# Contribution 1

## Contents |

# Abstract

Fatigue is one of the most damaging mechanisms in structural components. With the development of structural nanomaterials, it is imperative to investigate the fatigue damage phenomena at the atomic scale. To study fatigue behavior at the nanoscale, one must apply non-continuum modeling frameworks, such as molecular statics (MS), molecular dynamics (MD), and Monte Carlo (MC) methods. To date, only MD and MS simulations using embedded atom method (EAM) and modified embedded atom method (MEAM) potentials have been conducted, and this paper reviews these simulations of the nanoscale fatigue-crack growth in nickel and copper including single crystals, bicrystals, and polycrystals. A nanoscale size middle tension (MT) specimen with the lateral side applied periodic boundary conditions was used to investigate the fatigue behavior in nickel and copper single crystals. Simulation results revealed that the cyclic plastic deformation at the crack tip was the main influencing factor for fatigue-crack growth. Two main nanoscale mechanisms of crack propagation were observed: (1) the main cracks linked with the voids nucleated in front of crack tip due to high dislocation density generated by the cyclic loading; and (2) the main cracks broke the atomic bonds in the crack plane without much plasticity. For the bicrystals and polycrystals, the grain boundaries exerted resistance to the crack propagation. To study the interactions between cracks and grain boundaries, four cases of grain boundary interfaces for copper and two cases of grain boundaries for nickel were simulated. In copper bicrystals, the crack path deviated and moved from one grain to another for high misorientations, while there were voids nucleating at grain boundaries in front of the crack tip that linked back with the main crack. Similar to macroscale fatigue, dislocation substructures were observed to develop in the atomic lattice during cyclic loading. In nickel bicrystals, for large misorientations, the cracks were stopped by grain boundaries. For small misorientations, the crack propagated through the grain boundary, but the growth rate was reduced due to the resistance of the grain boundary. Fatigue-crack growth rates for nanocracks were computed and compared with growth rates published in the literature for microstructurally small cracks (micron range) and long cracks (millimeter range). A nanostructurally small crack (NSC) was introduced in terms of the CTOD. The quantified NSC growth rates in copper single crystals were very similar with those experimentally measured for small cracks (micron range) and with those at stress-intensity-factor ranges lower than the threshold for long cracks (millimeter range). The atomistic simulations indicated that reversible plastic slip along the active crystallographic directions at the crack tip was responsible for advancing the crack during applied cycling. In the case of single or double plastic slip localization at the crack tip, a typical Mode I fatigue crack arose along a slip band and then grew into a mixed Mode I + II crack growth mechanism. For crystal orientations characterized by multiple slip systems concomitantly active at the crack tip, the crack advance mechanism was characterized by nanovoid nucleation in the high density nucleation region ahead of the crack tip and by linkage with the main crack leading to crack extension. To facilitate observations of fatigue-crack growth, the simulation of a copper polycrystal was performed at low temperature 20 K as well. The crack propagated along persistent slip bands within the grain. The crack propagated along grain boundaries when the angle between the direction of crack propagation and the grain boundary was small, while it was impeded by the grain boundary when the angle was large. The results obtained for the crack advance as a function of stress intensity amplitude are consistent with experimental studies and a Paris law exponent of approximately two.

**Keywords**: Embedded atom method; Fatigue; Nanostructurally small crack; Atomistic modeling; Structural nanomaterials.

Authors: Mark F. Horstemeyer, Diana Farkas, Sungho Kim, Tian Tang, Gabriel Potirniche

Corresponding Author: Mark F. Horstemeyer

Digital Object Identifier (DOI): [15]

# Introduction

Fatigue in metallic materials subject to repeated cyclic loading has been an active research area since last century and continues to be a focus of structural materials study. Most investigations of fatigue have been performed at the microscale and/or macroscale. Hence, the mechanism of fatigue failure is fairly well understood with the crack lengths ranging from a few microns to millimeters/centimeters. The classic continuum approaches are very powerful tool to solve the problems at the macro- or microscale. However, several reasons exist for performing atomic scale simulations. First, due to the development of nanomaterials and nanostructures, fatigue will become, at some point, an issue for the designers. Without the knowledge base for nanoscale fatigue, the designs will not be as robust. Second, macroscale fatigue models, particularly ones with microstructural sensitivities do not cover the length scale in the atomic region. To garner nanoscale crack growth rules in order to get more accurate macroscale predictive tools, atomistic simulations are warranted to provide mechanism understanding of grain boundary effects, crystal orientation effects, and driving force versus material resistance effects. Finally, nanoscale fatigue simulations such as those provided in this review can give insight into the fidelity of higher scale mathematical models and micromechanical finite element simulations by either providing the pertinent equations or at least the parameters for already developed equations.
In light of the aforementioned comments, the focus of this paper is to present a review on the atomistic modeling of fatigue crack growth in single crystals, bicrystals, and polycrystals of copper and nickel reported in ^{[1]} ^{[2]} ^{[3]} ^{[4]}. We try to sum up the works performed in this field so far. The survey of the literature is included in Sections 2–4. We also added some new atomistic fatigue-crack growth results in nickel bicrystals and copper polycrystals, which had not been published in the literature. The summary is presented in the final section with thoughts regarding new areas of research regarding atomistic modeling of fatigue.

# Simulation Method

### Interatomic potentials

In this paper, MEAM potential was used to simulate the fatigue-crack growth in copper and nickel single crystals and copper bicrystals. The EAM potential was applied to simulate the fatigue-crack growth in nickel bicrystals and copper nanocrystalline materials. The simulations utilize an EAM potential for nanocrystalline Ni ^{[5]} and has been tested as part of our previous work dealing with fracture under monotonic loading ^{[6]}.
Also, if one were to use molecular dynamics to perform fatigue cycling on polymers and/or ceramics, one would probably not need to use EAM or MEAM potentials. A good summary of the class of bond order formalisms that has proven valuable for covalently bonded systems was given by Brenner ^{[7]}. Stoneham et al. ^{[8]} summarized the shell model, which is a modification of a pair potential, used for ceramics.

### Atomistic model set-up

#### Single crystals

To simulate the fatigue-crack growth in copper and nickel single crystals, five specimens with different orientations were employed to study the nanoscale fatigue-crack growth in copper and nickel single crystals. These orientations, shown in Fig. 1, are [1 1 1], [1 0 0], [1 1 0], [1 0 1] and [1 2 2], respectively.

#### Bicrystals

In the bicrystals studied in this work, the notion of small misorientation angles and large misorientation angles were analyzed after the large strain work of Hansen and Hughes ^{[9]}. Clearly, more important grain boundaries or ones that are maybe more prevalent in copper and nickel should be studied, but this present study only examines some idealizations to show the connections to higher scale models.

##### Copper bicrystals

Four cases of grain boundary interfaces were simulated in the study of fatigue-crack growth in copper bicrystals. The general configurations of these bicrystals follow the types experimentally analyzed by Li ^{[10]}, and they are shown in Fig. 2. The first case AA, shown in Fig. 2a, corresponds to a [0 1 2]–[1 1 1] tilt grain boundary.

##### Nickel bicrystals

The bicrystals of nickel comprised two symmetrically rotated single crystals, as shown in Fig. 3, which are demarcated by a tilt grain boundary. The orientation of the [1 1 0] axes of the two single crystals were inclined by equal angles to the [0 0 1] axis.

#### Nanocrystalline FCC metals

##### Copper nanopolycrystalline

Voronoi construction was employed to build the initial atomic configuration ^{[11]}. All of the columnar grains in the specimens were constructed using the common axis [0 0 1] around which each grain rotated random angles. We made three specimens containing 5, 20, and 40 grains, respectively. After the initial generation, the specimens were equilibrated at a temperature of 20 K. To facilitate the propagation of fatigue crack, the simulation was performed at 20 K also. Each specimen had an initial edge crack on the left as shown in Fig. 4.

##### Nickel nanopolycrystalline

Just like the copper polycrystalline specimens, the initial nickel atomic configurations used in our studies were generated using a Voronoi construction ^{[12]} as well. The columnar grains in the sample were generated by using a common [1 1 0] axis for all grains and a random rotation angle around this axis for the various grains. The sample contained 36 grains with an average grain size of 6 nm. The periodicity along the [1 1 0] axis common to all grains was kept at the lattice periodicity along that direction. The grain boundaries present in these samples were of pure tilt character and had random misorientation angles. After their initial creation, the samples were fully relaxed using a conjugate gradient technique. The initial relaxation included the simultaneous energy minimization with respect to the total sample volume. The relaxed configuration was then used as a starting configuration in both the molecular statics and the dynamics techniques to study crack propagation using the same EAM interatomic potential and cyclic loading.

### Discussion of effects of boundary conditions

The major difficulty with the nanofatigue simulations is the strong constraint imposed by using periodic boundary conditions in the z-direction with a relatively small through-thickness dimension. Some have studied these effects based upon monotonic loads ^{[13]}.
To verify the effect of thickness in z-direction, we performed the investigations on the specimen with Orientation C as shown in Fig. 1c. The thickness varied from four unit cell to twenty unit cells. The length of one unit cell was about 0.35 nm, and a uniaxial tension strain in the y-direction was applied up to a total strain of ε = 0.06.The averaged stress–strain curves of the different thickness specimens are shown in Fig. 5. We observed that the stress dropped once the dislocations nucleated at the crack tips. The initial slope of the stress–strain curve and dislocation nucleation stress were independent of the thickness in the z-direction.

# 3. Simulation Results

### Fatigue-crack growth in single crystals

#### Copper single crystals

The loading applied was in a strain-controlled mode with the maximum strain level of εmin = 0.01 and an applied load ratio of R = εmin/εmax. This high load ratio for the tension–tension fatigue loading was chosen to prevent the crack faces from contacting each other, otherwise partial rewelding of the crack faces occurred, leading to difficulty in growing the crack. All atom velocities were initiated in the required direction at the beginning of each loading and unloading half cycle to alleviate the stress wave overlap that could arise from the high rate of deformation. Before applying the fatigue loading, the temperature in the specimens was equilibrated at 300 K. During fatigue loading for the first 10–15 cycles, we did not observe the mean stress relaxation. However, for most crystal orientations analyzed, the mean stress stabilization was recorded. Fig. 6 ; Fig. 7 show fatigue-crack growth in the specimens indicated in Fig. 1a–e. These figures show contour plots of the plastic deformation around the crack tip at various loading cycles for all five orientations considered. In the case of Orientation A, shown in Fig. 6, the crack grew in the plane (1 1 1), propagating along the [View the MathML source 1 0] direction, and the view plane is (1 1 View the MathML source). The two slip bands forming at the crack tip correspond to the directions [1 3 2] and [3 1 2], representing the traces the slip planes (1 View the MathML source 1) and (View the MathML source 1 1), respectively on the view plane (1 1 View the MathML source). The slip bands were oriented at an angle of 22.2° with respect to the vertical [1 1 1] orientation.

#### Nickel single crystals

Dislocation substructures that arose during the cyclic loading at constant strain amplitude were analyzed, and the main results are presented in Fig. 8; Fig. 9. The fatigue loading applied had a maximum strain of εmax = 5 × 10−3 and a ratio of R = εmin/εmax = 0.5. The high load ratio was chosen in order to prevent the inner faces of the void from contacting each other during unloading. From previous simulations, it was observed that contact of the crack surfaces led to the welding of the crack faces which led to difficulty in propagating the crack.

#### Comparison between fatigue-crack growth in copper and nickel single crystals

Comparison between all orientations for nickel and copper are shown in Fig. 10. The results indicated that the crack growth rate was highly dependent on the orientation of both crystals. For the first six cycles when the crack was small, and it did not enter yet the zone where the edge effect was dominant, the largest crack growth rate was experienced by Orientation C [1 1 0]. In this case, the rapid crack growth can be associated with the very little crack tip blunting observed. The smallest crack growth rate for both nickel and copper is shown by Orientation D [1 0 1].

### Fatigue-crack growth in bicrystals

Grain boundaries play an essential role in determining the resistance of fatigue crack propagation in materials ^{[14]}. In this section, the influences of grain boundary on the fatigue-crack growth in copper and nickel bicrystals were investigated. The MEAM potential was used for copper, and the EAM potential was used for nickel.

#### Fatigue-crack growth in copper bicrystals

Fig. 11 illustrates the fatigue-crack growth process for the first tilt grain boundary AA. Fig. 11a shows the initial configuration of the specimen. Starting at Cycle 2, the crack tip blunted and developed two distinct slip bands at the crack tip. The upper slip band was formed by plastic shear on the primary slip system [1 0 View the MathML source](1 1 1). The angle between the trace of this slip system on the view plane (1 View the MathML source 1) and the vertical tensile axis [0 1 2] is 50.76°. The lower slip band is due to plastic slip on the conjugate slip system [1 0 View the MathML source](1 1 1). The trace of this slip system on the plane of view (1 View the MathML source 1) is the [3 2 1] direction and forming a 61.41° angle with the vertical axis. At Cycle 9, the crack deviated along the primary slip band into a mixed Mode I + II crack growth. At Cycle 10, due to the stress concentration effect of the grain boundary, the two slip bands active at the crack tip became more diffuse, with the crack tip plasticity being spread out more evenly around the crack tip. At Cycle 11, near the grain boundary void nucleation was observed in front of the crack tip in the region with a fairly high density of dislocations. The nucleated void joined back with the main crack and induced the crack propagation into the adjacent grain. Large crack tip blunting was observed at the passage of the crack into the second grain. During loading from Cycle 2 to Cycle 15, the development of grain substructures occurred in the specimen. SBs formation along the primary slip system was observed in the first grain. The SBs were parallel to the primary slip directions. Also, regions with high and low dislocation densities were observed in the specimen after Cycle 10. The second grain also experienced substructuring in dislocation cells, mainly in the beginning of the fatigue process while the crack was still in the first grain.

### Fatigue-crack growth in nanocrystalline

#### Fatigue-crack growth in nanocrystalline copper

Fig. 12 shows the centrosymmetry parameter, and Fig. 13 shows the associated uniaxial Green strain illustrating the fairly large local strains (50%) at the grain boundaries near the crack tip; Fig. 12a also shows the initial configuration of the specimen. The crack propagated along persistent slip bands within the grain. At the sixth cycle, it encountered the grain boundary. Then the crack propagated along the grain boundary and traversed into another grain.

#### Fatigue-crack growth in nanocrystalline nickel

The crack propagation of nanocrystalline nickel is shown in Fig. 14, where the position of the crack tip is plotted as a function of the number of cycles. The slopes in this type of plot were computed for each of the stress intensity amplitudes considered. Fig. 15 shows the resulting rates of crack advance calculated using molecular dynamics for the three values of the stress intensity amplitude. The results show that crack advance rates are not very sensitive to the technique as the molecular dynamics and molecular statics giving similar results. This suggests that the molecular dynamics results do not have major spurious effects due to the unrealistically high loading rates. To further establish confidence in these results, we studied the basic mechanisms of crack advance using both techniques. For both cases, we observed a continuous increase in the number of dislocations present in the crack tip region as the crack advanced, and the formation of nanovoids or vacancy clusters ahead of the main crack.

# Summary and future directions

In this paper, we reviewed the research, which is fairly recent, related to atomistic modeling of fatigue-crack growth in FCC metals at the nanoscale and have added some new simulation results to provide understanding of nanostructurally small cracks (NSC). In single crystals, fatigue damage was caused by persistent slip band formation, and crack growth was simulated using center crack specimens. Clearly, MD and MS fatigue simulations have recently been used to reveal mechanisms at the nanoscale but for greater impact several new research endeavors are warranted:

1. To date, only FCC metals have been investigated. BCC and HCP materials need to be examined since many structural materials such as iron, steel, magnesium, titanium might find their way into nanostructural materials.

2. Because of the “welding” that occurs in the R = −1 simulations, an increasing applied load was used to characterize the nanoscale crack growth. Clearly, different R-values and boundary conditions for that matter are needed to examine the different stress state effects that could be observed in nanostructural materials.

3. Temperature dependent nanoscale cyclic plasticity and fatigue has yet to be studied. MD and MC could be used to quantify the temperature effects at different applied strain rates (frequencies).

4. The bridge between atomistic scale fatigue and dislocation density fatigue simulations needs to be defined and used for multiscale materials modeling. This in turn, could motivate the development of microstructurally small crack modeling efforts from a physical basis.

5. Including molecular dynamics studies with probabilistic distributions of microstructures and grain boundaries that lead to macroscale fatigue predictive tools can also be a fruitful endeavor. The single crystal, simple bicrystal, and simple polycrystalline studies conducted to date provide only a starting point for NSC behavior.

# Acknowledgements

MFH, TT, and SK would like to acknowledge the Center for Advanced Vehicular Systems (CAVS) at Mississippi State University for supporting this work.

# References

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