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
It turned out that experimentally measured values of
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 , 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 , 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 .
MGs possess noncrystalline disordered structure that can be characterized by a random potential relief with randomly distributed atoms in the potential minima . 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) . 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)  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 , 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 .
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.
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.
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
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
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 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: dσe/dΔ0=0. Using (8), (10), and (11), we obtain the equation for instability points in the form:
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
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
2.3 Thermally activated sliding
At low stress regime, when σe does not exceed the yield stress
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 . As was expected, the curves for different temperatures meet at some stress point
The temperature Tg was estimated as Tg=0.012μ0va/kB . The parameter of local activation volume va was taken to be about 4.5va . As was expected, the curves for different temperatures meet at same stress point
We can calculate the activation volume of sliding according to the definition :
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
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 dψ(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
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
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,
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,
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 Cb≈σa/lcl≈0.15 and
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 ) 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 . Omitting the long list of authors holding a similar concept, we refer to a more recent Gleiter’s model of nanoglass MG structure , 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 . 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 , 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 . 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
We thank H. Wollenberger, C. Abromeit, M.-P. Macht, V. Naundorf, and N. Wanderka for fruitful discussions.
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:
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
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
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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Published Online: 2013-11-06
Published in Print: 2013-11-01