Accelerated rejuvenation in metallic glasses subjected to elastostatic compression along alternating directions
AAccelerated rejuvenation in metallic glasses subjected toelastostatic compression along alternating directions
Nikolai V. Priezjev , Department of Mechanical and Materials Engineering,Wright State University, Dayton, OH 45435 and National Research University Higher School of Economics, Moscow 101000, Russia (Dated: August 11, 2020)
Abstract
The influence of static stress and alternating loading direction on the potential energy and me-chanical properties of amorphous alloys is investigated using molecular dynamics simulations. Themodel glass is represented via a binary mixture which is first slowly annealed well below the glasstransition temperature and then subjected to elastostatic loading either along a single direction oralong two and three alternating directions. We find that at sufficiently large values of the staticstress, the binary glass becomes rejuvenated via collective, irreversible rearrangements of atoms.Upon including additional orientation of the static stress in the loading protocol, the rejuvenationeffect is amplified and the typical size of clusters of atoms with large nonaffine displacements in-creases. As a result of prolonged mechanical loading, the elastic modulus and the peak value of thestress overshoot during startup continuous compression become significantly reduced, especially forloading protocols with alternating stress orientation. These findings are important for the designof novel processing methods to improve mechanical properties of metallic glasses.Keywords: metallic glasses, elastostatic loading, mechanical processing, yield stress, moleculardynamics simulations a r X i v : . [ c ond - m a t . s o f t ] A ug . INTRODUCTION The development of novel fabrication techniques and processing methods that suppresscrystallization and oxidation of metallic glasses is important for various biomedical andstructural applications [1]. Although metallic glasses are known to possess exceptionallyhigh yield stress and relatively large elastic strain limit due to their amorphous structure,upon further increasing stress, the plastic deformation becomes sharply localized withinnarrow regions, called shear bands , leading to material failure [2]. Notably, it was recentlydemonstrated that upon triaxal compression at room temperature, metallic glasses can berejuvenated and exhibit strain-hardening , which inhibits the formation of shear bands andcatastrophic failure during uniaxial tension or compression tests [3]. In general, commonmethods to rejuvenate amorphous alloys and improve their plasticity include irradiation,cold rolling, high-pressure torsion, flash annealing [4–7], elastostatic loading [8–17], andmore recently discovered, cryogenic thermal cycling [18–20]. It was originally shown that aparticularly simple way to induce rejuvenation and improve mechanical properties of metallicglasses is to apply a static stress slightly below the macroscopic yield stress [8]. In particular,the prolonged static loading within the elastic range induces irreversible rearrangements ofatoms in a disordered solid and relocates the system into a higher potential energy state witha lower density [13]. More recently, it was found using atomistic simulations that binaryglasses undergo a relaxation-to-rejuvenation transition upon increasing uniaxial static stressat about half the glass transition temperature [14, 17]. However, the influence of alternatingloading direction, multiaxial constraint, and hydrostatic pressure and temperature duringelastostatic loading of amorphous alloys on their energy states and mechanical propertiesremains largely unexplored.During the last decade, a number of molecular dynamics simulation studies have beencarried out to investigate the atomic structure and mechanical properties of amorphous ma-terials subjected to stress- or strain-driven periodic deformation [21–44]. Interestingly, itwas demonstrated that at zero temperature atomic trajectories in a disordered solid underlow-amplitude oscillatory shear eventually fall into the so-called ‘limit cycles’ and becomeexactly repeatable after one or more shear cycles [24, 26]. In the presence of thermal fluctu-ations, rapidly quenched glasses subjected to periodic shear strain below the yielding pointgradually relax towards low energy states with nearly reversible dynamics, typically during2housands of cycles, in a process termed ‘mechanical annealing’ [32, 33, 38, 40, 44]. Morerecently, it was shown that at low temperatures, the relaxation process can be accelerated inperiodically driven metallic glasses if the shear orientation is alternated along two or threespatial directions [38]. It was later found that when an additional shear orientation is intro-duced in the loading protocol, the number of transient cycles before the yielding transitionis reduced but the critical strain amplitude remains unchanged [42]. Despite this progress,however, the optimization of thermomechanical processing methods requires further investi-gation of the operating conditions and details of loading protocols in order to access a widerrange of energy states and improved properties of amorphous alloys.In this paper, the influence of the static stress and variable loading direction on thepotential energy and mechanical properties of disordered solids is investigated using molec-ular dynamics simulations. We study the binary mixture slowly annealed below the glasstransition temperature and then loaded statically ether along a single direction or alongtwo and three spatial directions. It will be shown that with increasing static stress, thepotential energy increases and the size of plastically deformed regions becomes larger. Re-markably, the effect of rejuvenation is enhanced when an additional orientation of the staticstress is included in the loading protocol. Furthermore, the results of mechanical tests afterelastostatic loading indicate that both the elastic modulus and yield stress are reduced inrejuvenated samples.The rest of the paper is divided into three sections. The details of molecular dynamicssimulations and the loading protocol are described in the next section. The time depen-dence of the potential energy and mechanical properties as well as the analysis of nonaffinedisplacements are presented in section III. The main results are briefly summarized in thelast section.
II. MOLECULAR DYNAMICS (MD) SIMULATIONS
In our study, the amorphous alloy is represented via the Kob-Andersen (KA) binarymixture of two types of atoms with strong non-additive interaction that suppresses crystal-lization upon cooling below the glass transition temperature [45, 46]. The parametrizationof the KA model is similar to the one used by Weber and Stillinger to study the amorphousmetal alloy Ni P [47]. In the KA model, the atoms of types α, β = A, B interact via the3ennard-Jones (LJ) potential as follows: V αβ ( r ) = 4 ε αβ (cid:104)(cid:16) σ αβ r (cid:17) − (cid:16) σ αβ r (cid:17) (cid:105) , (1)with the standard parametrization for the energy and length scales: ε AA = 1 . ε AB = 1 . ε BB = 0 . σ AA = 1 . σ AB = 0 . σ BB = 0 .
88, and m A = m B [45]. To speed up thecomputation, the cutoff radius of the LJ potential is set to r c, αβ = 2 . σ αβ . The systemconsists of N = 60 000 atoms. Throughout the study, the physical quantities are reportedusing the LJ units of length, mass, energy, and time, as follows: σ = σ AA , m = m A , ε = ε AA , and, correspondingly, τ = σ (cid:112) m/ε . The MD simulations were carried out usingthe LAMMPS code [48] with the integration time step (cid:52) t MD = 0 . τ [49].The sample preparation procedure is similar to the one employed in the previous studyon elastostatic loading along a single direction [17]. Specifically, the binary mixture was firstequilibrated in a periodic box at the temperature T LJ = 1 . ε/k B and zero pressure. Here, k B denotes the Boltzmann constant. In all simulations, the temperature was regulated viathe Nos´e-Hoover thermostat [48, 49]. After equilibration, the system was linearly cooledwith the computationally slow rate 10 − ε/k B τ at zero pressure to the reference temperatureof 0 . ε/k B . In the previous study, the glass transition temperature of the KA model wasfound to be about 0 . ε/k B , when the mixture was cooled with the rate 10 − ε/k B τ at zeropressure [6].The loading procedure includes heating the glass to the annealing temperature T a andat the same time increasing the normal stress during 5000 τ , then annealing the systemduring the time interval t a at a constant value of the normal stress, followed by subsequentrelocation of the glass to the reference state ( T LJ = 0 . ε/k B and P = 0) during 5000 τ ( e.g. ,see Fig. 1). It was previously shown that rejuvenation is enhanced during elastostatic loadingalong a single direction at the annealing temperature T a = 0 . ε/k B [17], and, therefore, inthe present study all simulations were carried out at this value of T a . Furthermore, thestatic stress was applied either along the same direction or alternated along two and threespatial directions. In all cases, the normal stress components perpendicular to the imposedstatic stress were set to zero. During production runs, the stress components, potentialenergy, atomic configurations, and system dimensions were stored at T LJ = 0 . ε/k B and P = 0. The data were accumulated only for one realization of disorder due to computationallimitations. For example, a typical run during t a = 2 . × τ using 28 processors required4bout 360 hours at the Ohio Supercomputer Cluster. III. RESULTS
It has long been realized that the rate at which a multicomponent mixture is cooledacross the glass transition point is an important factor that determines the atomic structureand the potential energy state of a glass [2]. Hence, a more slowly annealed glass is settledat a deeper energy level, whereas rapidly quenched glasses typically have higher potentialenergy [50]. Moreover, depending on the amplitude of periodic strain deformation, the glasscan be either rejuvenated or relaxed [51]. In the case of elastostatic loading, the glass canbe rejuvenated via collective rearrangements of atoms, predominantly near soft spots, if theimposed stress is sufficiently large and the annealing temperature is well below the glasstransition temperature [17]. Interestingly, a transition from relaxed to rejuvenated statesupon increasing static stress at about half the glass transition temperature was reportedin the recent MD study [17]. In principle, the potential energy can be further increasedduring elastostatic loading if the orientation of the static stress is occasionally changed,thus allowing different clusters of atoms to rearrange irreversibly. In the present study, weexplore the effect of rejuvenation in a well annealed binary glass for three loading protocols;namely, when the static stress is applied along a single direction or changed along two orthree spatial directions.The results of the previous MD study on relaxation and rejuvenation of binary glassessubjected to elastostatic loading [17] provide guidance on the choice of parameters for theloading protocol with alternating orientation of the static stress. Thus, it was shown thatrejuvenation is enhanced if the annealing temperature is sufficiently below the glass transi-tion temperature T g ≈ . ε/k B at P = 0 [17]. In the present study, all simulation resultsare reported for the annealing temperature T LJ = 0 . ε/k B , at which the aging effects arenegligible during t a = 2 . × τ [17]. Moreover, it was also found that during compressionof a well annealed glass at the constant strain rate ˙ ε = 10 − τ − at T LJ = 0 . ε/k B , the peakvalue of the stress overshoot is σ Y ≈ . εσ − [17]. Therefore, the loading protocol in thepresent study includes four values of the static stress below σ Y ; namely, 0 . εσ − , 1 . εσ − ,1 . εσ − , and 1 . εσ − . Furthermore, the static stress was applied either along a single (ˆ z )direction or alternated along two (ˆ z and ˆ x ) and three (ˆ z , ˆ x , and ˆ y ) directions. In the last5wo cases, the orientation of the static stress was changed every 10 τ .The dependence of the potential energy on the annealing time and loading protocol isshown in Fig. 2 for the indicated values of the static stress. Note that the data in Fig. 2 aretaken at the reference state, T LJ = 0 . ε/k B and P = 0, whereas the static stress was appliedat the temperature T a = 0 . ε/k B . As shown in Fig. 2, the glass becomes rejuvenated whenthe imposed static stress is sufficiently large (greater than 0 . εσ − ) and t a (cid:46) . × τ .Note that the increase in potential energy with respect to the dashed line is larger for highervalues of the static stress. This is consistent with the results of the previous MD study onelastostatic loading where the static stress was applied along a single direction [17]. It can beclearly seen in Fig. 2 (c, d) that in the case of alternating orientation of the static stress, theeffect of rejuvenation is significantly amplified. This trend can be rationalized by realizingthat upon changing the orientation of the static stress, the stress in the system becomesredistributed and a different set of localized rearrangements can be facilitated (during thetime interval 10 τ ), leading to faster rejuvenation. As shown in Fig. 2 (c, d), the largestincrease in energy is attained when the static stress is alternated along all three spatialdirections.We further comment that during elastostatic loading at 1 . εσ − along alternating di-rections, the samples were extensively compressed along the direction of the applied stress.Therefore, the data are not reported for t a (cid:38) . × τ in Fig. 2 (d). For example, the timedependence of the potential energy and the system size for alternating loading at 1 . εσ − is presented in Fig. 3. It can be seen in Fig. 3 (a) that, following a gradual compression for t a (cid:46) . × τ , the system becomes significantly deformed during a relatively short timeinterval ≈ × τ . This plastic strain is unrecoverable, and the corresponding change in thepotential energy is shown in Fig. 3 (b). In general, we find that the loading time until plasticflow is reduced for larger values of the static stress and loading protocols with alternatingstress orientation.The spatial organization of stress-induced, irreversible rearrangements of atoms can beanalyzed via their nonaffine displacements [52]. More specifically, the nonaffine measure fora particular atom can be computed by using the transformation matrix J i , which linearlytransforms positions of neighboring atoms, and at the same time minimizes the following6xpression: D ( t, ∆ t ) = 1 N i N i (cid:88) j =1 (cid:110) r j ( t + ∆ t ) − r i ( t + ∆ t ) − J i (cid:2) r j ( t ) − r i ( t ) (cid:3)(cid:111) , (2)where ∆ t is the time interval between two successive configurations and the sum is taken overthe neighboring atoms within 1 . σ from the position r i ( t ). It was originally demonstratedby Falk and Langer that the nonaffine measure is an excellent indicator of irreversible sheartransformations in deformed amorphous solids [52]. More recently, the appearance of clustersof atoms with relatively large nonaffine displacements was connected to rejuvenation ofelastostatically loaded binary glasses [17]. In recent years, the collective rearrangements ofatoms in disordered solids were also reported during steady [53–57] and periodic [27–34, 38–40, 42, 44] deformation and thermal cycling [58–60].The effects of loading time and stress orientation on plastic rearrangements of atomsare presented in Figures 4–6. In all cases, the value of the static stress is fixed to 1 . εσ − ,and the lag time in Eq. (2) is set ∆ t = 6 × τ . The reference times t = 0, 6 × τ ,1 . × τ , and 1 . × τ are chosen within the rejuvenation regime, 0 (cid:54) t a (cid:54) . × τ ,as shown in Fig. 2 (c). The loading protocol consists of the static stress applied either alonga single (ˆ z ) direction (see Fig. 4) or alternated along two (ˆ z and ˆ x ) directions (see Fig. 5)or all three directions (see Fig. 6). It is clearly seen in Figs. 4–6 that the size of clustersof atoms with large nonaffine displacements increases when an additional spatial dimensionfor the stress orientation is included in the loading protocol. Note also that upon loadingalong a single direction, a large fraction of the system deforms plastically during the first∆ t = 6 × τ , but only a few isolated clusters are present at longer times, as shown in Fig. 4.This observation correlates with a steep increase and subsequent slow growth of the potentialenergy, as indicated by green circles in Fig. 2 (c). By contrast, when the stress orientationis alternated along all three spatial directions, most of the atoms undergo large nonaffinedisplacements (larger than the typical cage size ≈ . σ ), and the number of rearrangementsincreases with time (see Fig. 6), which corresponds to the enhanced rejuvenation shown byblue triangles in Fig. 2 (c).We next present the elastic modulus, E , in Fig. 7 and the yield stress, σ Y , in Fig. 8 forthe subset of data points shown in Fig. 2. The mechanical properties reported in Figures 7and 8 were computed during startup continuous compression with the constant strain rate7 ε = 10 − τ − . More specifically, the elastic modulus was calculated from the best linear fitto the data at small strain, ε (cid:54) .
01, whereas the yield stress is defined by the peak valueof the stress overshoot at about 5% strain. The samples were strained along the ˆ x , ˆ y , and ˆ z directions, and then the data for E and σ Y were averaged over three spatial directions. Thehorizontal dashed lines in Figs. 7 and 8 indicate the values of E and σ Y before elastostaticloading was applied. As expected, the general trend for the time dependence of E and σ Y is inversely correlated with the variation of the potential energy shown in Fig. 2. Inparticular, it can be observed that the yield stress is only slightly reduced for the staticloading at 0 . εσ − and 1 . εσ − , while the effect of alternating loading becomes significantat larger values of the static stress, 1 . εσ − and 1 . εσ − (see Fig. 8). Thus, the maximumdecrease of the yield stress due to elastostatic loading at 1 . εσ − along three directions isabout 15% at t = 1 . × τ , as shown in Fig. 8 (c). Altogether, these results demonstratethat both elastic modulus and yield stress become reduced upon loading at sufficiently largevalues of the static stress; and the effect is amplified if the loading orientation is periodicallyalternated along two or three spatial directions. IV. CONCLUSIONS
In summary, we investigated the effects of elastostatic loading and alternating orientationof the static stress on the potential energy states and mechanical properties of amorphousalloys using molecular dynamics simulations. The well annealed alloy was prepared byslowly cooling a binary mixture at zero pressure from the liquid state to a temperature wellbelow the glass transition temperature. After annealing, the binary glass was subjected toprolonged elastostatic compression either along a single direction or along two and threealternating directions. It was demonstrated that elastostatic loading at sufficiently largevalues of stress induces collective plastic events and significant rejuvenation. Moreover,upon introducing an alternating stress orientation in the loading protocol, the potentialenergy is further increased with respect to the untreated sample, and the typical size ofplastic rearrangements becomes comparable with the system size. Finally, the mechanicalproperties were probed by continuously compressing the treated samples at a constant strainrate. We found that both the elastic modulus and the peak value of the stress overshoot arereduced with increasing loading time or alternating stress orientation.8 cknowledgments
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