## Abstract

Deformation and fracture of metallic glasses are often modeled by stress-based criteria which often incorporate some sorts of pressure dependence. However, detailed mechanisms that are responsible for the shear-band formation and the entire damage initiation and evolution process are complex and the origin of such a pressure dependence is obscure. Here, we argue that the shear-band formation results from the constitutive instability, so that the shear-band angle and arrangements can be easily related to the macroscopic constitutive parameters such as internal friction and dilatancy factor. This is one reason for the observed tension-compression asymmetry in metallic glasses. The free volume coalescence leads to precipitous formation of voids or cavities inside the shear bands, and the intrinsic “ductility” is therefore governed by the growth of these cavities. Based on a generalized Stokes–Hookean analogy, we can derive the critical shear-band failure strain with respect to the applied stress triaxiality, in which the cavity evolution scenarios are sharply different between tension-controlled and shear/compression-dominated conditions. This is another possible reason for the tension-compression asymmetry. It is noted that diffusive-controlled cavity growth could also be the rate-determining process, as suggested by the recent measurements of shear-band diffusivity and viscosity that turn out to satisfy the Stokes–Einstein relationship. This constitutes the third possible reason for the tension-compression asymmetry.

## 1 Introduction

*τ*

_{s}=

*σ*sin

*θ*cos

*θ*, the normal stress on the shear-band plane is

*σ*

_{n}=

*σ*sin

^{2}

*θ*,

*τ*

_{Y}is the material yield strength, and

*μ*is the coefficient of internal friction. While the left-hand side reaches a maximum at an inclined angle that deviates from

*π*/4 by $\xb1[\pi 2\u221212tan\u22121(1/\mu )]$ for tension and compression, respectively, it only predicts the same deviation from

*π*/4. Besides, Eq. (1) in geomechanics originates from the homogenization of collective behavior of many microcracks (or called shear faults) that are subjected to frictional sliding of the crack surfaces. There is no such counterpart mechanism in metallic glasses. What’s more, the representation in Eq. (1) has been blindly adopted for the failure analysis of metallic glasses. The shear-band formation and the subsequent evolution into a shear failure indicate that a proper analysis should be based on the boundary value problem, rather than from a constitutive model like in Eq. (1). Consequently, in this work, we attempt to build up a mechanics model that capture the physical processes and provide the accurate analysis of the pressure dependence.

*θ*

_{0}is very close to 45 deg, Rudnicki and Rice [3] give its dependence on the macroscopic parameters, including internal friction

*μ*and dilatancy factor

*β*, by

*ν*is Poisson’s ratio, $Nmin=3\sigma III\u2032/\sigma e$, $N=3\sigma II\u2032/\sigma e$, $Nmax=3\sigma I\u2032/\sigma e$, $\sigma I\u2032,\sigma II\u2032,\sigma III\u2032$ are the principal deviatoric stresses, $\sigma e=32sijsij$ is the Mises stress, and

*s*

_{ij}are the deviatoric stress tensor. Summation convention is implied for repeated Latin subscripts. The theoretical prediction in Eq. (2) can successfully reproduce the tension-compression asymmetry in the shear-band angle under uniaxial loading, if $\nu =0.3\u223c0.4$ and $\mu +\beta =0.1\u223c0.5$ [4].

*c*is an unknown dimensionless parameter to be calibrated experimentally, and

*σ*

_{m}=

*σ*

_{kk}/3 is the mean stress. While these approaches have achieved great successes in material failure analysis, they cannot be directly adopted for metallic glasses, because the failure process should be described in Fig. 2(b). With a pre-existing crack, the failure of metallic glasses is governed by the competition between crack-tip shear-band formation and cleavage crack [2,12]. Without any pre-existing cracks, failures always start from shear bands by cavity nucleation and growth [13,14]. As shown by experimental observation in Li et al. [5] and theoretically analyzed in Wright et al. [6], shear-band formation in metallic glasses results into the free-volume oversaturation, which quickly leads to the nucleation of numerous nanometer-sized cavities. Consequently, it is the cavity growth and the subsequent microcrack formation that dictate the intrinsic “ductility,” i.e., the maximum shear strain when an individual shear band can sustain prior to fracture failure. Qu et al. [14] employed 3D X-ray tomography to examine the evolution of shear-band cracking during compression test of Zr

_{65}Fe

_{5}Al

_{10}Cu

_{20}metallic glass. Due to resolution limit, they only observed the evolution of microcracks on the shear-band plane before it fractures. These microcracks are elongated and then slightly kinked out of the normally flat shear-band plane, until they coalescence into larger microcracks. These observations provide the salient features on what could be the possible governing mechanisms for shear-band failure.

Based on the above discussions, this work aims to determine the critical condition for the catastrophic failure of an array of cavities (circular or lenticular or other shaped) lying on a narrow planar band. This problem resembles the intergranular creep fracture in polycrystalline materials at elevated temperatures [15–19]. As shown in Fig. 3(a), these cavities can grow due to the creep flow of the surrounding material or because of the diffusive process along the grain boundaries. In our problem, we have a shear band that has much higher shear strain rate than outside the shear band, and oftentimes the temperature inside is high so that the shear band behaves liquid like. How and what knowledge from intergranular creep fracture analysis can be transferred here will be thoroughly discussed in this paper.

## 2 Prediction Based on Creep-Constrained Cavity Growth

As explained in Introduction, the problem that dictates the failure process will be the growth of a planar array of cavities on the shear band, subjected to faraway applied stress fields. These cavities are nucleated from the unstable free volume field and quickly coalesce into feature sizes that are comparable to the shear-band thickness. Therefore, their growth should be controlled by the creeping response of the nearby material.

*f*

_{cavity}=

*a*

^{2}/

*b*

^{2}, is found to be

*α*

_{n}and

*β*

_{n}are obtained from fitting to finite element simulations, given by

The modified Sham–Needleman equation in Eq. (5) clearly shows the dependence on the stress triaxiality, *T* = *σ*_{m}/*σ*_{e}. We have added the term 1 − *f*_{cavity} to the left-hand side of Eq. (5), without which the original version in Ref. [16] only works when *f*_{cavity} is less than about 0.6.

*T*remains unchanged, the failure condition can be obtained by integrating Eq. (5),

*f*

_{0}and

*f*

_{final}, respectively. If the mean stress is negative, i.e.,

*T*< 0, these cavities on the shear-band plane never grow so that the material becomes immortal, which is clearly not the reality. Nevertheless, we have a prediction of the shear-band failure strain, as given by

*c*

_{1}that needs to be calibrated with experiments. Here we write down the function

*G*

_{tension}, which is reciprocal to

*F*

_{Sham-Needleman}, for the sake of easy comparison to Eq. (12) as will be explained shortly.

If the applied loading condition is pure shear (*T* = 0) or shear/compression-dominated (*T* < 0), the modified Sham–Needleman model fails. Their simulations were performed mostly for *T* > 1/3, noting that the uniaxial tension corresponds to *T* = 1/3. Analytical solutions for two-dimensional circular or elliptical holes exist for linear elastic Hookean solid or Newtonian viscous material [20–23] and for perfectly plastic solids [24]. Even if we can follow the same solution approaches, these solutions are for small deformation and will not change the cavity closure result if *T* < 0. In shear/compression-dominated situations, as explained in Fig. 3(b), the cavities will be flattened out to micro-cracks, and then rotate and elongate until they interact with the neighboring micro-cracks. This is a large-deformation result that shows the long-term evolution of these cavities, which are only amenable to detailed finite element simulations.

*T*, especially when the loading condition is shear or compression dominated. This difficulty, however, can be bypassed by noting (i) the analogy between the initial value problem of a pure creeping solid under loads and the boundary value problem of a nonlinear elastic solid under the same loading conditions [25], and (ii) the extensive earlier works on computational cell model by Tvergaard and Nielsen [9]. If we replace all the rate measures in the creeping constitutive law by the non-rate ones in nonlinear elasticity (e.g., $\epsilon \u02d9ij$ by $\epsilon ij$), the governing equations for these two sets of constitutive models are identical. When the stress exponent

*n*= 1, we see the analogy between Newtonian viscous material and Hookean elastic solid, which is also called Stokes–Hookean analogy. Furthermore, a nonlinear elasticity is the same as the deformation plasticity (i.e., proportional loading and no unloading). Tvergaard and Nielsen [9] have performed extensive finite element simulations of the cell model in Fig. 3(b) with the power-law plasticity, given in one-dimensional form by

*σ*

_{Y}, the Young’s modulus

*E*, and the working hardening exponent

*N*. The above analogy leads to the same response of the cell model for a creeping solid with

*n*and reference stress and strain rate,

*σ*

_{0}and $\epsilon \u02d90$, respectively. The analogy dictates that

*n*= 1/

*N*. Based on the above discussion, we denote this method as the generalized Stokes–Hookean analogy.

*T*is fixed as a constant, and the peak load corresponds to the flattening and rotation of the initial cavity, followed by the quick elongation into slit like cracks in the precipitous load drop stage. Although the generalized Stokes–Hookean analogy makes the correspondence between strain rates and strains, we note that the stress-triaxiality dependence of the failure process should be the same in both problems. In other words, there is no need for us to extend Sham–Needleman problems to the stress triaxiality range investigated by Ref. [9] A simple reading of the failure conditions with respect to the stress triaxiality will suffice. However, Tverggard and Nielsen [9] presented extensive results for

*n*= 10, but only with a few data points for other

*n*values. They also compared their computational cell model with respect to the modified Gurson model proposed by Nahshon and Hutchinson [8]. Although the agreement is only moderately satisfactory, this does provide additional results that complete a parametric study. These results should also agree with the Sham–Needleman results for

*T*> 1. Our detailed fitting leads to the following results:

*α*

_{n}and

*β*

_{n}given in Eq. (6). When using this criterion, it should be noted that the critical strain should be assessed for the shear band, and the unknown dimensionless parameter

*c*

_{2}should be calibrated from experimental comparisons.

Predictions in Eqs. (9) and (13) are plotted in Fig. 5 for representative *n* values, which provides an explanation of the tension-compression asymmetry of “intrinsic ductility” in metallic glasses. It should be reminded that the shear band angle only shows a very weak dependence on the stress state, as explained in the Introduction. In contrast, metallic glasses show little ductility under tension, but moderate ductility under compression. According to Fig. 5, under tension-dominated loading conditions (i.e., *T* > 1/3), there is an inverse dependence of shear-band failure strain on *T*. When the applied stress is shear/compression dominated (e.g., *T* < 0), the complex evolution of cavities from flattening, rotating, elongating, and coalescing leads to considerable critical strain for failure. This can only be properly explained in Eq. (13), as opposed to the modified Sham–Needleman model in Eq. (9).

## 3 Prediction Based on Diffusion-Controlled Cavity Growth

*σ*

_{n}with the atomic volume Ω. This causes a gradient in chemical potential and thus drives the surface and interface diffusion that enlarges the cavity volume. Following Hull and Rimmer [26] and Cocks and Ashby [27], we write down the evolution equation as

*D*

_{sb}is the self-diffusivity along the shear band,

*δ*

_{sb}is the shear band thickness,

*k*

_{B}is the Boltzmann constant, and

*T*

_{K}is the absolute temperature in Kelvin. It is a daunting task to measure the shear-band diffusivity. One of such rare successes is presented by the work of Bokeloh et al. [28], which employs the tracer measurements by using a radioactive isotope. For Pd

_{40}Ni

_{40}P

_{20}metallic glass, they found out that the shear band diffusivity is $Dsb=(1.13\u223c1.96)\xd710\u221217m2/s$ and the shear band viscosity is $\eta sb=(0.11\u223c5.3)\xd7105Pa\u22c5s$. Song and Nieh [29] and Wright et al. [30] used high-speed camera and captured the dynamic response of shear-band propagation, and found out that the shear-band viscosity is $\eta sb=(2\u223c7)\xd7104Pa\u22c5s$ for Vitreloy 1 and $\eta sb=(0.11\u223c5.3)\xd7105Pa\u22c5s$ for Zr-based metallic glasses. While shear band diffusivity is not known in these works, their viscosity measurements agree well with theoretical models by Ref. [7], which suggests the existence of high temperature and liquid-like behavior inside the shear bands. These measurements actually satisfy the classic Stokes–Einstein relationship

*R*is atomic radius. Therefore, the doubts on the order of magnitude of their measured viscosity by Bokeloh et al. [28] are not warranted, because of this relationship.

^{−17}m

^{2}/s and external viscosity of 10

^{11}Pa · s, this length scale will be about 20 nm. While diffusive process inside the shear band is of atomic scale as justified by the Stokes–Einstein relationship, the above Needleman–Rice length scale amplifies the diffusive contribution to cavity growth for cavities of tens of nanometers in size. Therefore, the early stage of cavity growth under tension-dominated stress state is governed by both creep and diffusive processes, while later stage is purely by creep-constrained cavity growth. Substituting Eq. (15) into Eq. (16) gives an alternative representation, $LNeedleman\u2013Rice=[\delta sb\Omega 2\pi R\u22c5\eta ext\eta sb]1/3$, so that the cavity growth equation in Eq. (14) can be rewritten as

*c*

_{3}needs to be calibrated from experiments.

The forms of *G*_{diffusion}(*T*) and *G*_{tension}(*T*, *n*) are similar, while *c*_{1} mainly depends on initial value of *f*_{cavity} and *c*_{3} additionally on the initial cavity size (i.e., the value of *a*^{3}/(*δ*_{sb}*R*^{2})). For this reason, *G*_{diffusion}(*T*) is not plotted on Fig. 5. When the cavities are small, the concerted contributions from both diffusive processes and creep deformation could lead to a very fast growth rate, so that the resulting intrinsic ductility can be very small. For shear/compression-dominated loading conditions, *T* < 0 and the applied normal stress is negative. The Hull–Rimmer process shall not contribute to the cavity growth; in fact, the compressive normal stress will lead to cavity closure—a process denoted as solid-state bonding [31]. Consequently, the diffusive-controlled cavity growth provides yet another reason for the tension-compression asymmetry of “intrinsic ductility” in metallic glasses.

## 4 Summary

Because the free volume field inside the shear band is highly unstable and quickly leads to the nucleation of nanometer-sized cavities, it is the cavity growth that dictates the intrinsic ductility, namely, the critical strain that the shear band can sustain prior to its transition into fracture failure. Prior studies on intergranular creep fracture are incomplete for small and negative *T* values. Noting the generalized Stokes–Hookean analogy, this work is able to present a comprehensive description of the dependence of the shear-band failure strain on *T*. The cavity evolution under shear/compression-dominated stress states provides a mechanistic explanation of the contrasting ductility under tension versus compression conditions. The contribution from diffusive processes offers another complementary scenario for cavity growth.

## Acknowledgment

The author would like to dedicate this work to Professor Kyung-Suk Kim at Brown University for the occasion of his 70th birthday and in this special issue that honors his contributions in nano-, bio-, and fracture mechanics across a wide range of spatiotemporal scales. Financial support is provided by the National Science Foundation (IIP 2052729).

## Conflict of Interest

The author declares that there are no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Data Availability Statement

Data contained in this paper are available upon request to the corresponding author.

## References

_{41.2}Ti

_{13.8}Cu

_{12.5}Ni

_{10}Be

_{22.5}Bulk Metallic Glass Over a Wide Range of Strain-Rates and Temperatures