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Journal of the Mechanical Behavior of Materials

Editor-in-Chief: Aifantis, Katerina

Managing Editor: Skoryna, Juliusz


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Volume 22, Issue 3-4

Issues

Theoretical strength and homogeneous sliding in metallic glass: exactly solvable model

Nikolai Lazarev / Alexander Bakai
Published Online: 2013-11-06 | DOI: https://doi.org/10.1515/jmbm-2013-0017

Abstract

At low temperature, T→0, the yield stress of a perfect crystal is equal to its so-called theoretical strength. The yield stress of nonperfect crystals is controlled by the stress threshold of dislocation mobility. A noncrystalline solid has neither an ideal structure nor gliding dislocations. Its yield stress, that is, the stress at which the macroscopic inelastic deformation starts, depends on distribution of local, attributed to each atomic site, critical stresses at which the local inelastic deformation occurs. We describe exactly solvable model of planar layer strength and sliding with an arbitrary homogeneous distribution of local critical stresses. The rate of the thermally activated sliding is closely related to parameters of the low-temperature strength. The sliding activation volume scales with the applied external stress as

where β<1. The proposed model accounts for mechanisms and the yield stress of the low-temperature deformation of polycluster metallic glasses, because intercluster boundaries of a polycluster metallic glass are natural sliding layers of the described type.

Keywords: activation volume; creep; polycluster metallic glasses; yield strength

1 Introduction

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 μcr. Therefore, the yield stress of a perfect crystal is proportional to μcr 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 σth=μcr/2π. At σ>σth, the cohesion between atomic layers breaks down and a sliding layer is formed. Then, improved estimations of σth were obtained (see, e.g., [2]). It appears that the reasonable estimate for σth 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, σP, occurs to be much less than σth. 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 μi depends on the atomic configuration at site i. Thus, μMG 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 Tg, where Tg 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 ∼102 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, μcr 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.

2 Model of homogeneous sliding in disordered atomic layer

In a nonmetallic glass, the elementary inelastic rearrangements of atomic configurations can be described as (a) splitting-recombination and (b) translation of the broken bonds, Figure 1A and B, correspondently. Rearrangements of the potential relief related to these inelastic deformations are shown in Figure 1 as well. This picture is also applicable to MG, although, in this case, the potential relief is much shallower, because metallic bonds are not as strong as covalent ones.

Elementary rearrangements under shear stress. (A) Reversible splitting of a coincident site into two noncoincident sites (left to right) and recombination of two opposite noncoincident sites with subsequent formation of coincident site. (B) Transposition of coincident and noncoincident sites. Below are shown corresponding rearrangements of the potential relief. • regular sites; , coincident sites; , noncoincident sites.
Figure 1

Elementary rearrangements under shear stress.

(A) Reversible splitting of a coincident site into two noncoincident sites (left to right) and recombination of two opposite noncoincident sites with subsequent formation of coincident site. (B) Transposition of coincident and noncoincident sites. Below are shown corresponding rearrangements of the potential relief. • regular sites; , coincident sites; , noncoincident sites.

In a polycluster, the intercluster boundaries are regions of weak cohesion; therefore, here, plastic deformation may occur by the sliding of one cluster over another. Figure 2 shows schematically the structure and the potential relief of the sliding layer consisting of coincident and noncoincident sites. Due to strong alternation of sites of different types, the formation of gliding boundary dislocations is mostly suppressed.

A fragment of sliding layer and a schematic potential relief for sites in this layer.
Figure 2

A fragment of sliding layer and a schematic potential relief for sites in this layer.

2.1 Basic equation

When a planar layer is subjected to the external shear stress σe, and atoms experience only elastic displacements, then external stress is homogeneous in the layer. If some atoms execute inelastic displacements, this causes stress redistribution, that is, stress relaxation at places of inelastic deformation and stress concentration at the elastically deformed areas. Thus, local stresses in sites are heterogeneously distributed and time dependent. We assume that the local stress relaxation occurs by independent single-jump atomic rearrangements resulting in the external stress concentration at nondisplaced sites

where Δ(t) is the fraction of displaced sites at time t.

The sliding velocity within the layer is a macroscopic quantity defined by the average frequency of inelastic displacements under the external stress

where <d> is the average site displacement during the elementary relocation and τsl is the average time for displacement of all sites of the slip layer per interatomic spacing.

The sliding time τsl is controlled by rates of individual inelastic displacements of atoms. Within a simple model of a particle in the two-level potential, see Figure 3 and Appendix 1, the probability of thermally activated jump under external stress σe can be expressed by

Two-level potential for sites rearrangements.
Figure 3

Two-level potential for sites rearrangements.

where v0 is the vibration frequency in the site i, α=va/kBT, va is the activation volume of elementary single jump that is assumed to be of about atomic volume, T is the temperature, kB is the Boltzmann constant, and the parameter σi=2Ed/va is the critical shear stress when the atomic configuration goes from a site to a neighbor site without thermal activation. The physical meaning of the parameter σi is similar to the shear strength of a perfect crystal (1). Unlike a crystalline lattice, an amorphous solid is characterized by a wide spectrum of local critical stresses. Denote the distribution function of σi by g(σi).

The fraction of displaced sites Δ(t) can be expressed in an explicit form via the probability of local inelastic displacements:

Equation (7) is an integral equation with respect to Δ(t). The fraction Δ(t) enters into the integrand P(t, σi) via (4) and (6).

In the case of high external stress when the local shear stress exceeds the critical one, σe/(1-Δ(t))>σi, the site i has become displaced without thermal activation for a time as short as ∼1/v0. Therefore, the integration in (7) is carried out not from zero, but from the lower limit

Combining (7) and (8), we obtain the master equation in the form:

The sliding time τsl can be found from the equation Δ(τsl)=1.

2.2 Low-temperature strength of deformation layer

In the low-temperature limit, T→0, the second term in the right-hand side of Equation (9) is equal to zero. The macroscopic yield stress

was found by solving combined Equation (8) subject to condition

Evidently,

is a functional of the distribution function g(σ). Because of the nonnegativity of g(σ), the functional

is the single-valued function of x. Hence, the function

is also the single-valued function of x. Equations (10) and (11) determine a single parametric relation between σe and Δ0, thereby giving the exact solution of the problem of low-temperature yield stress.

To illustrate the solutions obtained, take as an example the distribution function g(σ) depicted in Figure 4. Two maxima of g(σ) are attributed to atom groups with different surroundings. The left peak is associated with noncoincident sites having critical stresses of the order of

the right peak corresponds to coincident sites with
The variances were taken approximately equal
The parameter p controls contributions of peaks. It was taken as p=0.7 for cluster boundaries. Estimates of the fraction of noncoincident sites are taken from [21, 22]. The characteristic binding energies in coincident sites are less than in the cluster body by about 0.2 eV [24]. The dashed line in Figure 4 serves to illustrate the distribution of critical stresses in cluster body where we put p=0.

Examples of distribution function of critical shear stresses. Solid line corresponds to g(σ) composed of two Gaussian distributions g(σ)=pg1(σ)+(1-p)g2(σ), with  p=0.7. Dashed line shows the same distribution function but with p=1.
Figure 4

Examples of distribution function of critical shear stresses.

Solid line corresponds to g(σ) composed of two Gaussian distributions g(σ)=pg1(σ)+(1-p)g2(σ), with

p=0.7. Dashed line shows the same distribution function but with p=1.

The solution of Equations (10) and (11) is shown in Figure 5 for the case of a trial distribution function g(σ) depicted in Figure 4 with a solid line. At some points σs, the value Δ(σe) changes in discrete steps with increasing σe. In case of everywhere differentiable distribution function g(σ), we can find points σs by solving the equation: e/dΔ0=0. Using (8), (10), and (11), we obtain the equation for instability points in the form:

Dependence of relative part of moved sites on external stress. Arrows show the changes in Δ(σe) when σe increases.
Figure 5

Dependence of relative part of moved sites on external stress. Arrows show the changes in Δ(σe) when σe increases.

The critical stresses σs are directly related to xs by (11). Analysis shows that, for any distribution function g(σ) decreasing faster than 1/σ2 at σ→∞, Equation (12) has at least one solution. The largest root of this equation defines the yield stress

Figure 5 shows that, for the above-mentioned distribution function, the yield stress

is approximately six times less than the maximum available critical stresses in distribution g(σ). At the symmetrical distribution function g(σ) depicted by dashed line in Figure 4, the yield stress is about

Because the variance of the critical stress values δ0 depends essentially on composition and thermal history of MG, it is important to investigate dependence of

on δ0. Taking g(σ) in the form of truncated Gaussian distribution shown in Figure 4 by a dashed line, we have obtained the dependence shown in Figure 6. One can see that
decreases fast when δ0 increases. At the variance δ0>σ0/2, the yield stress is nearly equal to σ0/2, that is, it is nearly four times less than the maximal value of critical stresses σmax=2σ0.

Dependence of the yield stress  on the variance of local critical shear stresses.
Figure 6

Dependence of the yield stress

on the variance of local critical shear stresses.

2.3 Thermally activated sliding

At low stress regime, when σe does not exceed the yield stress

Equation (9) for the fraction of displaced sites is conveniently rewritten in the form

where Z(t)=1-Δ(t). The parameter Z0 is defined by the relation (8).

The sliding time τsl can be found from the equation Z(τsl)=0.

To solve nonlinear integral equation (13), first, we transform it to the parametric form

Differentiating F(t), we directly get dF/dt=ν0exp(ασe/ Z(F)) or in the differential form:

Integration of Equation (15) gives the implicit solution for the function F(t)

With F→∞, Equation (16) determines the sliding time of inelastic shear strain on interatomic spacing:

Here, the functional Z(F) is defined by Equation (14).

Equations (17) and (5) provide the analytical solution of the problem of thermoactivated sliding in the disordered layer.

Figure 7 shows the results of numerical integrations of Equation (17) for the distribution functions g(σ) shown in Figure 4. Solid and dashed lines correspond to two- and one-peak distributions g(σ), respectively. The estimation of glass temperature Tg can be found in [27]. As was expected, the curves for different temperatures meet at some stress point

where the sliding time tends to zero. This point has a simple physical meaning, namely, the threshold of athermal boundary sliding at T→0.

Dependence of the strain rate on external stress at several temperatures: T1=0.8·Tg, T2=Tg, and T3=1.2·Tg.
Figure 7

Dependence of the strain rate on external stress at several temperatures: T1=0.8·Tg, T2=Tg, and T3=1.2·Tg.

The temperature Tg was estimated as Tg=0.012μ0va/kB [27]. The parameter of local activation volume va was taken to be about 4.5va [28]. As was expected, the curves for different temperatures meet at same stress point

where the sliding time tends to zero. This point has a simple physical meaning, namely, the threshold of athermal planar sliding at T→0. The deformation map shown in Figure 7 qualitatively resembles the behavior of a bulk MG [29].

We can calculate the activation volume of sliding according to the definition [30]:

Stress dependence of Vact is shown in Figure 8. It is not a constant. Evidently, Vact at low stresses is much larger than the atomic volume va and scales with external stress as

Dependence of activation volume on the external stress. Solid and dashed lines correspond to two- and one-peak distributions g(σ), respectively. Short dashed line corresponds to more narrow one-peak distribution.
Figure 8

Dependence of activation volume on the external stress.

Solid and dashed lines correspond to two- and one-peak distributions g(σ), respectively. Short dashed line corresponds to more narrow one-peak distribution.

The exponent β depends weakly on temperature and lies in the range 0.5<β<1. Whereas stress is growing, Vact(σe) decreases to the value that reaches the atomic volume when σe approaches the yield stress

Integration of Equation (17) can be performed numerically, but in many cases it can be solved analytically. Making change of variable in Equation (17), F=exp(αx), we obtain

where, according to (14), the function Z(x) is given by

Function ψ(x) has a maximum at the point x0 where (x)/dx=0 and d2ψ(x)/dx2>0. The equation for x0 takes the form

At low temperatures (α>>1), this equation is simplified to

where Z(x0) is defined by (21).

The second derivative of ψ(x) in the point x0 is

Upon integrating (20) subject to (23) and (24), we get

Equation (25) was obtained assuming that the distribution g(σ) is a smoothly varying function in the vicinity of x0. Using Equations (20) and (22), it can be shown that ∂ψ(x0)/∂σe=1/Z(x0). Therefore, it follows from Equation (23) that Z(x0) depends on σe as

This result validates relation (19) for activation volume at low stresses.

3 Boundary sliding, dislocation-like edge of sliding zone, and heterogeneous plastic deformation of MG

Boundary sliding in polycrystals is blocked in the triple joints of crystalline grains. Similarly, boundary sliding in a polycluster is blocked in triple joints of clusters. In a polycrystal, the sliding layer can propagate into the neighboring grain in the form of a dislocation channel. In a polycluster, the boundary sliding layer can also propagate into the neighboring cluster overcoming the triple joint blocking, but the sliding zone structure within the cluster body essentially differs from that of the dislocation channel due to lack of the translation invariance of the amorphous structure. In essence, the edge of the sliding zone in cluster is Volterra dislocation with core of irregular size and structure. The local order within the dislocation sliding layer in MG is violated because it contains many fault matches. Its structure is similar to that of intercluster boundary (Figure 2). Microscopically, the edge of sliding zone within the cluster body is a quasi-one-dimensional structure defect. We call it dislocation-like edge of sliding zone (DLESZ). DLESZ is a carrier of heterogeneous inelastic deformation within the cluster body. When it reaches the cluster boundary, the cluster body becomes divided into two parts. In other words, it cuts the noncrystalline grain into two parts.

Boundary sliding and propagation of DLESZ are elementary processes of the heterogeneous plastic deformation resulting in the shear band initiation.

Initial stages of the plastic deformation of a polycluster are shown schematically in Figure 9. A fragment of the two-dimensional section is depicted in Figure 9A. Boundaries demarcate clusters. At the initial stage (Figure 9B), boundary sliding at

takes place. As a result, the STZ of size ld is formed. It is bounded by DLESZ that is triple junction with neighboring clusters. The shear stress on the DLESZ is [31, 32]

Stages of the plastic deformation of a polycluster. (A) Initial polycluster structure; cluster boundaries are shown. (B) Under external stress, a DLESZ is formed as a result of boundary layer sliding. This STZ is shown as bold line. The loop is blocked at triple joints of clusters. (C) DLESZ climbed into the neighboring clusters. Inset, its microscopic structure. Noncoincident sites are marked by semicircles. (D) Nanocracks along the intercluster boundary are initiated.
Figure 9

Stages of the plastic deformation of a polycluster.

(A) Initial polycluster structure; cluster boundaries are shown. (B) Under external stress, a DLESZ is formed as a result of boundary layer sliding. This STZ is shown as bold line. The loop is blocked at triple joints of clusters. (C) DLESZ climbed into the neighboring clusters. Inset, its microscopic structure. Noncoincident sites are marked by semicircles. (D) Nanocracks along the intercluster boundary are initiated.

With σ increase, two modes of the inelastic deformation occur. First (Figure 9C), DLESZ is propagating into the neighboring cluster body when stress becomes larger than the theoretical strength of the cluster body,

where μcl is the mean value of the shear modulus within the cluster body. As it follows from (26), this mode takes place at

Second (Figure 9D), cracks on the jointed boundaries can be initiated if the Griffits condition

is fulfilled. Here, E is the Young modulus of the cluster; εs and εb are the surface energy and the boundary energy, respectively; and lc is the crack size that is comparable with the cluster size.

Using (26), the criterion (28) can be presented as follows:

Relations (27) and (28) determine the necessary conditions of the shear band initiation and propagation. Because ld is increasing at DLESZ propagation, an overshoot of the stress-strain curve occurs after the shear band initiation.

The formulated criteria (27) and (29) have to be supplemented with equations describing the macroscopic plastic deformation process (similar to those formulated by Langer et al. [10–13] to explain the localization of the shear transformations within shear bands of width ∼102a).

Finally, let us estimate the yield stress of the polycluster MG,

“Soft” boundaries diminish the mean value of the macroscopic shear modulus of MG. As a crude estimate, we can write

where Cb is the fraction of atoms belonging to the boundary layers. If the boundary layer width is about 2a and the cluster size lcl≈40a then Cbσa/lcl≈0.15 and

With
we have
This estimate agrees with experimental evidences [4].

4 Discussion

The obtained exact solution of the formulated mean-field model describes the theoretical strength and sliding within STZs of MGs. STZs initially belong to the boundary layers. These defects determine the macroscopic strength of MG and are responsible for its low-temperature plastic deformation.

The intercluster boundary layer can have something different in structure than that shown in Figure 2. Because they are forming at the continuous process of the liquid solidification, it is natural to assume that, at the glass formation, the solid-like clusters are growing and amount of the fluid-like fraction decreases. This scenario of glass transition is described within the framework of the model of heterophase fluctuations developed in [33–35]. Grown solid-like clusters are contacting in glass and the structure of their interfaces depends on the local composition and solidification kinetics. In essence, the boundary layers are forming from residual fluid-like fraction. Along with narrow boundaries with structure shown in Figure 2, formation of more thick layers with a poor short-range order (like in nanocrystalline materials [36]) is expected, too. Thus, the sliding layer can consist of intermitting areas of different local microstructure. The field emission microscopy clearly demonstrates existence of the boundary layers of width ∼1 nm [19, 24, 37] but does not allow to resolve details of the boundary microstructure. Therefore, the microscopic structure of boundaries is an issue of further structural investigations and computer simulations.

It has to be noted that the idea of a mosaic structure of glasses is not a new one. Initially on a qualitative level, it was formulated by Lebedev in the form of a crystallite hypothesis [38]. Omitting the long list of authors holding a similar concept, we refer to a more recent Gleiter’s model of nanoglass MG structure [39], which is similar to the nanocrystalline structure but is obtained by consolidation of noncrystalline clusters. This model was reproduced in molecular dynamic simulations to examine mechanical properties of these materials [40]. It was shown that the plastic deformation of the simulated nanoglasses starts from the boundary sliding. Overshooting of the stress-strain curve was observed as well.

The elementary processes initiated by the dislocationless sliding considered here play decisive role in the shear band initiation. Shear band is forming due to propagation, branching, and merging of the sliding layers and nanocracks. However, the shear band formation, localization, and propagation dynamics is out of the scope of this publication. To solve this problem, a new approach taking into account both thermodynamic and dynamic aspects has to be developed. On the contrary, more systematic experimental data on the sliding and the shear band initiation in MGs are desirable. It has to be noted that the acoustic emission experiments [41–45] give a valuable additional information about the sliding dynamics in MGs. As it was pointed out in [46], the shear bands kinetics in bulk and ribbon-shaped MG obtained at different cooling rates has the same features. It can be considered as indirect evidence of existence of the sliding layers of the same structure in different MGs.

It is worth noting that the model of dislocationless sliding with random potential relief of the infinite sliding layer can be also applied to the description of the frictional force of a friction couple with known surface roughness and adhesion bond. Protrusions on rough sliding surfaces are distributed in a wide range [47]. Therefore, to calculate the sliding velocity, see Equation (5), we have to find both the effective time τsl and the effective scale of elementary displacement <d>.

In spite of some unclear features of structure of nanocrystalline metals, their mechanical properties have much in common with MG properties. Furthermore, the plastic deformation mode of nanocrystals depends essentially on grain size dG. At

∼15 nm the yield stress depends on dG in accordance with the Hall-Petch relation. However, at
the yield stress becomes proportional to dG [36, 48]. We assume that, at small dG, the deformation mode is controlled by grain boundary sliding during grain rotations. In this case, boundary reconstructions play a role of lubricant redistribution that diminishes the inner friction of the grain boundary creep. The detailed description of this mode will be given elsewhere.

We thank H. Wollenberger, C. Abromeit, M.-P. Macht, V. Naundorf, and N. Wanderka for fruitful discussions.

Appendix

Let us consider the model of a particle in the two-level potential that is characterized by two barrier parameters, Ed for the direct jump and Er for the reverse one (Figure 3). In a random potential relief, the parameters Ed and Er are the random variables. The transitions of the particle from a site to another site are assumed to represent a Poisson process in time and described by the kinetic equation

where Pi is the probability to find a particle in the site i=1, 2; ν0 is the vibration frequency, β=1/kBT; T is the temperature, and kB is the Boltzmann constant.

It is worthwhile to introduce the following parameters:

and σinva=Ed-Er, where
is the average energetic barrier and the parameter σin can be interpreted as an internal stress and va is the local activation volume of elementary displacement.

Applied external shear stress σe changes barrier heights Ed and Er,

Because P1+P2=1, we can set P2=P and P1=1-P. Then, Equation (31) subject to (32) becomes

where

When the external stress is a slow-changing function of time,

the solution of Equation (33) reads

If the external stress σe does not depend on time and

the atomic configuration goes from a site to a neighboring site without thermal activation. The parameter σi is the critical shear stress at site i.

Combining Equations (4) and (34), we have

Here,

Note that parameter
has the same distribution function as parameter σi, that is, g(σ(-))=g(σ).

The general solution (36) can be used for time-dependent stressing, for example, reciprocating sliding. Here, we assume that external stress σe is time-constant and

σe/(1-Δ)>σin. These assumptions give ω(t)=Ω(t) and the relation (6) for P(t).

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About the article

Corresponding author: Nikolai Lazarev, NSC Kharkov Institute of Physics and Technology, 61108 Kharkov, Ukraine, e-mail:


Published Online: 2013-11-06

Published in Print: 2013-11-01


Citation Information: Journal of the Mechanical Behavior of Materials, Volume 22, Issue 3-4, Pages 119–128, ISSN (Online) 2191-0243, ISSN (Print) 0334-8938, DOI: https://doi.org/10.1515/jmbm-2013-0017.

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