Difficulties in description and interpretation of peculiarities of plastic deformation mechanisms of metallic glasses (MG) arise because of lack of sufficient knowledge of their structure and structural defects. However, at least in one point, all experts have reached a consensus: at low temperatures, diffusional mobility of single atoms is depressed and cannot play a significant role. Therefore, just sliding is responsible for the low-temperature plastic deformation of MG. This mode of deformation is known as heterogeneous plastic deformation controlled by shear bands initiation and propagation. A typical width of a shear band in MG is about 20 nm, that is, it is much larger than the mean interatomic distance. Meanwhile, shear band initiation is controlled by microscopic shear transformations depending on the local critical stresses that are needed for inelastic rearrangements of atoms.

Considering a crystalline metal as a reference system, we have to note that, due to structure translational invariance, only one value characterizes the local strength. It is the local critical stress needed for the inelastic shearing of one atom. In a perfect crystalline lattice, this quantity depends on the shear modulus *μ*_{cr}. Therefore, the yield stress of a perfect crystal is proportional to *μ*_{cr} and the proportionality coefficient depends on the lattice geometry and interatomic potential relief under shear. The yield stress of a perfect crystal

was treated by Frenkel [1], who approximated the periodic potential landscape of atoms by a sinusoidal function. He found that critical stress, which is known as theoretical shear strength, is

*σ*_{th}=

*μ*_{cr}/2

*π*. At

*σ*>

*σ*_{th}, the cohesion between atomic layers breaks down and a sliding layer is formed. Then, improved estimations of

*σ*_{th} were obtained (see, e.g., [2]). It appears that the reasonable estimate for

*σ*_{th} is given by

It turned out that experimentally measured values of

are close to the theoretical strength only for the defect-free crystals (e.g., whiskers). The yield stress of real crystalline solid containing dislocations is determined by the stress threshold of dislocation mobility. The local critical stress of atoms within the dislocation core, determining the threshold of dislocation mobility,

*σ*_{P}, occurs to be much less than

*σ*_{th}. As was estimated by Peierls [3], for metals, it reads

This quantity determines the threshold of the shear bands initiation in real crystals.

MG structure is random and has no translational invariance. Therefore, it does not contain gliding dislocations, as crystal does, and we lose a simple guideline for description and interpretation of the local and macroscopic shear plastic deformations. The randomness of the potential relief has to be properly accounted and dislocationless mechanisms of the shear plastic deformation leading to shear band formation and propagation have to be considered.

The low-temperature yield stress of an MG has been found to be also proportional to the macroscopic shear modulus [4], but the proportionality factor differs from those in (1) and (2)

Whereas, in a crystal, the macroscopic value of *μ* is equal to its microscopic value [5], the shear modulus in MG is a random quantity. Its local value *μ*_{i} depends on the atomic configuration at site *i*. Thus, *μ*_{MG} is a mean value. The macroscopic shear modulus of an amorphous alloy is typically down to 30% lower than the shear modulus of crystal of the same composition [4].

MGs possess noncrystalline disordered structure that can be characterized by a random potential relief with randomly distributed atoms in the potential minima [6]. To find the yield stress of MG, one has to consider the problem of strength and thermally activated inelastic rearrangements of atoms within a shear layer where each atomic site is characterized by a random local critical stress for inelastic rearrangements.

Different models of the inelastic shear strain in MG were developed. The first model of this type belongs to Argon [7–9]. According to Argon, at low temperatures (<0.8 *T*_{g}, where *T*_{g} is the glass transition temperature), MGs are deforming due to inelastic shear rearrangements of atomic groups composed of about 10 atoms. Then, these carriers of plastic deformation were termed shear transformation zones (STZs) [10]. Having regarded STZs as carriers of the plastic deformation, a set of equations of motion (roughly analogous to the Navier-Stokes equations for fluids) were deduced [10–13]. STZ velocity, density, and orientation are dynamic variables. During deformation, STZs appear and annihilate persistently. It has to be noted that, in the Argon model, MG contains STZs, but its other structural properties are not specified. They are implicitly accounted in the model parameters. To connect STZs with shear band formation, it was assumed that the free volume [14–16] is properly redistributed and created in the deformed MG.

Then, these ideas were utilized in similar approaches. For example, the cooperative shear model (CSM) [17] postulated that STZ involves ∼10^{2} atoms and that the shear transformation is a cooperative process similar to the α-relaxation in liquids. In computer simulations [18], the size of STZs was estimated to be about 1.5 nm. It should be noted that CSM is focused on the initial stage of anelastic and inelastic shear transformations.

Important features of models based on the STZ concept are as follows. First, sliding is the main mode of the inhomogeneous inelastic and viscous-plastic deformation at low temperatures. Second, up to hundreds of atoms can be involved in STZ, and the linear size of an STZ is estimated to be of several nanometers or less. It means that peculiarities of MG structure on the scale of ∼10 nm play a key role in the shear transformation and initiation of shear bands. Recent experimental investigations on tensile strength [19, 20] revealed a strong size effect when the specimen size is <100 nm. These results indirectly point out the structure heterogeneities of about 10 nm in size.

The polycluster model of MG structure, developed in [21–23], is based on the idea that the majority of atoms possess “perfect” noncrystalline local order. Groups of atoms with different locally preferred configurations (subclusters) are associated in noncrystalline clusters with narrow and stable intercluster boundaries. Validity of this model is confirmed by many direct and indirect experimental investigations. Direct examination of the MG structure by means of the high-resolution field emission microscopy shows that both rapidly quenched amorphous ribbons and bulk MGs, obtained at relatively small cooling rate, possess a fine polycluster structure with characteristic sizes of clusters and subclusters about 10 and from 1 to 3 nm, respectively [24, 25]. The boundary width occurs to be comparable with the atomic size. The mean binding energy of atoms within the boundary layer is lower than in the cluster body by the 10th of eV. Because the boundary density was found to be extremely high (∼10^{-5} –10^{-6} cm^{-1}), they have to play a dominant role in the plastic deformation of MG.

The mean value of the shear modulus of the cluster body is comparable with the shear modulus of a crystal and the strength of the cluster body is expected to be nearly equal to the theoretical strength [in this case, *μ*_{cr} in Equation (1) has to be replaced by the average on the cluster body value of the shear modulus]. The local value of critical stress required for inelastic relocation of an atom within the boundary layer is a random quantity, being less than that in the cluster body. Therefore, the intercluster boundaries are natural STZs. Shear transformations under applied stress appear first of all due to sliding within the boundary layers. Refraining from discussion of existence and stability of boundary dislocations in MG (partially this issue is considered in [21–23]) we will be focused here on the problem of dislocationless sliding localized in a layer with a random distribution of local critical stresses. First, this problem was considered in [21–23]. Approximate solutions of equations deduced there were obtained under the assumption that the distribution function of the local critical stresses is a spatially homogeneous piecewise constant function. The obtained approximate solutions gave qualitative and, in some cases, reasonable quantitative description of the sliding process and allowed to predict the mechanism of the shear band formation on a qualitative level. Then, this approach was applied for the description of MG hardening during partial crystallization [26].

In this paper, we consider the problem of the sliding in a localized layer and strength of MG possessing the polycluster structure. The model of [21–23] is developed further. The theoretical treatment is based on the exactly solvable model of the homogeneous sliding in layer with randomly distributed local strengths. This model is used for description of inelastic processes in MGs.

The paper is organized as follows. We start with the formulation of homogeneous sliding model in a disordered structure. Then, kinetic equations of sliding in a planar layer with arbitrary distribution of local critical shear stresses are formulated. Rigorous solutions of the sliding master equations are obtained. It is shown that these solutions determine the macroscopic strength of the sliding layer, sliding velocity, and effective activation volume as functionals of the distribution function of the local critical stresses. Consideration of the heterogeneous plastic deformation of MGs is presented in Section 3. The last section contains a brief discussion and conclusions.

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