configurations near the intragranular crack tip at different moments

Full size image

In order to characterize the crack tip blunting and sharpening quantitatively, we introduce the atomic crack angle. As a new concept, it is defined as the angle between two lines connecting two crack-tip atoms at the outermost boundary of the crack surface and without a common atom, as shown in Fig. 3. In this figure, the light blue circles represent the boundary atoms and the dark blue circles represent the inner atoms. For case 1, the connection line between atoms A1 and A2 has a common atom A2 with the connection line between atoms A2 and A4, so their angle is not the atomic crack angle. Similarly, the connection line of atoms A2 and A4 has a common atom A4 with the connection line between atoms A3 and A4, so their angle is also not the atomic crack angle. Different from the above two angles, the angle between the A1-A2 line and the A3-A4 line is the atomic crack angle. For case 2, because the A1-A2 line and the A3-A4 line both pass through the " × " atom, this atom should be discarded when calculating the atomic crack angle. In general, an increase of the atomic crack angle means that a crack opens further. Therefore, the crack tip would be gradually blunted when the atomic crack angle increases. Conversely, a decrease in the atomic crack angle indicates that the crack is closing and its tip is sharpened gradually.

Fig. 3
figure 3

Schematic diagram of atomic crack angle

Full size image

According to traditional understanding, a crack tip is gradually blunted in the loading stage but sharpened in the unloading stage. Figure 4 shows the atomic configurations near the crack tip at four different moments during the loading stage. In the lower left corner of each subgraph, the crack tip configuration indicated by the red circle is magnified to measure the atomic crack angle easily. Figure 4a, b are for the new cracks forming in the grain boundaries parallel to the X-axis. At 50.8 ps and 52 ps, the atomic crack angle at the crack tip is 26.7° and 80.0° respectively, which means that the atomic crack angle at the new crack tip increases during this process. As a result, the crack opens and the crack tip is blunted. Figure 4c, d are for the original crack. At 74.8 ps and 77.6 ps, the atomic crack angle at the crack tip is 12.3° and 62.5°, respectively. It means the crack opening and crack tip blunting. Accordingly, the above observations from the atomic crack angle are consistent with the traditional understanding. In addition, it can also be found by comparison that crack tip blunting is always accompanied by dislocation emission.

Fig. 4
figure 4

Atomic configurations and atomic crack angles near the intragranular crack tip at different loading moments

Full size image

The dislocation emission and motion in the loading stage can be observed more carefully by the local DXA structures. Figure 5a shows the DXA structure near the intragranular crack tip at 84.6 ps. It can be found that Shockley partial dislocations represented by the green dots obviously exist at the two ends of the stacking faults, and their Burgers vector is \(\frac{1}{6} < 121 >\). They slip in the \(< 112 >\) direction of the \(\{ 111\}\) slip plane, drive the stacking faults to move and provide an alternative crack propagation direction, as marked by the red arrows. Sung et al. [6] also obtained similar results in the crack growth simulation of single crystal Ni. When the crystal orientation is < 100 > , the Shockley partial dislocation with Burgers vector of \(\frac{1}{6}[2\mathop 1\limits^{\_} 1]\) slips on the \((11\mathop 1\limits^{\_} )\) slip surface. Luque et al. [19] found that a Shockley partial dislocation with Burgers vector of \(\frac{1}{6}\{ 111\} < 112 >\) was emitted at the crack tip located at the grain boundary of bicrystal Cu.

Fig. 5
figure 5

Local DXA structures in the intragranular crack model at 84.6 ps and in the intergranular crack model at 57.6 ps

Full size image

3.2 Intergranular Crack

Figure 6 shows the local atomic configurations at the crack front of the intergranular crack model at different moments. At the beginning of the loading stage, the crack opens gradually and there are microcracks to initiate in the horizontal grain boundary in front of the crack, as shown in Fig. 6a, b. These microcracks propagate forward along the grain boundary. At the same time, some intrinsic stacking faults are emitted from the grain boundaries of the front grains away from the crack. When the strain reaches 0.06, the crack grows along the grain boundary toward the triple junction ahead of it. In the upper and lower grain boundaries perpendicular to the load direction in front of the crack, there are microcracks or microvoids forming, as shown in Fig. 6c. When the strain further increases to 0.1, due to the obstruction of the grain, the crack stops growing. However, in the upper and lower grain boundaries perpendicular to the load direction, the microcracks or microvoids gradually connect with each other to form new cracks almost through the whole grain boundaries, as shown in Fig. 6d. Accordingly, different from the intragranular crack case, the intergranular crack propagation would be possibly kept if the loading continues. In addition, it is found that a few intrinsic and extrinsic stacking faults are formed at the grain boundaries of the front grains near the crack. After entering the unloading stage, as the strain decreases, both the original crack and the new cracks begin to shrink and narrow, as shown in Fig. 6e. However, even when the strain returns to zero, the original crack is not shortened and the new cracks do not close completely, but the stacking faults from the forward grain boundaries becomes much denser, as shown in Fig. 6f. They are induced by more disordered atoms at the grain boundaries during the unloading process.

Fig. 6
figure 6

Local atomic configurations around the intergranular crack at different moments

Full size image

Figure 7 shows the atomic configurations near the intergranular crack tip at four different moments during the loading stage. Figure 7a, b are for the original crack. At 30.4 ps and 33.2 ps, the atomic crack angle at the crack tip is 72.2° and 104.0°, respectively. This is due to the severe atomic misarrangement in the front grain boundary during the crack propagation along the grain boundary, resulting in the exaggerated atomic crack angle and the fully blunting of the crack tip near the triple junction. Figure 7c, d are for the new crack forming in the lower grain boundary parallel to the X-axis. At 66.6 ps and 68.2 ps, the atomic crack angle at the crack tip is 77.8° and 45.2°, respectively. Apparently, the crack tip is sharpened in this process, which is not consistent with the traditional understanding. It seems that a crack tip may also be sharpened temporarily in a loading stage.

Fig. 7
figure 7

Atomic configurations and atomic crack angles near the intergranular crack tip at different loading moments

Full size image

In the above process of loading and unloading, there is no dislocation emission and stacking fault observed near the intergranular crack tip. Figure 5b shows the local DXA structure of the grain in front of the crack tip at 57.6 ps. At this moment, new cracks have started to appear at the upper and lower horizontal grain boundaries in front of the original crack. Within the grain, there are Shockley dislocations emitted in \(\{ 111\}\) slip plane and \(< 112 >\) slip direction, as marked by the red arrows. Similar to the intragranular crack case, they induce the stacking faults to emit and slip. At the same time, Shockley partial dislocations appear at both ends of the stacking faults, so that extended dislocations also form here.

4 Conclusions

The molecular dynamics simulations were performed on nanocrystalline Ni with an intragranular or intergranular edge crack under single-cycle mode I loading. Their atomic configuration evolutions were observed and their atomic crack angle variations were analyzed. The following main conclusions were obtained.

  1. (1)

    Intragranular crack propagation could be induced by stacking faults at the crack tip and change its direction. During the loading stage, microcracks or microvoids would be formed in the grain boundary perpendicular to the loading direction and develop towards the original crack, so the intragranular crack propagation tends to transform into intergranular crack propagation.

  2. (2)

    The intergranular crack perpendicular to the loading direction would propagate along the grain boundary and encounter strong resistance at the triple junction. New cracks would form in the upper and lower grain boundaries of the forward grain, so the intergranular crack propagation could be kept.

  3. (3)

    For an intragranular or intergranular crack case, the dislocation density increases even in the unloading process. The crack tip is possibly sharpened temporarily during loading, which was observed in the intergranular crack propagation. These are different from traditional understanding.