Atomistic simulations of ductile failure in a b.c.c.

high entropy alloy

F. Aquistapace∗a, N. Vazquez∗a, M. Chiarpottia, O. Deluigib,c, C.J. Ruestesa,b, Eduardo M.
Bringaa,b,c,d*
a
Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Cuyo, Mendoza 5500, Argentina
b
CONICET, Mendoza 5500, Argentina
c
Facultad de Ingeniería, Universidad de Mendoza, Mendoza, 5500 Argentina
d
Center for Applied Nanotechnology, Universidad Mayor, Santiago, Chile 8580745
*Corresponding author: ebringa@yahoo.com

1. Abstract
Ductile failure is studied in a bcc HfNbTaZr High Entropy Alloy (HEA) with a pre-existing void.
Using molecular dynamics simulations of uniaxial tensile tests, we explore the effect of void radius
on the elastic modulus and yield stress. The elastic modulus scales with porosity as in closed-cell
foams. The critical stress for dislocation nucleation as a function of the void radius is very well
described by a model designed after pure bcc metals, taking into account a larger core radius for the
HEA. Twinning takes place as a complementary deformation mechanism, and some detwinning
occurs at large strain. No solid-solid phase transitions are identified. The concurrent effects of
element size mismatch and plasticity lead to significant lattice disorder. By comparing our HEA
results to pure tantalum simulations, we show that the critical stress for dislocation nucleation and
the resulting dislocation densities are much lower than for pure Ta, as expected from lower energy
barriers due to chemical complexity.

Keywords: High Entropy Alloys; void growth; plasticity; dislocations; Molecular Dynamics.

2. Introduction

High-entropy alloys (HEA), a class of multi-principal element alloys, were first studied by
Cantor et al. [1] and Yeh et al. [2]. These alloys typically contain four or more principal elements in
nearly equiatomic compositions and have received significant attention due to their exceptional
mechanical properties, such as high fracture resistance, ductility and strength [3, 4, 5, 6]. In
addition, their mechanical properties and deformation mechanisms can vary significantly as
temperature changes from cryogenic to high temperature regimes [4, 7, 8]. These properties are
derived from the interplay of several effects: enthalpy and entropy mixing effect, sluggish diffusion,
severe lattice distortion, and cocktail effect [9]. In addition to excellent mechanical and thermal
properties, HEAs can display radiation resistance [10, 11, 12], and their fabrication in the form of
nanoporous structures has also revealed good catalytic properties [13]. Due to a variety of potential
applications, the mechanical properties of HEAs are of great interest and several reviews on HEAs
applications and their mechanical properties are available [5, 6, 14, 15, 16].

*
These authors contributed equally to this work

Preprint submitted to High Entropy Alloys and Materials March 29, 2022
 Like many alloys and metals, HEAs are prone to ductile failure by void initiation, growth, and
coalescence. Gludovatz et al. [4] reported ductile fracture by void growth and coalescence in an fcc
FeNiCoCrMn alloy, while Gao et al. [17] presented an in-situ TEM investigation on void
coalescence for the same alloy. Unlike their fcc counterparts [5], refractory bcc HEAs are, in
general, less studied and have received important attention recently [18, 19, 20, 21, 22, 23, 24].
Atomistic simulations have provided valuable insights in the understanding of the exceptional
properties of HEAs [25, 26, 27]. These include studies on grain boundary structure [28], lattice
strain [29], dislocation structure [30], tension/compression of nanowires [31], and the role of
compositional fluctuations on the deformation behavior [32, 33]. In fact, molecular dynamics
simulations have been used to provide valuable insights into the void growth mechanisms for fcc
HEAs [34, 35, 36]. For example, a penny-shape void was recently studied for an fcc HEA, leading
to dislocation emission from the void [37]. Gludovatz et al. [4] revealed ductile fracture related to
pore nucleation and coalescence in the fcc Cantor alloy CoCrMnFeNi. Zhang et al. [38] found that
nanopore nucleation did not require second-phase particles in the Cantor alloy.
HEA with bcc structure have not been the subject of the same number of studies. One of the main
reasons is the difficulty of implementing adequate interatomic potential functions for these
materials, let alone that they should also work properly at the large stresses and strains reached
during void deformation. Related to the present work, we can mention the contribution of Mishra et
al. [20], using a combined hybrid Monte Carlo and molecular dynamics (MC/MD) simulations to
study microstructural aspects of the TaNbHfZr alloy. In turn, Huang et al. [39] presented a modified
embedded-atom method potential for HfNbTaTiZr and used it to explore chemical short-range order
in HfNbTaZr HEA.
More studies are needed in order to shed light into ductile failure by void growth in refractory bcc
HEAs. The purpose of the present work is to systematically explore the effect of void radii on the
elastic modulus, yield stress and deformation mechanisms for a prototypical bcc HEA (HfNbTaZr)
under uniaxial tension. This includes a thorough characterization of the interatomic potential on the
aspects relevant to elasticity and plasticity. The results are then compared with analytical models for
the aforementioned mechanical properties. In addition, by comparing our HEA results to pure
tantalum simulations, we provide insights into dislocation nucleation and the effects of chemical
complexity on dislocation activity.
3. Methods

3.1. Samples and simulation protocol

The simulations were performed using LAMMPS [40]. Body-centered cubic (bcc) HfNbTaZr
and bcc Ta single crystals were modeled with an embedded atom (EAM) potential [41, 42]. We
used bcc single crystals with periodic boundary conditions in all directions and studied uniaxial
tensile strain loading along the [100] (x-direction). The simulation domain was initially set up as a
cubic sample containing 1003 bcc unit cells (Ta lattice parameter of 0.3303 nm, HfNbTaZr lattice
parameter of 0.363 nm). For the HfNbTaZr, each atom was distributed randomly in a bcc lattice
considering an equiatomic composition. A single spherical void was created at the center of each
sample, by removing atoms within the void volume. In order to study the influence of the void
radius, several simulations were carried out using different radii R. In particular R/b =
5,7,10,15,20,25, b being the Burgers vector modulus for each material (b = 0.303 nm for the
HfNbTaZr alloy). After generation, each sample was energetically minimized using the conjugate
gradient method coupled to a box resizing strategy to allow the simulation box to reach zero
pressure.

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 This minimization (at zero temperature), was followed by an equilibration of 2 ps at 300 K and
zero pressure. Thermostating during this equilibration was achieved with velocity re-scaling. An
anisotropic Nose-Hoover barostat was used during the equilibration to zero pressure, with a
relaxation time of 0.3 ps and a 0.2 drag term to aid in damping pressure oscillations, within the NPH
ensemble.
A uniaxial tensile strain rate of 109 s−1 was applied in the [100] direction for 200 ps, resulting in
a total uniaxial strain of 20%. Lateral strains were impeded. In order to explore strain-rate effects,
selected cases were run at a strain-rate of 10 8 s−1. These strain rates are characteristic of shock
spallation experiments [43], where the dissipation of local heating is severely impeded, as well as
the expansion in the directions perpendicular to the shock direction, and the use of a NPT ensemble
would lead to artificial results. Because of this reason, many previous studies of void growth [35,
44] have used the NVE ensemble during loading, as it is done here. Uniaxial tension was applied
changing the periodic box size, with a 1 fs timestep. Since no temperature control was imposed, we
were able to measure heating effects produced by plastic activity during deformation.
Post-processing was performed using OVITO [45]. Defect identification was done by means of
the Polyhedral Template Matching (PTM) algorithm [46] and the dislocation extraction algorithm
(DXA) [47], together with the Crystal Analysis Tool (CAT) [48]. Shear strain calculations were also
performed using OVITO, by calculating the atomic-level deformation gradient and the strain tensor
at each particle, from which a von Mises local shear strain can then be obtained [49].

3.2. Characterization of the interatomic potential

The reliability of MD simulations is strongly dependent on the reliability of the interatomic
potential used to describe interactions. For studies concerning plasticity, a potential should
demonstrate the absence of unphysical behavior such as non-valid slip systems or solid–solid phase
transitions not expected from a phase diagram [50]. This section is devoted to the evaluation of the
potentials chosen in the present study.
We employed a recently developed model EAM potential for a HfNbTaZr high entropy alloy
[42]. This potential predicts a stable bcc structure for the four-component alloy and it has shown a
satisfactory behavior in predicting well-known aspects of dislocations in HEAs, such as the wavy
nature of the dislocation core [51]. The elastic behavior of structures with cubic symmetry is
described by their elastic constants, C11, C12, and C44 [52]. We calculated these elastic constants for
HEA and pure Ta using the Maiti and Steurer potential [42]. The orientation-dependent elastic
modulus for (100) single crystals was calculated by means of the following equation [53]:

. (1)
B is the bulk modulus, defined as:

. (2)
The shear modulus G was defined as the arithmetic mean over the Voigt-averaged and Reuss-
averaged shear moduli

. (3)
According to Ziegenhain et al. [54], GReuss [55], is defined as

. (4)

GVoigt, the Voigt-averaged shear modulus [56], is defined as

3
 . (5)
ν is the Poisson ratio, defined as

. (6)
The Elastic modulus, E, is

E = 2 G (ν + 1) (7)

The elastic anisotropy can be described by the Zener anisotropy factor, X, as

. (8)

Predictions from eqs. (1) through (8) are summarized in Table 1, showing good agreement with
experimental values and with previous simulation results using a different interatomic potential.
High strain-rate tension, such as the one developed under spall conditions, leads to large stresses
up to tens of GPa and it is important to establish the pressure dependence of the elastic constants;
this is shown in Fig.1. The HEA potential behaves smoothly under pressure and in the range
considered here, without discontinuities in the quantities or their derivatives. With respect to the
stability of the HfNbTaZr bcc phase under pressure, Figure 2(a) presents enthalpy versus pressure
for the bcc, fcc and hcp phases. Figure 2(b) presents the dependence of the enthalpy difference of
fcc and hcp phases with respect to the bcc phase. In short, it can be seen that the bcc has lower
enthalpy than fcc or hcp phases for hydrostatic pressures higher than -14 GPa.
We calculated the generalized stacking fault energy for Ta and the HEA (HfNbTaZr) (See
Figure 3). The values obtained for Ta are in reasonable agreement with those obtained in [57] using
other potentials. For the HEA we use 20 different random samples and found variations of
approximately 2 percent, reaching values typical of bcc metals. The stable stacking fault energy is
between 6 and 12 mJ/m2, which is similar to some fcc materials [58]. We find unstable stacking
fault energy values close to 600 mJ/m−2, which can be compared to values around 400-500 mJ/m−2
for FeTiNiCo bcc alloys [59].
We note that HEA are often characterized by their lattice mismatch δ, which is a function of
atomic radii [60]. Using the atomic radii from Senkov and Miracle [61], we obtain δ = 3.3 for this
alloy. As a reference, this value is similar to that of the CoCrFeMnNi (fcc) HEA but smaller than
the size mismatch for a TaNbHfZrTi (bcc) HEA [62]. Still, the resulting lattice distortion can lead to
large stresses and energy fluctuations.

4. Results

4.1. Stress-strain curves

Figure 4 presents the stress strain plots for three different void radii, namely R/b = 7, R/b = 15
and R/b = 25 studied at a strain rate of 10 9/s and 108/s. Supplementary Figure S2 presents the stress
strain curves for all HEA simulated cases, while Figure S1 in Supplementary Material shows the
same for Ta, for strain rates of 109/s. We note that Ta presents much higher stress than the HEA in
all cases, as expected.
The plots are characterized by a linear elastic regime at low strains. As strain exceeds ∼0.025,
the elastic behavior enters a non-linear regime, until the onset of plasticity indicated with arrows on
each curve. This onset is detected by the appearance of dislocations by means of DXA in OVITO.
After the onset of plasticity, the stress strain curves present a rather rounded peak followed by an

4
abrupt fall of stress until stabilizing for strains larger than 15%, at a flow stress in a range of 0.2 to
0.4 GPa. At the higher strain rate the peak is wider and the plastic nucleation stress is higher, as
expected from previous results for Ta [63]. Stress shows an oscillating behavior at large strain, after
plasticity from the central void has reached and crossed the periodic boundaries. Table 3 presents a
summary of the main quantitative features determined after Figure 4 and Supplementary Figure S2.
As R/b increases, so does porosity ϕ. As expected, as porosity increases, the elastic modulus E
decreases, and so does the peak stress and the stress corresponding to the onset of plasticity. The
trends in elastic modulus and stress at the onset of plasticity are analyzed in the following sections.

4.1.1. Elastic modulus

In order to assess the effect of the void on the elastic response, the elastic modulus E of the
porous samples was determined for each simulation. The slope of the linear elastic region up to a
strain of 0.01 was determined by least-square fitting, and results are presented in Table 3. As a first
approximation, these values can be compared with the Gibson-Ashby scaling law for the elastic
modulus of closed-cell foams [64],

(9)
where c is a fitting parameter, typically in the range of (0.1−1.0). Sometimes, the solid volume
fraction φ is used instead of porosity, φ=(1-ϕ). Figure 5 presents a comparison of eq. 9 and the
elastic moduli of our tests. It must be noted that the elastic modulus E0 on eq. 9 was calculated
considering the crystallographic orientation of the sample. The fitting factor c, c = 0.728 ± 0.007
falls within the range reported by Gibson and Ashby. Interestingly, the variation on the elastic
modulus with porosity is adequately captured by means of a closed-cell Gibson-Ashby model [64]
through eq. (9).

4.2. Plastic Stress Model

The Von Mises stress at the onset of plasticity (σY ) can be compared to predictions using an
analytical model after Tang et al. [44]:

(10)
with G<111> the shear modulus corresponding to <111> planes, γ the surface energy, b the Burgers
vector, and R1 the dislocation loop radius. ν<111> is a directional Poisson ratio [44]. The dislocation
core in units of b, ρ, is taken as a fitting parameter on eq. (10). For these simulations we have used
R1 = R/2. A recent study of surface energy in many quaternary HEA, including for instance
TiZrHfNb and NbTaMoW, but not TaZrHfNb, found surface energies in the range 2-3 J/m2 [65],
and we choose a value of 2.5 J/m2 for our fit. Values of G<111> and ν<111> obtained from the elastic
constants and used for the fits are shown in Table 2. The MD and model results are presented on
figure 6. Instead of maximum global shear stress, the Von Mises stress is used in eq. (10) [66].
In contrast to previous works on voids in pure fcc and bcc metals [44, 63, 67], where the critical
stress for dislocation nucleation strongly depends on void size, Fig. 6 indicates that the critical stress
for dislocation nucleation in the HEA varies relatively little in the range for 5 < R/b < 25. The
model by Lubarda et al. [68] was designed for shear loops in fcc metals, and does not capture as
well the behavior of voids emitting prismatic loops in bcc metals. Despite its simplicity, the
analytical model by Tang et al. [44] allows to capture the magnitude of the stress and the void-size
dependence very well for both the HEA and Ta. This is achieved by using ρ = 4.5 to take into
account the large dislocation core size observed in our HEA simulations (core radius ≈ 3 – 5 b, as

5
shown in Figure 7). Similarly, large dislocation cores in bcc HEA have been reported [69]. We note
that the fit to Ta results used a dislocation core of 2.5b, which is consistent with some cores
observed in our MD. This is larger than the value of 1b used by Tang et al. [44] for a different Ta
interatomic potential.
In general, lattice distortion facilitates dislocation nucleation [33], but impedes dislocation
motion [30]. One can assume that yield strength is normalized with shear modulus. For
experimental samples with pre-existing dislocations, the normalized yield strength will then
increase with lattice distortion. However, for samples where dislocation nucleation dominates the
early stages of plasticity, as in our samples with voids, single crystal nanopillars [31, 70] and
nanofoams [71], the normalized yield strength will decrease with lattice distortion. There are other
aspects that influence the yield stress and deformation behavior in HEAs, these include several
aspects of the GSFE curve [72] that have been demonstrated to play a role in HEAs mechanical
behavior [73,74]. For instance, Fig. 3 shows that, compared to Ta, the HEA crystals do have a lower
generalized stacking fault energy curve, which also explains its lower yield strength. Adding an
extra degree of complexity, the GSFE curve can be influenced by local lattice distortion fluctuations
[75].
Plasticity initiation can also be glanced from the temperature evolution of the samples, observed
in Supplementary Material Fig. S3. At low strain, there is adiabatic cooling due to tension, and then
a sudden temperature rise due to plasticity, which after approximately 10% strain reaches a steady
increase, similar for all void radii. Larger void radii clearly leave the elastic regime earlier, but
produce so many dislocations that they start forming junctions and get pinned, stopping their
contribution to heating due to dislocation motion. For a detailed analysis on plastic heating on bcc
metals due to dislocation-mediated void growth, the reader is referred to [76].

4.3. Deformation mechanisms

Void growth by dislocation emission and propagation has been observed in pure fcc and bcc
metals [44], but this behavior was also recently verified in simulations of fcc HEA alloys with
voids, CoCrFeMnNi [34-36] and AlCrCuFeNi2 HEA [77]. Plasticity in this HEA proceeds by
nucleation of dislocations and twinning at higher strains, but the large lattice distortions lead to an
increasing number of atoms which are identified as having fcc or hcp structure by CNA and PTM.
Figure 8 presents an evolution of the total dislocation density versus strain for our 10 9/s simulations.
Figure 8 shows that, as the R/b ratio increases, the onset of dislocation-mediated plasticity
decreases. Such an event is followed by a rapid increase in total dislocation length and density,
coincident with a sharp decrease in tensile stresses, until a saturation value is reached, in the range
of 1-3 1016 m−2, consistent with previous studies on bcc metals [63]. Similar results are observed for
Ta, as observed in Figure S4. At 12% strain dislocation density starts to decrease.
Kink-pair nucleation in screw dislocations controls strength in elemental bcc metals, but screw
dislocations already present many kinks in HEA [78, 79]. New results indicate that edge
dislocations might play a dominant role in HEA at high temperatures [80]. In our simulations we
find a mixture of edge and screw components according to DXA, with a qualitative predominance
of screw segments. In pure Ta, dislocation loops extend away from the void, with the screw
segments growing perpendicular to the void surface due to the glide of the edge components, whose
length is nearly constant at the front of the loop.
We also analyzed the average dislocation segment length for Ta and HEA for a void size of R/b
= 10 (See Figure S5). As expected, the HEA has shorter dislocation segments. Due to the relatively
high lattice distortion and the chemical complexity of the HEA, it is more difficult for dislocations
to glide.
To better understand the deformation mechanisms, we use the Crystal Analysis Tool (CAT) [48]
on samples of R/b: 7, 15 and 25 for HEA and Ta samples at a strain rate of 10 9/s. Tables S2 and S3
6
present the results for HEA and Ta samples, respectively. There are no bcc twins for those cases at
low strain, but there is a large amount of atoms identified as fcc by CAT, which uses CNA for atom
filtering. The formation of fcc and hcp domains is not unexpected, since the formation enthalpy of
those phases was found to be close to that of the initial bcc phase. However, using a clustering
algorithm from OVITO, it was verified that these fcc atoms are mostly isolated and do not represent
a true phase transition, but only the limitation of the structure detector to deal with a strained lattice
environment. Something similar occurs with hcp atoms. PTM gives results similar to CNA, see
Figure 9. This high lattice distortion was also observed in bcc medium entropy alloys composed by
NbZrTi [81]. For the case of R/b = 10, we observe some twins, as depicted in Figure S6, but these
defects decrease at higher strain.
The main deformation mechanisms for Ta samples depend on the void radius. For R/b = 7 and
R/b = 10 there are dislocations, with some limited twinning. For R/b = 15 and R/b = 25 there are
dislocations and significant twinning. Figure S7 shows a typical nanotwin in Ta. Twinning tends to
decrease at strains higher than 10%, and this detwinning is particularly large for R/b = 10. A similar
process was reported by Wehrenberg and co-workers in their In situ X-ray diffraction measurement
of shock-wave-driven twinning and lattice dynamics on Ta [82]. We can observe such behavior in
Figure S5. Twins start to increase at 8% strain and to decrease at 11% strain; after that dislocation
plasticity is again dominant, and dislocation length increases. Similar results were obtained by Wei
et al. [83]. Finally, for R/b = 25 the main deformation mechanism is twinning. In this case, twinning
also decreases at higher strain. These results are in good agreement with results using an EFS Ta
potential [44]. Using the methods presented in [63], we obtained approximate edge dislocation
velocities for Ta that are similar to the ones obtained with the EFS potential by Tang et al. [63].
Unfortunately, this method is difficult to apply in the HEA where dislocations are wavier and
dislocation segments are shorter, as seen in Fig. S5.
Finally, Figure 10 presents a detailed analysis of deformation mechanisms for our R/b = 25
simulations; similar figures are shown for cases R/b = 15 and 20 in the Supplementary Material (see
Figure S8 and S9). Dislocations and twins nucleate on the void surface and propagate, determining
the edge of the shear strain region. As mentioned before, in the region around the void, according to
CNA and PTM, there is a large amount of fcc and hcp atoms, but these do not represent a new
phase. These non-bcc atoms may represent twins, dislocation cores and lattice distortions. These
domains correspond to regions of shear strain in the range of 0.02-0.06 and dislocations keep
accumulating as shear strain builds up.

5. Summary and Conclusions

Experimental evidence suggests that HEAs are prone to ductile failure by void initiation,
growth, and coalescence. Such a scenario is expected under a variety of loading conditions,
including spallation [43]. To shed light into elusive aspects of void growth in HEAs under tensile
stresses and high strain-rate conditions, we systematically explored the effect of a void immersed in
a bcc HfNbTaZr HEA using molecular dynamics simulations. The influence of the void radius on
the elastic modulus, yield stress and deformation mechanisms was studied and the results were then
compared with analytical models. Our main findings are:

● The EAM potential we used for the HEA reproduces reasonably well the elastic
constants of the individual elements at zero pressure [42], and we have verified
reasonable behavior for compressive and tensile stress, together with phase stability and
stacking fault energy.
● The elastic modulus was found to be dependent on the porosity and the dependence is
adequately captured by a Gibson-Ashby type equation for closed-cell foams.
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 ● The stress required for the onset of plasticity is lower for the HEA than for Ta, which is
expected from the chemical complexity in the HEA. Similar behavior was reported when
comparing bcc TaTiZrV to Ta [84], and also for fcc NiCuCoFeCr HEA compared to Ni
[31, 33]. The lower nucleation stress is also consistent with the unstable SFE being much
lower in the HEA than in Ta.
● This stress decreases monotonically as the void radius increases, but this decrease is
lower than for pure Ta.
● Both features above can be explained very well by an analytical model based on
dislocation plasticity [44]. A large dislocation core radius for the HEA is required to fit
the MD results, and such core size was also observed in the MD simulations.
● Simulations at a lower strain rate show lower yielding stress and strain, as expected.
● Plasticity initiates at the void surface. Ledges act as nucleation sites. Dislocation
mediated plasticity was clearly identified.
● Twinning appears after dislocation activity, and twin growth is observed. At a large
strain, however, detwinning is found.
● Atomic structure analysis indicates the presence of fcc and hcp atoms in the plastic
region. However, they do not represent a local phase transition. Close inspection of
atomistic configurations indicates that most non-bcc atoms are isolated, and some non-
bcc clusters are simply highly distorted bcc clusters at the detection limit of the structure
analyzers.

At high temperatures, the mechanical behavior of HEAs could be affected not only by changes in
dislocation mobility due to temperature [80], but also due to the HEA composition [85]. Future
studies could assess the influence of initial temperature on the mechanical behavior and
deformation mechanisms of the HfNbTaZr alloy considered here.
Recent spallation studies of an fcc HEA alloy show a spall strength much higher than for any fcc
metal [43]. Our results for a single crystal bcc HEA alloy would suggest a lower spall strength than
for pure Ta. Because the stress required to nucleate plasticity from a void in the HEA is much lower
than the one for Ta, the resulting dislocation density is significantly lower, which might lead to an
increase in ductility at high strains. Chemical complexity in the HEA leads to significantly larger
dislocation core radii which facilitate dislocation nucleation from the pre-existing voids. Something
similar may occur for HEA where plasticity is controlled by dislocation nucleation from surfaces, as
in single crystal nanowires, nanopillars [31, 70] and nanofoams [71]. This interplay between
dislocation nucleation from surfaces and chemical complexity could help designing improved
materials for technological applications.

6. Acknowledgements

EMB thanks support by PICTO–UUMM-2019-00048 and SIIP-UNCuyo grant 06/M104. CJR

thanks support by Agencia I+D+i PICT-2018-00773 and a SiiP-UNCuyo grant. MC and NV thank
an EVC-CIN Scholarship for scientific vocations. The simulations were run on the Toko-FCEN-
UNCuyo computer cluster, part of SNCAD-MinCyT, Argentina. This work used computational
resources from CCAD – Universidad Nacional de Córdoba (https://ccad.unc.edu.ar/), which are part
of SNCAD – MinCyT, República Argentina.

7. CRediT author 

FA, NV and MC: Matias: Formal analysis, Investigation, Writing - Original Draft, Visualization,
Data Curation. OD: Formal analysis, Investigation, Writing - Review & Editing, Visualization, Data

8
Curation. CJR: Methodology, Validation, Formal analysis, Investigation, Writing – Review &
Editing, Supervision, EMB: Conceptualization, Methodology, Validation, Formal analysis,
Investigation, Supervision, Writing – Review & Editing, Project administration.
8. Data Availability

The datasets generated during and/or analysed during the current study are available from
the corresponding author on reasonable request.

9. Conflict of interest: The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence the work reported in this
paper.

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11. Figures and Tables

C11 C12 C44 G ν X E E100

(GPa) (GPa) (GPa) (GPa) (GPa) (GPa)

HfNbTaZr 166 119 52.8 53 0.418 1.66 138.72 66.62

Ta-EAM 240 162 74 74 0.402 1.6 196 70

Ta-EFS 230.8 143.5 91.3 67.9 0.325 2.09 180 120

Ta-Exp. 264 160 82 - - - -

Table 1: Elastic properties of bcc HfNbTaZr and bcc Ta at zero pressure using the EAM potential
by Maiti and Steurer [42]. For Ta, the results are compared to that of Dai et al. [86] EFS potential
and with experimental values by Stewart et al. [87]. Elastic constants Cij, bulk modulus B, average
shear modulus G, average Poisson ratio ν, elastic anisotropy X, average elastic modulus E, and
elastic modulus for the crystallographic orientation studied.

Parameter HEA Ta

G<111> 28.8GPa 46.3GPa

γ 2.49 mJ m−2 2.49 mJ m−2

b 0.303 nm 0.287 nm

ν<111> 0.327 0.325

m 2.2 2.2

ρ 4.5 2.5

Table 2: Parameters used in the analytical model for the plasticity threshold [44]. The values for γ
for Ta was obtained from [44], and for HEA was estimated based on values for several other
quaternary HEA [65] . G<111> and ν<111> were obtained from the Cij in Table 1.

16
 R/b ϕ(%) E(GPa) σVM(max)(GPa) εmax σVM(Υ) εΥ(%)

5 3.5×10−4 38.7 1.4882 7.6 1.4428 9.3

7 9×10−4 38.8 1.5177 7.1 1.4834 8.3

10 2.44×10−3 38.7 1.4717 6.9 1.4758 7.5

15 8.44×10−3 38.1 1.2803 6.2 1.4156 6.2

20 1.95×10−2 37.5 1.2064 5.2 1.3161 6.2

25 3.80×10−2 36.6 1.1915 5.1 1.2185 4.4

Table 3: Summary of porosity (ϕ) elastic moduli (E), peak stress (σVM(max)), strain associated with
peak stress (εmax), stress at the onset of plasticity (σVM(Υ))) and associated strain εΥ.

17
Figure 1: Pressure dependence of the elastic constants Cij for the HEA and Ta using the potential by
Maiti and Steurer [42]. Negative pressure indicates tension.

18
Figure 2: Enthalpy under tension for different structures (a) and enthalpy difference for different
selected structures (b) of the HfNbTaZr HEA as predicted using Maiti and Steurer EAM potential.
[42].

19
Figure 3: Generalized stacking fault energy for Ta and the HfNbTaZr HEA calculated using Maiti
and Steurer EAM potential. [42].

20
Figure 4: Stress-strain plots for selected cases of HEA samples. Stress values are lower than Ta pure
samples. Arrow indicates the onset of plasticity. Stress was calculated using the solid volume
obtained using ConstructSurfaceMesh from OVITO.

21
Figure 5: Normalized elastic modulus E/E0 as a function of porosity ϕ (blue) and fitting of Gibson-
Ashby model (64). The R-squared value for the fit was R2=0.98. E0 corresponds to the elastic
modulus E100 = 66.62 GPa of the sample’s crystallographic orientation.

Figure 6: Critical Von Mises stress for dislocation nucleation versus R/b, obtained from MD
simulations. The fits using the model by Tang et al. [63], equation 10, are also shown. Values used
for the fits are included in Table 2. The dislocation core ρ (in units of Burgers vector b, was 4.5 for
HEA and 2.5 for Ta in the model. Dislocation loops were assumed to have an initial radius R1 = R/2.

22
 (a)

(b)

(c)

(d)

Figure 7: Selected dislocation cores for HEA (a-b) and Ta (c-d) samples. Atoms are colored using
PTM: blue (bcc), green (fcc), and withe (other). Dislocations are obtained with DXA. A slice 5 nm
thick is shown. Note that frames have slightly different length scales. Dislocation cores are roughly
approximated by circles, with radii: 4.1 b (a), 3.5 b (b), 1.7b (c), and 2.4 b (d).

23
Figure 8: Evolution of total dislocation density for 109/s HEA simulations.

Figure 9: Slice of 0.3 nm of HEA sample. Atoms are colored according to their PTM structure: bcc
(blue), hcp (red) and fcc (green).

24
Figure 10: Snapshots for simulations with R/b = 25. Rows correspond to strains 𝜺 = 4%, 5.2%,
5.6%, respectively. All bcc atoms were erased.

25
Supplementary Material
The Supplementary Material includes Figures S1-S9, and Tables S1-S3.

R/b ϕ(%) E(GPa) σVM(max)(GPa) εmax σVM(Y ) εΥ(%)

7 0.0009 72.90 4.16 8.1 3.79 6.7

10 0.003 72.71 3.81 7.2 3.32 5.6

15 0.0092 71.1 3.37 6.4 3.09 5.2

25 0.043 68.08 2.76 4.7 2.76 4.8

Table S1: Summary of porosity (ϕ) elastic moduli (E), peak stress (σVM(max)), strain associated with peak
stress (εmax), stress at the onset of plasticity (σVM(Υ))) and associated strain εΥ.

Sample HCP FCC FCC-ISF FCC COHERENT TWIN BCC TWIN

R/b = 7 0.68 11(5) 15205 (15205) 0 0 0

0.72 18(9) 19063 (19063) 2 (1) 2 (2) 0

0.76 23 (11) 23216 (23216) 7 (3) 0 0

R/b = 15 0.40 13 (6) 2390 (2390) 0 0 0

0.44 10(5) 3321 (3321) 0 0 0

0.48 85(42) 4554 (4554) 9(4) 13(13) 0

R/b = 25 0.36 10(5) 1849 0 0 0

0.40 20(10) 2881 0 1 0

0.44 199(99) 4420 26(13) 28(13) 0

Table S2: Structural analysis of HEA samples using the Crystal Analysis Tool.
 Sample 𝜺 HCP FCC FCC-ISF FCC COHERENT TWIN BCC TWIN

R/b = 7 0.64 0 55 0 0 0

0.68 0 148 0 0 12

0.72 0 240 0 0 0

R/b = 15 0.48 0 77 0 0 0

0.52 0 192 0 0 192

0.56 0 454 0 0 2715

R/b = 25 0.40 0 26 0 0 0

0.44 0 151 0 0 1

0.48 0 288 0 0 1085

Table S3: Structural analysis of Ta samples using the Crystal Analysis Tool.

Figure S1: Von Misses stress as function of strain for all Ta samples. Arrows indicate the onset
plasticity. Stress was calculated using the solid volume obtained using ConstructSurfaceMesh from
OVITO.

2
 Figure S2: Von Mises stress as function of strain for all HEA samples at a strain rate of 109/s.

Figure S3: Effective temperature as a function of strain for HEA samples.

3
Figure S4: Evolution of total dislocation density for Ta samples at strain rate 109/s.

Figure S5: Average segment length as function of strain for void radio of R/b = 10 for HEA and Ta
samples.
4
Figure S6: Twins observed in HEA sample with void size of R/b = 10. (a) Atoms are colored according
to their PTM structure: bcc (blue), hcp (red) and fcc (green). (b) Atoms colored according to
orientation.

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Figure S7: Twins observed in Ta sample with void size of R/b = 10. Atoms colored according to
orientation.

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 Local atomic environment Shear strain Dislocations

Figure S8: Snapshots for simulations with R/b = 15. Rows correspond to strains 𝜺 = 4%, 6%, 7.6%,
respectively. bcc atoms were erased.

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 Local atomic environment Shear strain Dislocations

Figure S9: Snapshots for simulations with R/b = 20. Rows correspond to strains 𝜺 = 4%, 6%, 7.6%,
respectively. bcc atoms were erased.