In the free volume theory, the mechanism of deformation is considered as a competing process between the forward jumps of the atoms driven by the applied shear stress τ from one state to another and the backward jumps through diffusion or annihilation. During this process, an excess amount of volume, or volume dilatation, is created which is required to accommodate the moving atoms. This theory provides us with the plastic strain rate equation, (1) which is related to the free volumes v_f with a rate equation: (2) where A₁ and A₂ are temperature-related constants, α is a geometrical factor of unit order, v^* the critical volume of atom, the free volume averaged by atom numbers, Ω the atomic volume, k the Boltzmann constant and T the temperature. is the effective elastic modulus given by Eshelby’s solution for elastic inclusion [24]. n_D is a constant related to diffusion with a value ranging between 3–10.

Equation 1 shows that plastic flow has a monotonic relationship with the free volume. Therefore, when free volume increases, the effective viscosity described as is reduced. Consequently, the material will be softened more at locations with high concentration of free volumes, which is the main reason perceived for strain localization. Note that this observation is based on the assumption that the material is viscoelastic and the deformation is more like a viscous flow than an elastic or a plastic deformation. In addition, in this model, only the shear stress τ is taken into account as the driving force of both the strain and free volume. In addition, with these two scalar functions, however, we are not able to deal with the second order stress tensor in multiaxial stress state. To simplify the problem, as shown below, idealized material models are introduced and different yield criterions are employed to give an effective stress.

In conventional elastoplastic material models, the total strain is additively decomposed into elastic and plastic parts and then treated separately: (3)

The elastic strain is related to the stress through the generalized Hooke’s law: (4) where is the elastic material matrix with K the bulk modulus and G the shear modulus in an isotropic solid.

To treat the plastic flow, one of the approaches is to employ a yield criterion to first define a state where the plastic flow starts and then give the corresponding flow rule. The most widely used criterion in metals is the von Mises criterion, in which the octahedral shear stress is the only reason for yield and thus is independent of the hydrostatic stress or normal stress. The yield function is given in the form of J₂, which is the second invariant of the deviatoric stress tensor. (5) where with s = σ − pI is the deviatoric stress tensor, is the hydrostatic stress, and I is the identity tensor. Note here that the hydrostatic stress p has a positive sign in tension and negative in compression, unlike that used in the engineering convention. The associated flow rule is then defined by (6) where λ is the proportional coefficient providing the magnitude for the flow. This flow equation indicates that (1) the principal axes of the stress and the plastic flow coincide, which is known as coaxial rule; and (2) because s is a traceless tensor, the plastic volume change will vanish. The effective stress is defined from yield function 5 therefore as (7) which will take the place of τ in Equations 1 and 2 as the driving force of the plastic flow and free volume.

However, deformation in metallic glasses involves volume changes as explicitly hypothesized in both the free volume model and shear transformation zone theory, and observed in experiments and atomistic simulations. One of the basic mechanisms of the free volume change is its dilatational barrier jumping, which obviously cannot be reflected by the von Mises rule alone. In the early work [17,18,19], free volume is only taken as an order parameter without the consideration of the dilatation and its coupling to hydrostatic stress. This situation will be dealt with in two separate cases in the following sections.

It has been expected by many researchers that the hydrostatic component of the stress should affect the deformation of metallic glasses. Because the hydrostatic stress only changes the volumetric part of the strain according to the elastoplastic assumption, we can imagine that the tensile hydrostatic stress would make the free volume generation easier, while the compressive one more difficult. Such an effect was mentioned by Steif [17] and later discussed in detail by Flores et al. [25]. Flores et al. further assumed that the hydrostatic stress only has effects on the initial free volume, and the effects are different for tension and compression following Equations 8 and 9, respectively: (8) (9) where v_f,0 is the free volume before increment, p the hydrostatic stress, K the bulk modulus. Note here that the effect for tension is much larger than that for compression because . The large difference is also confirmed recently by Guo and Li [26] in an atomistic modeling of the equation of state of a model of NiZr metallic glasses. As a modification to the von Mises model discussed in section 2.2, the free volume change will also contribute partly to the volumetric strain increment, which is assumed to follow Hooke’s law, (10)

One of the consequences of inclusion of the hydrostatic stress is the asymmetry of tension and compression since the effect differs between them as shown above. We shall discuss this model in detail later in section 4.2.

The attempt to incorporate hydrostatic stress or normal stress into yield criterion was done largely in granular materials [27] and also other materials such as polymers [28] where volumetric dilation is more obvious than in metallic systems due to the larger degree of incompressibility of the materials. To explain the shear band angle inclination, the most intuitive criterion is the Coulomb-Mohr model, in which the yield shear stress τ_y is a linear function of the normal stress σ_n perpendicular to the shear plane, (11) where c is the shear resistance and μ = tan ϕ is the internal friction coefficient with ϕ being the internal friction angle.

Figure 1 shows a sample under a uniaxial applied tensile stress σ_app, the resolved normal and shear stress on the shear plane can be expressed as (12) (13) where A₀ is the initial cross sectional area of the sample and θ is the inclination angle of the shear plane with respect to the loading axis. Substitute the above two equations into Equation 11, and we can get the yielding condition: (14)

The shearing angle θ with respect to the loading axis can be determined by maximizing the left side of Equation 14. Thus, (15)

The Coulomb-Mohr model can explain the shear band angle deviation from θ = 45° as compared with that predicted from the von Mises type of theories. Obviously, the simplicity makes the model very appealing for bulk metallic glasses where why the SBA inclination has not been rationalized clearly [20]. However, it is not fully convincing to define a preferred shear plane, as shown in Equation 11, before the yielding point when the sample is mostly under homogeneous deformation. Anand [23] chose the principal directions of stresses as the potential slip directions, which made it possible to apply the Coulomb-Mohr criterion locally. Nevertheless, there is no necessary connection between the principal stress directions and the orientations of shear planes. The finite numbers of potential slip systems obtained in this manner may reduce the possibilities of flow under complex stress state. In addition, the Coulomb-Mohr model does not consider the volumetric effects at all; and thus the effect needs to be implicitly inserted, which is often done in ad hoc fashion through the material parameters rather than in the mechanics equations, as in Equation 11.

An alternative way to consider the hydrostatic stress or the volume dilatation effects is through the Drucker-Prager (DP) model based on Drucker and Prager’s work on soil plasticity that appeared in 1952 [29]. The yield function suggested by this model is an extension of the von Mises model by adding a term of hydrostatic component of the stress, (16) where I₁ = tr(σ) is the first invariant of the stress tensor. The Drucker-Prager model can also be looked upon as a smooth generalization of the Coulomb-Mohr criterion in the yield surface [30]. From the yield function (Equation 16), we can similarly define an effective stress, or DP stress as (17)

For the von Mises criterion, the flow rule is obtained by directly taking the stress derivative of the yield function, namely associated flow. However, there is no evidence showing that the flow vector must be connected with yield function. Another function called plastic potential function Q is introduced to give the general case [30], (18)

Then the flow rule is (19)

a and b are two parameters which will be discussed later. If a ≠ b the flow is called non-associated; otherwise it is associated. From Equation 19, we can similarly calculate the volumetric strain rate as , which is not zero, provided that b ≠ 0. This property means that in the Drucker-Prager model, the dependence on hydrostatic stress will introduce an accompanying volume change during deformation, or vice versa [30]. This effect is physically consistent with the dilatational feature of the free volume theory, as well as the experimental results. Moreover, also from the data of molecular dynamics (MD) simulation, volume dilatation is believed to be the reason for shear softening [12].

As compared to the Coulomb-Mohr model, the volume dilatation in the Drucker-Prager model does not appear immediately apparent in its contribution to the shear band angle inclination. However, we show that contrary to this view, the nonzero plastic volume increment in the Drucker-Prager model could indeed lead to shear band angle different from 45°. This result is proved by by Zhao and Li [31] using Hill’s zero extension rate theory and Mohr’s circle.

For the constitutive models, we can describe the displacement boundary problem in general by the following governing equations, (20) (21) where u is the displacement vector and f_b represents the body force which is assumed to be zero here.

For the Drucker-Prager model, by substituting Equation 19 into Equation 4, we can get the constitutive equations relating the stress to the strain in the rate form, (22) (23) where is the deviatoric part of the strain tensor with ε_v = tr(ε) the volumetric part of the strain.

Note that the proportional coefficient λ is used to be determined by assuming a work-hardening law, for example, for crystalline metals. However, work hardening has seldom been observed in the deformation of metallic glasses. In contrast, the volume dilatation model is used to capture the strain-softening mechanism, as done by Steif [32], Huang [18], and Gao [19]. Here, we also assume that λ will follow the trend of the plastic flow shown in Equation 1. The free volume evolution is provided by Equation 2. Following the normalization scheme [32] and replacing the shear stress τ with effective stress σ_e defined in Equation 17, we get the expressions of λ and free volume rate, (24) (25) where , σ₀ = 2kT/Ω, and β = v^* / Ω. Note that the constitutive equations derived here are for the Drucker-Prager model. By simply setting the parameters a = b = 0, the DP equations will reduce to the case of the von Mises criterion.

By now, the boundary problem is fully described through Equations 20–25. To solve the problem of deformation and shear localization, a UMAT subroutine [33] is developed and incorporated with ABAQUS finite element software using the return mapping algorithms [19,34,35].

In order to give a direct comparison of the constitutive theories developed in the above session, we will set up three elastoplastic models for each of the three theories:

The model material that we will use in this paper is Zr_41.25Ti_13.75Ni₁₀Cu_12.5Be_22.5 bulk metallic glass, or Vitreloy 1. The mechanical constants are available from a number of experimental tests [5,21]: the Young’s modulus E = 96 GPa, Poisson ratio ν = 0.36, and the average atomic volume Ω = 16.4 ų. The initial average free volume can be estimated through thermal expansion equation, , where ξ is the thermal expansion coefficient with the value of 4.0 × 10⁻⁵ K⁻¹ for this Zr-based BMG. Thus, after normalization, . β = v^* / Ω = 0.8, the geometrical factor α = 0.105. σ₀ = 2kT/Ω is used to normalize the stress with the order of 1 GPa. n_D is taken to be 3.

For the DP model, there are two more parameters that need to be determined, that is, the coefficients in yield function and plastic potential function a and b. In this simulation, we assume that the flow is associated. It is shown that non-associated flow can predict an accurate plastic volume expansion by choosing a proper value of b. However, the comparison is beyond the scope of this paper [36]. Therefore, we have only one parameter a left now.

As mentioned earlier, the DP model is a smooth extension of Coulomb-Mohr criterion in the yield surface. Thus a can be obtained by matching the yield surfaces of these two models, (26) where ϕ is the internal friction angle defined in the Coulomb-Mohr model (Equation 11). ϕ is related to the shear band inclination angle θ through Equation 15. But, for the DP model, Equation 15 is no longer valid. However we can still estimate the coefficient a from Equations 15 and 26 and use it as our input. For shear band inclination angle θ = 48°, a = 0.045 and for θ = 51°, a = 0.087. The influence of coefficient a on the yielding has been discussed in Zhao and Li [26] in detail.

Two types of samples and different loading will be used in this work. The first is a simple shear of one CPE4 [33] element. The sample is subject to plane strain restriction. The basic material behaviors of the three models could be obtained from this simulation. Then plane strain tension and compression will be conducted on a rectangular sample with the height-to-width-ratio of three shown in Figure 2. The sample contains 8,517 CPE4 elements. All the material properties and parameters are described in section 3.2. More details on simulation can be found in [31,36].

As mentioned above, the reason of shear banding is believed to be a result of the volume dilatation, which has been elaborated in detail in the free volume theory [8]. Recent MD simulation of MG samples in simple shear also demonstrated this connection [12]. The first step of our simulation is therefore to model a simple shear in one CPE4 element using the three theoretical models. For the DP model, the coefficient a is taken to be 0.045. The results for simple shear are shown in Figure 3.

From Figure 3a,b we find that all three models give a similar trend in the shear stress strain relationship and free volume evolution. The material first deforms linear elasticity until the yield point, which is the maximum allowable stress state by the criteria. The material shows little plasticity approaching the yield point but an obvious strain softening right after the yielding. This softening is caused by the increase of the free volume as shown in Figure 3b. The free volume increases slightly at the elastic region. However, when it gets close to the yield point, it will gain an abrupt raise, which, as discussed by Spaepen [8], Steif [32], Huang [19], and Gao [19], softens the material through decreasing viscosity. Among the three models, the DP model gives a higher steady state shear stress noted from Figure 3a than the other two models, which results from the explicit hydrostatic stress dependence.

The hydrostatic stresses for these three models are plotted in Figure 3c. Here we note that during the simple shear, the boundary conditions only allow strain in the shear, or xy-direction. In other words, the volume is preserved because the trace of strain tensor is zero. However, if the volume would tend to change, the material will feel an internal hydrostatic stress imposed from the boundary conditions. Basically, the volume contraction will make the material under tensile hydrostatic stress, while the expansion will result in compressive hydrostatic stress. Following our convention, the sign of the former is positive and the latter negative. Figure 3c shows that in the elastic region, these three models predict little volume change. This is consistent with Hooke’s law used in dealing with this regime. The difference, however, emerges around the yielding point. The deformation in J₂ model continues with no volume change. The J₂P model gives a compressive hydrostatic stress, as the pressure in Equations 8 and 9 are external, has the opposite sign to that of the internal pressure. The hydrostatic stress shows a similar trend to the free volume, which can be explained by Equation 10. According to the discussion given above, the surge of the hydrostatic stress indicates that the material is ready to expand without the boundary condition restriction. This response is the same as that predicted from the volume dilatation suggested by the free volume model [8,9] and the observation from the MD simulations [12]. On the other hand, the DP model gives an internal conjugate hydrostatic stress with a negative sign for a compressive internal hydrostatic stress, indicating a volume expansion during the simple shear deformation.

The formation of the shear band under plane strain tension and compression was first simulated by using the J₂ model. The sample is shown in Figure 2a. During the deformation, the bottom boundary is fixed and the upper one is displaced along y-direction at a constant rate of 1 × 10⁻⁶ s⁻¹. For the initial free volume configuration, we randomly distribute the free volumes in each element with the mean value estimated to be 0.05 and the variance of 0.0001, which is consistent with experimental estimation. The initial free volume configuration is plotted in Figure 2b.

Figure 4 and Figure 5 show the results of the plane strain tension and compression. Stress strain curves and free volume evolution are plotted in Figure 4a,b respectively. Shear banding occurs at a very narrow time range. Basically, it starts when the material gets to the yield point and ends at the initiation of the shear region, leaving no macroscopic plasticity as seen in experiment. The compression shows a little higher yield stress but the difference is relatively small, that is, within the statistic error of the computation. The contours of strain are plotted in Figure 5, showing the process of shear banding. Before the yielding, the deformation is homogeneous in the entire sample. When the stress state reaches the yield point, strain starts to concentrate on some small regions as shown in Figure 5a,e. Then these localized regions start to propagate and connect to each other, forming the shape of narrow and extended bands. The strain inside the shear bands is calculated and is obviously larger than that in the regions outside of the bands by about two orders of magnitude, showing a severe localization phenomenon. After yielding, there is no more new shear band generated.

For uniaxial tension (Figure 5a–c), the continuous tensile loading after localization will make the material show early signs of “necking” at the intersection of the shear band and the surfaces with a slight narrowing/shrinking in the region. Also, the upper part of the material appears to slide relative to the lower part along the shear plane as illustrated by the arrow, creating a surface offset. For compression (Figure 5d–f), the compressive loading will introduce a bending torque when shear banding occurs, which leads to buckling in the samples with the aspect ratio larger than unity. Reducing the aspect ratio will help to prevent the buckling instability from happening.

The angles between the shear plane and the loading axis are all at or very close to 45°, same as the maximum resolved shear stress direction predicted by the von Mises criterion. The corresponding free volume spatial distributions at the yield point are also plotted in Figure 5d for tension and Figure 5h for compression. We can see that the regions with higher free volume, or more “flow defects,” coincide with the strain localized regions. This connection indicates again that the increase of the free volume softens the materials and leads to localized deformation as hypothesized in the free volume model.

There has been intense discussion of the mechanisms of shear banding. Experimental observations suggest a correlation between the localization and the volume expansion during the deformation, which is manifested through such phenomena as the vein pattern on the fracture surfaces and the void formation inside and around the shear bands [37,38]. The shear localization appears to be the consequence of some excess volume and its evolution. In the continuum point of view, this localized deformation is simply represented macroscopically by the strain-softening, which we saw in Figure 5. But these observations cannot decisively distinguish whether the liquid-like patterns and voids are the results of shear banding or the cause, or whether the volume dilation is caused by local heating or geometric incompatibility of the hard cores of atoms under stress. This issue, therefore, needs to be resolved with more detailed, atomistic scale modeling and quantitative constitutive modeling. As shown in the MD work by Li and Li in an extensive MD simulation [12], the softening is due to the loss of elastic rigidity in the sense of the decreasing of material’s shear modulus caused by the volume dilatation close to the yielding point. The constitutive modeling performed here demonstrated that it is likely the latter is responsible for the initiation of the shear banding. Our recent modeling using a coupled thermal-mechanical model indicates that local heating induced localization is not the dominant mechanism, at least at low strain rate as normally seen in experiments [39].

In this section, we will further investigate the shear banding by all the three models with the particular emphasis on the comparison of the shear band angles and the strength differential effect, i.e., through J₂P and DP models. It is observed in experiments that the shear band angle of metallic glasses usually do not follow the directions of the maximum resolved shear stress predicted by the von Mises model [40,41]. When the von Mises model is modified through different ways to incorporate the pressure component of the stress, such as in the J₂P model, no significant change is observed in the shear band angle deviation. Instead, we observe only a change in the strength asymmetry between tension and compression. To represent this asymmetry, a variable called strength differential effect defined as (27) is used, where σ_c and σ_t are yield stress for compression and tension, respectively.

As we see in Figure 4a, the stress difference between plain strain tension and compression are very minimum for the J₂ model. For the DP model, it is not hard to find that the SD is associated with the pressure dilation coefficient a. Thus SD was calculated based on a series of simulations using different pressure dilation coefficient a. We found that the stress difference increases with the pressure dilation coefficient a. And a = 0.17 gives a reasonable SD = 24%, which agrees with the experimental investigation.

As discussed in section 2, the deviation of the shear band angles is due to the plastic dilatation (Equation 19). As shown in section 4.1, the DP model that we developed is capable of representing the plastic dilatation. Therefore, we would expect that the DP model could also predict the correct shear band inclination angle. Here we will compare the von Mises model with J₂P as well as the DP models on this point. For the sake of consistency, we use exactly the same initial free volume configuration in the samples for the simulations by assigning each element with the free volume of (28) where NOEL is the number to label the element, which is randomly assigned when the sample is freely meshed [33]. This method can give an approximate randomly distributed free volume with mean value of 0.05 and variance of 0.0001. Moreover, the configuration is kept the same when we deform the same sample.

Figure 6 shows the contours of the strain for plane strain tension by (a) J₂ model, (b) J₂P model, (c) DP model with a = 0.045 and plane strain compression, (d) J₂ model, (e) J₂P model, (f) DP model with a different a = 0.087. All the contours are plotted at the time when the processes of shear banding are completed, that is, right after the yielding.

Firstly, we compare the shear band angles on these SB contours. Because we have multiple “bands” on each configuration, the quantitative measurements will contain certain fluctuations. For clarity, we first visualize the major shear bands by drawing the parallel lines along each band to indicate the orientations of shear bands and then make a direct measurement of the angles. The quantitative analysis will be considered later on. No matter for tension or for compression, the J₂ and J₂P models show nothing but 45°, whereas the DP model gives obvious changes. For tension, the DP model shifts the shear band up and gives an angle greater than 45°, while for compression the shear band is shifted lower and the angle is less than 45°. These angle deviations given by the DP model capture the trend qualitatively as found by extensive experimental measurements [41].

The reason we used a larger coefficient a (0.087) for compression than for tension is that compression was found not as sensitive to the coefficient a as in tension. The shear band angle change for compression by the DP model with a = 0.045 is too small for visualization in this round of comparisons. However, we show that it does give SB angle change, although it is small. [31]

Although the J₂P model cannot predict the shear band angle change, it does have some effect on the shear bands. From Figure 6b,e, J₂P model seems to generate more shear bands than J₂ model in tension and less in compression. If we calculate the ratios of the maximum and minimum strain, the strain in shear bands from the J₂P model is much less localized than those from J₂ and DP model, which means that including pressure effect via Equations 8 and 9 gives the material a tendency to deform more uniformly, or increase the shear band density. As discussed above, in J₂P model the tensile pressure will make the free volume production easier, resulting in more shear bands as shown in Figure 6b, whereas the compressive pressure will diminish the shear bands more effectively, as shown in Figure 6e. We also noticed that this effect is less pronounced in compression than in tension, which is consistent with the discussion in section 2.3.