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Simulations of Shear bands in Metallic Glasses with Mesoscale Modeling

  • Chih-Jen Yeh
  • Yu-Cheng Chen
  • Pierre Hamm
  • Chang-Wei Huang
  • Yu-Chieh LoEmail author
Original Research
  • 49 Downloads

Abstract

Metallic glasses (MGs) have many attractive advantages such as high yield strength, high hardness, high wear resistance, high fracture toughness, and low friction coefficient. However, the localized deformation and poor ductility due to shear bands prohibit the further applications of MGs. The localized deformation could be controlled by the residual stresses in MGs. In this study, a mesoscale model which combines the kinetic Monte Carlo algorithm (kMC) and the finite element method (FEM) is constructed to investigate the effects of residual stresses on deformation behaviors of MGs. The developed computational framework is implemented by integrating in-house Matlab program for kMC and commercial finite element software, Abaqus. In the mesoscale model, a shear transformation zone (STZ) with nanoscale volume is the basic deformation unit and is represented by an element in the FEM model. Each element whether is transformed into an STZ is determined by the kMC algorithm according to its energy state. The developed mesoscale model can simulate the deformation process of MGs with the time and length scales which are greater than those by atomistic modeling. Finally, a sanity check was performed to verify the accuracy of FEM in solving Eshelby’s inclusion problem. Then the developed computational framework is applied to simulate the uniaxial tension test of the metallic glass specimen with different residual stresses. Simulation results demonstrate that the residual stresses could dominate the occurrences of STZs and change the stress–strain curve of MGs.

Keywords

Metallic glasses Shear bands Mesoscale Kinetic Monte Carlo Finite element method 

Introduction

The yielding and plasticity behaviors of MGs mainly depend on the formation of shear bands. Shear bands can be viewed as the macroscopic flow defects and are observed in most bulk metallic glasses (BMGs) with uniaxial strain about 2% [1]. Mechanical properties of metallic glass depend on the rich diversity of shear banding process from nucleation to propagation. To improve the mechanical properties of MGs, it is important to thoroughly understand the initiation, growth, propagation, and consequences of shear bands during the deformation process. Shear bands have shapes of thin bands with about 20 nm in width, at which localized high shear strains occurred. It is well-accepted that the elementary process of high shear strain in amorphous materials, such as MGs, is the shear transformation [2, 3]. A shear transformation is composed of hundred atoms and is often referred to a shear transformation zone (STZ). As a result, STZs is the smallest flow defects in bulk metallic glasses (BMGs) with a feature size around 1–2 nm3. So far, atomistic simulations are frequently applied to deepen our understanding of the nucleation of a shear band via percolation of STZs. However, their intrinsic spatial–temporal scale limits place certain restrictions. For example, the time and length scales accessed by classical molecular dynamics (MD) are typically ~ 1 μs and ~ 100 nm. Due to such limitations, the strain rate in MD simulations is usually in the range of 106–8 s−1. This super high strain rate may pose effects on the kinetics of material behaviors. As such, the mechanical behaviors from MD simulations may show deviation compared with experiments. Therefore, it is necessary to validate atomistic simulation results with clear physical interpretation.

On the other hand, the regular finite element method (FEM) or constitutive equation model cannot perform the stochastic nature of flow defects. For example, FEM has suffered attacks in dealing with the nucleation of a shear band through the percolation of STZs in metallic glass. To overcome the limitations in continuum methods, Bulatov and Argon [4] developed a mesoscale model combining the kinetic Monte Carlo (kMC) algorithm and a real-space Green’s function to account for the features STZs and shear band formation. Homer et al. [5] expanded upon their approach by employing a finite element method (FEM), instead of the Green’s function of Bulatov and Argon, to solve stress–strain redistribution after inserting an eigenstrain induced by an STZ. Later, a similar method in which the redistribution of stress and strain is determined by a phase-filed microelasticity theory was proposed by Zhou et al. [6], and a generation-dependent softening is designed in their model to provide a more realistic distribution of STZ features. In contrast with the FEM solver, the phase-field microelasticity theory using Fourier spectral method to solve the induced elastic stress/strain field in reciprocal space can provide a faster convergence than in real space [7]. It has an advantage over a real-space approach when the model size becomes large. Besides that, the feature of the discrete strain mode in Zhou’s model can easily extend into crystalline system, which makes the study of metallic-glass-matrix composites by such method also be possible. However, the mathematical complexity of this elastic-strain field treatment may make its promotion in academic research become difficult and restricted.

From the viewpoint of education, we would prefer to reduce the barrier during the methodology learning even though we have to sacrifice the efficiency of computation a bit. As a result, a computational framework which adopts FEM to replace the Fourier spectral method in Zhao’s model but keeps others the same is proposed in this study. To achieve this goal, the commercial FEM software, Abaqus [8], is adopted as the inhomogeneous stress solver to estimate the induced stress–strain redistribution in the model. The kMC algorithm which calls Abaqus package as the stress solver determines the occurrence of STZ according to the energy state of each element with stochastic nature was implemented in Matlab. To verify the accuracy of the computational framework, uniaxial tensile tests of metallic glass sheets are simulated to study the formation of shear bands. Furthermore, it is well-known that the annealing, cold-working, and surface treatment in the fabrication process could induce residual stresses in MGs. Such residual stresses are up Mega Pascal and likely to change the mechanical behaviors of MGs [9, 10]. It is worthy to investigate what mechanisms induced by residual stresses change the mechanical properties of MGs. As a result, the effects of residual stresses on the mechanical properties of MGs are investigated using the developed mesoscale model.

Theory and Methodology

Methodology

Regarding the dynamics of shear transformation zone in the kinetic Monte Carlo (kMC) algorithm, we applied a heterogeneously random shear transformation model to explore the shear band behavior observed in experiments. The main idea is that we use a voxel to represent the deformation behavior of an STZ. The STZ is theoretically composed of hundred atoms with a size of 1–2 nm in space and is replaced by a voxel which is the basic unit in this model. The shear transformation of each voxel has corresponding activation energies Q g ( m) and strain modes m, at which the subscript g represents the generation of the transformation and the superscript m is the m-th transformation mode, as shown in Fig. 1. We constructed an event list for STZs in the kinetic Monte Carlo algorithm. In this study, the number of modes is set to 20 for each voxel.
Fig. 1

Illustration of the heterogeneously randomized STZ model replotted from Ref. [6]. The images on the right illustrate a shear transformation on voxel A from generation-0 to generation-1 with their corresponding activation barriers (Q) and modes (m). The image on the left represents the collective behavior of STZs developing the shear band

In addition, we follow Zhou to conduct the generation-dependent softening in deformation of metallic glass. This method is able to perform the initiation of the shear band based on the information collected from atomistic calculations or experiments. Which area will trigger transformations depends on an event catalog for the transformation modes and the corresponding free energy barriers. The transformation mode implies how many kinds of eigenstrains an STZ can have and the expectation value of eigenstrain is given to 0.1 according to an atomistic study [11]. The free energy barrier for an STZ mode has a form as:
$$Q_{g} = \Delta F \cdot \exp ( - \eta_{g} ) - \frac{1}{2}V_{g} \cdot \sigma_{ij} \cdot \varepsilon_{ij} ,$$
(1)
where ∆F is the Helmholtz free energy with a typical range around 1–5 eV, ηg is a softening factor at each generation change from g to g + 1. In addition, ηg is based on two empirical parameters, \(\kappa_{t}\) and \(\kappa_{p}\), where \(\kappa_{t} /\kappa_{p} = 3\) was suggested by Zhao et al. [6]. Vg is the volume of an STZ, σij is stress tensor, and εij is strain tensor. The remarkable feature of generation-dependent softening is that the evolution of STZ events on an area relies upon the plastic history happened on the place. Once an area has already undergone a shear transformation event, next time to trigger a shear transformation on the same place will be easier.
The evolution of microstructure in the model is a state-to-state dynamics. As a result, the configuration at generation-g, expressed as Sg(x), is a state function of local inelastic strain εg(x), stress σg(x), local softening ηg(x), and elapsed time of shear transformation t g elap (x). It can present as Sg(x) = {εg(x), σg(x), ηg(x), t g elap (x)}. The time increment for each event is determined by the transition state theory:
$$t^{\prime} = \frac{1}{{v_{0} }}\exp \left( {\frac{{Q_{g} }}{{k_{B} \cdot T}}} \right),$$
(2)
where v0 is a trial frequency. When the stress in the model is big enough to lower the activation barrier to a certain level, the random number worked in the kMC algorithm will implement an eigenstrain for an STZ on a proper area. Here the eigenstrain is treated as thermal strain in FEM. After FEM resolves stress and strain distributions of a new configuration, the data will be gotten back to Matlab for next round.

Procedures

To implement the computational framework of the mesoscale model, different software packages and coding languages are integrated. The main program and the corresponding wrapper is developed by Matlab to control the whole process and to deal with the eigenstrain according to the kinetic Monte Carlo algorithm. Abaqus is adopted to performed linear analyses in solving Eshelby’s inclusion problem while FORTRAN is used to code the user subroutines (uexpan.f in Abaqus) which works as an interface to read the eigenstrains from Matlab and provides the information to Abaqus.

The flowchart of the proposed computational framework is shown in Fig. 2. After generating a model in Matlab, Abaqus INP file should be built, and element type, periodic boundary condition, material parameters like elastic modulus and Poisson’s ratio and etc. are given. Besides, the most important thing is making Matlab call Abaqus to solve the stress field, and the new stress field is sent back to Matlab to calculate the strain energy of the next step.
Fig. 2

Flowchart of the proposed computational framework

For a specified deformation, the activation energy at each voxel, Qg, is equal to the Helmholtz free energy difference minus the strain energy. The kMC algorithm determines the activation rate and time increment for the present step according to the activation energy. The activated events are categorized into three groups: athermal events, plasticity events, and elasticity events, as shown in Fig. 2.

While the athermal event is activated, and the transition is completed immediately, that is, the transition time is not counted. On the other hand, the athermal event is activated when there is at least one negative activation energy among all strain modes.

For example, as shown as Fig. 1, an STZ model with mode = 20 is divided by 4 × 4 plane stress elements, and each element (voxel) is with 20 modes, that is, there are 320 (4 × 4×20) activation energies we need to check whether is negative. If any negative activation energies among 320 modes, the athermal event is activated. However, when 320 activation energies are all positive, plasticity events and elasticity events are preferred. On the condition such that t′< t, the plasticity event for an STZ is activated by inserting an eigenstrain into the model, and also the corresponding elastic strain on the uniaxial tensile direction is applied, where t′ is the time increment for the transition of an STZ event, and t is the time increment corresponding to the controlled strain rate. On the contrary, on the condition such that t′> t, the elasticity event at which only the uniaxial tensile strain is applied is activated.

In this study, Abaqus solves the elasticity problem for a new mechanical status, and the corresponding stress field is got back to the kMC algorithm. Then the kMC algorithm computes the corresponding activation energy in each voxel base on the new stress field for the present step. Repeat previous steps until the average strain is equal to 0.1. The algorithm will decide the response of the three activated events, i.e., athermal plasticity, thermal plasticity, or pure elasticity, according to values of the activation energy Q g ( m) . If in the case of Q g ( m)  < 0, athermal plasticity will be preferred. When in the case of Q g ( m)  > 0 and time increment t′< t, thermal plasticity will be triggered. However, when in the case of Q g ( m)  > 0 and time increment t′ > t, pure elasticity will be selected. An event is determined by the algorithm at each step, and after that, the new stress filed is calculated by Abaqus. Moreover, the new stress filed leads to a new distribution of activation energy Q g + 1 ( m) at the next step. The procedure is repeated, until the average strain of model is 0.1. The flowchart in Fig. 2 is implemented by a combination of Matlab script and commercial FEM software Abaqus in this study.

Results and Discussion

Sanity Check

A sanity check is carried out to confirm that the numerical solution from FEM can satisfy the required accuracy, in consistent with the analytical solution. Imagine an infinite cylindrical inclusion embedded in an infinite square column. The cylindrical with radius R undergoes a spontaneous transformation strain \(\gamma_{12}^{*} = 2e_{12}^{*} = 0.02\). The width and height of the square column, B and L, are equal to 1, respectively, while the radius of the cylinder, R, is 0.02 as shown in Fig. 3. The elastic modulus and Poisson’s ratio are 88.6 (without unit) and 0.371, respectively.
Fig. 3

Geometries of the inclusion and matrix of sanity check

A typical plane is chosen and the problem is solved by a two-dimensional plane strain model. The mesh of the unit square consists of four-node quadrilateral elements (CPE4 in Abaqus) outside the inclusion with a few triangle elements (CPE3 in Abaqus) in the circular inclusion. Periodic boundary conditions are applied on the four edges of the square. Given the eigenstrain, the corresponding stress and strain fields can be obtained from Abaqus. The analytical solution of the shear stress field outside the inclusion is given by [13]:
$$\sigma_{12} (x,y) = \frac{{\mu e_{12}^{*} R^{2} (x^{4} - 6x^{2} y^{2} + y^{4} )}}{{(x^{2} + y^{2} )^{3} (1 - \nu )}},{\text{ for }}x^{2} + y^{2} > R^{2} ,$$
(3)
where μ and ν represent the shear modulus and Poisson’s ratio, respectively. Figure 4 demonstrates the corresponding shear stress distribution along y = 0.05 and y = 0.10 (as shown in Fig. 3) from numerical simulations and the analytical solutions. One can find that the numerical results coincide well with the analytical results. As a result, this example demonstrates that FEM can be a good solver to study Eshelby’s inclusion problem.
Fig. 4

The distribution of shear stress σ12 for a cylindrical inclusion transformation with the modulus remains the same as the matrix along y = 0.5 and y = 1.0

BMG Tensile Test

To verify the functions of the developed framework, uniaxial tensile tests of BMG sheets are performed to investigate the formation of shear bands. A square plate which has dimensions of 217.6 nm × 217.6 nm and is divided by 128 × 128 plane stress elements (CPS4R in Abaqus) in the FE model. Each element has dimensions with 1.7 nm × 1.7 nm, which agrees with the typical size of STZ. Periodic boundary conditions are applied on the four edges of the plate. The strain rate and strain increment are 1 × 10−4 s−1 and 1 × 10−4, respectively. In “Procedures”, we mentioned t is the time increment corresponding to the controlled strain rate. Because the simulations are under strain control, a applied strain increment \(\Delta \bar{\varepsilon }\) and strain rate \(\dot{\bar{\varepsilon }}\) at each simulation step need to be assigned in order to determine the time increment \(t = {{\Delta \bar{\varepsilon }} \mathord{\left/ {\vphantom {{\Delta \bar{\varepsilon }} {\dot{\bar{\varepsilon }}}}} \right. \kern-0pt} {\dot{\bar{\varepsilon }}}}\) during which the current configuration can maintain. Following Zhao’s work, the strain rate and strain increment are 1 × 10−4 s−1 and 1 × 10−4, respectively [6].

The temperature is assumed at 300 K. Material properties, adopted from typical MG systems in the literature, are listed in Table 1 [6].
Table 1

List of simulation parameters

Parameter

Value

E

88.6 GPa

v

0.371

v 0

1 × 1013 Hz

F

5 eV

Q act

0.37 eV

\(\Delta \bar{\varepsilon }\)

1 × 10−4

\(\dot{\bar{\varepsilon }}\)

1 × 10−4 s−1

Figure 5 demonstrates the equivalent stress–strain curves with different numbers of STZ modes, M, from numerical simulations. From Fig. 5, one can observe that each curve is linear in the beginning. After the peak stress value which is the “yielding point” in the view of macroscopic elasticity, these stress–strain curves turn out an obvious stress drop for mode M = 20 and 16, respectively. In addition, a large number of STZ mode, M, produces a stress drop after the peak stress. After the stress drop, the trend on stress–strain curves is jagged and steadied with different modes. It is worth noting that the case with mode M = 12 does not have an obvious stress drop after peak stress value, and the trend on the stress–strain curve is more steady and less jagged than the others. The obtained stress–strain curves in Fig. 3 are similar to those from Zhao et al. [6] in which they solved Eshelby’s inclusion using Fourier spectral method. In addition, these stress–strain curves also have the same characteristics as those from Utz et al. [12] who modeled MGs produced by different quenching rates.
Fig. 5

Equivalent stress–strain curve with different modes

Figures 6, 7 and 8 demonstrate the von Mises strain distribution at different applied strain with different STZ mode (12, 16, and 20). Shear bands can be found in these models, which are the advantages of the mesoscale modeling method due to the statistical characteristics. It is obvious that shear bands are divergent for a small value of the STZ mode (M = 12) and are convergent for a large value of the STZ mode (M = 20). For the case with small STZ mode (M = 12), shear bands display a homogenous deformation distribution mainly, and the width of plasticity deformation is not only shorter but also more divergent than the others. In contrast, for cases with the greater STZ mode come shear bands more intensive. Moreover, comparing Fig. 5 with Figs. 6, 7 and 8, one can find that the stress drop occurs when the presences of shear bands. When the shear bands occur frustrated, the stress–strain curve after the drop has less jagged. On the other hand, the stress–strain curve after the drop decreased obviously when the shear bands concentrated in the specimen.
Fig. 6

von Mises strain distributions at different applied strain with STZ mode = 12

Fig. 7

von Mises strain distributions at different applied strain with STZ mode = 16

Fig. 8

von Mises strain distributions at different applied strain with STZ mode = 20

Residual Stress

Stress triaxiality may have an impact on the percolation of STZs and lead to a transition of strain softening to hardening [14]. It motivates us to explore whether the development of shear bends can be influenced through a design of residual stress distribution. Figure 9a, b are the distribution of the residual stress with average magnitudes of 200 MPa and 1000 MPa, respectively. The pattern of distribution is designed as a T-junction shape. In general, the percolation of STZs will follow a 45° angle along the tensile loading direction. Two branches below the junction will attempt to guide the percolation of STZs forming a V shape. The other branch is used to disturb the development of shear band along the 45° direction. If possible, we will expect to see either shear band branching or blunting in the result. Their process of deformation was of interest. Initially, the residual stress did play a role in that most STZs appear in the area of residual stress. This is because the residual stress in this work is designed to reduce the activation energy barrier of STZ. Later, an inhomogeneous-to-homogeneous transition started to happen in the case of the mode of 200 MPa. The homogeneous deformation would continue until the end of the tensile deformation, as shown in Fig. 9c. In contrast, Fig. 9d shows that two shear bands quickly formed and touched together to be a V shape in the model of 1000 MPa. Interestingly, shear bands after their junction obviously became blunting and broadening. Furthermore, the shear bands were tilted after their junction. The results suggest that the residual stress indeed posed a certain effect on the development of shear bands. The effect clearly responded to the stress–strain curves shown in Fig. 10. A comparison among three curves shows that shear softening, in the beginning, dominated the plastic deformation of metallic glass for the cases without residual stress and with a residual stress distribution of 200 MPa but was not apparent for the case of 1000 MPa. Significantly, the strain hardening gradually occurred in the case of 200 MPa during the deformation and raised the flow stress to the same level to the case of 0 MPa. Although there is no significant softening in the case of 1000 MPa, the level of yield stress and flow stress are much lower than both others. This can be attributed to the big magnitude of residual stress that may lower the average stress of the model. Nonetheless, the effects of residual stress in this work achieve our expectations that can clearly influence the development of shear bands.
Fig. 9

a, b Distributions of residual stress for two different magnitudes; c, d von Mises strain distributions corresponding to (a) and (b), respectively

Fig. 10

The stress–strain curves corresponding to the magnitudes of the residual stress of 0 MPa, 200 MPa, and 1000 MPa, respectively

Conclusions

In this study, we have constructed a new mesoscale modeling computational framework which integrates the in-house program of the kinetic Monte Carlo algorithm (kMC) and the commercial software finite element method (FEM). The developed computational framework which is based on shear transformation zone dynamics is applied to study the mechanical behaviors of metallic glasses. A heterogeneously randomized STZ model is adopted to study strain localization and extreme value statistics during deformation of metallic glasses. The computational framework which combines the kinetic Monte Carlo (kMC) method and the finite element method (FEM) is very flexible and can be extended into solving related problems. In addition, we applied the mesoscale model to explore the effects of residual stress on the development of shear bands which have not been well investigated using such a method formerly. Despite the simplification of our tests, the results prove that the blunting/branching behaviors of shear bands will become possible by applying well-designed residual stress conditions. This issue deserves an in-depth investigation in the future in order to enhance the ductility of bulk metallic glasses.

Notes

Acknowledgements

The authors want to acknowledge the financial support of the Ministry of Science and Technology (MOST 107-2218-E-008-015), Republic of China (Taiwan). The authors also like to thank Simutech Solution Corporation for providing computational resources.

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Copyright information

© Korean Multi-Scale Mechanics (KMSM) 2019

Authors and Affiliations

  1. 1.Department of Civil EngineeringNational Cheng Kung UniversityTainanTaiwan
  2. 2.Department of Materials Science and EngineeringNational Chiao Tung UniversityHsinchuTaiwan
  3. 3.Department of Civil EngineeringChung Yuan Christian UniversityChungliTaiwan

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