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Journal of Nuclear Materials 571 (2022) 154 002
Contents lists available at ScienceDirect
Journal of Nuclear Materials
journal homepage: www.elsevier.com/locate/jnucmat
Nanoindentation experiment and crystal plasticity study on the
mechanical behavior of Fe-ion-irradiated A508-3 steel
Pandong Lin, Junfeng Nie
∗, Meidan Liu
Institute of Nuclear and New Energy Technology, Key Laboratory of Advanced Reactor Engineering and Safety of Ministry of Education, Tsinghua University,
Beijing 10 0 084, PR China
a r t i c l e i n f o
Article history:
Received 13 February 2022
Revised 5 August 2022
Accepted 29 August 2022
Available online xxx
Keywo rds:
Nano indentation
CPFEM
Hardness
Irradiation
Dislocation loop
a b s t r a c t
This study investigates the mechanical properties of A508-3 steel irradiated with Fe ions to 0.1, 0.4, 2.0,
and 5.0 dpa at 20, 100 , and 300 °C using a nanoindentation experiment and crystal plasticity finite
element model (CPFEM). The Nix–Gao model is applied to obtain the hardness H
0 based on the mea-
sured data. The hardness of the steel increases with increasing radiation damage at all temperatures. The
simulations of the nanoindentation process by CPFEM loaded in the [001], [110], [ 111], [112], and [123]
directions concurred with the corresponding experimental data in all the studied cases, validating the
CPFEM results. The dislocation loop expedites the increase of mobile dislocation and retards the decrease
of immobile dislocation, resulting in larger von Mises stresses of the irradiated samples with a flatter
shape, especially near the irradiated region. Furthermore, the area of the von Mises stress shrinks with
an increase in temperature. This study could help understand the macroscopic deformation behavior of
irradiated steel based on experimental and microstructural analyses.
© 2022 Elsevier B.V. All rights reserved.
1. Introduction
Considered an indispensable structural component of the nu-
clear fission reactor, the reactor pressure vessel (RPV) is ex-
posed to prolonged radiation that causes irradiation harden-
ing/embrittlement [1] . Irradiation hardening/embrittlement occurs
because of matrix defects, including clusters, dislocation loops, and
vacancy-solute atom complexes, due to the low Cu content in the
RPV steel A508-3 [ 2 , 3 ]. However, the mechanisms of defect evolu-
tion are not clear. For example, it is unclear how the irradiation-
induced defects hinder the movement of dislocations and result in
reduced ductility and increased strength [4] . Therefore, to illumi-
nate the effect of irradiation on the mechanical properties of A508-
3 steel, the various methods used to study the irradiation effect
and defect evolution must be investigated systematically.
Various micromechanical testing methods, including micropillar
compression, in situ tensile tests, and nanoindentation, have been
developed to investigate the micromechanical response of materi-
als [5–10] . For example, Heczel et. al [5] used micropillar compres-
sion to investigate the effect of chemical inhomogeneities on the
local mechanical behavior. The stress–strain behaviors were found
to be identical despite the differences in the chemical composition.
Li et. al [7] tested the Q235 specimen by the in-site tensile method.
∗Corresponding author.
E-mail address: niejf@tsinghua.edu.cn (J. Nie) .
The results indicate that there was a decrement in the value of
the tensile strength with the increase in the depth and scratch an-
gle. Among these methods, nanoindentation is convenient for the
rapid estimation of mechanical properties, especially at the micro
or nanoscales, when it is difficult to obtain bulk material samples
suitable for the other methods.
The nanoindentation test could be used to determine the
hardness and elastic stiffness modulus of materials. Zhang et al.
[11] studied the irradiation behavior of Fe-ion-irradiated Chinese
A508-3 steel at room and high temperatures by nanoindentation.
The steel had become harder after high-temperature irradiation
because of the formation of solute clusters. Su et al. [12] studied
the influence of grain boundary on plastic deformation by nanoin-
dentation near the grain boundaries. They found that boundaries
with higher angles resulted in a decrease in the pile-up topog-
raphy. Liu et. al [13] found that the average hardness increased
with the irradiation dose for China A508-3 steel, which was ir-
radiated by protons. Based on the Kasada method, the relation-
ship between the positron annihilation parameter and hardness
was determined. Ding et. al [14] evaluated the irradiation hard-
ening of Chinese RPV steel A508-3, which was irradiated by 3.5
MeV 56Fe3 + ions. The hardening phenomenon was observed after
irradiation, and a power relationship between the bulk equivalent
hardness and the damage level was proposed.
However, the nanoindentation method alone produces insuf-
ficient information on local mechanical response and evolution.
https://doi.org/10.1016/j.jnucmat.2022.154002
0022-3115/© 2022 Elsevier B.V. All rights reserved.
P. Lin, J. Nie and M. Liu Journal of Nuclear Materials 571 (2022) 154002
Tabl e 1
Chemical composition of Chinese A508-3 steel (wt%).
Element C Mn P S Cu V Si Mo Ni Cr
Content 0.18 1.41 0.008 0.003 0.041 0.003 0.22 0.50 0.66 0.17
The combination of the crystal plasticity finite element method
(CPFEM) and nanoindentation has garnered research interest be-
cause it overcame this drawback. The new approach quantitatively
reveals the microstructural evolution, mechanical responses, and
deformation process under varying loads. Unlike the traditional fi-
nite element method, CPFEM, which is based on the crystal plas-
ticity theory, considers the plastic slip and lattice rotation and is
suitable for simulating the available experimental data.
Numerous studies were conducted on this topic. Zhou et al.
[15] developed a CPFEM and used it to simulate the complex local
lattice process in an equiaxed TC6 titanium alloy under nanoinden-
tation loading. They numerically predicted the nucleation and con-
tinuous growth process of the subgrains by virtually tracking the
misorientation angle map at different indentation depths. Based
on the crystal plasticity theory, Cackett et al. [16] analyzed the
spherical nanoindentation of single-crystal Cu using two different
indenter tips. They also discovered that geometrically necessary
dislocations contributed to the size effect. Liu et al. [17] devel-
oped a CPFEM to investigate the effect of the coefficient of fric-
tion (COF) on the evolution of the crystallographic texture and
mechanical behavior of an initially oriented Al single crystal dur-
ing nanoindentation. These studies led to the inference that the
deviation of indentation hardness with COF is significantly small,
and the piling-up curve decreases as the COF increases. Further-
more, Xiao et al. [18] proposed a strain-gradient crystal plasticity
theory accounting for the irradiation effect for the surface nanoin-
dentation of ion-irradiated FCC metals. The simulated force–depth
relationship with/without the irradiation effect agrees well with
experimental data. Das et al. [19] developed a crystal plasticity
finite element irradiated material. It captured irradiated-induced
hardening followed by strain-softening through the interaction
of irradiation-induced defects and gliding dislocations. Nie et al.
[20] used CPFEM to study the irradiation effect in the stress–strain
response during nanoindentation. Based on the model, numerical
results are challenged by experimental data and are found to be
in excellent agreement with the reported results. Note that CPFEM
could be applied on the indentation process for ion-irradiated ma-
terials.
In this study, A508-3 steel samples were irradiated with 3.5-
MeV Fe
3 + at 20, 10 0 , and 300 °C to increasing damage levels
(radiation dose) of 0.1, 0.4, 2.0, and 5.0 dpa. Their mechanical
properties were analyzed by nanoindentation by which the spe-
cific values of irradiation hardening could be obtained. Simulations
were then performed using a physical CPFEM. The simulated load–
indentation-depth and hardness–indentation-depth curves were
compared with the experimental data to validate the model. Fi-
nally, the microstructure evolution of A508-3 steel was investi-
gated using the proposed CPFEM. The relationships not only be-
tween experiment and simulation but also between microscale and
macroscale were derived, which promotes the study of the irradia-
tion effects of RPV steels.
2. Materials and experimental method
2.1. Materials, irradiation, and nanoindentation
The chemical composition of A508-3 steel samples used in this
study—the material constituting the RPV at the Shandong Shidao
Bay 200-MWe high-temperature gas reactor (HTGR)—are listed in
Fig. 1. Depth distribution of damage calculated by SRIM 2013.
Table 1 , as provided by the manufacturer. Several samples with the
dimensions of 15 mm ×15 mm ×1 mm were prepared according
to the size requirement of nanoindentation and ion-implantation
beam. Next, the samples were mechanically polished with an SiC
paper (up to 800 grit), a diamond suspension (2.5 μm ), and a col-
loidal silica suspension (0.05 μm ). The surface roughness require-
ment of the sample was less than 0.01 μm . The surface was chemi-
cally polished to remove the residual stress layer produced by me-
chanical polishing. An optical microscope was used to check the
surface of the polished sample for scratches.
The samples were irradiated with Fe ions at 3.5 MeV in the ter-
minal chamber of the Sector-Focused Cyclotron (SFC) at the Heavy
Ion Research Facility in Lanzhou (HIRFL) in the Institute of Mod-
ern Physics (IMP), Chinese Academy of Sciences (CAS). A508-3 steel
sample was irradiated at 20 °C, 100 °C and 300 °C with irradiation
doses of 0.1, 0.4, 2.0, and 5.0 dpa. The damage distribution calcu-
lated by SRIM 2013 [21] with rapid calculation and E
d _ Ni
= 40 eV is
illustrated in Fig. 1 . As per the figure, the damage ranges from the
surface of incidence up to a depth of 1,50 0 nm, peaking at approx-
imately 1,0 0 0 nm.
Hardness tests were performed using the XP nanoindenter at
Tsinghua University. The profiles were measured as a function of
depth up to 2,0 0 0 nm using the continuous stiffness measurement
method with a diamond Berkovich tip with a radius of 20 nm, a
strain rate of 0.05 s
−1
, and a frequency of 45 Hz. The hardness
for each sample was obtained by averaging five measurements. The
measurement points were sufficiently spaced, and the distance be-
tween consecutive points exceeded 1 mm so as to not affect each
other. Note that all the nanoindentation experiments were initi-
ated at 20 °C, while the irradiation temperatures were 20, 100 , and
300 °C.
2.2. Experimental results and discussion
2.2.1. Elastic modulus (based on data between the indentation depths
of 500 and 1,5 0 0 nm)
Table 2 lists the Young’s moduli E of the samples at different
temperatures. Note that the temperature is the irradiation tem-
perature not the indentation test temperature. The values were
2
P. Lin, J. Nie and M. Liu Journal of Nuclear Materials 571 (2022) 154002
Fig. 2. Elastic moduli–indentation depth curves of A508-3 steels at: (a) 20 °C, (b) 10 0 °C, and (c) 30 0 °C.
Tabl e 2
Experimental elastic moduli (50 0–1,50 0 nm).
Temperature ( °C) 20 100 300
E (GPa) 231.38 223.32 218.28
Standard deviation (GPa) 3.09 8.67 11.94
directly determined using an XP nanoindenter. The value of the
specific irradiation temperature was determined by averaging all
results of both the unirradiated and irradiated cases because the
elastic properties after irradiation remained the same, within the
error bars, as with those before the irradiation [22] , as shown
in Fig. 2 (a)–(c). Furthermore, it was found that the indentation
depth has negligible effect on the elastic moduli in the range from
500 nm to 1500 nm. The method to obtain the elastic moduli is
reasonable. Hence, the elastic modulus has a negative relationship
with the irradiation temperature from Table 2 . This tendency is
echoed by the literature [23] . Furthermore, the experimental elas-
tic moduli were used as the inputs of the CPFEM in Section 3 .
2.2.1. Hardness depth profiles of A508-3 steel
To exclude uncertain factors from the surface, we disregarded
data from regions shallower than 50 nm. Fig. 3 illustrates the typ-
ical profiles of average hardness versus indentation depth of the
A508-3 steel samples before and after Fe-ion irradiation. For a spe-
cific sample, the indentation hardness significantly decreased at
shallower depths and then slightly decreased in deeper regions.
The samples were significantly hardened by irradiation, causing a
strong increase in indentation hardness with the dpa. This trend
could be attributed to the formation of irradiation-induced defects.
The 5-dpa sample irradiated at 300 °C exhibited the largest peak-
hardness value. The temperature also affected the hardness of the
A508-3 steel. For example, an increase in the hardness with the
temperature was observed when the indentation depth was less
than 10 0 0 nm, which is echoed with Ref. [11] . Subsequently, the
value of hardness reaches the adjacent level.
2.2.3. Nix–Gao model analysis
In the indentation size effect (ISE), small indents yield higher
hardness readings than large indents. This trend is particularly pro-
nounced in nanoindentation [ 24 , 25 ]. The Nix–Gao model describes
the ISE, in which the hardness depth profile is given by the follow-
ing equation [26] :
H = H
0
1+
h
∗
h
1 / 2
(1)
where H is the hardness, H
0 is the hardness limit when the in-
dentation depth becomes indefinitely large, h
∗is the characteristic
length, and h is the indentation depth. Fig. 4 illustrates H
2 vs. 1 /h
plot. Note that H
0 is the square root of the intercept for the linear
fitting of hardness data on the surface near region. Furthermore,
the value of H
0 is not influenced by the size effect and can be
used as a parameter to characterize the hardening effect in materi-
als under irradiation. As illustrated in Fig. 4 , the H
2 vs. 1 /h plot in
the measured depth region exhibits a strong linear relationship for
the unirradiated sample and a bilinear relationship for the Fe-ion
irradiated sample. The results with the irradiated sample deviate
3
P. Lin, J. Nie and M. Liu Journal of Nuclear Materials 571 (2022) 154002
Fig. 3. Hardness–indentation depth curves of A508-3 steels at: (a) 20 °C, (b) 100 °C, and (c) 300 °C (d) 20, 100 and 300 °C with 5.0 dpa.
Tabl e 3
Bulk equivalent hardness H
0
of A508-3 steel at different irradiation temperatures and radiation doses.
Bulk equivalent hardness H
0
(GPa) Characteristic length h
∗(nm) Irradiation dose (dpa)
0 0.1 0.4 2.0 5.0
Irradiation temperature ( °C) 20 H
0 2.52 3.40 3.41 3.53 3.59
h
∗85.7 83.3 149.7 95.6 103.2
100 H
0 2.51 3.89 4.20 3.98 4.01
h
∗62.9 34.8 29.9 151.0 117.5
300 H
0 2.59 2.68 3.63 5.52 5.53
h
∗51.7 308.7 63.1 33.5 160
from the linear relationship in deeper regions, which could be due
to the softer substrate effect [27] .
When the plastic zone extends to the lower damaged or un-
damaged region, the linear fitting in the corresponding depth re-
gion deviates from the previous one, yielding a bilinear relation-
ship. Table 3 lists the values of H
0
and h
∗for all samples. At
first, an obvious irradiation hardening was obtained for the A508-
3 steel, which is consistent with the results of previous studies
on V-4Ti after H and He irradiation [28] and Fe-based ferritic al-
loys after Fe-ion irradiation [27] . For a given material and indenter
geometry, H
0
depends on the dislocation density [26] , indicating
that an increase in the dislocation density after irradiation can re-
sult in increased hardness in the irradiated material. This tendency
is similar with those of irradiated ODS steel, MA956, and V-4Ti
[ 28 , 29 ]. Note that the primary defect responsible for hardening is
the dislocation loop, which impedes the dislocation glide [ 30 , 31 ].
This is plausible because a higher dpa level would produce more
defects, which can act as obstacles and harden the surface. This is
discussed in Section 3 . Apart from that, the depth at which data
start to deviate from the linear fitting is in negative correlation
with dpa. The SRIM simulation results show that the correspond-
ing thickness of the damage layer is approximately 1500 nm. More
defects are generated with increasing dpa; consequently, the plas-
tic zone extends to the undamaged region more quickly, leading
to a lower indentation depth. Regarding the unirradiated sample, a
good linear fitting of H
2 versus 1/ h can be found in the measured
depth region. Finally, the effect of irradiation on the h
∗is ambigu-
ous based on the experiment.
3. Crystal plasticity finite element model (CPFEM)
3.1. Crystal plasticity theory
According to classical plastic plasticity theory, the deformation
of metals consists of elastic and plastic stages [32–34] . The defor-
4
P. Lin, J. Nie and M. Liu Journal of Nuclear Materials 571 (2022) 154002
Fig. 4. H
2
–1/ h curves of A508-3 steel samples at different temperatures: (a) 20 °C, (b) 100 °C, and (c) 300 °C.
mation gradient tensor F is expressed as
F = F
∗·F
P
, (2)
where F
∗is the elastic term indicating the stretching and rotation
of the lattice, and F
P is the plastic term associated with crystallo-
graphic slip.
The velocity gradient tensor is determined by:
L =
·
F
·F
−1
= L
∗+ L
p (3)
L
∗= ˙
F
∗·F
∗−1 (4)
L
p
= ˙
F
p
·F
p−1
=
N
α=1
˙ γαn
∗αm
∗α, (5)
where superscripts
∗, p, and αdenote the elastic and plastic com-
ponents and the αth slip system, respectively; ˙ γα, n
∗α, and m
∗α
represent the plastic shear rate, glide plane normal, and glide di-
rection, respectively; and N is the total number of slip systems.
The elastic constitutive relationship proposed by Hill and Rice
[34] is described by
∇∗
σKi
= C : D
∗=
·
σKi
−W
∗·σKi
+ σKi
·W
∗, (6)
where
∇∗
σKi
represents the Jaumann derivative of the Kirchhoff stress
tensor σKi
in the intermediate configuration, C denotes the stiff-
ness tensor, D
∗denotes the stretching rate tensor, and W
∗de-
notes the rotation rate tensor.
Based on Orowan’s formula [35] , the plastic shear rate is de-
termined by the density of mobile dislocations ρα
M
, the norm of
Burgers vector b , and the mean velocity of the mobile dislocations
in the slip system v
α, as follows:
˙ γα= ρα
M
b
¯
v
α, (7)
where v
αwas defined by Kothari and Anand [36] in Kocks-type
activation form [37] . ˙ γαcan be obtained as follows:
˙ γα= 0 if
|
τα|
< g
α
˙ γα= ˙ γα
0
exp
−Q
0
kT
1 −|
τα|
−g
α
ˆ ταp
q
sgn
(
τα)
if
|
τα|
≥g
α, (8)
where ˙ γα
0
is the reference rate; Q
0
is the activation energy for glide
under zero stress; k is the Boltzmann constant; T is the tempera-
ture in Kelvin; ˆ ταis the maximum thermal resistance to slip (equal
to ˆ τα
0
G/G
0
, where ˆ τα
0
is the initial value of thermal resistance to-
ward slip); G
0 and G are the elastic shear modules at 0 K and cur-
rent temperature T , respectively; ταis the resolved shear stress
that can be determined by Schmid’s law [38] as τα= σ: μα; p
and q are parameters related to the shape of the activation en-
thalpy curve; and g
αis the thermal resistance to slip, which is
characterized by two components: (i) the interactions between dis-
locations and (ii) the interactions between dislocations and dislo-
cation loops. The hardening equation based on Taylor’s hardening
law [ 33 , 39 ] is expressed as
g
α= Gb
q
ρ
N
β=1
A
αβρβ
dis
+ q
i
N
α
i
d
α
i
, (9)
5
P. Lin, J. Nie and M. Liu Journal of Nuclear Materials 571 (2022) 154002
Fig. 5. (a) Geometrical model for the nanoindentation of A508-3 steel with an indenter. The indented section is a cylinder with a height of 30 μm and a radius of 15 μm .
(b) Meshes of the sample’s computational domain.
Fig. 6. Nonuniformly distributed dislocation loop near the irradiated sample surface. (a) Number density of dislocation loop N and (b) diameter of dislocation loop d .
Fig. 7. The distribution of number density and diameter of dislocation loop with 1, 2, 4 and 8 layer.
where q
ρdenotes the dislocation barrier strength parameter. A
αβ
is the matrix of dislocation interaction coefficients between differ-
ent slip systems, ρβ
dis
is the total dislocation density of the βth slip
system, which is linearly related to the mobile dislocation density
ρβ
M
and the immobile dislocation density ρβ
I
( ρβ
dis
= ρβ
M
+ ρβ
I
). q
i
is
the dislocation loop strength parameter [40] , and N
α
i
and d
α
i
are
the number density and mean size of the dislocation loop, respec-
tively.
The evolution of a mobile dislocation ρα
M
is expressed as [41] :
˙ ρα
M
=
k
mul
bl
d
|
˙ γα|
−1
bλ|
˙ γα|
−2 R
c
b
ρα
M
|
˙ γα|
. (10)
The first term on the right-hand side of Eq. (10) represents the
dislocation product [42] , where k
mul
is the multiplication coeffi-
cient of mobile dislocation and l
d
is the mean length of the mobile-
dislocation fragments. The second term represents mobile disloca-
tions trapped as immobile ones [43] , where λis the mean trapping
distance. The last term represents dislocation annihilation occur-
ring in opposite directions [44] .
The evolution of the immobile dislocation density ρα
I
is mod-
eled as [41] :
˙ ρα
I
=
1
bλ|
˙ γα|
−k
dyn
ρα
I
|
˙ γα|
, (11)
6
P. Lin, J. Nie and M. Liu Journal of Nuclear Materials 571 (2022) 154002
Fig. 8. Mises stress contour at 10 0 °C in loading direction of [001] with indentation depth of 10 0 0 nm at (a) 1-layer, (b) 2-layer, (c) 4-layer and (d) 8-layer.
Fig. 9. Load–indentation-depth and hardness–indentation-depth curves for samples with different layer at 20 °C. (a) Load–indentation-depth curve and (b) hardness–
indentation depth curve.
The last term on the right-hand side of Eq. (11) denotes the
dynamic recovery of immobile dislocation [45] , and k
dyn
is the dy-
namic recovery coefficient of immobile dislocation.
Based on the molecular dynamics study of dislocation loops
[ 46 , 47 ], the loop was divided into two parts: a full-absorption and
partial-absorption dislocation loops, which are defined as follows:
N = f N
F
+ ( 1 −f ) N
P
. where fis the fraction of full-absorption
dislocation loops.
Based on the absorption formula proposed by Patra [40] , the
equation of evolution of full-absorption dislocation, where the di-
ameter of the dislocation loop dis constant and the number den-
sity of the dislocation loop Ndecreases, may be modified as fol-
lows:
˙
N
αd
α= ξαR
α
i
b
(
N
αd
α)
c
(
ρα
M
)
1 −c
|
˙ γα|
, (12)
where R
α
i
is the absorption radius, c is a material constant, and ξα
is the absorption probability, which is determined by the molecular
dynamics method [48–51] and is equal to 1 −exp ( −ω
αt
1
) . Here,
t
1 is the dislocation loop-dislocation interaction time, and ω
αis
the absorption frequency of the dislocation loop and follows the
Arrhenius-type equation:
ω
α=
b
d
α
i
v
D
exp
−G
k
B
T
, (13)
where v
D is the Debye frequency, and G is the activation energy,
which is determined by
G = Q
0
1 −|
τα|
−g
α
ˆ ταp
q
. (14)
For a given dislocation loop with a diameter d, a small dislo-
cation fragment will remain if d ≥d
C
( d
C is the critical diameter),
where the number density of the dislocation loop Nis constant
and the diameter of the dislocation loop ddecreases, whereas the
loop can be completely absorbed into a superjog on the mobile
dislocation when d < d
C
, where the diameter of dis constant and
the number density of Ndecreases.
If d ≥d
C
, the evolution of the dislocation loop is expressed as
follows:
N ˙
d = N V
s
V
d
t
= N d
π−d
2
d
b
|
˙ γ|
, (15)
where dis the variation of d, tis the interaction time, V
s
(=
L
dis
¯
v t d) is the total volume of the swept region for a dislo-
cation with total length L
dis
and average velocity ¯
v [52] , and V is
the total volume of the object element. Assuming that the inter-
action annihilates half of the dislocation loop [47] , the equivalent
diameter variation of the dislocation loop after the interaction is
d = d /π−d / 2 . Moreover, ρM
(= L
dis
/V ) and ¯
v (= | ˙ γ| / ( b ρM
)) .
7
P. Lin, J. Nie and M. Liu Journal of Nuclear Materials 571 (2022) 154002
Fig. 10. Load–indentation-depth and hardness–indentation-depth curves for samples at 20 °C. (a) Load–indentation-depth curve for unirradiated sample, (b) hardness–
indentation depth curve for unirradiated sample, (c) load–indentation-depth curve for sample with 0.1 dpa, (d) hardness–indentation-depth curve for sample with 0.1 dpa,
(e) load–indentation-depth curve for sample with 0.4 dpa, (f) hardness–indentation-depth curve for sample with 0.4 dpa, (g) load–indentation depth curve for sample with
2.0 dpa, (h) hardness–indentation-depth curve for sample with 2.0 dpa, (i) load–indentation-depth curve for sample with 5.0 dpa and (i) hardness–indentation-depth curve
for sample with 5.0 dpa.
8
P. Lin, J. Nie and M. Liu Journal of Nuclear Materials 571 (2022) 154002
Fig. 11. Load–indentation-depth and hardness–indentation-depth curves for samples at 100 °C. (a) Load–indentation-depth curve for unirradiated sample, (b) hardness–
indentation-depth curve for unirradiated sample, (c) load–indentation-depth curve for sample with 0.1 dpa, (d) hardness–indentation-depth curve for sample with 0.1 dpa,
(e) load–indentation-depth curve for sample with 0.4 dpa, (f) hardness–indentation-depth curve for sample with 0.4 dpa, (g) load–indentation-depth curve for sample with
2.0 dpa, (h) hardness–indentation-depth curve for sample with 2.0 dpa, (i) load–indentation-depth curve for sample with 5.0 dpa and (i) hardness–indentation-depth curve
for sample with 5.0 dpa.
9
P. Lin, J. Nie and M. Liu Journal of Nuclear Materials 571 (2022) 154002
Fig. 12. Load–indentation-depth and hardness–indentation-depth curves for sam-
ples at 300 °C. (a) Load–indentation-depth curve for unirradiated sample, (b)
hardness–indentation-depth curve for unirradiated sample, (c) load–indentation-
depth curve for sample with 0.1 dpa, (d) hardness–indentation-depth curve for sam-
ple with 0.1 dpa, (e) load–indentation-depth curve for sample with 0.4 dpa, (f)
hardness–indentation-depth curve for sample with 0.4 dpa, (g) load–indentation-
depth curve for sample with 2.0 dpa, (h) hardness–indentation-depth curve for
sample with 2.0 dpa, (i) load–indentation-depth curve for sample with 5.0 dpa, (j)
hardness–indentation-depth curve for sample with 5.0 dpa, (k) the enlarged pic-
ture of load–indentation-depth curve for sample with 5.0 dpa and (l) the enlarged
picture of hardness–indentation-depth curve for sample with 5.0 dpa.
If d < d
C
, the expression of the evolution of the loop is similar
to that of a full-absorption dislocation loop and is written as
˙
N d =
R
i
b
(
N
i
d
i
)
c
(
ρM
)
1 −c
|
˙ γ|
, (16)
Based on Eqs. (15) and (16) , the evolution of the partial-
absorption dislocation loop can be obtained as:
N ˙
d = N d
π−d
2
d
b
|
˙ γ|
d ≥d
C
˙
N d =
R
i
b
(
N
i
d
i
)
c
(
ρM
)
1 −c
|
˙ γ|
d < d
C
, (17)
3.2. CPFEM setup
3.2.1. Nano-indentation model and boundary conditions
The crystal plasticity model from the previous section was ap-
plied to simulate the nanoindentation of the A508-3 steel sam-
ples under Fe-ion irradiation. The constitutive equations were im-
plemented in the UMAT subroutine of ABAQUS. The sample was
considered a cylinder with a height and radius of 15 μm each
( Fig. 5 (a)). Fig. 5 (b) illustrates the mesh structure of the computa-
tional domain. The sample contains 28,434 elements of C3D8. The
bottom of the sample was fixed, and the remaining surfaces were
unconstrained. The indenter was considered a rigid body, and an
arc with a radius of 20 nm encircled the tip region. The interac-
tion between the indenter tip and the sample surfaces was set to
frictionless because it was too weak to affect the load–depth rela-
tionship [53–55] . Deo [56] indicated that both the dislocation loop
number density and the size at the initial stage of defect accumu-
lation were proportional to the square root of the cumulative dpa,
i.e., N = A dpa
1 / 2 and d = B dpa
1 / 2
, where A = 5 ×10
13
mm
−3 and
B = 3 . 7 ×10
−6
mm . We adopted Deo’s calculation method for this
simulation. Fig. 6 illustrates a typical graph of the defect density
distribution. Note that this is an empirical relation fit to the ex-
perimental observations and does not hold for high ( > 10) dpa lev-
els; it only satisfies the irradiation condition of this experiment.
The hardness can be calculated as a function of indentation depth
from the load–indentation-depth relationship following the stan-
dard method of Oliver–Pharr [57] , i.e., H = F
max
/A
c
, where F
max and
A
c
(= 24 . 56 h
2
c
) are the maximum load and true contact area, re-
spectively.
3.2.2. The effect of the irradiated layer numbers on simulated result
As have been noted above, the depth distribution of ion irradi-
ation damage is heterogeneous. The number density and diameter
of dislocation loop is related to dpa. In order to portray this, the ir-
radiated region could be divided some layers where the dpa in the
same layer is uniform. In this part, the effect of the irradiated layer
numbers is investigated. Fig.7 is the distribution of number density
and diameter of dislocation loop with 1, 2, 4 and 8 layer where the
value of total orange-yellow area is the same. It could be concluded
that the shape of Mises Stress contour is similar to each other from
the Fig.8 . But the transition region of different stress level in irra-
diated region is more flat in 8-layer case. Apart from that, the dif-
ference of load–indentation-depth and hardness–indentation-depth
curves between different layer is minor in Fig.9 . Furthermore, the
8-layer is the closest to the actual distribution of ion irradiation
damage versus depth among them. Such being the case, 8-layer is
accepted in the next study which is adequate to portray the effect
of ion irradiation.
3.2.3. Simulation parameters selection
As mentioned earlier, we simulated single BCC A508-3 steel.
The material constants and parameters in the constitutive equa-
10
P. Lin, J. Nie and M. Liu Journal of Nuclear Materials 571 (2022) 154002
Fig. 13. Numerical distribution of von Mises stress in different loading directions at 20 °C under an irradiation damage of 2.0 dpa for an indentation depth of 300 nm. (a)
[001]; (b) [110]; (c) [ 111 ]; (d) [112]; (e) [123].
tions are as follows. G is dependent on temperature and is es-
timated from the following expression [58] : G = −3 ×10
−5
T
2
−
5 . 6 ×10
−3
T + 87 . 6 ( G in GPa and T in K). The initial value of the
critical resolved slip resistance was 390 MPa [59] for all the slip
systems. The initial densities of mobile and immobile dislocations
are 7 ×10
6
mm
−2
. The other parameters used in this calculation are
presented in Tables 4 and 5 .
3.3. Simulation results and discussion
3.3.1. Load–indentation and hardness–indentation-depth relationships
of A508-3 steel
The proposed CPFEM was applied to simulate the nanoinden-
tation behavior of unirradiated and irradiated samples when the
perspective was loaded along the [001], [110], [111], [112], and
11
P. Lin, J. Nie and M. Liu Journal of Nuclear Materials 571 (2022) 154002
Fig. 14 . Evolutions of the von Mises stress (MPa), mobile and immobile dislocations (mm
−2
), and full- and partial-absorption dislocation loops (mm
−2
) at different indenta-
tion depths (mm
−2
) under 0.4 dpa at 20 °C.
[123] directions. As illustrated in Figs. 10 , 11 , and 12 , the nu-
merical results for both the load and hardness-indentation depth
curves matched the corresponding experimental data under differ-
ent irradiation damage levels at 20 °C, 100 °C and 300 °C, respec-
tively. In fact, the experimental result is the average values. The
final simulated values should be averaged by five crystal orienta-
tions. The final simulated result is not displayed in the form of fig-
ure which is due to the fact that the numerical results for both
the load and hardness-indentation depth curves at different orien-
tation matched the corresponding experimental data well and is
aiming to display the crystal anisotropy more clearly. During the
loading stage, the load applied to the indenter increased with the
indentation depth, which is primarily attributed to the increasing
contact area and hardening behavior induced by the dislocation
and dislocation loop ( Eq. (9) ). For simplification, the unloading part
was omitted as it was not the focus of this study. Moreover, crys-
tal anisotropy could be observed clearly by comparing the load–
indentation depth and hardness-indentation depth relationships in
different loading directions, which concurs with a previous study
[ 18 , 62 ]. The model does not account for the cross-slip of screw
dislocations. Cross-slip of screw dislocations will be investigated
by MD method, which is the major job in the future. The results
without cross-slip is acceptable. For example, we could see crystal
anisotropy from the enlarged picture in Fig.12 (k) and (i). Accord-
ing to the definition of the relationship between hardness and load
in Section 3.2 , hardness is inversely proportional to the square of
the indentation depth but directly proportional to the load. There-
fore, in the region near the surface, the difference in load could
12
P. Lin, J. Nie and M. Liu Journal of Nuclear Materials 571 (2022) 154002
Fig. 15 . Evolution of von Mises stress (MPa) at different indentation depths.
Tabl e 4
Model parameters.
Parameter Value Units Used in Re ference
p0.47 - Eqs. (8) and (14) [36]
q 1.1 - Eqs. (8) and (14) [36]
˙ γα
0
1 ×10
7 s
−1 Eqs. (8) and (14) [60]
G
0 87,600 MPa Eqs. (8) and (14) [60]
G 82,534 MPa Eqs. (8) , (9) and (14) [60]
ˆ τα
0
390 MPa Eqs. (8) and (14) [59]
Q
0 1 . 21 ×10
−19 J Eqs. (8) and (14) [60]
k 1 . 38 ×10
−23 J/KEqs. (8) and (14) Constant
T 293 K Eqs. (8) and (14) Constant
q
ρ0.06 - Eq. (9) [60]
q
i 1 - Eq. (9) [60]
A
αα 1 - Eq. (9) [40]
A
αβ 0.2 - Eq. (9) [40]
k
mul
0.0955 - Eq. (9) [60]
R
c 1 . 5 ×10
−6 mm Eq. (9) [60]
k
dyn
275 - Eq. (11) [60]
be magnified by the proportional coefficient 1 /h
2
c
. Once the plastic
zone has extended the unirradiated region, the difference in load
can be small, as the unirradiated substrate contributes to harden-
ing.
3.3.2. Microstructure evolution of sample with and without
irradiation
Fig.13 depicts the numerical distribution of the von Mises stress
in five indentation directions, [001], [110], [111], [112], and [123]
for an indentation depth of 300 nm. All the directions displayed
strong crystal anisotropy, indicating the significant influence of
crystal orientation on the stress-field distribution. Moreover, the
von Mises stress was symmetrically distributed in the indenta-
tion directions [001], [110], and [111], which concurs with a pre-
vious study [63] . The largest plastic zone occurred in [111], indi-
cating that the crystallographic orientations that can drive plastic
deformation in the direction of the indentation axis through active
slip systems are more sensitive to the substrate effects. It is pos-
sible that dislocation glide in the in-plane slip direction resulted
in the varied stress field. The overall stress values and distribu-
tion in the region near the indenter with an indentation depth
of 300 nm were similar, regardless of the indentation direction.
Consequently, the differences in the load–indentation-depth and
hardness–indentation-depth were obscure. When the indentation
depth was less than 200 nm, the stress field near the indenter var-
ied, which is reflected in the considerable difference in the load–
indentation-depth and hardness–indentation-depth curves for dif-
ferent crystal orientations.
Tabl e 5
Parameters for dislocation loop evolution.
Parameter Value Units Used in Re ference
c 0.8 - Eqs. (12) , (16) and (17) [60]
R
α
i
1 . 5 ×10
−6 mm Eq. (12) [61]
d
α
i
3 . 0 ×10
−6 mm Eq. (13) [61]
v
D 1 ×10
13 /s Eq. (13) Constant
t
1 20 °C 1 . 0 ×10
−6 s Eq. (13) Obtained
by
MD
method
100 °C 8 . 0 ×10
−7
300 °C 6 . 0 ×10
−7
d
c 20 °C 2 . 5 ×1 0
−6 mm Eq. (17) Obtained
by
MD
method
100 °C 3 . 0 ×1 0
−6
300 °C 3 . 5 ×1 0
−6
13
P. Lin, J. Nie and M. Liu Journal of Nuclear Materials 571 (2022) 154002
Fig. 16. Evolutions of mobile and immobile dislocation densities under 0.1, 0.4, 2.0, and 5.0 dpa, respectively. The indentation depth is 20 0, 30 0, and 40 0 nm in the [110]
direction.
Fig. 17. Evolutions of full- and partial-absorption dislocation loop densities under 0.1 and 5.0 dpa. The indentation direction is [001], and the indentation depth is 1,4 0 0 nm.
14
P. Lin, J. Nie and M. Liu Journal of Nuclear Materials 571 (2022) 154002
Fig. 18. Effect of tem perature on the distribution of the von Mises stress in the unirradiated sample. The indentation depth is 1,0 0 0 nm, and the indentation direction is
[123].
The evolution of macroscopic behavior is closely related to
the microstructure during the indentation process. Fig. 14 (a)–(y)
presents the evolution of the von Mises stress, mobile and im-
mobile dislocation densities, and full- and partial-absorption dis-
location loop densities at different indentation depths under 0.4
dpa at 20 °C, respectively. To highlight the contrast, the evolu-
tion of the von Mises stress of the sample without irradiation at
20 °C is presented in Fig. 15 . Hence (1) when h = 0 nm, the mo-
bile and immobile dislocations were constant for all the samples
according to the definition in Section 3.2 . The full- and partial-
absorption dislocation loops were initialized as with the experi-
mental conditions and were non-uniformly distributed in the irra-
diated layer. The plastic deformation around the indenter triggered
the evolution of the mobile and immobile dislocations and full-
and partial-absorption dislocation loops based on the constitutive
laws in Eqs. (10) , (11) , (12) , and (17) . The density of mobile dislo-
cations increased with an increase in the indentation depth be-
cause dislocation multiplication extended dislocation annihilation
and pinning, and the increase was evident near the indenter tip.
In contrast, the density of immobile dislocations decreases.
Note that the total dislocation density gradually increased dur-
ing the process. The densities of the full- and partial-absorption
dislocation loops gradually decreased due to the varied absorption
mechanisms. (2) The von Mises stress values of the irradiated sam-
ple were larger and the shape flatter, especially near the irradi-
ated region. This is closely related to the evolution of the dislo-
cation and dislocation loop density. Expecting a distinct increase
in the mobile density near the indenter tip, a larger mobile dis-
location density could also be observed in the irradiated region.
The molecular dynamics study of the interaction between disloca-
tion and dislocation loop indicated the generation of a superjog,
leading to an increase in the dislocation length. Therefore, the ef-
fect of the dislocation loop on the evolution of the mobile disloca-
tion density is considered in the mean length of the mobile dislo-
cation fragments l
d
, namely l
d
= 1 /
βρβ
M
+ N
β
i
d
β
i
. Consequently,
the contour lines extended laterally perpendicular to the indenta-
tion direction.
Distinct stratification was observed at the irradiated and unirra-
diated boundaries of the immobile dislocation-density distribution.
The effect of the dislocation loop works on the mean trapping dis-
tance, λα( 1 / λα= 1 / λα
ρ+ 1 / λα
i
= βρ
ρα
M
+ ρα
I
+ βi
N
α
i
d
α
i
), on the
immobile dislocation density. Therefore, the immobile dislocation
density decreased more gradually in the irradiated layers. The den-
sity decreased rapidly in the unirradiated region, which is the rea-
son for the distinct stratifications. Finally, the dislocation loop it-
self increased the slip resistance according to Eq. (9) , leading to
an increased stress field in the irradiated region. For the unirradi-
ated region, the superposition of mobile and immobile dislocations
caused the von Mises stress to plateau along the direction perpen-
dicular to the indentation depth. These factors contributed to the
overall value and shape of the von Mises stress contour.
Fig. 16 illustrates the evolutions of the mobile and immobile
dislocation densities under 0., 0.4, 2.0, and 5.0 dpa, respectively.
The indentation depth is h = 20 0, 30 0, and 40 0 nm in the indenta-
tion direction [110]. Hence (1) at a specific indentation depth, both
the mobile and immobile dislocation densities increase with the
increase of damage level. Based on the typical model proposed by
Nix and Gao [26] , H
0
is proportional to mobile density, whereas
h
∗is inversely proportional to it. Consequently, H
0
increases and
h
∗decreases, implying a reduced size effect with the increase in
dpa. In other words, the theoretical, simulated, and experimental
results were consistent with one another, indicating the validity of
the CPFEM; (2) A similar tendency of the evolutions of the mobile
and immobile dislocation densities in [111] is observed in [110].
Note that, at the same indentation depth, the change in the value
of the mobile dislocation density is also higher than that of the
immobile one. When h = 200 and 400 nm, the maximum value
of the dislocation loop density near the indenter tip along [111] is
marginally higher than that along [110] ( Figs. 13 and 15 ).
We analyzed the full- -and partial-absorption mechanisms at an
indentation depth h of 1,4 0 0 nm in [001], as shown in Fig. 17 . Re-
gardless of the mechanism, the dislocation loop density decreased
as the indenter was pressed to the sample. When the irradia-
tion damage was 0.1 dpa, where the total diameter of the dislo-
cation loop in the irradiated layer was less than d
C
. Partial absorp-
tion is equivalent to full absorption, according to Section 3.1 . Con-
sequently, the contours of the two cases were almost the same.
When it comes to 5.0 dpa, the diameter of dislocation loop extends
d
C as the depth ranging from 0 nm to 140 0 nm. The difference be-
tween the two loops is evident; the distribution of evolution of the
full-absorption dislocation loop was more concentrated in the re-
gion near the indenter. Note that the partial-absorption dislocation
loop density of a specific point is greater than that of full absorp-
tion, especially in the irradiated layer.
Fig. 18 (a), (b), and (c) present the distributions of von Mises
stress in the unirradiated sample at 20, 100, and 300 °C and an
indentation depth of 1,0 0 0 nm in the [123] direction. For better
comparison, the von Mises stress contours of the three tempera-
tures are drawn in the same image in Fig. 17 (d). The area of the
von Mises stress shrinks with an increase in temperature, which is
consistent with a previous study [64] . Furthermore, the existence
15
P. Lin, J. Nie and M. Liu Journal of Nuclear Materials 571 (2022) 154002
Fig. 19 . Von Mises stress for samples under different conditions. The indentation direction is [123].
of the irradiation layer leads to a higher stress level, as seen in
Fig. 19 . The distribution of stress has jagged stratification in the
irradiated layer. In contrast, a conical shape was obtained in the
unirradiated layer.
4. Conclusions
The mechanical properties of A508-3 steel irradiated with Fe
ions to 0.1, 0.4, 2.0, and 5.0 dpa at 20 °C, 100 °C, and 300 °C were
investigated by a nanoindentation experiment and CPFEM simula-
tion. The measured data was input to the Nix–Gao model to obtain
the hardness H
0
and characteristic length h
∗. Constitutive equa-
tions were implemented in ABAQUS to simulate the nanoinden-
tation process. The deformation behavior of the Fe-ion-irradiated
A508-3 steel was investigated through detailed discussions about
the microstructure evolution. The conclusions are as follows:
•The irradiation dose did not influence the elastic modulus of
A508-3 steel. The hardness of the A508-3 steel increased with
increasing irradiation damage level at all temperatures. This
indicates that irradiation-induced defects, mainly dislocation
loops, have a significant influence on dislocation gliding and
plastic behavior. Based on the Nix–Gao model, H
0
increases,
which indicates the increasing hardening with an increase in
the dpa.
•A CPFEM based on the dislocation density and non-uniformly
distributed irradiation-induced dislocation loop density for BCC
A508-3 steel was proposed. Four dominant hardening contri-
butions are involved in the framework of the classical crys-
tal plasticity theory: mobile and immobile dislocations and
full-absorption and partial-absorption dislocation loops. Load–
indentation-depth and hardness–indentation-depth curves were
simulated using the proposed CPFEM. The simulations loaded
along the [001], [110], [111], [112], and [123] directions con-
curred with the experimental data for all cases, validating the
CPFEM.
•The microstructural evolution mechanisms were investigated to
understand the macroscopic deformation behavior. Evidence for
crystal anisotropy was discovered along different directions, in-
dicating the significant influence of the crystal orientation on
the distribution of the stress field. Furthermore, the von Mises
stress is symmetrically distributed along [001], [110], and [111].
The dislocation loop accelerates the increase of mobile disloca-
tion and retards the decrease of immobile dislocation, resulting
in greater von Mises stress values of the irradiated sample with
a flatter shape, especially near the irradiated region. Comparing
the two types of dislocation loops, the partial-absorption dislo-
cation loop density of a specific point was larger than that of
the full-absorption under 5.0 dpa. As for the temperature ef-
fect, the area of the von Mises stress shrank with an increase
in temperature.
( Eqn 1, 2, 3, 4, 5, 6 and 7 )
Declaration of Competing Interest
The authors declare that they have no known competing finan-
cial interests or personal relationships that could have influenced
the work reported in this paper.
CRediT authorship contribution statement
Pandong Lin: Conceptualization, Formal analysis, Investigation,
Methodology, Validation, Visualization, Data curation, Writing –
original draft, Writing –review & editing. Junfeng Nie: Funding
acquisition, Resources, Software, Conceptualization, Methodology,
Project administration, Supervision, Validation, Writing –review &
editing. Meidan Liu: Data curation.
Data Availability
The authors do not have permission to share data.
Acknowledgments
The support of National Key Research and Development Plan
of China under Grant No. 2020YFB1901600 , National Major Science
and Technology Project of China under Grant No. 2017ZX06902012
and No. 2017ZX06901024 are gratefully acknowledged.
Supplementary materials
Supplementary material associated with this article can be
found, in the online version, at doi: 10.1016/j.jnucmat.2022.154002 .
16
P. Lin, J. Nie and M. Liu Journal of Nuclear Materials 571 (2022) 154002
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