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We present and compare three elastoplastic models currently used for deformation of metallic glasses, namely, a von Mises model, a modified von Mises model with hydrostatic stress effect included, and a Drucker-Prager model. The constitutive models are formulated in conjunction with the free volume theory for plastic deformation and are implemented numerically with finite element method. We show through a series of case studies that by considering explicitly the volume dilatation during plastic deformation, the Drucker-Prager model can produce the two salient features widely observed in experiments, namely, the strength differential effect and deviation of the shear band inclination angle under tension and compression, whereas the von Mises and modified von Mises models are unable to. We also explore shear band formation using the three constitutive models. Based on the study, we discuss the free volume theory and its possible limitations in the constitutive models for metallic glasses.

Metallic glass, also called amorphous alloy, is a relatively new material characterized by the random, disordered atomic arrangement which is different from the ordered crystalline structure of metals and alloys. The first metallic glass, an Au-Si eutectic alloy, was synthesized by Duwez and his coworkers in 1960 through rapid cooling the liquid to prevent crystallization [

Extensive experimental works have been done to study the mechanical properties of BMGs. For example, tensile loading showed that BMGs can have 2% or more elastic strain limit and therefore higher yield strength than the conventional crystalline materials [

As known [_{g}), and inhomogeneous deformation at low temperature (<0.7_{g}), where _{g} is the glass transition temperature. In the inhomogeneous mode, the plastic strain is highly localized into thin bands called shear bands (SBs), which is the main reason for the brittleness. On the continuum level, the constitutive behaviors of metallic glasses at low temperature are rather simple: a nearly linear elastic regime is interrupted by either fracture, when the samples are subject to unconfined deformation loading, such as tension; or limited plastic flow, when the samples are subject to confined deformation loading, such as compression. The simplicity in the mechanical behaviors makes it difficult to formulate physically sound constitutive relations for metallic glasses from the experimental inputs alone. The additional information needed includes detailed structural and micromechanical properties of any potential defects that promote yielding and plastic deformation. However, due to the lack of the long-range translational symmetry, there is no Bragg diffraction in metallic glasses, which is a main source to detect structural defects in deformation. As a result, the atomic scale mechanism of the plastic deformation of BMGs still remains open. Nevertheless, several atomic-scale models were proposed to explain the underlying microscopic deformation mechanism of metallic glasses [

The development of constitutive models for continuum modeling of the deformation behaviors has followed a path intimately connected to the free volume model. From Spaepen’s free volume model, Steif _{2} invariant of the stress deviator [

The constitutive models mentioned above are all capable of producing inhomogeneous deformation or shear localization [

In this paper, we conduct a comparative study on three types of elastoplastic models for metallic glasses, namely the von Mises model (J_{2}), the von Mises model modified by hydrostatic stress effect (J_{2}P), and the Drucker-Prager (DP) model. As in the previous studies [

The paper is organized as follows. In the next section, we shall present three elastoplastic constitutive models. The sample preparation and modeling procedures are described in

There are two different ways to set up the constitutive relationship for a material. One is based on fitting the curves obtained from experiments phenomenologically and obtaining the corresponding constitutive equations. Although this method can capture exactly the same features as the experiments show, it is less able to provide the fundamental physics and thus lacks the generality. The other is to generalize the microscopic mechanism into continuum regime. This approach could be more physically consistent but often suffer from lacking of critical details in connection between the two length scales. The free volume model [

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,
_{f}_{1} and _{2} are temperature-related constants, α is a geometrical factor of unit order, ^{*} the critical volume of atom, _{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

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

The elastic strain is related to the stress through the generalized Hooke’s law:

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 J_{2}, which is the second invariant of the deviatoric stress tensor.

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 [

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 [_{f}_{,0} is the free volume before increment,

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

The attempt to incorporate hydrostatic stress or normal stress into yield criterion was done largely in granular materials [_{y}_{n}

The two-dimensional illustration of the Coulomb-Mohr criterion.

_{app}_{0} 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:

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

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 [

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 [_{1} =

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

Then the flow rule is

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 [

For the constitutive models, we can describe the displacement boundary problem in general by the following governing equations,
_{b}

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,
_{v} =

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 [_{e}_{0} = 2^{*} / Ω. Note that the constitutive equations derived here are for the Drucker-Prager model. By simply setting the parameters

By now, the boundary problem is fully described through Equations 20–25. To solve the problem of deformation and shear localization, a UMAT subroutine [

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:

(a) The von Mises model (J_{2} model): For the Equations 20–25, set

(b) The von Mises model plus hydrostatic stress effect (J_{2}P model): In addition to model (a), Equations 8–10 will be also included. The procedure is that before solving Equations 20–25, we calculate the initial free volume modified by hydrostatic stress from Equations 8 and 9, then run the time integration to obtain the solutions at that time step, and finally apply Equation 10 to update the hydrostatic stress in addition.

(c) The Drucker-Prager model (DP model):

The model material that we will use in this paper is Zr_{41.25}Ti_{13.75}Ni_{10}Cu_{12.5}Be_{22.5} bulk metallic glass, or Vitreloy 1. The mechanical constants are available from a number of experimental tests [^{3}. The initial average free volume can be estimated through thermal expansion equation, ^{−5} K^{−1} for this Zr-based BMG. Thus, after normalization, ^{*} / Ω = 0.8, the geometrical factor α = 0.105. σ_{0} = 2_{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

As mentioned earlier, the DP model is a smooth extension of Coulomb-Mohr criterion in the yield surface. Thus

Two types of samples and different loading will be used in this work. The first is a simple shear of one CPE4 [

A rectangular sample under plane strain tension and compression: (

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 [

From

The results from a simple shear by the three constitutive models: (

The hydrostatic stresses for these three models are plotted in _{2} model continues with no volume change. The J_{2}P 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 [

The formation of the shear band under plane strain tension and compression was first simulated by using the J_{2} model. The sample is shown in ^{−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

The results from plane strain tension and compression using the J_{2} model: (

For uniaxial tension (

Shear band formation in the plane strain tension (_{2} model. Strain contours for tension are shown at (^{4}, the initiation of shear localization, (^{4}, the yielding point, and (^{4}, at the end of the shear localization. Strain contours for compression are shown at (^{4}, the initiation of shear localization, (^{4}, the yielding point, and (^{4}, the end of the shear localization. The corresponding spatial distributions of the free volume at the yield point are plotted in (

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

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 [

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, _{2}P 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 [_{2}P 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
_{c}_{t}

As we see in _{2} model. For the DP model, it is not hard to find that the SD is associated with the pressure dilation coefficient

As discussed in _{2}P 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

_{2} model, (b) J_{2}P model, (c) DP model with _{2} model, (e) J_{2}P model, (f) DP model with a different

Shear band formation under plane strain tension by (_{2} model, (_{2}P model, (_{2} model, (_{2}P model, (

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 J_{2} and J_{2}P 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 [

The reason we used a larger coefficient

Although the J_{2}P model cannot predict the shear band angle change, it does have some effect on the shear bands. From _{2}P model seems to generate more shear bands than J_{2} 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 J_{2}P model is much less localized than those from J_{2} 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 J_{2}P model the tensile pressure will make the free volume production easier, resulting in more shear bands as shown in

Based on the free volume theory, three elastoplastic models with different yield criteria have been examined and compared in describing deformation of bulk metallic glasses. The conclusions are summarized here:

Shear banding as the inhomogeneous deformation mode of metallic glass was first simulated by the J_{2} model for both plane strain tension and compression. The results show the detailed dynamic process of formation of shear bands. Starting from the randomly distributed initial free volume configuration, severe strain localization in the form of shear bands was observed when the material yields, which is accompanied with abrupt increases of free volume.

Shear band angles and SD effect are two commonly observed phenomena. The shear band angles were first compared among the three models. The DP model gives a larger than 45° shear band angle in tension and a smaller than 45° in compression, all in agreement with experimental findings, while J_{2} and J_{2}P models predicted the same 45° in both tension and compression. The results show that the shear band inclination angle change is related to the hydrostatic stress during the localized deformation of metallic glasses.

While the J_{2} does not predict SD effect at all, the J_{2}P and DP models do. The SD effect described by DP model was also found to show increasing dependence of coefficient

Despite the success of the DP model as compared with the J_{2} and J_{2}P models, there might be limitations imposed by various parts of the constitutive models. One is the free volume model on which all these constitutive models are based. Although some key features can be captured including shear localization, shear band inclination angle, and tension-compression strength differential effect, we are still not able to see serrated flow. This is due partly to the rapid increase of free volumes. Another potential limitation is from the transition state theory where only the activation barrier is considered. As a result, after generalization into continuum simulation, the integration method is only performed at the single Gaussian point. It is difficult for this kind of local plasticity model to fulfill the strain compatibility requirements especially when the strain is highly localized. These issues, along with several numerical optimization methods, are currently being addressed by the authors.

We would like to thank the financial support of this work provided by an NSF EAger grant under the contract number NSF-1193590.

The authors declare no conflict of interest.

_{41.2}Ti

_{13.8}Cu

_{12.5}Ni

_{10.0}Be

_{22.5}

_{40}Ni

_{40}P

_{20}metallic-glass

_{41.25}Ti

_{13.75}Ni

_{10}Cu

_{12.5}Be

_{22.5}bulk amorphous-alloys