supplementary materials: the plastic yield and...

SUPPLEMENTARY MATERIALS: The plastic yield and flow behavior in

metallic glasses

SET-UP OF MOLECULAR DYNAMIC SIMULATIONS

All numerical metallic glass samples are prepared in a similar way which is summarized as following: the molec-ular dynamics (MD) simulations are performed using LAMMPS (Large-scale Atomic/Molecular Massively Par-allel Simulator)1. As a first step, we create a box with an approximated size based on the number of atoms(∼ 500 000 − 700 000 atoms) and their lattice sizes. The atoms are placed randomly within the box according tothe alloy composition. Further, we created MD systems based on B2 stacking by replacing the corresponding percent-age of the initial material by the remaining alloy composition. This did not influence the final stress-strain responses.Therefore, all results presented in this work are based on a random placement of the atoms within the box. As afirst step, we apply an energy minimization followed by an initial equilibrium calculation at 300K at zero pressure.Then, the material is heated up to 2500K at a heating rate of 11K/ps, at zero external pressure (NPT ensemble,NoseHoover thermostat). The system is kept at 2500K for 20 ps to allow the system to relax and to be in equilibrium.Afterwards, the system is cooled down to 300K at different cooling rates. The box is allowed to expand or shrinkduring heating, holding and cooling. The heating temperature is well above the melting temperature of the systems(see e.g.,2,3). The effect of the different cooling rates is discussed later within this document.Periodic boundary conditions are imposed on all faces so that shear offsets would not develop in the samples.

Uniaxial tensile/compression loading is applied in 1-direction at an absolute constant strain rate of ε̇ = 4×108 s−1. For

plane-strain loading, the deformation is applied in 1-direction at an absolute constant strain rate of ε̇ = 2√3×108 s−1

and the deformation in 2-direction is constrained. The remaining directions are pressure free. The plane strainboundary conditions are correctly applied which was verified by a study of the elastic constants under uniaxial andplane strain loading. Isothermal conditions at 300K are achieved for the system during deformation (NPT ensemble,NoseHoover thermostat), where we observed fluctuations in the small range of 1K. The time step size in the MDsimulations is set to 0.002 ps. The stress is obtained from the components of the pressure tensor of the total systemprovided in LAMMPS. The generation of the tensile/compression test was inspired by the tutorial provided online byMark Tschopp4 (see also5). The stress-strain curves are plotted with MATLAB R©. The visualization of the MD datais done with AtomEye6.In this study, we modeled one binary system Ni80Al20

7 (with θg ≈ 800 K8) and the ternary systems Fe-1.13Cu-1.36Ni9 (with θg ≈ 900 K10) and Cu46Zr47Al7

11. For the Cu46Zr47Al7 (which is taken as the example in the paper),we are using the EAM potential for ZrCuAl from11. The EAM potential for FeCuNi is taken from9.12 provided theEAM potential for aluminum-nickel alloys.

CHECK OF HOMOGENEOUS DEFORMATION

Classical measures from molecular dynamics, e.g., as centrosymmetry, cannot be used to visualize an inhomogeneousor localized deformation behavior in the amorphous metallic glass sample. We investigate as measures of the localdeformation the atomistic local von Mises shear strain invariant η

sto see the presence of homogeneous deformation

within the sample. The atomistic local von Mises shear strain invariant ηsis implemented as standard measure

in AtomEye6, and shown in Figure S1. No clear deformation localization or shear band development is visible.Additionally, the local shear strain ηMises (also named least-square atomic strain)13 was investigated. The calculationof ηMises requires two atomic configurations - one representing the current and the other the reference configuration.Again, no obvious inhomogeneity was observed and the results are skipped for brevity. Hence, our samples haveundergone homogeneous deformations in the MD simulations.

INFLUENCE OF COOLING RATE

The influence of the cooling rate on the subsequent mechanical test is shown in Figure S2. The sample preparation,especially the cooling process, has a significant influence on the material’s response. If the cooling rate is very slow,the metallic glass sample has enough time to find an equilibrium state, leading to a small amount of annealed freevolume after cooling. Increasing the cooling rate results in a higher annealed free volume. The free volume saturatesduring deformation. Therefore, a low initial free volume leads to a pronounced stress overshoot in the stress-straincurve until a steady-state value is reached. In accordance with MD results13 as well as experimental observations14,

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TABLE SI. Steady-state stresses of different MD simulations (obtained from values |ε| > 0.2), including standard deviationand maximal fluctuation from the mean value.

loading case component mean value standard deviation maximal fluctuation

Ni80Al20 (cooling rate 100K/ps)

tension LMD 2.5699GPa 0.0255GPa ±0.0801GPa

plane-strain tensionLMD

a 2.9376GPa 0.0348GPa ±0.0854GPa

CMDb 1.5435GPa 0.0222GPa ±0.0839GPa

compression |LMD| 2.8481GPa 0.0293GPa ±0.1082GPa

plane-strain compression|LMD| 3.2879GPa 0.0462GPa ±0.1571GPa

|CMD| 1.5896GPa 0.0301GPa ±0.0963GPa

Cu46Zr47Al7 (cooling rate 1K/ps)


plane-strain tensionLMD 1.5610GPa 0.0255GPa ±0.0612GPa

CMD 0.7960GPa 0.0121GPa ±0.0364GPa



|CMD| 0.8345GPa 0.0158GPa ±0.0449GPa

Fe-1.13Cu-1.36Ni (cooling rate 100K/ps)


plane-strain tensionLMD 2.3206GPa 0.0360GPa ±0.0867GPa

CMD 1.2217GPa 0.0169GPa ±0.0508GPa



|CMD| 1.2351GPa 0.0196GPa ±0.0675GPa

a LMD: loading stressb CMD: constraint stress

a very high cooling rate yields an initial free volume reaching its saturation right away and no stress overshoot isobserved. From the simulated stress-strain curves shown in Figure S2, note that although the sample cooling ratesignificantly affects the stress overshoot, the steady-state flow stresses are identical regardless of the sample coolingrate.

RESULTS FOR DIFFERENT ALLOY SYSTEMS

To support our results for the exemplary Cu46Zr47Al7 metallic system, we investigated two further alloy systems.The steady state yield stress properties are independent of the cooling rate. To save computational time, a coolingrate of 100K/ps is applied to all systems. The stress-strain responses for Ni80Al20 are shown in Figure S3: the steadystate stresses are quickly reached and no stress-overshoot is visible. For the ternary Fe-1.13Cu-1.36Ni system, theresults are displayed in Figure S4. Here, a stress overshoot is seen, indicating the feasibility for higher free volumedistributions. For completeness, the steady state values as well as the deviation and range of the obtained stressesfor all three systems are given in Table SI. The resulting fits for the von Mises and Mohr–Coulomb flow rule aresummarized in Table SII.All systems show the same trends as Cu46Zr47Al7. First, the loading stress in plane-strain tension (compression) is

markedly higher than the loading stress in simple tension (compression). Second, the constraint stress in plane-straintension (compression) is very near one-half of the loading stress in plane-strain tension (compression). The results inTables SI and SII indicate that the maximal error of our theoretical expectation is obtained for the constrained stresscomponents at approximately 7%. This error is slightly higher than for the result of Cu46Zr47Al7 presented in thepaper. This error definitely is in an acceptable range, taking into account the simplicity of the model. The resultsfor both additional systems support our statement that a yield criterion of pressure-sensitive von Mises type is moreappropriate to accurately describe the plastic yield and flow behavior in metallic glasses.

1S. Plimpton, Journal of Computational Physics 117, 1 (1995).2F. Jiang, Z. Zhang, L. He, J. Sun, H. Zhang, and Z. Zhang, Journal of Materials Research 21, 2638 (2006).3A. Cao, Y. Cheng, and E. Ma, Acta Materialia 57, 5146 (2009).

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TABLE SII. The steady-state stresses obtained from the molecular dynamics (MD) simulations for different metallic glasssystems. Further, the theoretical steady-state stresses determined from the von Mises plastic yield criterion and flow rule andthe Mohr–Coulomb yield criterion and the double-shear plastic flow rule are listed.

molecular dynamics von Mises Mohr–Coulomb

LMDa CMD

b rMDc LvM CvM rvM LMC CMC rMC

Ni80Al20 (ζ = 0.089, s = 1.56GPa)

ST 2.57GPa NAd NA 2.57GPa NA NA 2.57GPa NA NA

SC −2.85GPa NA NA −2.85GPa NA NA −2.85GPa NA NA

PST 2.94GPa 1.54GPa 0.53 2.87GPa 1.43GPa 0.5 2.57GPa IDe ID

PSC −3.29GPa −1.59GPa 0.48 −3.42GPa −1.70GPa 0.5 −2.85GPa ID ID

Fe-1.13Cu-1.36Ni (ζ = 0.081, s = 1.21GPa)

ST 2.01GPa NA NA 2.01GPa NA NA 2.01GPa NA NA

SC −2.21GPa NA NA −2.21GPa NA NA −2.21GPa NA NA

PST 2.32GPa 1.22GPa 0.53 2.25GPa 1.12GPa 0.5 2.01GPa ID ID

PSC −2.53GPa −1.24GPa 0.49 −2.64GPa −1.32GPa 0.5 −2.21GPa ID IDa L•:loading stressb C•:constraint stressc r• = C•/L•:ratio of constrained to loading stressd NA:not applicablee ID:indeterminable

4M. Tschopp, “Uniaxial tension,” (2014), https://icme.hpc.msstate.edu/mediawiki/index.php/Uniaxial Tension.5D. E. Spearot, M. A. Tschopp, K. I. Jacob, and D. L. McDowell, Acta Materialia 55, 705 (2007).6J. Li, Modelling and Simulation in Materials Science and Engineering 11, 173 (2003).7D. Z. Chen, D. Jang, K. M. Guan, Q. An, W. A. Goddard, and J. R. Greer, Nano Letters 13, 4462 (2013).8L. Wang, C. Peng, Y. Wang, and Y. Zhang, J. Phys. Condens. Matter 18, 7559 (2006).9G. Bonny, R. Pasianot, N. Castin, and L. Malerba, Philosophical Magazine 89, 3531 (2009).

10T. Egami, S. Poon, Z. Zhang, and V. Keppens, Phys. Rev. B 76, 024203 (2007).11Y. Q. Cheng, E. Ma, and H. W. Sheng, Phys. Rev. Lett. 102, 245501 (2009).12G. Purja Pun and Y. Mishin, Philosophical Magazine 89, 3245 (2009).13S. O. Futoshi Shimizu and J. Li, Materials Transactions 48, 2923 (2007).14P. De Hey, J. Sietsma, and A. van den Beukel, Acta Mater. 46, 5873 (1998).

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|εeng| ≈ 0.08 |εeng| ≈ 0.16 |εeng| ≈ 0.24tension

ηs0.3

0

plane-strain

tension

compression

compression

plane-strain

compression

plane-strain

compression

FIG. S1. Local strain plots of Cu46Zr47Al7 metallic glass system. Results for the local von Mises Shear Strain Invariantηsin AtomEye obtained for Cu46Zr47Al7 system (view in 1 − 3 plane) consisting of ∼ 700 000 atoms at nominal strains

|εeng| ≈ {0.08, 0.16, 0.24}.

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true strain [−]

truestress

[GPa]

cooling rate 1K/ps

cooling rate 2K/ps

cooling rate 100K/ps

0 0.1 0.2 0.30

0.5

1

1.5

2

2.5

FIG. S2. Influence of cooling rate. The absolute-valued stress-strain curves under compression for the Cu46Zr47Al7 sample of∼ 700 000 atoms after cooling rates of 1K/ps, 2K/ps and 100K/ps.

true strain [−]

truestress

[GPa]

simple tensionsimple compressionplane-strain tensionplane-strain compression

0 0.1 0.2 0.30

0.5

1

1.5

2

2.5

3

3.5

LMD

CMD

FIG. S3. Ni80Al20 system. The absolute-valued stress-straincurves for various loading conditions. System with ∼ 500 000atoms, cooling rate 100K/ps. LMD: loading stress; CMD:constraint stress in the molecular dynamics simulation.

true strain [−]

truestress

[GPa]

simple tensionsimple compressionplane-strain tensionplane-strain compression

0 0.1 0.2 0.30

0.5

1

1.5

2

2.5

3

3.5

LMD

CMD

FIG. S4. Fe-1.13Cu-1.36Ni system. The absolute-valuedstress-strain curves for various loading conditions. Systemwith ∼ 700 000 atoms, cooling rate 100K/ps. LMD: load-ing stress; CMD: constraint stress in the molecular dynamicssimulation.

supplementary materials: the plastic yield and...

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