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Metallic glasses are known for their outstanding mechanical strength. However, the microscopic mechanism of failure in metallic glasses is not well-understood. In this article we discuss elastic, anelastic and plastic behaviors of metallic glasses from the atomistic point of view, based upon recent results by simulations and experiments. Strong structural disorder affects all properties of metallic glasses, but the effects are more profound and intricate for the mechanical properties. In particular we suggest that mechanical failure is an intrinsic behavior of metallic glasses, a consequence of stress-induced glass transition, unlike crystalline solids which fail through the motion of extrinsic lattice defects such as dislocations.

Metallic glasses show high mechanical strength with the yield strain as high as 2%, comparable to those of the strongest crystalline materials [

In this article we discuss the nature and mechanisms of elastic, anelastic and plastic deformation of bulk metallic glasses mainly from the atomistic point of view, covering simulation as well as diffraction experiments, but excluding macroscopic tensile or compression mechanical testing. The subjects treated here are not new problems. For elastic behavior the effect of structural disorder was first discussed in the seminal work by Weaire,

The elasticity theory used in mechanical or civil engineering is the elasticity theory of a continuum body, developed before the existence of an atom was confirmed. For instance elastic deformation is defined by

Therefore we do not expect affine deformation in a glass at the atomic level, even though a metallic glass deforms just as a crystalline solid, following the Hook’s law at the macroscopic level.

This point was recognized early in the simulation of deformation in metallic glasses by Weaire _{i}_{i}_{i}

Stress-strain curve of glassy iron by simulation for uniaxial tension. Compared to the curve expected for affine deformation the apparent shear modulus is significantly lower.

A part of this softening originates from spatial variation in the elastic moduli. It is known that if the local shear elastic constant,

Indeed the atomic-level elastic moduli have a wide distribution. The atomic-level stresses and elastic moduli are defined as the local response of energy to affine deformation [

It is easy to do so for a pair-wise potential _{ij}^{αβ}, where α and β are Cartesian coordinates. The energy response defines the atomic level stress, σ_{i}^{αβ}, and the atomic level elastic modulus, _{i}^{αβγδ};
_{i}^{αβ} is applied the local strain ε_{i}^{γδ} cannot be given simply by σ_{i}^{αβ}/_{i}^{αβγδ}, because atoms are connected to each other and each atom cannot be displaced independently. In continuum mechanics this interdependence is expressed as the elastic compatibility condition. For this reason calculating the local strain in an inhomogeneous body is a very difficult theoretical problem. Analytically it is difficult to go beyond the variational calculation as was done first by Hashin and Shtrikman. Formally the Green’s function method by Kröner [

On the other hand Suzuki

In order to evaluate the relative contributions from these two effects, elastic heterogeneity and local plasticity, we computed the apparent shear modulus of a model amorphous iron with the modified Johnson (mJ) potential [_{s}_{red}_{red}_{1}_{C}_{C}_{s}_{1}_{red}_{1}_{1}^{−4}) at 100 K. The shear modulus calculated for affine deformation (Born modulus) is 69 GPa. The initial value of the macroscopic shear modulus, 57 GPa, is reduced from the Born modulus due only to the softening by the inhomogeneous local shear modulus, amounting to 18% softening. When ε_{s}_{1}

Apparent shear modulus, _{affine}

Then, why some authors [^{−3}). On the other hand the LJ glass shows almost continuous softening with a very small critical strain (ε_{red}^{−5} << ε_{1}_{red}

The Lennard-Jones potential compared with the Modified Johnson potential for iron and the Dzugutov potential.

It should be noted that in metals the interatomic potentials are dominated by the Friedel oscillation [

As shown above metallic glasses are inherently inhomogeneous at the atomic level when it comes to elastic deformation. The atomic-level elastic moduli have wide distributions reflecting significant distribution in local atomic environment. Surprisingly details of the interatomic potential influence the nature of deformation. The potentials which distinguish the first neighbors from the second neighbors, such as the mJ and Dz potentials, result in the intrinsic anelasticity even at small strains, whereas this effect is not clearly seen for the LJ potential.

Elastic strains in crystalline solids can be readily observed through the shifts in the position of the Bragg peaks. In glasses as well strain can be related to the shifts in the structure function, _{0} is the atomic density. Paulsen

There is, however, a minor problem before we discuss the implications of these results. In all of these measurements, with the exception of [_{ℓ}^{m}_{ℓ}^{m}_{ℓ}^{m}_{ℓ}_{0}(_{0}(

Spherical Bessel function, _{ℓ}

For affine deformation it can be readily shown that

Anisotropic pair-density function (PDF) of glassy Zr_{52.5}Cu_{17.9}Ni_{14.6}Al_{10}Ti_{5} under compressive stress of 1.2 GPa (red solid line) compared to the derivative of the isotropic PDF (dashed line) [

A majority of researchers interpret these results in terms of distance-dependent strain [_{zz,aff}

Reference [_{c}^{−1}, and thus the core of the clusters are not deformed. The basis of their argument is their observation that the strain-induced anisotropy in _{c}.^{−1} does not exist, in clear disagreement with the assertion in [

_{52.5}Cu_{17.9}Ni_{14.6}Al_{10}Ti_{5} glass under tension at 1.2 GPa. Note that they do not decay even beyond 6 Å^{−1}, and

On the other hand the difference between the observed _{zz,anel}_{zz,aff}_{app}_{zz,anel}_{zz}_{,aff}/ε_{app}

_{52.5}Cu_{17.9}Ni_{14.6}Al_{10}Ti_{5} after creep deformation at 300 °C for 30 min. under the stress of 1.2 GPa, compared to

When a solid is heated to a temperature below the glass transition temperature and is subjected to a stress below the elastic limit, slow deformation (creep) is induced and the sample changes its shape. If the sample is later annealed without a stress the sample recovers a part of the deformation. So creep deformation is composed of recoverable anelastic deformation as well as unrecoverable plastic flow. Such a behavior is observed in many crystalline as well as glassy materials, including metallic glasses [

Indeed the structure after creep has the anisotropy mainly with the ℓ = 2 symmetry [

Formation of bond-orientational anisotropy under stress [

Anisotropic PDF,

Incidentally, microscopically the glass never recovers the same atomic arrangement after anelastic recovery. The glassy state is characterized by extremely high degree of structural degeneracy. Once the atomic structure is deformed it never comes back to the exactly same atomic connectivity network when the stress is removed. Thus strictly speaking there is no true anelasticity in glasses. This situation is the same as the case of apparently elastic deformation discussed above. In that sense every deformation in metallic glasses has a plastic component. However, when the plastically deformed region is localized and surrounded by the matrix which is only elastically deformed, the matrix applies back-stress to recover the original state. When the sample is heated without a stress the original state may be recovered. Thus operationally we can say that the system exhibits anelasticity.

At room temperature metallic glasses fail by forming shear bands, where plastic strains are strongly localized. Failure behavior is controlled by nucleation of shear bands and interaction among multiple shear bands [_{g}_{∞}_{M}

(_{80}Si_{20} [

The initial report that the transition from homogeneous to inhomogeneous flow depends on the sample size [

On the other hand the results by the nano-indentation measurements appear to be real, and relevant to the discussion on the mechanism of deformation. It was found that in the nano-indentation measurement the pop-in stress is much higher than the yield stress for macroscopic samples (

(_{52.5}Cu_{17.9}Ni_{14.6}Al_{10}Ti_{5} as a function of indenter radius [

The same argument was advanced by Tian _{49}Zr_{51}) with a dog-bone shape prepared by FIB, they found that the elastic limit was about 4%, twice as much as that for bulk metallic glass (2%). They argue that the surface imperfections, such as oxide inclusions, initiate the shear bands, and thus the mechanical strength is controlled by the flow stress. In sub-micron samples, however, imperfections can be avoided, resulting in a higher elastic limit. This explains why metallic glasses all show more or less the same strength, 2% in strain [

An equally important role is played by the structural parameter ϕ. In crystalline materials deformation leads to multiplication of dislocations which accelerates deformation, until dislocations start to become entangled resulting in work-hardening. Thus we may use the dislocation density as ϕ. In glasses, however, the nature of the structural parameter ϕ is not obvious, as discussed in detail later. Here, it is sufficient to note that the effective temperature, _{eff}_{f}_{∞} is the viscosity extrapolated to _{eff}_{eff}_{a}_{eff}_{eff}_{a}_{eff}_{f}_{eff}_{α}.

_{eff}_{Y}_{f}_{Y}_{eff}_{eff}_{eff}_{Y}_{f}

Simulated stress-strain curve of glassy Zr_{50}Cu_{40}Al_{10} at a low temperature, showing stress over-shoot for yielding before attaining the steady-state flow [

However, experimentally observed stress-strain curves [

Because a glass is obtained usually by cooling the liquid, the structure of the glass, which can be described by the fictive or effective temperature, _{eff}_{eff}_{eff}_{g}_{g}_{eff}_{g}_{g}_{g}_{eff}

Temperature dependence of the volume of a liquid through the glass transition at _{g}_{eff}

The volume is reduced by structural relaxation, and is usually explained by removal of excess free-volume which was quenched-in during fast cooling. However, the amount of reduction in volume is small, just a fraction of a percent [_{eff}_{60}Ni_{15}P_{25} the total volume reduction when annealed at 15K below _{g}

Most metallic glasses show no or limited ductility [

Ductility is a complex property, and relates to the ability of the material to enumerate shear bands and absorb the energy of deformation, which goes back to the constitutive law, Equation (16). Because the parameters in Equation (16) depend on temperature, ductility also depends on the measurement temperature. Indeed as in crystalline materials the presence of the ductile-to-brittle transition (dbt) is reported [_{g}

The first realistic computer simulation of plastic deformation was done by Kobayashi _{57}Zr_{43} composed of 1533 atoms [

Simulation of deformation in Cu_{64}Zr_{36} glass (288,000 atom model) at 300 K showing the formation of a shear band [

It is well-known that the applied stress can accelerate local atomic rearrangement leading to deformation. For instance in the Eyring’s rate theory [

Now for the steady-state flow it is informative to plot the equal-viscosity lines for the _{0}(η) is the temperature where viscosity is η at σ = 0, and σ_{ 0} is the stress where viscosity is η at

(_{50}Cu_{40}Al_{10}. (

More importantly, this result directly connects the glass transition and mechanical failure, and suggests that mechanical failure is caused by the stress-induced glass transition. Therefore it is an intrinsic property of a glass, not involving defects. It is well-known that crystals deform through the motion of defects, such as dislocations. Naturally there have been numerous attempts to define defects which are responsible for the mechanical failure of metallic glasses, such as the free-volume [

Actually if defects are controlling failure, the fracture strength should vary depending on the preparation of the sample. That is exactly what is observed for crystalline materials where the yield stress can vary by more than two orders of magnitude, and various metallurgical processes were invented to strengthen the materials. In comparison the fracture strengths of metallic glasses are remarkably uniform, always around 2% in strain [

However, the defects are important in initiating failure through yielding as discussed above. Just before yielding, when the applied stress is already higher than the flow stress (see

The constitutive law of flow, Equation (16), is a strong function of temperature as shown in _{g}_{g}

(_{50}Cu_{40}Al_{10} at various temperatures [_{g}

Interestingly the η-σ data can be collapsed into a universal curves by scaling;
_{g}

The scaling behavior of viscosity η, and flow stress σ, simulated for Zr_{50}Cu_{40}Al_{10} [_{g}

Now in the glassy state s does not depend strongly on _{∞}_{M}_{∞}_{M}_{M}_{α}. Of interest here is the competition between the strain rate _{M}_{α}. At low temperatures τ_{α} is longer than τ_{M}_{α} is comparable to τ_{M}

The flow stress is a weak function of the strain rate (

A number of theories have been proposed to explain the deformation behavior of metallic glasses. For instance the most widely used theory is the free-volume theory of Spaepen [

In the free-volume theory a strong case is made for expressing the effective temperature, _{eff}_{f}_{liq}_{glass}_{g}_{g}_{f}_{f}_{g}^{−2}. So the ratio, ν_{f}_{g}_{f}_{f}

In the STZ theory _{eff}_{∞}_{0} is the atomic (molecular) volume, χ = _{B}T_{eff}/e_{Z}_{Z}_{f}

In general the constitutive law may be written as
_{0} is the attempt frequency, γ_{0} is the strain unit which is of the order of unity, _{±}_{±} is the density of defects at forward or backward position. In the free-volume theory [^{m}

Now a non-linear behavior is caused by the feedback from strain to _{g}_{0} was found to be the yield stress at low temperatures. In either theory the constitutive law and the feedback equation have to be solved self-consistently to produce the answer.

These theories have been successful in reproducing the deformation behavior. _{α}., to change ϕ, in this case the STZ density _{α} is slower than _{α} is faster than

Theoretical stress-strain curves by the shear-transformation-zone (STZ) theory [

The mode-coupling theory (MCT) was formulated in order to explain the rapid rise in the relaxation time as the glass transition is approached from high temperatures [_{σ} is a coupling constant [^{2}(_{q}

In the free-volume theory the value of _{f}

Crystalline solids mechanically fail through motion of lattice defects, such as dislocations at low temperatures and vacancies and grain boundaries at high temperatures. Naturally various defect models were proposed as the deformation mechanism in metallic glasses. Free-volume [

A glass, on the other hand, is just an extremely slow liquid. The glass transition is not a phase transition, but merely a crossover point where the relaxation time of the system exceeds the experimental time-scale. Therefore flow can be induced simply by reducing the relaxation time by increasing the effective temperature. A liquid state is characterized by distributed local topology of atomic connectivity, such as the local coordination number, as discussed below. The distribution can be changed continuously, which corresponds to varying the effective temperature continuously.

Because the local structure varies from an atom to another the local response to external stress is heterogeneous; some atoms are more easily moved than others. In that sense it is reasonable to call more mobile atoms as “defects”. However, these “defects” are not separated from “non-defects” in a clearly distinguishable way. We have to define some cut-off in the continuous distribution of some structural parameter to define them. For instance in the free-volume model the minimum size of the space where an atom can move in, ν*, is the cut-off to define the free-volume [_{eff}_{eff}

Volume is one of the physical properties easier to measure, and is the most intuitive structural parameter to express the effective temperature. It has long been recognized that volume increases more rapidly with temperature in liquids compared to glasses (

Spaepen [_{0} ratio is ~0.1, smaller by an order of magnitude compared to the hard-sphere systems [

For instance, the effect of the externally applied pressure on the strength is very small, so the strengths in compression and tensile tests are practically the same [

The small value of the ν*/ν_{0} ratio is consistent with the observation that a vacancy is unstable in metallic glasses, and breaks up into smaller volumes [_{0} ratio. Now when the ν*/ν_{0} ratio is small the original idea that an atom can move into this space becomes unrealistic, because such an action will require a large amount of energy. So in the “distributed free-volume” picture the volume aspect is not important any more. It is just a region which is easier to shear. So the concept evolved into the idea of shear-transformation-zone, STZ [

Another problem is that the fraction of the free-volume, ν_{f}_{0}, is small, and amounts only to a few percent. Free-volume is reminiscent of a vacancy in the lattice. In crystalline solids, the density of lattice defects is very low, but because they are topologically very different from the matrix and highly mobile they totally control the mechanical deformation. In glasses, however, defects are different from the matrix only quantitatively, but not qualitatively. Thus it is unlikely that a small number of defects can carry the entire load of deformation. The density of “defects” estimated by the topological fluctuation theory [_{def}_{0} = 0.1 [_{f}_{def}_{0} ratio for metallic liquids [

In spite of various problems in practice the free-volume theory works well, because, as we pointed out above, the theoretical structure of the free-volume theory is similar to other theories, such as the STZ theory, even when different microscopic mechanisms are assumed. When parameters are chosen appropriately it can describe the experimental results quite well. In converse, however, the success of the free-volume theory in reproducing experimental results does not justify the physical reality of the model.

For the STZ concept the “volume” character is less important, but weakness of resistance to shear, or low local shear modulus, is the main characteristics. Johnson and Samwer [

The STZ theory by Langer and Falk [_{g}_{M}

(_{∞}_{∞}

As mentioned above the mode-coupling theory can be extended to account for the flow behavior of a liquid under shear stress. Indeed the results shown in

It is very difficult to characterize the structure of liquids and glasses because of the absence of symmetry and their strong disorder. The most common method is to consider the topology of atomic connectivity, by defining the nearest neighbors [

Typical changes in the interatomic distance as a bond between

The nearest neighbors are usually defined by the Voronoi construction [

In covalent glasses, such as amorphous silicon or silicate, chemistry places strong restriction on local topology. In such a case it is natural to define defects as deviations from dominant topology, for instance from the ring statistics [_{C}_{A}_{B}_{A}_{B}

Nevertheless, local topology of atomic connectivity was found to be the key in defining the local deformation event. It was found that during the shear flow right after a bond oriented in the compressive direction (~45° away from the direction of the shear flow) is cut, a new bond is formed in the perpendicular direction (inset of ^{−13} s, much shorter than the Maxwell relaxation time, and is comparable to the time for a sound wave to travel from one atom to the nearest neighbor (_{LT}

(_{0} = 0.761 × 10^{−13} s [_{∞}

Such local configurational excitations are the elementary unit of STZ. However, LCE distorts the glass locally, resulting in strong stress field. This stress field triggers another LCE in the immediate neighborhood, and produces cascade of LCE actions as shown in _{LT}

Location where atomic bonds are cut during the shear flow for _{0}^{−1} at 300 K, within two intervals of 250 fs (= 3.29 τ_{0}, τ_{0} = 0.761 × 10^{−13} s). The progress in time is shown by color. The cut bonds are clustered suggesting cascade chain reactions. Here the bond lifetime is 81 τ_{0}. The size of the box is 59.34 × 10 × 59.34 Å^{3}. This figure was generated from the model used in [

Because a liquid under shear flow can sustain a shear stress, its structure cannot be isotropic, but should resemble that of a solid under stress. This is indeed the case, and as shown in _{M}_{T} = 10.56 Å at 300 K, 10.85 Å at 500 K, 11.06 Å at 700 K, and 11.12 Å at 900 K. But ζ, depends significantly on the shear rate at high rates, and ζ extrapolates to zero at a critical shear rate,

(_{zx}

At a first look the mechanical properties of metallic glasses do not appear to reflect the disordered nature of the atomic structure, because they seem to be superficially similar to those of crystalline metals. Upon a closer look, however, we recognize that the structure has profound and intricate effects on the mechanical properties. This is because at the atomic level the local response of a metallic glass to the external stress field is strongly heterogeneous. Even in the case of elastic response to a uniform stress the local compliance is not uniform. And yet each local portion of the system cannot independently respond to the stress, because they are connected to each other, and thus have to satisfy the elastic compatibility condition. Similarly in plastic deformation local structural changes create long-range stress fields and affect each other, as a consequence of satisfying the elastic compatibility condition. In addition local structural changes alter the effective, or fictive, temperature. Here we reviewed recent advances in our understanding of such unique features of elastic, anelastic and plastic deformation of metallic glasses, focusing on the atomic-level phenomena. A large amount of researches on the mesoscopic behavior, such as the effect of shear bands, are not included in this review, even though they are important in understanding the actual behavior of a macroscopic sample.

Many issues remain poorly understood, and several competing theories are used by different researchers. Different theories define the effective temperature in its own way. In the free-volume theory it is expressed as free-volume (Equation (24)), whereas in the STZ theory it represents the STZ density (Equation (26)). In the MCT the structure factor, S(Q), plays that role. They are nearly linearly related to each other, thus they are practically equivalent. In each case it is a phenomenological parameter in the equation, and the connection to the microscopic details is not obvious. This point has to be clarified by further simulations and experiments.

One issue we particularly emphasized in this article is the role and nature of the defects. Because crystalline solids fail due to the motion of lattice defects much effort has been directed to define the defects in metallic glasses. In glasses, however, defects are not topologically distinct. They are those at the edges of wide distribution of the local states, with the cut-off in some parameter which is chosen by taking the properties into account. They are a part of the structure which is in equilibrium at the effective temperature. Therefore the critical parameter is the effective temperature, and the physical reality of the defects is of secondary importance to the success of the theory in reproducing the data. That is why various theories, such as the free-volume theory and the STZ theory, are equally successful, even though they are based on quite different microscopic mechanisms. Conversely, successful reproduction of the data for macroscopic experiments does not guarantee the correctness of the microscopic picture in the theory. After all microscopic details can be determined only either by microscopic experiments or by atomic-level simulation. Because of the complex nature of the atomic structure a truly microscopic theory of deformation in metallic glasses is yet to be developed.

On the other hand macroscopic defects, such as inclusions, play an important role in initiating the local plastic flow and formation of shear bands. That is why the flow stress, not the local yield stress, determines the strength of a bulk metallic glass. Ironically because the flow stress is an intrinsic property, not affected by extrinsic defects, mechanical failure of a metallic glass is an intrinsic behavior, a consequence of the stress-induced glass transition.

The authors are grateful for useful discussions with J. S. Langer, M. L. Falk, J.-L. Barrat, S. Yip, E. George, H. Bei, J. R. Morris, E. Ma, M. W. Chen and K. Kelton. This research was supported by the U.S. Department of Energy, Basic Energy Sciences, Materials Sciences and Engineering Division. X-ray diffraction experiments were carried out at the 1-ID beamline of the APS which is funded by the U.S. Department of Energy (DOE), Office of Science, under Contract No. DE-AC02-06CH11357.

The authors declare no conflict of interest.

_{41.2}Ti

_{13.8}Cu

_{12.5}Ni

_{10.0}Be

_{22.5}

_{64.13}Cu

_{15.75}Ni

_{10.12}Al

_{10}bulk metallic glass investigated by

_{32}Ni

_{36}Cr

_{14}P

_{12}B

_{6}

_{40}Ni

_{40}P

_{14}B

_{6}studied by energy-dispersive X-ray diffraction

_{60}Ni

_{15}P

_{25}bulk metallic glass

_{57}Zr

_{43}

_{81}Zr

_{19}

_{g}

^{2/3}temperature dependence