Molecular dynamics simulation of crack propagation behaviors at the Ni/Ni₃Al grain boundary

Abstract

The primary purpose of this paper is to study intergranular and transgranular crack propagation behavior at the grain boundary (GB) of Ni/Ni₃Al by Molecular Dynamics (MD) simulation. Cracks are loaded in tension mode I. The stress–strain curves, the changes of dislocation network, the crack propagation speeds and the changes of interfacial energy are compared with those of two models. The results show that the yield stress of the intergranular crack model is greater than that of the transgranular crack model. It is found that the intergranular crack propagation does not extend along the GB, but the crack extends into the γ′ phase. In addition, the transgranular crack extends at an angle of 45° and the dislocation moves on the slip system {11̄1} [1̄01]. The transgranular crack model breaks completely under a small strain. Moreover, the tranagranular crack propagation results from deformation damage of the model, and its propagation mechanism is shear rupture along the slip plane. The intergranular crack initiates propagation at the most dense position of the slip bands.

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1. Introduction

Nickel based single crystal superalloys have been widely used for high-temperature applications because of their good creep resistance.^1,2 The main structure of nickel based single crystal superalloys is the precipitation of the ordered γ′-Ni₃Al phase with L1₂ structure, which is coherently embedded in a solid solution matrix of the γ-Ni phase.³ These alloys are far stronger than pure γ or γ′ single phase materials due to the presence of γ/γ′ interfaces which prevents dislocation motion at high temperatures.⁴ The GB between the reinforcement and the matrix plays a crucial role in determining the mechanical properties of the Nickel based single crystal superalloys. Crack propagation is strongly influenced by the specifics of the GB.⁵

Computer simulation plays an important role in investigating the mechanism of materials' strength at the atomic length scale, and it has to be taken into account for understanding the propagation of a crack at an atomically sharp crack tip at an interface as well as dislocation emission at the crack tip. Hocker et al.⁶ have conducted a MD study on the brittle/ductile interfaces of Ni/B₂–NiAl under mechanical loading. They report that interfaces have an influence on strain induced material failure by nucleation of defects. Liu et al.⁷ explored the influence of Re on the propagation of a (010) [101] crack in the Ni/Ni₃Al interface using MD, and showed that γ_us was not affected by Re–Ni atomic interaction, and that Re–Re atomic interaction has some effect on γ_us. Wu et al.⁸ investigated the microstructure evolution and stress distribution characteristics of a pre-cracked nickel single crystal at different temperatures using MD, and showed clearly that the crack propagation process and stress distribution characteristics are closely related to the change of temperature. Xie et al.⁹ used MD simulations to investigate the mechanisms of low-temperature impact toughness of the ultrafine grain structure steel. The simulation results suggest that the sliding of the {001} [110] type and {110} [111] type GB can improve the impact toughness. However, there still exist three very important questions to discuss: the difference between intergranular crack propagation and transgranular crack propagation, the origin of the crack, and the formation mechanism of these cracks, so further research at an atomistic scale is necessary.

In this work, the mechanism of the intergranular and transgranular crack propagation at the GB of Ni/Ni₃Al will be carefully explored by MD simulations. The change of dislocation network is also analyzed based on the observation of the atomic structure evolution at the GB. Moreover, we use the crack length to predict the degree of crack growth. A comprehensive understanding of the behaviors of interfacial crack propagation is achieved.

2. Computational methods

The MD simulation is performed by using the embedded atom method (EAM) potential developed by Yamakov,¹⁰ Daw and Baskes,¹¹ which could provide an effective description of the transition metals with the face-centered cubic (FCC) structure, considering that this potential has a good ability to account for dependence of the strength of individual bonds on the local environment, such as surface and defects so that a reasonable simulation of fracture and damage could be obtained.⁸ Two types of mode I cracks are studied, including types of intergranular and transgranular crack propagation as shown in Fig. 1.

Fig. 1 (a) MD model of Ni/Ni₃Al GB with a crack at the left of the model, which represents a model of intergranular crack propagation. (b) A crack at the top of the model, and it represents a model of transgranular crack propagation.

The nanowire is constructed as a regular FCC lattice with initial surface orientation of [100], [010], and [001] in the X, Y and Z directions respectively. In order to minimize the residual stresses originating from the lattice misfit the dimensions of the GB have to be chosen so that na_γ′ = ma_γ with natural numbers n and m.¹² The lattice parameters of Ni and Ni₃Al are 3.520 Å and 3.573 Å, respectively. These imply that n = 66 and m = 67. It indicates that, within this range of the misfit, a phase interface formed by 66a_γ′ and 67a_γ should relax the stress induced by the different lattice parameters. The system is modeled by two grains: 66a_γ′ × 66a_γ′ × 12a_γ′ cubic box with the Ni₃Al lattice constant a_γ′ and the 67a_γ × 67a_γ × 12a_γ cubic box with the Ni lattice constant a_γ for the case of a [001] GB.

Since the lattice misfit creates a stress field, the coherent phase interface is unstable. Based on the principle of minimum energy, the atoms on the phase interface will rearrange to minimize the elastic stress field between the γ and γ′ phase.⁴ Therefore, a full relaxation must be performed before the tensile load is applied. The Ni/Ni₃Al model is first relaxed for 100 ps, by running 100 000 steps with a constant time step of 1 fs, to get an initial equilibrated state. The misfit dislocation networks are found to be formed on the phase interface.

For intergranular crack model, shrink-wrapped and fixed boundary conditions are applied perpendicular to the phase interface in the x direction, periodic boundary condition is applied in the y direction, and the z direction is simulated considering the shrink-wrapped boundary condition. The transgranular crack model is constructed by applying the shrink-wrapped boundary condition only in the x direction, periodic boundary condition is applied in the y direction, and shrink-wrapped and fixed boundary conditions are applied perpendicular to the phase interface in the z direction. Three layer atoms at the most bottom and top are fixed as a plane in the model. And displacement with a movement rate 0.5 Å ps⁻¹ is applied to the top plane which is a reasonable value for MD tension simulations.¹³ A Nose–Hoover thermostat is applied to maintain the system temperature at a constant value of 300 K. The stresses are calculated using the viral theorem, which is controlled by the Berendsen barostat algorithm.^14,15 All simulations are performed using an MD code called LAMMPS.¹⁶ The visualization tools AtomEye¹⁷ are used in the atomistic simulations.

To investigate the stress fields around the crack tip in the process of stretch breaking, the bulk stress tensor σ_αβ of a nanowire is defined to be the strain derivative of the total energy per unit volume.

where E is the total energy of the nanowire, ε_αβ is the strain tensors α, β ∈ (x, y, z), Ω₀ is the undeformed atomic volume in the perfect crystal and σ_αβⁱ is the local stress at atom site i. Using the embedded-atom potential, σ_αβⁱ can be expressed as^18,19
E_i is the energy per atom and is the velocity component in the j direction of atom α.

3. Results and discussion

3.1. Deformation behavior and dislocation propagation

Fig. 2 shows the stress strain curves of intergranular and transgranular crack model. External crack propagation is not obvious in intergranular crack model at the elastic range, however, internal crack propagation speed is fast (Fig. 2a and b) which is mainly because the crack extends along the slip plane at specific direction. External crack extends rapidly with the increase of strain during the yielding stage (Fig. 2a and c–e). As we can see in Fig. 2a, the crack expansion does not extend along the GB, but crack extends into the γ′ phase which is consistent with the ref. 4.

Fig. 2 Stress–strain curves for the model of intergranular and transgranular crack propagation.

The yield stress of the intergranular crack is greater than that of transgranular crack. Transgranular crack breaks completely under the small strain. External crack is obvious in transgranular crack model at the elastic range which is different from intransgranular crack model (Fig. 2b). We find that the crack does not vertical cross the GB, but extends at an angle and the dislocation moves on the slip system {11̄1} [1̄01]. This is mainly because crack propagation is affected by shear stress. The dislocation density increases with the increasing strain. When the dislocation pile-up achieves a certain level, the crack initiates at the most dense position of the slip bands.

Dislocation core occurs at the Ni/Ni₃Al GB in the wake of imposing force in the z direction, as shown in Fig. 3. Xie et al.³ have shown that these dislocations slip along the two slip planes and form a slip band at each side of the crack. With elapse of the simulation time, more and more dislocations are emitted from the crack tip and slip away in the two slip bands, leading the crack to propagate by a ductile manner. The model of this paper has Ni/Ni₃Al GB which is different from the above paper. The significant feature of Ni/Ni₃Al model is that the GB will appear dislocation nets. The Ni/Ni₃Al models are far stronger than pure γ or γ′ single phase materials due to the presence of γ/γ′ interfaces which prevents dislocation motion at high temperatures. The interface between the γ/γ′ dislocation structure determines the mechanical properties of the Ni/Ni₃Al. With the increasing strain, dislocations slip into the γ matrix channel. The original two edge dislocations are constantly on the move and increasing continuously. When a large number of dislocations move to γ/γ′ phase interface, the interface appears the phenomenon of stress concentration. Dislocations cut into γ′ precipitates as the stress exceeds the threshold (Fig. 3b and c). Dislocations move gradually to the crack tip as the strain increased, and then crack begins to extend rapidly.

Fig. 3 Dislocation core and dislocation emission in the model of intergranular crack. (a) dislocation core at Ni/Ni₃Al GB, ε = 0.00299, (b) the slip plane and the slip direction, ε = 0.05656, (c) the motion of the dislocations, ε = 0.07442.

Edge dislocations occur at the Ni/Ni₃Al GB in transgranular crack model, as shown in Fig. 4a. Dislocations move along the interface in the x direction which is different from the intergranular crack model. When the dislocations move to the GB, they still extend in [110] and [01̄1]directions (Fig. 4b). With the increase of the strain, dislocations begin to appear at the crack tip (Fig. 4c), which is in conformity with ref. 20. With the increase density of dislocations in the matrix phase γ, a large number of dislocations move to γ/γ′ interface. Dislocations pile up at the interfaces and interact with each other. We find that Dislocations cut into γ′ precipitates where the dislocation network is damaged. Dislocation direction forms an angle of 45° relative to the interface (Fig. 4d). The crack propagates along the slip direction inside of the γ′ precipitates in Fig. 4e. By observing the shear stresses of the y and z directions, we can find that they have the same changing trend, as shown in Fig. 5. The summation of forces in y and z directions may be 45°, this is why dislocation direction forms an angle of 45° relative to the interface.

Fig. 4 Dislocation emission and dislocation movement in the model of transgranular crack. The simulation times (a–e) are corresponding the strain ε = 0.00366, ε = 0.00791, ε = 0.06110, ε = 0.06749 and ε = 0.10579, respectively.

Fig. 5 Shear stress of the y and z directions for the model of transgranular crack.

We adopt the critical shear stress to have a deeper analysis of the angle between the slip plane and the GB, which is in order to better understand the inherent mechanisms of slip system.

here A is the cross-sectional area, φ is the angle between slip surface normal and centerline, λ is the angle between slip direction and force F, σ is the magnitude of the applied tensile stress. Critical shear stress is the component of shear stress, resolved in the direction of slip, necessary to initiate slip in a grain. It is a constant for a given crystal.

Ni/Ni₃Al crystals are sliding on the {111} plane along [011] and [01̄1] directions respectively. Therefore, the shear stress along the sliding direction of slip plane is a real contribution to dislocation glide. When the normals of slip plane, glide direction and axial force are coplanar. When λ and φ are close to 45°, orientation factor (cos λ cos φ) gets the maximum value, as shown in Fig. 6. The minimum σ_s is called soft direction location. Ni/Ni₃Al crystal easily plastic deform under the external force.

Fig. 6 Sketch of shear stress, slip plane and slip direction.

3.2. Dislocation networks evolution and crack growth

To better understand the movement of the dislocation network at interface, we extract the dislocation networks at GB in intergranular and the transgranular crack model, respectively, as shown in Fig. 7. Interface dislocation network is gradually coarsening when the dislocation of intergranular crack model moves along the loading direction. With the applying load, interface dislocation network is gradually damaged and finally destroyed. When dislocations pile up at the interface, dislocations move on the slip system {111} [011] and {111} [01̄1].

Fig. 7 The evolution of dislocation networks in the model of intergranular and transgranular crack.

The shape of the dislocation network is hexagon. The hexagon will gradually decrease during the stretching in Fig. 7a and c. This is because the dislocation network interface is perpendicular to the loading direction. As the load is applied, dislocations move to the place of the concentration of stress, which results in the accumulation of dislocations. Fig. 7b and d show hexagon will gradually increase during the stretching. This is mainly because that the network interface dislocation is parallel to the loading direction. The two sides along the direction of force will be stretched, while the other four sides will be compressed.

Kinetics of crack extension helps us to better understand the crack propagation and fracture. To quantify the length of the crack with the changing strain, the crack length is calculated by the change of the surface energy during run time. Generally, it is believed that the stress and atomic energy of crack tip is greater than other parts, so the crack moves along the direction of the lowest surface energy.⁸ The relationship between the crack length and strain is shown in Fig. 8. At the beginning of crack extension, the strain of transgranular crack is greater than that of intergranular crack. Transgranular crack model begins to appear the crack surface when the strain is 0.056 and the crack extends to the border as strain is 0.071. In the following process of stretching, the length of crack remains unchanged, while the atoms of the model boundary are still in action until the model breaks.

Fig. 8 Crack length as a function of strain in the model of intergranular and transgranular crack.

The strain does not reach the threshold when the strain is less than 0.025, then the crack extends rapidly. When the strain is between 0.06 and 0.075, the crack length keeps unchanged, and the stress drops slowly in Fig. 2. In order to explain this phenomenon, we try to explore the dislocation of intergranular crack model. At this stage, dislocation move intensely, but external force can only support the dislocation motion, it can't provide sufficient energy for extending the crack. By contrasting two changing curves of crack length with strain, we can get the conclusion that transgranular crack propagation is more difficult than intergranular crack propagation. This is also mentioned in the ref. 21. At the same time it also illustrates that the GB hinders crack propagation.

The change of interface energy is the main indicator to value crack extension. The GB energy (E) is the reversible free energy change for making free surfaces from GBs. The GB energy E of a GB can be easily determined using the following equation:^7,22E = (E_γ + E_γ′ − E_γ/γ′)/S where E_γ/γ′ is the total energy of a fully relaxed Ni/Ni₃Al GB system. E_γ and E_γ′ are the energy of a fully relaxed pure Ni₃Al and pure Ni, respectively. S is the GB area between Ni₃Al and Ni as shown in Fig. 1.

Fig. 9 presents the average GB energy as a function of the strain. The average GB energy of intergranular crack model changes quickly, while the average GB energy of transgranular crack model changes slowly. The initial GB energy of intergranular crack model is greater than that of transgranular crack model. The area of the GB reduces with the expansion of the crack, which results in interface energy becoming lower. On the one hand, transgranular crack model gradually increases the surface energy, on the other hand, it remains the area of the GB unchanged. Therefore, the average GB energy of transgranular crack model is more than −0.2 eV Å⁻². Transgranular interfacial energy is zero when the strain is 0.07, which illustrates the fracture of transgranular crack model combined with Fig. 8.

Fig. 9 The average GB energy as a function of the strain.

4. Conclusions

In summary, MD simulations have been performed to study the influence of GB on the crack propagation of two types of mode I crack, including types of intergranular and transgranular crack propagation. The simulation result shows that the yield stress of the intergranular crack model is greater than that of transgranular crack model. The transgranular crack model breaks completely under a small strain. The intergranular crack propagation does not extend along the GB, but crack extends into the γ′ phase. In addition, the transgranular crack propagation does not vertical cross the GB, but extends at an angle of 45° and the dislocation moves on the slip system {11̄1} [1̄01]. Moreover, the tranagranular crack propagation results from deformation damage of the model, and its propagation mechanism is shear rupture along slip plane. The intergranular crack initiates propagation at the most dense position of the slip bands.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (no. 51210008).

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