Solid materials, whether crystalline or glasses, are characterized by their elasticity. Generally, elasticity is independent of the probing strain if it is not exceeding the yielding point. Here, by contrast, we experimentally capture a pronounced strain-dependent elasticity in metallic glasses, as manifested by nonlinear mechanical damping in the apparent elastic deformation regime (∼1/100 of the yielding strain). Normal damping behaviors recover at higher temperatures but still below the glass transition. Atomistic simulations reproduce these features and reveal that they could be related to avalanche-like local structural instabilities. Our findings demonstrate that the standard elasticity is not held for metallic glasses at low temperatures and plastic events can be triggered at small perturbations. These results are consistent with previous simulations of model glasses and a scenario of hierarchical free-energy landscape of mean-field theory.
I. INTRODUCTION
Glasses are disordered materials that lack the long-range order of crystals but behave mechanically like solids.1 One defining feature of solids is their elasticity—the ability to resist a deformation by applying a force and to return to its original shape when that force is removed. Glassy materials like metallic glasses (MGs) are generally known for their large elasticity. However, recently, through simulations of model glasses, Procaccia and co-workers2–4 reported that the elasticity would not be strictly held in glasses. Unlike the normal crystalline solids in which plastic responses occur only when the external stress (or strain) exceeds a critical threshold (i.e., the yielding point), a tiny small perturbation would lead to the non-reversible plastic events in glasses.5–7 This intriguing phenomenon is therefore called “elastic breakdown.”8
Meanwhile, the elasticity breakdown has also been proposed in a mean-field theory.8 It advocates that deep in the glassy state, there might be a new phase transition called the Gardner transition,9,10 which separates high temperature normal elasticity and the elastic-breakdown at low temperatures.11 These findings provide new opportunities to explain these anomalies in a unified way.8,12–14
Despite the theoretical advances, it remains unknown whether the elastic breakdown is indeed applicable to real glassy materials. Although it might be revealed by simulations,11,15,16 generalizing the related methods to real glassy materials is not easy due to difficulty of tracking the trajectories for individual atoms. Consequently, experimental evidence for these predictions is still scarce.5,17
In this connection, it would be desirable to detect the breakdown of elasticity in real glasses or, to a lesser extent, to scrutinize any possible anomalies of elastic properties. Mechanical damping as measured by dynamical mechanical spectroscopy (DMS) could be useful in this regard. In crystalline materials, almost all the structural defects and dynamical processes have their fingerprint-like damping features in mechanical spectra.18–20 Likewise, it is anticipated that mechanical spectroscopy could be revealing in glasses as well.21,22
In this work, using dynamical mechanical spectroscopy, we capture a previously unidentified anomalous behavior that might be of relevance to the theory-predicted elasticity breakdown in glasses. We experimentally probed a pronounced strain-dependent nonlinear mechanical damping in the elastic deformation regime in a wide range of metallic glasses (MGs) having different chemical compositions and thermal histories. Combining atomistic simulations, we demonstrate that the anomalous nonlinear damping can be related to the avalanche-like local structural instabilities.
II. MATERIALS AND METHODS
We used nine MGs with different chemical compositions for experiments, as reported in Table SI (supplementary material). The mechanical damping of them was studied by the dynamical mechanical analysis (DMA, TA Q800) with the discrete testing frequencies of 0.5, 1, 2, and 4 Hz.
The temperatures and enthalpies of the transitions are determined by using the conventional differential scanning calorimeter (Mettler-Toledo DSC2) with a heating rate of 20 K/min.
We carried out molecular dynamics (MD) simulations of a model Cu65Zr35 MG (containing N = 256 000 atoms) that interact with an embedded atom method potential.23,24 The glassy samples were prepared by quenching a liquid from 3000 K to 10 K at a rate of 1 K/ps (or 1012 K/s). Mechanical damping of the model is studied by an approach of MD simulations of dynamical mechanical spectroscopy (MD-DMS) that numerically reproduces the protocol of real DMA experiments.25,26 Specifically, at a temperature T, we apply a sinusoidal strain with a period tω (related to frequency f = 1/tω) and a strain amplitude εA along the x direction of the model MG and the resulting stress σ(t) and the phase difference between stress and strain δ are measured and fitted with .
III. RESULTS
A. Anomalous nonlinear damping
Figure 1 presents our key experimental findings: the nonlinear damping effects (NDEs) in elastic deformation regimes of MGs. As a typical example, Fig. 1(a) shows the temperature dependence of mechanical damping tan δ of a Fe78Si9B13 MG at seven different strain amplitudes ranging from 0.029% to 0.163%. Here, tan δ is the tangent of the phase angle δ between the applied sinusoidal strain and the stress as a response. It measures the time-dependent elastic behavior (i.e., anelasticity) of a solid. We used small strain amplitudes which are about 10-100 times smaller than the yielding points (∼2%) of the MG27 so that all the measurements are within the macroscopic linear elastic response regime. Thus, tan δ is expected to be independent of the strain amplitudes if that anelasticity is the underlying mechanisms for damping.
However, at odds with this expectation, the measured tan δ in Fig. 1(a) clearly shows pronounced strain amplitude dependence over a wide range of temperature: the larger the strain amplitudes, the larger the values of tan δ. This trend is more prominent with decreasing temperatures. Obviously, such a nonlinear effect cannot be accounted for in terms of the anelastic mechanism. Instead, there must be certain low energy defects (or structural instabilities) in MGs that can be activated under very small strain perturbations.6,7,28
Figures 1(b)–1(d) report the similar findings in other three MGs with widely different chemical compositions. They suggest that the NDEs could be universal to MGs. We note that the pronounced peaks in Figs. 1(c) and 1(d) over the intermediate temperature range are due to the β relaxations.29–31 We find that larger strain amplitudes enhance the peak intensity of β relaxations but do not influence the peak temperature or the activation energy, as shown in Figs. 1(c) and 1(d) and Fig. S10 (supplementary material). The NDEs persist to higher temperatures than that of β relaxations. These results suggest that the atomic motions governing β relaxations are compatible with that of NDEs.
With further increasing temperature, Figs. 1(a)–1(d), the NDEs gradually become weak and eventually vanish at about 0.7–0.9 Tg (slightly dependent on the chemical composition and testing frequencies, see Table SI in the supplementary material) where the values of tan δ are eventually independent of strain amplitudes, as can be seen in, e.g., Fig. 1(a) around T = 650 K for the Fe–Si–B MG (whose Tg ∼ 873 K). Because damping in MGs at high temperatures is largely from structural α relaxation, this suggests, α relaxation would disrupt or change the related mechanism of the NDEs in MGs. This is in contrast to β relaxation at low temperatures, as found in Figs. 1(c) and 1(d). Overall, these results demonstrate that the standard linear elasticity is not strictly held at low temperatures in MGs, but it gradually recovers with increasing temperatures.
Figures 1(e) and 1(f) present measurements of tan δ against continuous strain amplitudes for two different MGs at constant temperatures. One can see that the NDEs are weak for the strain amplitudes smaller than 0.01% but become pronounced for the strain amplitudes larger than 0.1% and a smooth crossover from weak to pronounce around strain amplitude 0.05%. Such a crossover shifts slightly to lower strain amplitudes with increasing temperature.
B. Influence of aging on non-linear damping
As MGs exist far from equilibrium, thermal histories usually have profound influences on properties. Figure 2 presents the results of the effect of aging on the NDEs for Cu46Zr44Al8Y2 and Pd77.5Ni6Si16.5 MGs. Figures 2(a) and 2(b) show the DSC curves for the as-cast and aged (573 K, 12 h) glasses; no crystallization was detected based on x-ray diffractions for both. One can see that there are large exothermic dips before the onset of glass transition for the cast MGs. While for the annealed glasses, such an exothermic dip disappears. These results contrast the different thermodynamics states of the two MGs. Figures 2(c) and 2(d) compare the NDEs for these two samples, and we find that except the difference in the absolute values of tan δ, the NDEs are essentially similar for both cases.
The exothermic dip in the as-cast MGs is usually attributed to the free volume. The presence of NDEs in both glasses indicates that free volume might not be the origin of NDEs in MGs, instead the NDEs could be intrinsic to MGs.
IV. DISCUSSION
To reveal the atomistic origin of NDEs in glasses, we conduct large-scale molecular dynamics calculations to simulate the amorphous structure and its response to mechanical vibrations in probing the damping and NDEs (supplementary material). Figure 3(a) shows a typical stress-strain curve of Cu65Zr35 model MG; it indicates that while the global yield strain is about 5%, the linear elastic strain limit is about 2.5%. We thus investigate the mechanical damping mainly in the linear elastic strain regime as shown by the discrete circles in Fig. 3(a) denoting for the strain amplitudes used in the MD simulations of dynamical mechanical spectroscopy (MD-DMS32). Figure 3(b) displays the measured tan δ by MD-DMS for different strain amplitudes. One can see that the simulations capture the prominent features of NDEs that are consistent qualitatively well with the experimental findings (Fig. 1).
We conducted a structural analysis based on the atomic displacements for every period of deformation during the MD-DMS. Figure 3(c) shows the distribution p(u) of non-affine atomic displacements u for different strain amplitudes at T = 20 K. Overall, p(u) consists of a pronounced peak at small u that indicates the most probable atomic displacements and exponential-decay tail33 at large u due to fast-moving atoms. Appreciable difference develops at the tail of p(u) for different strain amplitudes: the higher the strain amplitudes, the more prominent the tails. This trend is consistent with the NDEs in Fig. 3(b), and it implies that only small fractions of large-distance atomic motions contribute to the NDEs. Figures 3(d) and 3(e) validate the similar observations for T = 300 and T = 600 K, respectively. Moreover, the difference between p(u) for different strain amplitudes becomes smaller as temperature increases; this is also consistent with the NDEs both in experiments (Fig. 1) and MD simulations [Fig. 3(b)].
Figures 4(a) and 4(b) show typical spatial distributions of the atoms contributing to NDEs for T = 20 and 600 K, respectively. They are the atoms with u > uc, where uc is defined as when p(u) starts to develop exponential-like behaviors, as shown in Fig. 3(c), for example. We find that the atoms contributing NDEs are heterogeneously distributed and a lot of them aggregate to form clusters with different sizes, suggesting their cooperative motions and dynamical heterogeneity. Statistical analysis of the number of atoms in clusters as shown in Fig. 4(c) indicates that they follow power-law like distributions with an exponent about 1.9 ± 0.3. Such a power law distribution, indicating the lack of a characteristic length scale (scale-free), bears the hallmark of catastrophe and avalanche behaviors as found in many different systems.34 Possibly, a critical point is reached until one super-cluster spans over the whole system (i.e., percolation).
To understand the implications of the NDEs, the free- (or potential-) energy landscape picture can be invoked. Supercooled liquids and glasses contain meta-basins characterizing for α relaxation and sub-basins for β relaxations in configurational space. For glassy states at low enough temperatures, recent mean field theory predicts that the sub-basins (β relaxations) break up in a hierarchy of marginally stable sub-basins.9 These marginally stable sub-basins can be crossed either by thermal fluctuations or by external perturbations. The pronounced NDEs at low temperatures indicate that the sub-basins must involve a wide range of energy barriers: the more the strain energy inputs, the more the barriers (with higher energy) are crossed and consequently the larger the magnitudes of mechanical damping. On the other hand, the gradual disappearance of NDEs with increasing temperature implies that the free energy landscapes become smoother at higher barriers. Therefore, the general trend of the NDEs agrees with the theoretical predictions on the emergence of hierarchical (fractal) free-energy landscape at low temperatures.
Furthermore, we note that the main features of NDEs in MGs are essentially similar to the dislocation damping in crystalline materials.18,19,35 As a representative example, Fig. 1(g) shows the strain amplitude dependent tan δ for a Cu92.4Ni7.6 single crystal35 (contains dislocations but no grain boundaries) at different temperatures; one can see the pronounced NDEs and their variations with temperatures, which are akin to those of MGs.
In crystalline materials, the NDEs were studied decades before.18–20 The underlying mechanism involves dislocations breaking away from pinning centers distributed along their length. In this scenario, higher strain amplitudes cause longer breaking length of the dislocations from more pinning centers (dissipate more energy) and thus higher capacity of damping. Moreover, at elevated temperatures, diffusive motions take place and the pinning centers are not as strong as at low temperatures. Since glasses and crystals are drastically different in structures and they are at the two extremes of orderings, the similarity between them in NDEs underscores some general mechanism yet to be discovered.
Finally, we note that the NDEs could be of particular relevance for the study of mechanical properties of metallic glasses. Currently, MGs are at the frontier of metals’ research because of their outstanding physical, chemical, and mechanical properties that are not achievable in crystalline metals.36–39 However, the lack of appreciable mechanical ductility is a serious bottleneck for their widespread applications.37,40,41 To understand the underlying mechanism of plastic deformation of MGs is of paramount importance for both fundamental inquiries and technological advances.42–44 The NDEs might provide an effective tool to explore the deformation mechanisms in glassy materials.
V. CONCLUSION
In summary, we discover that the magnitudes of mechanical damping in metallic glasses are sensitive to the probing strains even they are far below the yielding strain. This anomalous nonlinear damping, supported by atomistic simulations, reveals that standard linear elastic is not held at low temperatures but always interweaved with avalanche-like structural instability. These findings also substantiate the theoretical predications about the potential energy landscapes established at the limit of infinite dimensions.
SUPPLEMENTARY MATERIAL
See supplementary material for additional DMA results.
ACKNOWLEDGMENTS
We thank Professor Bao-An Sun and Professor Yu-Liang Jin for helpful discussions. The computational work was carried out at the TianHe-1(A) of the National Supercomputer Center in Tianjin, China, the TianHe-2 of the National Supercomputer Center in Guangzhou, China, and the Gesellschaft fur wissenschaftliche Datenverarbeitung, Göttingen (GWDG), Germany. H.-B.Y. acknowledges the support from the National Science Foundation of China (No. NSFC 51601064) and the National Thousand Young Talents Program of China. K.S. acknowledges the support from the German Science Foundation within the FOR 1394, P1.