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A stochastic discrete slip approach to microplasticity: Application to submicron W pillars

April 2024
International Journal of Plasticity 176(1):103965

DOI:10.1016/j.ijplas.2024.103965

Authors:

Carlos Ruestes

Universidad Politécnica de Madrid

Javier Segurado

Universidad Politécnica de Madrid

A stochastic discrete slip approach is proposed to model plastic deformation in submicron domains. The model is applied to the study of submicron pillar (D <= 1 um) compression experiments on tungsten (W), a prototypical metal for applications under extreme conditions. Slip events are geometrically resolved in the specimen and considered as eigenstrain fields producing a displacement jump across a slip plane. This novel method includes several aspects of utmost importance to small-scale plasticity, i.e. source truncation effects, surface nucleation effects, starvation effects, slip localization and an inherently stochastic response. Implementation on an FFT-spectral solver results in an efficient computational 3-D framework. Simulations of submicron W pillars (D <= 1 um) under compression show that the method is capable of capturing salient features of sub-micron scale plasticity. These include the natural competition between pre-existing dislocations and surface nucleation of new dislocations. Our results predict distinctive flow stress power-law dependence exponents as well as a size-dependence of the strain-rate sensitivity exponent. The results are thoroughly compared with experimental literature.

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A stochastic discrete slip approach to microplasticity:

Application to submicron W pillars

Carlos J. Ruestesa,b,∗, Javier Seguradoc,a

aIMDEA Materials Institute, C/Eric Kandel 2, Getafe, 28906, Madrid, Spain

bInstituto Interdisciplinario de Ciencias B´asicas (ICB), Universidad Nacional de Cuyo

UNCuyo-CONICET, Facultad de Ciencias Exactas y Naturales, Padre Contreras

1300, Mendoza, 5500, Mendoza, Argentina

cUniversidad Polit´ecnica de Madrid, Department of Materials Science, E.T.S.I.

Caminos, Madrid, 28040, Madrid, Spain

Abstract

A stochastic discrete slip approach is proposed to model plastic deformation

in submicron domains. The model is applied to the study of submicron pil-

lar (D≤1µm) compression experiments on tungsten (W), a prototypical

metal for applications under extreme conditions. Slip events are geometri-

cally resolved in the specimen and considered as eigenstrain ﬁelds producing

a displacement jump across a slip plane. This novel method includes several

aspects of utmost importance to small-scale plasticity, i.e. source truncation

eﬀects, surface nucleation eﬀects, starvation eﬀects, slip localization and an

inherently stochastic response. Implementation on an FFT-spectral solver re-

sults in an eﬃcient computational 3-D framework. Simulations of submicron

W pillars (D≤1µm) under compression show that the method is capable

of capturing salient features of sub-micron scale plasticity. These include the

natural competition between pre-existing dislocations and surface nucleation

of new dislocations. Our results predict distinctive ﬂow stress power-law de-

pendence exponents as well as a size-dependence of the strain-rate sensitivity

exponent. The results are thoroughly compared with experimental literature.

Keywords: plasticity, simulation, slip localization, stochasticity, tungsten

∗Corresponding author: Carlos J. Ruestes e-mail:carlos.ruestes@imdea.org

Preprint submitted to International Journal of Plasticity April 17, 2024

arXiv:2404.10430v1 [cond-mat.mtrl-sci] 16 Apr 2024

1. Introduction

Progress towards fulﬁlling the promise of nanoscale solutions to engineer-

ing problems hinges on the development of computational and character-

ization techniques for an adequate assessment of the mechanical response

of the resulting materials and parts. For instance, micropillar compression

testing (Uchic et al., 2004; Uchic and Dimiduk, 2005) can provide critical

data for the development of a number of technologies at diﬀerent engineering

scales, from the micro/nanoscale (e.g micro/nano electromechanical systems

-MEMS / NEMS-, microsensors and actuators and microscale energy harvest-

ing and storage systems) to the macroscale. The technique is instrumental

in establishing correlations between the microstructure and the mechanical

properties of materials (Greer and De Hosson, 2011; Dehm et al., 2018). The

analysis and interpretation of the experimental data obtained often relies on

advanced computational simulations at diﬀerent length scales, which aim at

determining the underlying deformation micromechanisms responsible of the

observed behavior.

At the lowest scale, molecular dynamics simulations (MD) were used to

study the interaction of dislocations with pillar surfaces, leading to the pro-

posal of a surface-induced cross-slip mechanism (Weinberger and Cai, 2008).

In addition, several studies have been published with diﬀerent focuses, such

as unraveling the tension-compression asymmetry observed experimentally

(Healy and Ackland, 2014) or analyzing the origin of size eﬀects observed

experimentally (Yang et al., 2021; Yaghoobi and Voyiadjis, 2016), to name

a few. Nevertheless, the detailed nature of atomistic simulations, together

with integration timesteps of the order of 1-3 fs, enforces the use of ultra-high

strain rates of 107/s and above and very small pillar sizes (D < 100 nm).

The comparison between MD simulations and micropillar compression ex-

periments is thus limited by the inherent drawbacks associated with the sim-

ulation technique. Still, MD has made fundamental contributions to the un-

derstanding of source nucleation in micropillars and nanowires (Weinberger

and Cai, 2008; Zhu et al., 2008), even allowing to estimate nucleation stresses

on pristine fcc nanowires within 20% of the experimental values (Jennings

et al., 2013).

At the mesoscale, discrete dislocation dynamics simulations (DDD) have

been used to provide insights into the mechanisms responsible for the ex-

perimentally observed staircase stress-strain behavior (Tang et al., 2008) as

well as athermal size-dependent strengthening (Rao et al., 2008). The tech-

nique has also been used to study the eﬀect of specimen size on plastic ﬂow

and work-hardening (El-Awady et al., 2009) and also to study the charac-

teristics of source truncation controlled ﬂow behavior for submicron FCC

single crystal micropillars (Cui et al., 2014). Moreover, in order to capture

surface-related eﬀects, MD simulation-derived models of surface nucleation

(Ryu et al., 2011; Zhu et al., 2008) and surface-induced cross-slip (Wein-

berger and Cai, 2008) have been incorporated into DDD frameworks (Ryu

et al., 2013, 2015; Hu et al., 2019), allowing for the exploration of pillar sizes

in the range of 100 nm to a few microns. The technique has also been used to

develop stochastic models for the onset of plasticity in micro- and nano-scale

structures (Shao et al., 2014). In spite of their advantages, DDD simulations

still suﬀer from timescale limitations (few ns to few µs) and typical strain

rates used are in the range of 103/s −106/s, though strain rates as small

as 0.1/s are possible depending on the computational resources (Fan et al.,

2021). Both techniques, MD and DDD, typically rely on an implementation

in the form of massively parallel solvers to obtain results in a reasonable

time, which in turn impose important hardware requirements.

At the continuum scale, several groups had opted for ﬁnite element simu-

lations (FE) with diﬀerent material model approaches. FE simulations using

simple isotropic plasticity have proven useful in providing recommendations

for experiments design and reliable testing (Zhang et al., 2006; Kiener et al.,

2009). Simulations using crystal plasticity constitutive equations (CPFE)

simulations have been successfully used to study the eﬀects of initial crystal-

lographic orientation, diameter-to-length ratio and friction on the response of

Cu single crystalline pillars (Raabe et al., 2007). CPFE has also been used

for the study of polycrystalline Ni-based superalloy micropillars (Cruzado

et al., 2015). The use of CPFE models has also extended to size eﬀects

in Al pillar compression, focusing on a continuum description of starvation

(J´erusalem et al., 2012). CPFE relies on homogenization of the evolution

of large dislocation ensembles and on scale-separation, hypotheses that can

be perfectly assumed when studying the deformation of large crystal grains

or specimens of several microns. However, in small-scale testing the number

of dislocations in the specimen can be very small (just a few dislocations

in the full specimen), such that their eﬀect on the deformation is localized

in speciﬁc planes and show an stochastic nature. These aspects cannot be

homogenized in a continuum-scale law.

In order to ﬁll gaps between DDD and CPFE models several attempts

have been made to extend standard CPFE models to account for some char-

acteristic features of small-size tests (i.e. the stochastic nature of the response

and slip-band localization). Regarding the stochastic response, Ng and Ngan

(2008) proposed a Monte Carlo model based on the survival probability and

the burst size vs. stress distributions but with no further connection with the

material structure. Later on, Konstantinidis et al. (2014) proposed a cellular

automaton, based on the gradient plasticity framework of Zhang and Aifan-

tis (2011), to model stochastic eﬀects in pillar compression. In addition, Lin

et al. (2015) presented a crystal plasticity model in which stochasticity was

introduced by deﬁning the plastic strain as composed of a series of strain

bursts whose sizes follow a power-law distribution function and whose rates

are determined by a constitutive equation. More recently, a general stochas-

tic CP formulation based on kinetic Monte Carlo, was proposed in which

the active slip systems were selected based on the strain rates dictated by a

mobility law depending on resolved stress and temperature (Yu et al., 2021).

However, the homogenized nature of the formulation prevents its direct use

in applications where slip localization is expected.

The slip-band localization observed in experiments, resulting from the

activation of discrete slip events in a particular plane, cannot be naturally

captured by CPFE models. Some attempts have been made to emulate this

localization in CP frameworks. Lin et al. (2016) developed a dislocation-

based crystal plasticity model in which, in order to trigger localization in a

single slip band, strain-softening was added to the slip resistance evolution

law and a void was introduced in the center of the pillar to trigger localiza-

tion. More recently, Wijnen et al. (2021) presented a CPFE model adapted

for simulating heterogeneous plastic deformation in single crystals. In this

model, only some regions of the domain under study are modeled using CP

keeping the others as elastic material. These slip areas are sampled from

a probability distribution based on characteristics of the dislocation sources

and plastic localization is accomplished by considering diﬀerent slip bands

and the weakest link principle (Norﬂeet et al., 2008). These extended CPFE

models allow to describe the strain localization in pillars but maintain a de-

terministic nature and do not explicitly consider plastic events by individual

dislocations.

Summarizing, the mechanical response of metals and alloys under micro

compression testing is typically characterized by certain features: i) samples

usually have a low dislocation density and exhibit source truncation harden-

ing; ii) Plastic deformation is usually localized and in the form of slip bands;

iii) For diameters below 150 nm, or starved samples such as those produced

by mechanical annealing, surface nucleation of dislocations is likely to occur.

As a consequence, plasticity takes place by events of discrete nature and the

mechanical response of such structures is typically stochastic. Despite all the

aforementioned computational advances, capturing all these features into a

single and eﬃcient computationally-cheap 3-D framework is still a challenge.

The purpose of this work is two fold. Firstly, we present a multiscale com-

putational framework capable of capturing the fundamental aspects of the

mechanical response of submicron metallic parts. In this model, the displace-

ment jumps along a cross sectional plane of the sample produced by the slip

of a dislocation are introduced as eigenstrains around that plane using the

Eshelby formalism. The framework also accounts for surface nucleation by

implementing a model derived from MD simulations. The stochastic nature

of the mechanical response is captured by means of a kinetic Monte Carlo

selection process. In this kMC approach, the events correspond to the sin-

gle slip events produced by dislocations, whose probabilities are dictated by

physically-based laws. This approach is implemented in an FFT-based solver

resulting in a very eﬃcient and computationally-cheap framework that allows

to model complex 3D geometries. Secondly, we revisit micropillar compres-

sion experiments of single crystalline bcc submicron pillars (D≤1µm) with

focus on tungsten (W). We show that not only our method can successfully

capture fundamental aspects of the mechanical response at these scales, but

also allows for detailed investigation of size-eﬀects, strain-rate eﬀects and

statistics of strain-bursts.

2. Methodology

The framework developed assumes dislocation-mediated plasticity as the

underlying inelastic deformation micromechanism. We consider pre-existing

dislocations as well as newly developed dislocations due to surface nucleation.

The amount of pre-existing dislocations Ndon a pillar of diameter D= 2R,

height H, and initial dislocation density ρ, is determined as

Nd=int(ρπ R2H

) (1)

where Ldis the average length of a dislocation segment, here taken as R

for simplicity. Then, Ndslip planes are randomly selected along the pillar

axis with normal directions consistent with the available slip systems.

Surface nucleation can potentially happen at any position on the pillar

surface. From a practical point of view, in order to consider a dislocation

nucleation and its subsequent slip in our framework, a large but ﬁnite number

of nucleation sites NSN is considered. In practice, 100 < NSN <200. The

positions of the resulting slip planes are randomly selected along the pillar

axis, and their orientations are also based on the available slip systems.

In our framework, dislocations are not explicitly considered, but the eﬀect

of their extension when reaching the surface of the pillar. This eﬀect consists

in a relative displacement between the upper and lower parts of their plane

and experimentally would deﬁne a slip trace. This description is equivalent to

the deﬁnition of a closed dislocation loop in phase-ﬁeld dislocation dynamics

as an inclusion with an eigenstrain (an Eshelby inclusion) (Wang et al., 2001;

Rodney et al., 2003). In our case, only the ﬁnal stage of the loop is considered

which consists then in a loop with a shape deﬁned by the intersection of the

pillar with the slip plane; see Sec. 2.1 for details. The diﬀerent parts of the

model are introduced in the next sections.

2.1. Kinematics of a slip event

First we derive the elastic displacement introduced by a dislocation loop

considering it as an equivalent Eshelbian inclusion. Under this representa-

tion, the loop is idealized by moving the upper side of surface S, denoted

by S+, by bwith respect to the lower side S−, as shown in Fig. 1.a. Let n

be the normal to the loop, then the relative displacement between the upper

and lower parts of the loop in direction nis given by

JuKn(x) = bδS(x),(2)

where bis the burgers vector of the loop and δS(x) denotes the surface Dirac

delta function, that is inﬁnite if x∈Sand zero elsewhere. If the loop is

fully embedded in a material, the resulting elastic ﬁelds of this incompatible

deformation can be obtained using Green’s function (Mura, 2013). In our

model, the only stage considered is when the loop has ﬁlled all the pillar

cross section.

The displacement jump due to a full plane slip has been considered in

the past using a strong discontinuity approach in the context of ﬁnite ele-

ments and discrete dislocation dynamics (Romero et al., 2008) . However,

introducing this discontinuity in other numerical frameworks as FFT imply

smoothing out the jump to make the model numerically tractable. This can

be done by distributing the jump along a thin band of size h, such that it

renders the same total displacement as in the initial ideal case. In a simpliﬁed

way, the line integral along a path parallel to ncorresponds to

Zz=h/2

z=−h/2

ε(x)dz=JuKn(3)

Figure 1: Dislocation eigenstrain formalism and the application to a single crystalline

pillar. (a) A dislocation loop Lacting as a boundary of a dislocation surface S. By

sweeping the upper part S+with respect to S−in an amount b, a dislocation is eﬀectively

introduced. (b) For a dislocation sweeping a pillar, the dislocation can be replaced by a

plate-like Eshelbian inclusion of thickness h.

Extending the use of the equivalent Eshelbian inclusion, Figure 1.b de-

picts a scenario in which the cross-section of cylindrical pillar has been fully

swept in an amount of bby a single dislocation loop. The associated displace-

ment is linearly distributed over a thickness hin the direction perpendicular

to the loop. As explained later on (Sec. 2.4), the simulation domain is dis-

cretized in a box of [Nx, Ny, Nz] voxels, so in practice, his taken as twice

the distance between voxels (Capolungo and Taupin, 2019). As presented in

the Supplementary Material ﬁle, the method is relatively insensitive to the

election of the voxel size. In consequence, the strain εEI G (i)(x) associated

with a slip event i, is taken as:

εEI G (i)(x) = (b

h(si⊗ni), if xϵ plastic region

0, elsewhere (4)

where bis the burgers vector modulus, his the thickness of the slip

band and siand nistand for the burgers vector direction (slip) and normal

direction to the slip plane, si⊗nibeing the Schmid tensor. Consequently,

for a pillar that underwent Mslip events, the total eigenstrain ﬁeld εEIG (x)

is the superposition of all the eigenstrains applied,

εEI G(x) =

i=1

εEI G (i)(x) (5)

2.2. Constitutive laws for the slip events

The laws linking the stress at a given plane with its slip rate will be

presented in next sections.

2.2.1. Pre-existing dislocations

The model assumes that the pillar contains a certain number of disloca-

tions which will drive the slip events. Two types of pre-existing dislocations

are considered, these are single-arm sources and screw dislocation segments.

The former aid in the introduction of source truncation eﬀects, while the

latter allow for source exhaustion eﬀects.

Single-Arm Sources. A dislocation with a pinning point in the interior of the

crystal and a free end on the surface acts as a source of dislocations in small

pillars and is commonly termed as Single-Arm Source (SAS) (Parthasarathy

et al., 2007). In the present model, these sources are randomly distributed

in the pillar volume, assigning to each source a particular slip plane. The

critical resolved stress for activating a SAS is determined by its arm length

λ,

τSAS

CRS S =αSAS µ b

λ+τ0(6)

where αSAS is a source-strength coeﬃcient that considers the nature of

the source, its length and material, here taken as 0.6 after (Rao et al., 2007).

µis the shear modulus, bthe Burgers vector and τ0the friction stress. In

their presentation of the SAS model, Parthasarathy et al. (2007) include a

Taylor-type term in eq. 6 to account for dislocation forest interactions. This

term is not considered here since, for D ≤1µm specimens and typical dislo-

cation densities of the order of 1012m−2−1013m−2, the number of dislocations

present in the pillar becomes too small to homogenize their eﬀect (Nd≤10).

Under such conditions, weak interactions are expected. Thus, we essentially

neglect hardening. For pillar diameters above 1 micron, hardening should be

incorporated, as explained in Sec. 4.

In order to generate a random arrangement of SAS, for each source a

random pining point is generated in the pillar interior. Then, a slip plane

among the characteristic active planes of the metal lattice is randomly as-

signed. The source is deﬁned as a straight line of length λconnecting the

pinning point with the nearest point in the elliptical boundary resulting from

the intersection of the plane with the pillar surface. For more details on this

model, the reader is referred to (Parthasarathy et al., 2007).

This model has been widely used on experimental studies (Abad et al.,

2016), DDD simulations (Hu et al., 2019), Crystal Plasticity simulations (Gu

et al., 2021), as well as on recent discrete slip plane models (Wijnen et al.,

2021). In the present work, it is assumed that single-arm sources remain

immutable after activation. Changes in SAS could be potentially included in

the model, for example the incorporation of SAS destruction as proposed in

the work by Cui et al. (2015) on coated submicron pillars.

Screw dislocation segments. In addition to SASs, we also consider the possi-

bility of pre-existing pure-screw segments. In essence, such dislocations are

not pinned and can sweep across the slip plane, exiting the cylinder as they

reach its surface. Under these operating conditions, screw segments allow

to capture the so-called ”dislocation-starvation” behavior (Greer and Nix,

2006). In this case, the resolved stress necessary to move the dislocation is

only the lattice friction,

τscrew

CRS S =τ0(7)

The position of the screw dislocation within the pillar cross-section is

randomly chosen. As shown in section 3.3, this allows capturing diﬀerent

cases of mechanical annealing observed in the literature, with variable stress

drop due to the activation of these sources. It is worth noting that in our

implementation, dislocation cross-slip due to image forces in the vicinity of

the free surfaces is not considered. This is further discussed in Sec. 4.

2.2.2. Surface nucleation model

For pillars and nanowires in the range of a few tenth of nanometers

to a few hundred of nanometers, virtually defect-free microstructures are

not unlikely. Under these conditions, mechanical testing of nanopillars and

nanowires reveal extremely high tensile stress and atomistic simulations point

to surface nucleation of dislocations as the source of plasticity (Zhu et al.,

2008). In consequence, the framework developed includes surface nucleation

eﬀects.

The surface nucleation model adopted here was initially proposed after

atomistic simulations (Zhu et al., 2008; Ryu et al., 2011) and later adopted

on discrete dislocation dynamics (DDD) simulations (Ryu et al., 2015; Hu

et al., 2017, 2019). Brieﬂy, within a time span ∆t, the probability of a surface

nucleation event in a pillar of surface Sis given by

P=ν0exp[−F(σ, T )

kBT]·S

b2·∆t. (8)

Here, ν0is an attempt frequency and kBthe Boltzmann constant.Thus, the

ratio S

b2stands for the number of potential nucleation sites. Finally, F(σ, T )

is an activation free energy that depends on stress and temperature through

F(σ, T ) = (1 −T

)·F0(σ) (9)

with

F0(σ) = A·(1 −σ

σathm

)αSN .(10)

Here, Ais the zero-stress, zero-temperature activation energy, while αSN

and Tdare constants corresponding to the stress and temperature dependen-

cies, respectively. Tdis usually taken as half the bulk melting temperature

(Zhu et al., 2008). Finally, σathm is an athermal nucleation stress, correspond-

ing to the tensile stress required for surface nucleation at zero-temperature.

The possibility of surface nucleation is considered once P= 1.

The parameters A,Td,αSN and σathm are material-dependent and their

calibration require dedicated studies, either based on atomistic simulations

(Zhu et al., 2008) or on experiments (Chen et al., 2015). To the best of the

authors’ knowledge, there are no such studies on W. Therefore, we opt to ﬁt

these parameters to the available experimental data, following suggestions

by Chen et al. (2015). Nanopillar compression and nanowire tensile test

for [100]-oriented W are extremely scarce in the literature. Srivastava et al.

(2021) report yield stresses of the order of 3.5 GPa in their 100 nm diameter

pillar compression tests, whereas C´ordoba et al. (2017) report yield stresses

of 6.6 ±1.2 GPa for their 93 nm suspended wires under bending. Therefore,

Symbol Property Value Unit

TdDisorder temperature 1800 K

A zero σ, zero Tactivation energy 6 eV

αSN Fitting exponent stress dependence 4 -

σathm Athermal nucleation stress 16 GPa

Table 1: Surface nucleation model parameters.

we opted to ﬁt the parameters of eqn. 8 to a 100 nm target yield strength

of 4.5 GPa, in between the mean value of Srivastava et al. (2021) and the

lower bound of the range reported by C´ordoba et al. (2017). The result-

ing parameters are included in Table 1 after the ﬁtting procedure presented

in the Supplementary Material ﬁle. The surface nucleated dislocations are

considered as pure screw segments, as described in the previous subsection.

2.2.3. Dislocation mobility and displacement rates

The displacement rate of the pillar in the slip direction sidue to the slip

produced in a plane iby the movement of a dislocation can be obtained using

a reasoning similar to Orowan’s equation,

p=bivi

Disi(11)

where biand viare the Burgers vector modulus, and dislocation velocity

corresponding to source i, respectively. Dicorresponds to the length in

direction siswept by a single dislocation when fully crossing the slip plane

i. If the unit vector estands for the pillar axis and Dfor its diameter, then

Di=D/(ni·e), and the vertical displacement rate corresponds to

˙ui

p=˙

p·e=bivi

D/ni·esi·e=biviMi

D(12)

where Miis the Schmid factor of slip system associated to source iprojected

in direction e⊗e.

In order to compute the displacement rate for each plane i, the associated

dislocation velocity vimust be determined. Dislocation-mediated plasticity

in bcc metals can take place in 48 bcc slip systems including the {110},{112}

and {123}families of slip planes. Interestingly, some works propose the

Symbol Property Value Unit

µ0Attempt frequency 9.1 1011 s−1

h′Distance between Peierls valleys √6/3a0

wKink pair width 11 b

ζMean dislocation segment width 25 b

τPPeierls stress 2.03 GPa

∆H0Kink-pair energy at 0 K 1.63 eV

pEnergy proﬁle parameter 0.86 -

qEnergy proﬁle parameter 1.69 -

BPhonon drag coeﬃcient 9.8 10−4Pa*s

Table 2: Parameters for the dislocation mobility law. Obtained from molecular dynamics

simulations (Cereceda et al., 2013; Stukowski et al., 2015; Cereceda et al., 2016; Po et al.,

2016)

decomposition of {112}and {123}slip planes on alternating {110}slip planes

(Christian, 1983; Marichal et al., 2013; Ruestes et al., 2014). Based on this,

and for the sake of simplicity, we consider bcc metals with only {110}slip

planes (¯

b= 1/2a0<111 >dislocations). That is, |¯

b|is unique and equal

to a0√3/2, with a0the bcc lattice parameter.

The mobility laws for bcc tungsten dislocations have been deeply studied

using atomistic simulations (Cereceda et al., 2013; Stukowski et al., 2015;

Cereceda et al., 2016; Po et al., 2016). For tungsten, and considering rela-

tively low temperatures, dislocation velocities can be estimated by:

v=(µ0h′(ζ−w)

bexp(−∆H(τeff )

kBT),∆H > 0

τeff b

B,∆H≤0(13)

∆H(τef f ) = ∆H0⌈1−(τeff

τ∗

)p⌉q(14)

In the expression, µ0is an attempt frequency, h′is the distance between

Peierls valleys, ζis the mean dislocation segment length for the slip system, w

is the kink-pair width and ∆H0is the kink-pair formation energy. Parameters

pand qallow for the adjustment of the ”tail” (τeff ≃0) and ”top” (τef f ≃τ∗)

of the energy proﬁle ∆H, respectively (Kocks et al., 1975). The parameters

for W are presented in Table 2.

The upper part of the deﬁnition (∆H > 0) corresponds to the kink-

pair thermally-activated regime, typical of bcc metals (Kocks et al., 1975).

The lower part, in turn, corresponds to the phonon-drag-dominated regime,

typical of high stress / high strain-rate conditions (Po et al., 2016). In Eq.

(14) τeff represents the eﬀective resolved stress on the plane i,

τeff =|τi| − τi

a(15)

where τiis the resolved shear stress, averaged over the plane, for source i

resulting from the Cauchy stress projected in the particular system to which

that source belongs to,

τi=σ: (si⊗ni).(16)

In Eq. (15) τais the resistance to slip due to athermal barriers, as dislocation

groups, precipitates, which are not considered in the present study (τi

a= 0).

Finally, τ∗represents the part of the resistance to slip due to thermally

activatable obstacles. In the case of pure W the only thermal barrier is the

Peierls resistance, hence τ∗=τP.

In essence, the use of eq. 13 for the calculation of the pillar displace-

ment rates (eq. 12) allows to incorporate the thermally-activated motion

character of screw dislocations at low stresses and temperatures, as well as

the phonon-drag-dominated regime for newly surface-nucleated dislocations

under high stresses. Equations 12-13 allow for the eﬀect of each plane iin

the displacement rate of the pillar.

2.3. Kinetic Monte Carlo procedure for stochastic slip events

In our model, the activation of the previously described microscopic slip

mechanisms is controlled by a kinetic Monte Carlo selection process. Con-

sidering a displacement-controlled micropillar compression experiment with

an applied strain rate of ˙ε0, the velocity of the upper part of the pillar cor-

responds to ˙u0=H˙ε0. This displacement is accommodated by either elastic

or plastic deformation. The plastic contribution to the displacement of the

pillar corresponds to the superposition of the displacement produced by slip

in the active planes

˙up=X

˙ui

with ˙ui

pgiven by eq. 12. The elastic displacement rate then corresponds to

˙uE= ˙u0−˙up.(17)

Yu et al. (2021) proposed a rejection-free kinetic Monte Carlo algorithm

to obtain the stress-strain evolution in a deforming single crystal in which

the strain rates of the diﬀerent slip systems provide the set of event rates.

In this work, this idea is generalized to a full deforming specimen solving

the macroscopic stress-strain response as well as the spatial distribution of

the ﬁelds involved. At time tn, each of the iescape pathways has a rate

constant ˙ui

p, that characterizes the probability per unit time that the system

escapes to that state i. These rates can be used to assemble an array of

partial sums representing the accumulated rate of all the objects up to and

including object j,

rj=

i=1

˙ui

p+r0(18)

with r0= ˙ε0Then, the total escape rate is the sum of all the rates

rtot =

i=1

˙ui

p+r0(19)

Nbeing the total number of sliding planes considered. In agreement with

rejection-free kMC theory, the next event to be executed is the kth process

satisfying

rk−1< ξ1rtot ≤rk(20)

If the event selected corresponds to the rate r0, the next event will cor-

respond to an elastic event, i.e. no inelastic deformation will be introduced

in the plastic layers. Otherwise, a plastic event is chosen, for which the

corresponding eigenstrain will be applied. The pillar geometry, array of dis-

placement rates, and the resulting pillar shape after the whole process are

exempliﬁed in Figure 2.

Then the next time step is computed as:

δtn+1 =−logξ2∆ε∗

rtot

(21)

such that the time is advanced as:

tn+1 =tn+δtn+1 (22)

ξ1and ξ2are random numbers uniformly distributed in (0,1] and ∆ε∗is a

normalization factor (∆ε∗<= 1) to adjust the time step within physical time

scale bounds (Yu et al., 2021). Once an event is executed, the strain ﬁelds

and stress ﬁelds are updated.

Figure 2: Schematic of stochastic selection process. a) a number of plastic regions are

randomly selected both for pre-existing dislocations (blue) and possible surface nucleation

sites (red). b) Upon loading and based on eqns. 12 through 19, a sampling array of events

in built. The total rate rtot is composed of the prescribed applied strain rate r0and the

vertical displacement rates of each plastic region. In general, the value of each vertical

displacement rate ˙ui

pis diﬀerent from the rest of the displacement rates during each time

step, as represented by the diﬀerent box sizes. c) After selection of a number of events

using rejection-free kMC, the ﬁnal deformed state of the pillar reﬂects the operation of

pre-existing dislocations as well as a surface nucleation site. Arrow indicates most probable

event.

Time scale bounds. The absolute time scale emanating from the sampling

of eq. 17 using eqns. 18 through 21 represents the maximum time step

compatible with the rate equation under consideration. As shown by Yu

et al. (2021), it is indeed possible to use an arbitrarily smaller time step

without invalidating the method. A physically-reasonable upper bound for

the time step is that of the total time for a single dislocation to sweep the

pillar section:

ti=Di

vi(23)

In addition, the existence of tialso poses a constraint. On each plastic

event, the eigenstrain applied cannot be larger than that of eq. 4, which

corresponds to a dislocation fully sweeping a pillar on a time ti. Thus, tiis

an upper bound for the time increment δtn+1. As consequence, δtn+1/ti≤1.

In practice, the eigenstrain applied once a plastic event is selected is:

εEI G (i)(x) = ((δtn+1

ti)( b

h) (ˆs⊗ˆn), if xϵ plastic region

0, elsewhere (24)

Weakest link principle and deformation localization. Under the weakest link

principle, plastic ﬂow occurs when the stress is high enough to activate the

weakest source available in the crystal (Norﬂeet et al., 2008). In our model,

weaker sources with a lower activation stress will tend to produce higher dis-

location velocities (eq. 13) and thus render higher plastic displacement rates

(eq. 12). In consequence, a sample with several SASs, out of which one is

signiﬁcantly longer than the others, will result in a higher probability of acti-

vation of such source, thus favoring localization in the corresponding plastic

region. In contrast, for a sample with several SASs of commensurate length,

the kMC selection process will translate into the sequential activation of sev-

eral SASs, favoring delocalization of deformation into several plastic regions.

These aspects are further discussed in Sec. 4 after the results presented in

Sec. 3.

2.4. Solving the mechanical problem with an FFT algorithm

In order to obtain the strain and stress distribution in the pillar, which

drives the activation of slip in the plastic regions, an elastic problem with

eigenstrains has to be solved at each time step. An FFT-based approach will

be used for this purpose.

The total strain is the sum of the eigenstrains (input) and the elastic strain

εe, which appears to enforce total strain compatibility and stress equilibrium,

ε=εe+εEI G.(25)

Under linear elasticity, stress is related to strain through

σ=C:εe=C: (ε−εEI G) (26)

where Cis the stiﬀness tensor of the crystal.

The mechanical equilibrium corresponds to

∇ · [C: (ε−εEI G)] = 0(27)

and rearranging terms, the result is a linear diﬀerential equation

∇ · [C:ε] = −∇·[C:εEIG].(28)

If the right-hand side of Eq. (28) is known, as in our case, the total strain ε

can be directly obtained by the convolution of this term with the correspond-

ing Green’s function derivative. Moreover, if the medium is homogeneous and

periodic, closed expressions for Green’s function exist in Fourier space, and

the solution is just a multiplication.

However, when considering the present problem in a FFT framework,

the actual geometry has to be embedded in a cuboidal periodic domain Ω

which also contains the outer free space. In this case, the domain can be

considered as an heterogeneous microstructure formed by two phases and

this microstructure can be described by

C=(C1, if xin Ω1

C2, if xin Ω2

(29)

In the simulation domain, the metallic pillar occupies the region (Ω1) and

is surrounded by an inﬁnite compliant medium which occupies the region

(Ω2). For the solution of Eq. 28, considering Eq. 29, we follow the Fourier-

Galerkin approach. The details of the method can be found in (Vondˇrejc

et al., 2014; Zeman et al., 2017) and here only the ﬁnal equations are recalled.

The problem is discretized in a box of [Nx, Ny, Nz] voxels which correspond

to the same number of frequencies in the discrete Fourier space. The value of

C, strain and stress ﬁelds are approximated by trigonometrical polynomial,

whose coeﬃcients are given by the FFT of the value of these ﬁelds at the

voxels. Let Gbe the projection operator that, by convolution, extracts the

compatible part of a tensor ﬁeld (Vondˇrejc et al., 2014). The mechanical

equilibrium of the discrete stress ﬁeld translates into,

G ∗ σ=0(30)

where ∗indicates a convolution. Introducing the value of σfrom eq. 26,

G ∗ (C(x) : (ε−εE IG)) = 0.(31)

The total strain εis decomposed into a homogeneous average strain ten-

sor, represented by the macro-scale applied strain ET, and a periodic ﬂuctu-

ating micro-scale strain ﬁeld ˜ε, which is the unknown.

ε=˜ε+ET(32)

Combining the last two equations results in a linear system in which the

strain ﬂuctuation at the voxels are the unknowns.

G ∗ (C(x) : ˜ε) = −G ∗ (C(x):(ET−εEI G)) (33)

This equation can be solved easily by transformation to Fourier space where

the operator has a closed expression and convolutions are transformed into

products. If a linear discrete operator GC(•) is deﬁned as

GC(•) = F−1(ˆ

G:F(C(x):(•))) (34)

with Fand F−1the discrete Fourier transform operations and ˆ

Gthe projec-

tion operator in Fourier space, the equation to solve is

GC(˜ε) = −GC(ET−εEIG)).(35)

To solve this linear problem eﬃciently, an iterative solver has to be used. The

convergence rate and quality of the solution depend on the phase property

contrast, which here is inﬁnite because one of the phases is empty. Following

Lucarini et al. (2022) this problem can be solved eﬃciently and with minimal

noise by using a discrete projection operator (here rotated discrete G-operator

(Willot, 2015)) and a minimum residual iteration method as linear solver.

It is important to highlight that the framework presented is thought to

study general geometrical domains in 3-D with heterogeneous microscopic

ﬁelds. While this may not be regarded as fundamental for a simple cylin-

drical pillar, it is essential for the modeling of complex geometries, such as

nanoporous metals and nanoarchitected meta-materials.

For a set of initialization parameters, our method can be summarized by

the algorithm 1 presented in the Appendix.

3. Application to single-crystalline W pillar compression

Tungsten is a prototypical refractory metal with a high melting point

and good corrosion resistance. Its outstanding mechanical properties at high

temperatures, together with its low sputtering rate, make this material an

ideal candidate for plasma-facing components in fusion energy devices (Rieth

et al., 2013). In addition, tungsten is one of the most studied bcc metals

by micropillar compression testing (Schneider et al., 2009; Kim et al., 2010;

Abad et al., 2016; Srivastava et al., 2021) and its dislocation mobility laws are

well-documented (Cereceda et al., 2013, 2016; Po et al., 2016). These aspects

make W an ideal candidate to test the applicability of our framework.

Single crystalline tungsten pillars were simulated using the framework

described in the previous section. The pillar dimensions were varied in the

range from 25 nm to 1 µm at a strain-rate of 10−3s−1. The crystalline orien-

tation and loading axis were consistent with the [100] direction. In all cases,

an initial dislocation density of ρ= 5·1012 m−2was taken. Note that such an

election renders an initial dislocation count Ndin the range of 1 for the small-

est pillars to 8 for the largest ones. The number of potential nucleation sites

was taken as NSN = 128. The parameters of the surface nucleation model

are presented in Table 1. The parameters of the physically-based dislocation

mobility law are presented in Table 2. A list of the slip systems considered

can be found in Table A.3. The models used were discretized in a grid of

128 ·128 ·128 voxels with a kMC normalization factor of ∆ε∗= 10−3. See

Supplementary Material ﬁle for sensitivity of the results to the discretization.

In particular, we focus on three important aspects: size eﬀects (Sec.3.1),

strain-rate eﬀects (Sec.3.2) and mechanical annealing eﬀects (Sec.3.3). For

validation purposes, the results of the model are later compared with exper-

imental results in Sec. 4.

3.1. Size eﬀects

Figure 3 presents the stress-strain curves for all the simulations performed

in this study. For each of the pillar dimensions probed, ten simulations were

conducted. Blue lines correspond to source truncation (SAS) dominated

cases, whereas red lines correspond to surface nucleation-dominated ones.

The envelope of the mechanical response for all the stress-strain curves is

illustrated by the corresponding colored areas. Black lines correspond to the

average of the results.

For the largest pillars, with a diameter of 1 µm, the plastic regime is

characterized by a rather smooth output with minimum scatter among the

curves. This is a consequence of having several single-arm sources distributed

along the sample, allowing the activation of several slip planes in a sequential

way. For 500 nm diameter pillars, the plastic regime is again smooth, but now

the curves appear more scattered, as judged by the upper and lower limits of

the blue-shaded region. For a smaller pillar diameter, the average number of

dislocations is close to one. As their pinning position is randomly selected,

signiﬁcant scatter is then translated to the loading curves. The mean ﬂow

stress is higher than before, as expected for a decreased pillar diameter.

For 200 nm diameter pillars, again a signiﬁcant scatter is seen for the load-

ing curves. Interestingly, one case rendered a signiﬁcantly high ﬂow stress,

associated to surface nucleation of dislocations instead of the activation of

the SAS mechanism. This corresponds to the red curve, for which a SAS

length of 8 nm was randomly generated. The short length thus renders an

activation stress that is higher than that of the SN model. This behavior is

further analyzed in the Supplementary Material.

For 100 nm diameter pillars, now several cases appear both for the ac-

tivation of SA sources (blue) as well as surface nucleation (red). Both the

mean ﬂow stresses for the SAS cases as well as the average (black) increase

with respect to the previous pillar diameter. For 50 nm diameter pillars, the

surface nucleation mechanism is dominant, with few cases of activation of SA

sources at higher stresses compared to previous dimensions. Finally, for 25

nm diameter pillars, surface nucleation was found to be the only operating

mechanism. Note that as surface nucleation becomes the dominant mecha-

nism, the curves show smaller scatter. This is consistent with experimental

observations (Kiener and Minor, 2011; Huang et al., 2011).

Figure 3: Stress-strain response for W micro/nanopillars under compression. For each

of the pillar dimensions probed, ten simulations were conducted. Shaded regions corre-

spond to the envelope of the mechanical response for all the stress-strain curves, where

blue corresponds to single-arm source dominated cases and red corresponds to surface

nucleation dominated cases. Blue and red lines correspond to the average of each subset

(SAS-dominated and SN-dominated, respectively), whereas black lines correspond to the

average of all the results, irrespectively of their nature.

3.2. Strain-rate eﬀects

In order to evaluate the strain rate sensitivity of the ﬂow strength, W

pillar simulations were conducted at diﬀerent strain rates (10−3/s, 10−2/s

and 10−1/s) and for diﬀerent pillar diameters (1 µm, 500 nm, 200 nm and

100 nm). Figure 4 presents the logarithm of the ﬂow stress as function

of the logarithmic applied strain rate obtained by the model for pillars of

diﬀerent diameter and compared to that of Srivastava et al. (2021) at room

temperature. The log-log ﬂow stress data was then ﬁtted with a power-law

function of the strain rate, σ∝˙ϵm. The strain rate sensitivity parameter m

is then m=∂(lnσ)/∂(ln ˙ϵ).

The resulting curves are approximately linear in the double-logarithm

diagram, indicating a strain rate sensitivity mapproximately constant in the

strain range explored. Comparison with Srivastava et al. (2021) shows not

only similar strain rate sensitivity parameters but also a similar transition

trend from positive to null values as the pillar diameter decreases.

Figure 4: Size-dependent strain-rate eﬀects on the ﬂow stress of W pillars. Empty circles

and dashed lines correspond to experimental results by Srivastava et al. (2021). For

comparison purposes, the colors are the same as the ones used in Srivastava et al. (2021)

and correspond to: blue - 1 µm; red - 500 nm; magenta - 200 nm; green - 100 nm.

3.3. Mechanical annealing

In addition to single-arm sources, pre-existing pure screw dislocations can

also be considered, as presented in the methods section. These are valuable

for capturing mechanical annealing eﬀects (Shan et al., 2008; Huang et al.,

2011; Shan, 2012) associated with dislocation starvation (Greer and Nix,

2006). Figure 5 presents three distinctive stress-strain curves corresponding

to 100 nm diameter pillar simulations where one pre-existing screw dislo-

cation has been considered. The red curve corresponds to a pillar without

SASs and a screw segment located in the middle of the cross-section (SN-

dominated - case 1), while the green curve corresponds to a pillar without

SASs and a screw segment located halfway between the center of the pillar

and its surface (SN-dominated - case 2). In both cases, after the opera-

tion of the screw dislocation, the source has exhausted. In consequence, the

simulated test continues elastically until activation of the SN mode. A simi-

lar scenario would have been found for a case in which a pre-existing screw

dislocation habits a pillar with a SAS of suﬃciently small length. In our

implementation, the position of the screw segment is randomly chosen in the

pillar cross section considering a uniform distribution. Speciﬁc knowledge

of the material under consideration would allow to modify such distribution

accordingly and without invalidating our methodology. Diﬀerences in load

drops for the activation of screw segments are discussed in Sec. 4. In con-

trast, the blue curve corresponds to a pillar with a screw dislocation in the

middle of the cross section and a suﬃciently long SA source. Upon loading,

and after an initial elastic stage, the screw dislocation is activated -recall

that the critical resolved shear stress is lower than that of SASs -. After

exhaustion of the screw source, the mechanical response proceeds elastically

until reaching a stress corresponding to the CRSS of the SA source, which

gets activated several times for the rest of the simulation.

Figure 5: Stress-strain response for 100 nm diameter pillar simulations containing a pure

screw dislocation. Red and green curves correspond to pillars without SASs and a screw

dislocation located in the middle of the cross section (red - case 1) and halfway between

the center of the pillar and its surface (green - case 2). Blue curve corresponds to a pillar

with a suﬃciently long SAS and a screw dislocation located in the middle of the cross

section.

4. Discussion

Comparison with other methods. Several aspects of our proposed frame-

work are worth discussing. The framework proposed aims at modelling the

full elastic-plastic response with 3D resolution of microﬁelds. Its particular

application to nano-pillars and submicron-pillars allows to obtain the size

eﬀect in the yield like other simple models (Parthasarathy et al., 2007; Jen-

nings et al., 2013), but here the microﬁelds and post yield behavior are also

recovered, with the aided beneﬁt of obtaining the 3-D deformation pattern

(see below). Our model can therefore be placed in between DDD approaches

and continuum descriptions, such as crystal plasticity. Regarding CP, con-

tinuum microﬁelds are used but deformation is localized here. In this aspect,

our method shares a similarity with the concept of thin discrete slip bands

used by Wijnen et al. (2021) but the nature of our model is stochastic and

based on the displacement discontinuity created by slip events. Like in DDD

approaches (El-Awady et al., 2009; Srivastava et al., 2013; Cui et al., 2014;

Ryu et al., 2013, 2015, 2020), important aspects of the dislocation mobil-

ity laws and underlying physics are considered. The selection of the plastic

events does not take place using the weakest-link principle, like in (El-Awady

et al., 2009; Wijnen et al., 2021), but a Monte Carlo selection process over

the displacement rates calculated using an approach similar to Orowan’s

equation.

However, unlike DDD approaches, dislocations are not modeled explicitly,

but their eﬀect is introduced by means of the concept of Eshelby inclusions,

such that the eﬀect of dislocation gliding is introduced through an equiva-

lent eigenstrain. The total strain is thus the superposition of two ﬁelds, (ε

and the inelastic strain obtained as a sum of eigenstrains due to slip events

εEI G). Therefore, the calculation of the strain and stress ﬁelds becomes lin-

ear, in contrast to models based on localized crystal plasticity as (Wijnen

et al., 2021). This aspect, together with an inherently fast FFT solver, re-

sults in an eﬃcient and computationally cheap stochastic framework capable

of simulating experiments with 3D spatial resolution at strain rates lower

than 10−3/s on nano and submicron samples within a few hours on an oﬀ-

the-shelf workstation. The present implementation is based on small strains

since we are focusing on the initial ﬂow of the pillars, but can be extended to

ﬁnite strains by assuming a non-linear elasticity energy and solving the FFT

problem in ﬁnite strains (Zeman et al., 2017). Other aspects for further ex-

tension of the model include non-Schmid eﬀects. This could be done following

steps already taken with recent advances on the development of a physically-

informed continuum crystal plasticity model for tantalum (Lee et al., 2023).

The framework would also allow for the incorporation of twinning as a com-

plementary deformation mechanism. Discrete dislocation plasticity models

have proven useful for the understanding of dislocation slip-mediated twin-

ning mechanisms (Wang et al., 2022) and could be used as input for the

extension of the framework presented here.

Flow stress size dependence. The proposed framework allows to obtain

the statistical distribution of the mechanical response of a pillar as a function

of its size, as shown in Fig 3. To analyze these results, in Figure 6 the

ﬂow stress for diﬀerent pillar sizes obtained with our model is compared to

selected experimental references (Schneider et al., 2009; Abad et al., 2016;

Srivastava et al., 2021; Kim et al., 2010) in a double logarithm representation.

Flow stress values presented correspond to the average stress in the plastic

regime of the curves presented in Fig. 3, while vertical bars correspond

to standard deviation of all the results presented in Fig. 3, capturing the

dispersion of results produced by the stochastic selection of the SAS length.

Schneider et al. (2009) performed load-controlled compression tests on [100]-

oriented W pillars spanning from the nm scale (200 nm) to the micrometer

scale (6 µm). Later on, the same group produced an experimental study

probing temperature eﬀects on W pillars with similar orientation (Abad et al.,

2016). Our model successfully captures these experimental measurements as

well as an overall σ∝1/λ dependence of the yield strength in the 200

nm - 1 µm range (black dashed line). Srivastava et al. (2021) performed

displacement-controlled compression tests on [100]-oriented W nanopillars in

the range of 100 nm - 1 µm. Their results are systematically lower than

those reported in Schneider et al. (2009); Abad et al. (2016) yet following a

similar yield strength trend increase with decreasing size (σ∝1/λ). Kim

et al. (2010) performed displacement-controlled compression tests on [100]-

oriented W nanopillars in the range of 200 - 900 nm. Their results represent

a lower bound for the experimental data set chosen. Yet their values again

follow similar trends (σ∝1/λ). To the best of the authors knowledge, there

are no reports on [100]-oriented W pillars with diameters below 100 nm,

probably due to the associated experimental diﬃculties. At the limit of a

100 nm, we predict higher yield stresses than those reported in Srivastava

et al. (2021), yet our estimations agree with their measurements if we take

into account the standard deviation.

For diameters below 200 nm, the exponential dependence σ∝1/λ yields

to a surface nucleation-dominated trend (red dashed line), indicating a crossover

between predictions from the SAS and SN models. The critical sample size

observed is approximately 120 nm. Below this threshold, the ﬂow stress de-

termined by the SN model is lower than that by the SAS model, highlighting

the comparative diﬃculty of activating a single-arm source over nucleating

a dislocation from a free surface, emphasizing a preference for surface nu-

cleation. In contrast, when dealing with samples that exceed this critical

size, plasticity is typically ruled by the activation of pre-existing disloca-

tion sources, such as those represented by the SAS mechanism. A handful

of previous studies had focused on the critical size for bcc and fcc metals

using both experimental and computational techniques. For fcc materials,

Hu et al. (2019) used DDD simulations to predict the the ﬂow stress and

dominant yielding mechanisms in Cu nanopillars (100 nm < D < 800 nm),

reporting on a cross-over from SAS activation to SN operation as D reaches

≈110nm. Similarly, Shan et al. (2008) utilized transmission electron mi-

croscopy (TEM) to investigate the behavior of pre-existing dislocations in Ni

pillars of diﬀerent diameters. The authors observed that in the case of a 160

nm diameter pillar, the pre-existing dislocations gradually exited the pillar,

followed by the emergence of new dislocations, which can be attributed to

the phenomenon known as SN. Conversely, in the case of pillars with a di-

ameter of 290 nm, dislocations persisted even after undergoing compression,

which can be tentatively associated with SAS. For bcc metals, experiments

conducted on bcc Mo pillars suggest a critical size of around 200 nm (Huang

et al., 2011; Shan, 2012). We conclude that our critical size prediction for

[100]-oriented W pillars is in agreement with previous observations for other

metals at these scales.

Figure 6: Log-log size-dependence of the ﬂow stress compared with experimental literature.

Black dashed line indicates 1/λ dependence, in agreement with a SAS dominated regime.

Blue dashed line indicates the estimated power law exponent, while light-blue shadow

indicates ±values (αW≈0.31 ±0.11). Red dashed line indicates a surface nucleation

(SN) trend. Its intersection indicates a cross-over transition size. Vertical bars correspond

to standard deviation.

The results can be interpreted through the well-known power law relation,

namely

σy=σ0+k1D−αW(36)

where σ0is the strength of the bulk material, k1is a constant and αWis the

power law exponent. In practice, σ0is usually neglected (Brinckmann et al.,

2008; Abad et al., 2016; Srivastava et al., 2021) as the second term in eq. 36

dominates at these small scales. As a reference, Srivastava et al. (2021) report

on exponent of 0.319 for their 10−3s−1experiments, close to the αW= 0.33

value obtained by Abad et al. (2016) for the same strain rate. In turn, Kim

et al. (2010) informed a power law exponent in the range of 0.33 < αW<0.55.

On a lower end, Schneider et al. (2009) report on α1= 0.21. By linear ﬁtting

eq. 36 on the SAS-dominated regime (D > 120 nm), we obtain an exponent

αW≈0.31±0.11, in agreement with most of the experimental evidence. This

further supports the ability of the single-arm source model to understand the

power-law relation on the strength vs size dependence of micropillars (Lee

and Nix, 2012). The red dashed line, connecting the average yield stress

values obtained for the smaller pillars, presents a slope that seems to be an

indication of a possibly weaker size dependence.

The underlying cause for the values of αWare worth a discussion. Brinck-

mann et al. (2008) used the screw dislocation cross-slip mechanism concept,

derived from high strain-rate MD and DD simulations (Weinberger and Cai,

2008; Greer et al., 2008), to provide an explanation for their bcc Mo pil-

lar compression test results. They argued that the relatively low power law

exponent of bcc metals (αM o = 0.45) could be attributed to such eﬀects.

Interestingly, Kim et al. (2010) performed pillar compression tests on W, re-

porting on strikingly similar power-law exponents (αW= 0.44 ±0.11). One

could then assume that the same mechanism takes place in bcc W. Yet, our

power-law exponent estimations are on the lower bound of this regime, while

our model does not incorporate a single characteristic of the cross-slip mecha-

nism presented in (Weinberger and Cai, 2008). Therefore, our results suggest

that screw dislocation mobility alone could indeed provide adequate power-

law exponents provided the underlying physics of the dislocation mobility

law are fully incorporated ( eq. 13).

It is worth highlighting that previous experimental observations on bcc

Mo suggest that size eﬀect itself has a strong size eﬀect (Huang et al., 2011;

Shan, 2012). In their in-situ TEM bcc Mo studies, Huang et al. (2011) found

a strong increase in the power law exponent as the pillar diameter decreases

beyond 200 nm. In contrast, we predict a break down in the power-law

exponent as the W pillar diameter decreases below 200 nm. This diﬀerence

in behavior could be attributed to diﬀerent activation energies between Mo

and W. Small variations in activation energy can strongly inﬂuence yield

stress at the nanoscale. A break down in exponent can also be inferred

from dislocation dynamics simulations on Cu (Hu et al., 2019). We attribute

the break down in power law exponent to the change of mechanism, from

SAS-dominated to surface nucleation dominated, for which a weaker stress

dependence on diameter is expected (Zhu et al., 2008; Huang et al., 2015;

Hu et al., 2019). Certainly, this alternative explanation of the results leaves

space for future research and informed discussion.

Strain-rate sensitivity. Our results show a size dependence of the strain

rate sensitivity parameter, which changes from positive mto nearly null val-

ues (Figure 4). Huang et al. (2015) and Srivastava et al. (2021) observed the

same positive to null trends for their bcc Fe and W pillar experiments, re-

spectively. The strain rate sensitivity exponent mdepends on the activation

volume Vaand on the resolved stress on the active plane τthrough (Kocks

et al., 1975):

m=kBT

τVa

.(37)

In the previous equation, both the critical shear τand the activation vol-

ume Vacan be size dependent, and the origin of size dependency of mmight

have both contributions. Regarding the eﬀect of activation volume, Huang

et al. (2015) rationalized that in Fe this might be the main contribution to size

eﬀect in m. This size dependent min Fe arises from a competition between a

size-independent friction term -that rules for large pillar diameters (D >500

nm)- and a size-dependent term -that rules for smaller pillar diameters-, with

larger activation volumes for increasingly smaller pillars (100 nm <D<500

nm). They also argue that as the pillar diameter further decreases (D <100

nm), the activation volume becomes again smaller. Still, Fe is known to have

an activation volume that may display larger variations with stress compared

to other bcc metals with higher melting point (Christian and Masters, 1964).

Tungsten, in constrast, and particularly for low homologous temperatures,

displays a very low activation volume (Va≤10b3) (Kiener et al., 2019), con-

sistent with the dominance of the kink mechanism at low temperatures. Our

calculations of the activation volume (see Suppl. Mat.) render that for the

kink-pair activated mechanism, the stress-dependent activation volume is in

the range of 4 b3< V kink

a<11 b3for the stress range of the SAS-dominated

regime, whereas for the surface nucleation mechanism is in the range of 2.5

b3< V SN

a<4.5 b3for the stress range of the SN-dominated regime.

Therefore, considering the relatively low variations in the activation vol-

ume Vain W, changes in the strain rate sensitivity exponent mcan be un-

derstood based on the size-dependence of the stress reached on the pillar due

to the size dependent strength of SAS, as well as on the stress-dependence

of the dislocation mobility law. Similar conclusions on the importance of τ

were reached by Srivastava et al. (Srivastava et al., 2021) for their strain-rate

sensitivity studies on W pillars.

Future investigations using our proposed framework could also focus on

temperature eﬀects, as a recently developed CP model using similar disloca-

tion mobility laws has proven useful in exploring the temperature dependence

of deformation localization in irradiated tungsten (Li et al., 2021).

Mechanical annealing. Earlier MD simulation studies predict that such

behavior cannot take place in bcc pillars. Such simulations suggested that

the combined eﬀects of image forces, together with the dislocation core struc-

ture, would facilitate dislocation multiplication by a surface-assisted cross-

slip mechanism (Weinberger and Cai, 2008). SEM observations of multi-

ple slip traces on compressed bcc pillars seem to support this mechanism

(Schneider et al., 2009). On the other hand, in-situ TEM compression of Mo

nanopillars (Huang et al., 2011; Shan, 2012) and Fe–3% Si pillars (Zhang

et al., 2012) show clear evidence that mechanical annealing by dislocation

starvation can indeed take place in bcc metals. Our proposed computational

framework takes into account such scenarios. Interestingly, load drops due

to mechanical annealing are, in general, less marked than in our Figure 5, as

shown in (Shan et al., 2008). Other works do report stress drops of the order

of 0.5 - 1.0 GPa (Huang et al., 2011; Zhang et al., 2012). Both situations

can be taken into account in our method by an adequate selection of the

statistical distribution of pre-existing screw segments.

Deformation patterns and localization. Figure 7 presents the typical

deformation patterns obtained in our simulations, together with a compari-

son with a selected reference (Srivastava et al., 2021). For our largest pillar

diameter (D= 1µm), the deformation pattern reveals the activation of sev-

eral SASs on a variety of slip systems (see Supplementary Material movie), in

agreement with scanning electron microscopy (SEM) analysis of W micron-

sized pillars (Schneider et al., 2009; Abad et al., 2016). As the pillar diameter

decreases to the submicron regime (D= 500 nm and D= 200 nm) and for

the dislocation density chosen, only one SAS is present. Upon activation,

signiﬁcant deformation takes place and concentrates on a single slip band,

in agreement with SEM micrographs of W pillars with comparable diame-

ters (Srivastava et al., 2021). Further decrease of pillar diameter into the

nanometer regime (D= 100 nm) triggers surface nucleation instead of SAS

activation. The former takes place on multiple slip systems.

As explained in Sec. 2.3, our model favors the activation of weaker

sources. Localization of plastic activity in one or a few number of planes

depends on the number of potential slip sites (e.g. number of SAS) and

the statistical distribution of the slip resistance on that planes (linked with

the length of each SAS). Our 1 µm diameter pillars with ρ= 5 ·1012 m−2

have multiple SASs of variable length. They show a tendency for a rather

delocalized (homogeneous) distribution of deformation. In contrast, 200 nm

- 500 nm diameter pillars with similar dislocation density have 1 - 2 SASs on

average. In such cases, localization is favored (Figure 7). Cui et al. (2018)

presented a 2D Monte-Carlo model of DD source activation coupled with

crossslip channel widening to reproduce and physically explain the transi-

tion in the mechanism of plastic ﬂow localization in irradiated materials from

irradiation-controlled to dislocation source-controlled. They showed that as

the size decreases, the spatial correlation of plastic deformation decreases due

to weaker dislocation interactions and less frequent cross-slip, thus producing

thinner dislocation channels. For a low irradiation damage, they show that

as the diameter decreases from 1.5 µm to 300 nm, deformation transitions

from rather homogeneous to highly localized. Our results broadly agree with

this picture.

Statistical analysis of burst displacement. In order to analyze the

statistics of the plastic bursts obtained in the simulations, the complementary

cumulative distribution function (CCDF) of the burst displacement magni-

tude ∆Uis obtained. The CDDF is computed using the method presented by

Cui et al. (2016) and its result for a 1 µm diameter pillar is represented in Fig-

ure 8. The Figure suggest that, under the displacement-controlled conditions

(strain control) explored here, the system does not exhibit power-law scaling,

in agreement with Cui et al. (2016). In addition, the data spans less than two

orders of magnitude. In contrast, typical load-displacement controlled com-

pression experiments usually span several orders of magnitude, with bursts

often exceeding 10 nm (Alcal´a et al., 2020; Srivastava et al., 2021). Detailed

tracking of the plastic strain rate (Inset of Figure 8) shows that the dynamical

behavior is quasiperiodic, in agreement with displacement-controlled simu-

lations (Cui et al., 2016) and experiments (Papanikolaou et al., 2012). In

contrast, experiments under load-controlled conditions lead to a completely

diﬀerent scenario, where plasticity does not lead to marked stress drops,

allowing to trigger the operation of several SAS simultaneously and thus

exhibiting power law scaling. As a result, avalanche events can develop,

leading to a highly correlated dynamical response, consistent with the con-

cept of self-organized criticality (Bak et al., 1988). The framework presented

here, inherently implies the activation of one source at a time, leading to a

quasiperiodic response that it is consistent with displacement-control condi-

tions.

On the incorporation of strain hardening eﬀects. In all the cases pre-

sented in Figure 3, the ﬂow stress keeps constant, in agreement with the

assumptions introduced in Sec. 2. This is consistent with an scenario in

which the slip planes swept by the SAS exhibit a scarcity of dislocations in

intersecting planes, and where the pinning point of the SAS is stable un-

der the prevailing stress levels. Consequently, the activated SAS does not

encounter other defects, hence preventing the formation of new junctions

or the reduction of the source length. In addition, experiments on W sin-

gle crystalline pillars (Schneider et al., 2009) do not show hardening for the

relatively low strains explored in Figure 3.

For larger samples, our method could be extended to incorporate harden-

ing eﬀects by including a Taylor hardening-like term in eq. 6 and a disloca-

tion density evolution law (Cui et al., 2014). The latter should incorporate

size-dependent terms and bulk-like terms. Size-dependent terms include the

generation of dislocations due to the operation of existing sources (∝1/λ)

and the escape of dislocations reaching a free surface (∝ −1/D) (Cui et al.,

2014). Bulk-like terms include dislocation multiplication due to forest dislo-

cations (∝√ρ) and the annihilation of closely spaced dislocations of opposite

signs (∝ −ρ) (Devincre et al., 2008). If self and pair interactions must be

taken into account (e.g. for micron scale pillars above 1 µm), the correspond-

ing hardening coeﬃcient matrix can also be incorporated (Yu et al., 2021;

Lee et al., 2023). It must be emphasized that for submicron pillars, simple

calculations show that the contributions of the bulk-like terms are nearly an

order of magnitude smaller than the size-dependent terms (Cui et al., 2014).

Thus, by neglecting the former and provided the dislocation generation rate

is balanced with the escape rate from free surface which, together with a

relatively low dislocation density, the results presented in Sec.3.1 would be

reproduced.

To wrap up, the results presented here focus solely on tungsten and a

relatively simple geometry. Yet, these are not restrictions. For instance,

FeCrAl alloys and refractory multi-principal element alloys (RMPEAs) are

also promising candidate materials for use under extreme conditions in the

nuclear industry. FeCrAl alloys have already been probed using discrete dis-

location dynamics (Pachaury et al., 2023) as well as with crystal plasticity

approaches using Arrhenius-type rate equations (Gong et al., 2023). With

respect to RMPEAs, kink-migration of screw dislocations have been shown

to play a dominant role in their deformation (Balbus et al., 2024) and it is

expected that developments in dislocation mobility laws (Shen and Spearot,

2021) would allow for systematic investigations on these promising alloys us-

ing upper scale methods. Our proposed framework oﬀers ﬂexibility to explore

the micropillar compression behavior of such alloys by the incorporation of

speciﬁc dislocation mobility laws. To further extend its potential, the use

of a 3D FFT solver oﬀers opportunities for the exploration of a variety of

complex geometries, including nanoporous metals and nanoarchitected meta-

materials.

Figure 7: Residual deformation patterns for diﬀerent pillar diameters explored, together

with SEM observations by Srivastava et al. (2021). Reprinted with permission from Else-

vier. Frame colors correspond to dominating deformation mechanism (Blue - Single-Arm

source, Red - Surface nucleation).

Figure 8: Complementary cumulative distribution function (CCDF) of burst displacement

for one example of our 1 µm diameter pillars. Inset, corresponding evolution of plas-

tic strain rate showing typical quasiperiodic strain bursts, consistent with displacement-

controlled compression conditions. Note the breakdown from power law scaling (dashed

line).

5. Conclusions

A novel approach is presented to simulate the deformation of submicron

specimens, accounting for the discrete events produced by the slip of in-

ternal dislocation and surface nucleated ones in a stochastic manner. The

framework considers the slip events as eigenstrain ﬁelds that produce a dis-

placement jump across a slip plane and whose activation is driven by a Monte

Carlo (MC) method. Physically-based laws are incorporated to account for

activation probabilities, dislocation mobility and surface nucleation.

Two factors of stochasticity are taken into account: a random selection

of the position of defects and a random selection of plastic events after a

MC process on a sampling array of possible displacement rates, computed

using a similar concept as in Orowan’s equation. Implementation on a fast

FFT solver, results on an eﬃcient, computationally-cheap algorithm which

allows to simulate accurately the deformation of a specimen in a wide range

of strain rates and sizes in a fraction of the time needed using techniques as

MD or DDD.

The framework developed has been applied to the compression of tung-

sten (W) on [100]-oriented pillars in the 25 nm to 1 µm diameter range.

We ﬁnd an adequate agreement between simulations and experimental data.

The study a size-dependence of ﬂow stress, characterized by distinct power-

law exponents for regimes dominated by source truncation (D >120 nm,

αW≈0.31 ±0.11) and a break down of the power law scaling as surface sur-

face nucleation becomes dominant (D <120 nm). Pre-existing dislocations

naturally compete with surface nucleation of new dislocations. The former

are favored for pillars above 120 nm diameter, whereas the latter is favored

below such critical size. In addition, strain-rate sensitivity eﬀects are ade-

quately captured, including a size-dependence of the strain-rate sensitivity

exponent. This is attributed not only to changes in activation volume and

source strength but also to the stress-dependent dislocation mobility law,

whose form considers both the kink-pair regime, dominating at the relatively

low stresses found in submicron pillars, as well as the phonon drag regime,

that rules for very high stresses found in pillars of 200 nm and below. The

framework inherently implies the activation of one source at a time, leading

to a quasiperiodic response that it is consistent with displacement-control

conditions.

Finally, the framework presented here opens the possibility to provide

valuable analysis and interpretations of other small-scale testing techniques,

such as microtensile testing and microbending. Future developments of dis-

location mobility laws for FeCrAl alloys and multi-principal element alloys

would allow for a systematic investigation of these relevant metals, while

the possibility to include other micro and nanoscale geometries further ex-

tends the potential applications map to include nanolattices and nanoporous

materials, among other nanostructures of interest.

6. Acknowledgements

This project has received funding from the European Union’s Horizon

Europe research and innovation programme under the Marie Sklodowska-

Curie grant agreement no. 101062254. Funded by the European Union.

Views and opinions expressed are however those of the author(s) only and

do not necessarily reﬂect those of the European Union. Neither the European

Union nor the granting authority can be held responsible for them.

Appendix A. Appendix Section

Algorithm 1 kMC discrete slip approach with eigenstrains

1: procedure Initialization

•Deﬁne pillar geometry, C(x), in the discrete domain.

•Set positions, eigenstrain ﬁelds and strength of each plastic region i

•Set loading conditions ET(t),∆ε∗.

2: procedure Solution

3: while t < tT OT do:

4: Get rnd no. ξ1,ξ2∈(0,1]

5: Solve the equilibrium (Eq. 35) using FFT solver to obtain σ(Eq.

26) , |τi|(Eq. 16)

6: for each region do

7: if |τi|> τCRS S then

8: Compute: P(SN sites only), vi(Eq. 13) , ti(Eq. 23) and

˙ui

p(Eq. 12).

9: else

10: vi= 0

11: Compute accumulated rates rj,rtot

12: δtn+1 =−logξ2∆ε∗

rtot

13: if ξ1<r0

rtthen

14: δtn+1 =dt0

15: next event is elastic: εE IG (i)= 0

16: else

17: next event is plastic

18: Compute eigenstrain: εEIG (i)(Eq. 24)

19:

20: Update total eigenstrain ﬁeld εEI G (Eq. 5)

The slip systems considered in this study are presented in Table A.3.

iSlip system sini

1 [1¯

11](011) [1¯

11] [011]

2 [¯

1¯

11](011) [¯

1¯

11] [011]

3 [111](0¯

11) [111] [0¯

11]

4 [¯

111](0¯

11) [¯

111] [0¯

11]

5 [¯

111](101) [¯

111] [101]

6 [¯

1¯

11](101) [¯

1¯

11] [101]

7 [111](¯

101) [111] [¯

101]

8 [1¯

11](¯

101) [1¯

11] [¯

101]

9 [¯

111](110) [¯

111] [110]

10 [¯

11¯

1](110) [¯

11¯

1] [110]

11 [111](¯

110) [111] [¯

110]

12 [11¯

1](¯

110) [11¯

1] [¯

110]

Table A.3: Slip systems considered in this study

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Brittle and Ductile Deformations in Uniaxial Compression of Si Micropillars

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ACTA MATER

Nanoporous metals under extremes

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MRS BULL

Tailored control of the pattern forming instability that is initiated through selective dissolution in metal alloys has enabled the production of high-performing nanoporous metals. Nanoporous metals are “forged in fire,” evolving from exposure to harsh environments. This makes them uniquely adapted to extreme conditions with the potential for service in applications that require tolerance to corrosive environments, high-rate mechanical loading, and high-energy irradiation. Nanoporous metals are metastable structures and successful integration into extreme applications requires an atomistic understanding of how nanoporous morphologies evolve under relevant conditions. In this article, we explore our existing understanding of the evolution of nanoporous metals within extreme environments and how that understanding can be used to design mitigation strategies to drive operational longevity. Graphical abstract

Discrete slip plane analysis of ferrite microtensile tests: Influence of dislocation source distribution and non-Schmid effects on slip system activity

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Mater Des

A time-discontinuous elasto-plasticity formalism to simulate instantaneous plastic flow bursts

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Adaptation and validation of FFT methods for homogenization of lattice based materials

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COMPUT METHOD APPL M

An FFT framework which preserves a good numerical performance in the case of domains with large regions of empty space is proposed and analyzed for its application to lattice based materials. Two spectral solvers specially suited to resolve problems containing phases with zero stiffness are considered (1) a Galerkin approach combined with the MINRES linear solver and a discrete differentiation rule and (2) a modification of a displacement FFT solver which penalizes the indetermination of strains in the empty regions, leading to a fully determined equation. The solvers are combined with several approaches to smooth out the lattice surface, based on modifying the actual stiffness of the voxels not fully embedded in the lattice or empty space. The accuracy of the resulting approaches is assessed for an octet-lattice by comparison with FEM solutions for different relative densities and discretization levels. It is shown that the adapted Galerkin approach combined with a Voigt surface smoothening was the best FFT framework considering accuracy, numerical efficiency and h-convergence. With respect to numerical efficiency it was observed that FFT becomes competitive compared to FEM for cells with relative densities above ≈7%. Finally, to show the real potential of the approaches presented, the FFT frameworks are used to simulate the behavior of a printed lattice by using direct 3D tomographic data as input. The approaches proposed include explicitly in the simulation the actual surface roughness and internal porosity resulting from the fabrication process. The simulations allowed to quantify the reduction of the lattice stiffness as well as to resolve the stress localization of ≈ 50% near large pores.

Mobility of dislocations in FeNiCrCoCu high entropy alloys

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Nov 2021
MODEL SIMUL MATER SC

Dislocations in high entropy alloys (HEAs) are wavy and have natural pinning points due to the variable chemical and energetic landscape surrounding the dislocation core. This can influence the critical shear stress necessary to initiate dislocation motion and the details associated with sustained dislocation glide. The objective of this work is to determine the relationship between Schmid shear stress and dislocation velocity in single phase FCC FeNiCrCoCu HEAs using molecular dynamics simulations, with comparisons made to dislocation motion in homogeneous Ni and Cu. Simulations are performed for four different dislocation character angles: 0° (screw), 30°, 60° and 90° (edge). Several key differences are reported, compared to what is previously known about dislocation motion in homogeneous FCC metals. For example, the drag coefficient B in the phonon damping regime for HEAs has a nonlinear dependence on temperature, whereas this dependence is linear in Ni. Mobility relationships between different types of dislocations common in homogeneous FCC metals, such as the velocity of screw and 60° dislocations being lower than edge and 30° dislocations at the same shear stress, do not necessarily hold in HEAs. Dislocation waviness is measured and is found to correlate with the ability of dislocations to glide under an applied shear stress, including the temperature dependence of the drag coefficient B. These results confirm that the influence of HEA chemical complexity on dislocation motion is important and this data can be used to guide development of analytical or empirical models for dislocation mobility in HEAs.

A stochastic solver based on the residence time algorithm for crystal plasticity models

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Dec 2021
COMPUT MECH

The deformation of crystalline materials by dislocation motion takes place in discrete amounts determined by the Burgers vector. Dislocations may move individually or in bundles, potentially giving rise to intermittent slip. This confers plastic deformation with a certain degree of variability that can be interpreted as being caused by stochastic fluctuations in dislo-cation behavior. However, crystal plasticity (CP) models are almost always formulated in a continuum sense, assuming that fluctuations average out over large material volumes and/or cancel out due to multi-slip contributions. Nevertheless, plastic fluctuations are known to be important in confined volumes at or below the micron scale, at high temperatures, and under low strain rate/stress deformation conditions. Here, we develop a stochastic solver for CP models based on the residence-time algorithm that naturally captures plastic fluctuations by sampling among the set of active slip systems in the crystal. The method solves the evolution equations of explicit CP formulations, which are recast as stochastic ordinary differential equations and integrated discretely in time. The stochastic CP model is numerically stable by design and naturally breaks the symmetry of plastic slip by sampling among the active plastic shear rates with the correct probability. This can lead to phenomena such as intermittent slip or plastic localization without adding external symmetry-breaking operations to the model. The method is applied to body-centered cubic tungsten single crystals under a variety of temperatures, loading orientations, and imposed strain rates.

Temperature dependent deformation localization in irradiated tungsten

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INT J PLASTICITY

Tungsten is considered as the plasma-facing material in the nuclear fusion device, which has to withstand coupled irradiation, mechanical and thermal conditions. To guarantee the safe operation of energy devices, it is essential to accurately predict its response. As a typical body center cubic material, the plasticity of tungsten is believed to exhibit the strong temperature dependence and non-Schmid effect. It is far from well understood how these plasticity features influence the behavior of irradiated tungsten. To disclose this mystery, a crystal plasticity model is developed, which considers the kink-pair mechanism of screw dislocations, the contribution of edge dislocations, and the interaction between dislocations and irradiation defects. The model is firstly verified through comparing with the available experimental results at different temperatures, and then used to disclose the temperature dependence of deformation localization in irradiated tungsten. It is found that post-irradiation mechanical tests at room temperature underestimate the occurrence of dislocation channel and deformation localization at higher temperatures, while in the low temperature regime, the thermal-activated dislocation mobility law may lead to a short-term deformation localization without clear dislocation channels. Conditions for the localization of deformation in irradiated tungsten are discussed, which is hoped to guide the reliable design of tungsten components used in fusion and fission reactors.

A discrete slip plane model for simulating heterogeneous plastic deformation in single crystals

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May 2021
INT J SOLIDS STRUCT

In small-scale mechanical tests, such as micropillar compression tests, plastic deformation is often localized in narrow slip traces. These slip traces result from a few dislocation sources with relatively low nucleation stresses that are present in the material. In order to accurately simulate such small-scale experiments, the stochastics of the underlying dislocation network must be taken into account, which is usually done by performing discrete dislocation dynamics simulations. However, their high computational cost generally restricts these simulations to small and simple geometries and small applied displacements. Furthermore, effects of geometrical changes are usually neglected in the small strain formulation adopted. In this study, a discrete slip plane model for simulating small-scale experiments on single crystals is proposed, which takes the most important characteristics of dislocation plasticity for geometries in the micrometer range into account, i.e. the stochastics and physics of dislocation sources. In the model, the properties of all lattice planes are sampled from a probability density function. This results in a heterogeneous ow stress within a single crystal, unlike the uniform properties assumed in conventional crystal plasticity formulations. Moreover, the slip planes can be grouped together in bands via a weakest-link principle. The resulting equations are implemented in a standard crystal plasticity finite element model, using a finite deformation formulation. Within this setting, only the collective dislocation motion on glide planes is modeled, resulting in a significantly lower computational cost compared to frameworks in which the dynamics of individual dislocations are considered. This allows for simulating multiple realizations in 3D, up to large deformations. A small case study on micropillar compression tests is presented to illustrate the capabilities of the model.

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Bridging microscale to macroscale mechanical property measurements of FeCrAl alloys by crystal plasticity modeling

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Deformation, dislocation evolution and the non-Schmid effect in body-centered-cubic single- and polycrystal tantalum

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Evolution of partial dislocation slip–mediated deformation twins in single crystals: a discrete dislocation plasticity model and an analytical approach

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Jan 2022
INT J PLASTICITY

Deformation twinning is an essential plastic deformation mechanism that realizes the trade–off between strength and ductility. Twins nucleate and grow by the coordinated slip of partial dislocations on consecutive {111}–type slip planes. The route of twin growth is responsible for the evolution of twin morphology, which affects the non–coplanar dislocation slips by adjusting the twinning–associated mean free path. Incorporating such twinning mechanisms is critical for the accurate modelling and simulation of deformation behavior. In this study, a discrete dislocation plasticity (DDP) model was developed by integrating the source introduction methods and source activation criteria of Shockley partial dislocation. In the model, two twin nucleation mechanisms, i.e., the internal source and surface source, were considered concurrently, and the additional effect of stacking fault energy on the motion of partial dislocations was introduced. The evolution of partial dislocation slip–mediated deformation twins in micron–sized pillars of twinning–induced plasticity steel under uniaxial compression was investigated. The predicted twin morphologies and stress–strain curves from DDP simulation both agreed well with the experimental results, highlighting the inherent characteristics of partial–dislocation–based twinning behavior. The simulation results showed that the formation of nanometer–sized sharp twin tips was caused by the strong interaction between the front and rear dislocations on adjacent slip planes. In addition, a novel analytical model verified with the DDP simulation was proposed by considering the kinetics of the newly formed twin embryos. The competition between new twin activation and near–twin merging in determining the evolution of twin thickness was analyzed using the analytical model. The dependence of flow stress and twin morphology on the density and distribution of internal sources was demonstrated by considering the new twinning route. This research thus advances the understanding of partial dislocation slip–mediated twinning mechanisms.

A stochastic discrete slip approach to microplasticity: Application to submicron W pillars

Abstract

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