Fatigue behavior of Cu-Zr metallic glasses under cyclic loading

The effect of oscillatory shear deformation on the fatigue life, yielding transition, and flow localization in metallic glasses is investigated using molecular dynamics simulations. We study a well-annealed Cu-Zr amorphous alloy subjected to periodic shear at room temperature. We find that upon loading for hundreds of cycles at strain amplitudes just below a critical value, the potential energy at zero strain remains nearly constant and plastic events are highly localized. By contrast, at strain amplitudes above the critical point, the plastic deformation is gradually accumulated upon continued loading until the yielding transition and the formation of a shear band across the entire system. Interestingly, when the strain amplitude approaches the critical value from above, the number of cycles to failure increases as a power-law function, which is consistent with the previous results on binary Lennard-Jones glasses.


I. INTRODUCTION
Understanding structure-property-function relationships in metallic glasses, which are known to possess exceptionally high strength and large elastic strain limit, is important for numerous structural applications [1,2].Under cyclic loading at sufficiently large stress or strain amplitudes, such amorphous alloys ultimately fail via the formation of nanoscale shear bands and propagation of microscale cracks [3][4][5].On the microscopic level, an elementary plastic event in disordered alloys involves a swift collective rearrangement of a few tens of atoms or a shear transformation [6,7].Using a mean field elastoplastic model with a distribution of local yield barriers, it was recently demonstrated that fatigue life of athermal amorphous solids follows a power-law divergence when the yielding point is approached from above [8].Furthermore, within two models of elastoplastic rheology, it was found that fatigue in amorphous materials proceeds via slow accumulation of low levels of damage and it is followed by a sudden failure and shear band formation after a number of cycles that, in turn, depend on the strain amplitude, working temperature, and degree of annealing prior to loading [9].Despite recent advances, however, the exact mechanism of fatigue failure in metallic glasses and an accurate prediction of the fatigue life remain not fully understood.
In the last few years, the effect of cyclic loading on rejuvenation, relaxation, and yielding phenomena in amorphous alloys was extensively studied using particle-based simulations .In particular, it was found that after a number of transient cycles, athermal glasses undergo strain-induced reversible deformation where trajectories of atoms become identical during one or more periods [11,13].Whereas poorly annealed glasses tend to relax when subjected to small-amplitude cyclic shear [12,22,24,27,29], well-annealed amorphous alloys can be rejuvenated during a number of cycles before yielding at strain amplitudes above a critical point [33,35,42].The yielding transition in periodically deformed glasses is typically associated with the formation of shear bands initiated either at open boundaries [14,15,26,36,39] or in the bulk of the computational domain in case of periodic boundary conditions [20,21,23,24,28,33,37,40,42].In the vicinity of a critical strain amplitude, the number of cycles to reach the yielding transition depends on temperature [33], preparation history [20,28,37,40], strain amplitude [15,20,21,33], deformation protocol [30], and loading frequency [15].Notably, it was recently demonstrated that fatigue failure in wellannealed binary Lennard-Jones (LJ) glasses occurs after a number of cycles that increases as a power-law function when the strain amplitude approaches a critical value from above [42].
In spite of significant progress, however, the role of loading protocol, processing history, and interaction potentials on the fatigue life and yielding transition in disordered solids is still being actively investigated.
In this paper, we report the results of molecular dynamics (MD) simulations of a Cu-Zr metallic glass subjected to oscillatory shear deformation at strain amplitudes near a critical value.We consider a well-annealed sample prepared by cooling at a slow rate from the liquid state to room temperature.It will be shown that after a certain number of shear cycles, the glass undergoes a yielding transition characterized by the formation of a system-spanning shear band and a steep increase of the potential energy.We find that upon reducing strain amplitude towards a critical value, the number of cycles to reach the yielding transition increases as a power-law function, which is in agreement with simulation results for lowtemperature binary LJ glasses.
The rest of this paper proceeds as follows.In the next section, we describe the preparation procedure and the deformation protocol.The results for the potential energy, shear stress, nonaffine displacements, and atomic configurations are presented in section III.A brief summary is given in the last section.

II. MOLECULAR DYNAMICS SIMULATIONS
In this study, the Cu 50 Zr 50 metallic glass was simulated using the embedded atom method (EAM) potentials [43,44] with the time step △t = 1.0 fs.The total number of Cu and Zr atoms is 60 000.The system was initially placed into a cubic box and equilibrated at temperature of 2000 K and zero pressure.Following the cooling protocol described by Fan and Ma [45], the liquid was first cooled to 1500 K at the rate of 10 13 K/s, then to 1000 K at 10 12 K/s, and finally to 300 K at 10 10 K/s.Hence, the effective cooling rate across the glass transition temperature (T g ∼ 675 K) was 10 10 K/s.This preparation procedure was carried out in the NPT ensemble using the Nosé-Hoover thermostat, zero external pressure, and periodic boundary conditions [46].After cooling to 300 K, the linear size of a cubic box was fixed to 101.8 Å, and the glass was subjected to oscillatory shear deformation along the xz plane at constant volume, as follows: where T = 1.0 ns (10 6 MD time steps) is the oscillation period and γ 0 is the strain amplitude.
The MD simulations were performed for strain amplitudes in the range 0.055 ⩽ γ 0 ⩽ 0.064.
The potential energy, shear stress, and atomic configurations at the end of each cycle were periodically saved for post-processing analysis.Due to computational limitations, the data were collected only for one realization of disorder.

III. RESULTS
It is generally realized that mechanical properties of metallic glasses strongly depend, among other factors, on preparation history [47,48].Thus, sufficiently slowly cooled glasses settle at lower energy states and upon start-up continuous deformation tend to exhibit a pronounced yielding peak characteristic of a brittle response, whereas more rapidly cooled glasses typically deform in a ductile manner [49].Moreover, it was recently found that well-annealed binary glasses under cyclic loading yield at strain amplitudes smaller than the location of the yielding peak [20,28].When a critical value is approached from above, the number of cycles until the yielding transition increases as a power-law function for binary LJ glasses at a temperature well below the glass transition temperature [42].In the present paper, these results are extended to the case of well-annealed Cu 50 Zr 50 metallic glasses at room temperature.We also comment that test runs of more rapidly cooled Cu 50 Zr 50 metallic glasses (with the effective cooling rate of 10 12 K/s) have shown that cyclic loading near the critical strain amplitude induces significant structural relaxation and a rapid decay of the potential energy during a number of transient cycles, similar to the results reported for poorly annealed LJ glasses [30].
We first present the dependence of shear stress along the xz plane as a function of time in Fig. 1 for strain amplitudes γ 0 = 0.057 and 0.064.Note that for the largest strain amplitude γ 0 = 0.064 considered in the present study, the glass gradually yields after about 40 shear cycles, followed by plastic flow within a shear band upon continued loading.It can be observed that the number of cycles until the yielding transition increases significantly when the strain amplitude is reduced to γ 0 = 0.057.Note that the amplitude of stress oscillations after the yielding transition is nearly the same in both cases, and it is determined by the maximum stress within a shear band.A similar behavior of the shear stress was observed for low-temperature binary LJ glasses under periodic shear, although the yielding transition in the previous study was associated with a sharp drop in stress during only one shear cycle [42].
Next, the variation of the potential energy per atom versus cycle number is displayed in Fig. 2 for the same strain amplitudes, γ 0 = 0.057 and 0.064, as in Fig. 1.Each of the curves in Fig. 2 clearly show three stages of deformation under cyclic loading; namely, a solid-like response, a gradual yielding transition, and plastic flow within a shear band.The relatively large scatter of the data is attributed to thermal fluctuations and finite system size.When loaded at γ 0 = 0.064, the potential energy at the end of each cycle (the lower envelope) starts to increase abruptly, indicating the occurrence of localized plastic events that eventually lead to the formation of a shear band and the corresponding plateau after about 60 cycles.By contrast, the initial stage of accumulation of plastic rearrangements at the strain amplitude γ 0 = 0.057 takes about 300 cycles, and it is followed by a gradual yielding transition during 150 shear cycles.
A summary of the data for the potential energy after each cycle at zero strain as a function of the cycle number for strain amplitudes in the range 0.055 ⩽ γ 0 ⩽ 0.064 is presented in Fig. 3.It is evident that the number of cycles until the yielding transition becomes larger when the glass is loaded at a lower strain amplitude.Notice that the system did not yield for at least 700 cycles when periodically deformed at strain amplitudes γ 0 = 0.055 and 0.056.
Therefore, it can be deduced that the critical strain amplitude for the yielding transition is between γ 0 = 0.056 and 0.057.Further, the results in Fig. 3 demonstrate that the wellannealed Cu 50 Zr 50 metallic glass can be rejuvenated via cyclic loading at strain amplitudes slightly above a critical value, which is similar to conclusions for binary LJ glasses [42].The number of cycles needed to increase the potential energy by 0.001 eV per atom varies from about 20 to 300 depending on the strain amplitude (see Fig. 3).
It is well known that plastic deformation in disordered solids involves the so-called nonaffine rearrangements of atoms [50].In particular, the nonaffine measure for displacement of an atom from r i (t) to r i (t + ∆t) can be computed using the matrix J i , which transforms positions of neighboring atoms and at the same time minimizes the following expression: where the sum is taken over N i atoms that are initially located within 4.0 Å from r i (t).Irreversible plastic rearrangements of atoms typically occur when their nonaffine displacements exceed a cage size [16].It was previously shown that the cage size of Cu 50 Zr 50 alloy near the glass transition temperature is about r c ≈ 0.6 Å [51].
In Fig. 4, we plot the nonaffine measure, D 2 [(n−1) T, T ], as a function of the cycle number n for strain amplitudes 0.055 ⩽ γ 0 ⩽ 0.064.The nonaffine quantity, Eq. ( 2), was evaluated for atomic configurations at the beginning and end of each cycle when strain is zero and then averaged over all atoms.A close comparison of the data for the potential energy in Fig. 3 and the nonaffine measure in Fig. 4 reveals a very similar functional dependence of U and D 2 on the cycle number during the yielding transition for each γ 0 .Notice, for example, that the number of cycles until yielding at U = −4.948eV/atom and D 2 = 0.5 Å2 are essentially the same for strain amplitudes 0.057 ⩽ γ 0 ⩽ 0.064.These results indicate that the potential energy of atomic configurations that evolve via plastic deformation under periodic shear strongly correlates with the average nonaffine displacements of atoms during each cycle.
A more direct way to account for plastic events is to compute a fraction of atoms whose nonaffine displacements during one cycle are slightly greater than the typical cage size.In Fig. 5, the fraction of atoms with relatively large nonaffine displacements, D 2 [(n − 1) T, T ] > 0.49 Å2 , is shown as a function of the cycle number for the same strain amplitudes as in Fig. 4. It can be observed in Fig. 5 that the fraction of atoms varies from about 0.02 in the elastic regime at γ 0 = 0.055 and 0.056 to about 0.45 when a shear band is formed at large strain amplitudes γ 0 = 0.063 and 0.064.As is evident, the function n f (n) clearly shows an accumulation of plastic events followed by the yielding transition and extended plastic flow for γ 0 > 0.056.
The same data for the fraction n f are replotted in Fig. 6 as a function of the the ratio n/n Y , where n = t/T is the cycle number and n Y is the number of cycles to reach the yielding transition.Although the data are somewhat noisy, it can be clearly seen that all curves approximately follow the same trend.In our analysis, the values of n Y were computed at n f = 0.2 for γ 0 > 0.056 and reported in the inset to Fig. 6.The dependence of the number of cycles until yielding on the strain amplitude can be well captured by the power-law function: where the critical strain amplitude is taken to be γ c = 0.056.The best fit to the data yields an exponent of −1.25 in Eq. ( 3), implying a possible divergence of n Y upon approaching γ c from above.In practice, the power-law function, Eq. ( 3), can be used not only to determine the number of cycles to failure at a given strain amplitude but also to estimate a number cycles needed to rejuvenate the glass by inducing small-scale plastic events.For example, when the glass is loaded for n = n Y /2 cycles, then the fraction of plastic displacements becomes n f ≈ 0.05 for γ 0 > 0.056 (see Fig. 6).
Note that a power-law dependence with a different exponent of −1.66 was reported for periodically sheared binary LJ glasses at a temperature well below T g [42].It should be commented that the power-law function, n Y (γ 0 ), describes the data for LJ and Cu-Zr glasses that were both prepared by cooling across the glass transition at a 'computationally' slow rate.However, it was found that the critical strain amplitude at zero temperature increases when a glass is relocated into deeper regions of its energy landscape by using either the swap Monte Carlo algorithm [29] or mechanical annealing [34].A recent study showed an example of how the yielding transition becomes increasingly delayed in more stable (mechanically annealed) glasses at low temperatures when cyclic loading is applied at a fixed strain amplitude [33].Moreover, MD simulations of periodically loaded LJ glasses at about half T g revealed that although the yielding transition is delayed in better annealed glasses, the critical strain amplitude remains unchanged [40].Therefore, it is still unclear how, in general, the critical strain amplitude and the power-law dependence, n Y (γ 0 ), are affected by the degree of annealing for cyclically loaded glasses at a finite temperature.
The spatial extent of plastic events in the metallic glass under cyclic shear is illustrated in Fig. 7 for the strain amplitude γ 0 = 0.064 and in Fig. 8 for γ 0 = 0.057.For clarity, only atoms with large nonaffine displacements during one cycle, D 2 [(n − 1) T, T ] > 0.49 Å2 , are shown when strain is zero.It can be seen in Fig. 7 (a) that during the fourth cycle at the strain amplitude γ 0 = 0.064, atoms with large nonaffine displacements form several isolated clusters.Upon further loading, the steep increase in U (see Fig. 2) and n f (see Fig. 5) is characterized by the formation of an extended cluster of atoms that percolates across the entire system, as shown in Fig. 7 (b, c).After 80 cycles, the shear band develops along the xy plane, shown in Fig. 7 (d), which is consistent with the plateau level in the potential energy reported in Fig. 2. A similar trend can be observed in Fig. 8 for the strain amplitude γ 0 = 0.057, except that the yielding transition is delayed by about 300 cycles and the shear band is formed along the perpendicular plane yz.The results in Fig. 8 (a) also confirm that after about n Y /2 cycles at γ 0 = 0.057, plastic deformation involves only small-scale clusters (n f ≈ 0.04).

IV. CONCLUSIONS
In summary, we studied the effect of periodic shear deformation on the yielding transition in metallic glasses using molecular dynamics simulations.We considered a Cu 50 Zr 50 metallic glass prepared by slow cooling from the melt to room temperature and then subjected to oscillatory shear at constant volume.It was shown that during loading for hundreds of cycles at strain amplitudes slightly below a critical value, the potential energy at the end of each cycle remains nearly unchanged and plastic deformation proceeds via localized clusters of atoms with relatively large nonaffine displacements.When loaded at larger strain amplitudes, the glass becomes gradually rejuvenated via collective irreversible displacements of atoms from cycle to cycle until the yielding transition and the formation of a systemspanning shear band.Remarkably, the number of cycles to reach the yielding transition approximately follows a power-law dependence on the difference between the strain amplitude and the critical value.Despite a different value of the power-law exponent, these conclusions are in agreement with the results for binary Lennard-Jones glasses periodically deformed at a temperature well below the glass transition temperature.