Nuclear Engineering and Technology 51 (2019) 501e509

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Nuclear Engineering and Technology
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Original Article

Study on the irradiation effect of mechanical properties of RPV steels
using crystal plasticity model
Junfeng Nie a, *, Yunpeng Liu a, Qihao Xie c, Zhanli Liu b
a

Institute of Nuclear and New Energy Technology, Collaborative Innovation Center of Advanced Nuclear Energy Technology, Key Laboratory of Advanced
Reactor Engineering and Safety of Ministry of Education, Tsinghua University, Beijing, 100084, China
b
Applied Mechanics Lab., School of Aerospace Engineering, Tsinghua University, Beijing, 100084, China
c
Data Science and Information Technology Research Center, Tsinghua-Berkeley Shenzhen Institute, Shenzhen, 518055, China

a r t i c l e i n f o

a b s t r a c t

Article history:
Received 3 August 2018
Received in revised form
5 October 2018
Accepted 21 October 2018
Available online 30 October 2018

In this paper a body-centered cubic(BCC) crystal plasticity model based on microscopic dislocation
mechanism is introduced and numerically implemented. The model is coupled with irradiation effect via
tracking dislocation loop evolution on each slip system. On the basis of the model, uniaxial tensile tests of
unirradiated and irradiated RPV steel(take Chinese A508-3 as an example) at different temperatures are
simulated, and the simulation results agree well with the experimental results. Furthermore, crystal
plasticity damage is introduced into the model. Then the damage behavior before and after irradiation is
studied using the model. The results indicate that the model is an effective tool to study the effect of
irradiation and temperature on the mechanical properties and damage behavior.
© 2018 Korean Nuclear Society, Published by Elsevier Korea LLC. This is an open access article under the
CC BY-NC-ND license ( />
Keywords:
Crystal plasticity
Dislocation evolution
Irradiation effect
Damage
RPV steel

1. Introduction
Reactor pressure vessel(RPV) is an important component of
nuclear power plant. It will be subjected to extreme conditions
such as high temperature, high pressure and high energy neutron
during its operation. This will lead to the effect of temperature and
radiation on the mechanical properties of RPV steel, which are the
critical factors for operating safely a nuclear power plant or for
extending its lifetime [1]. Irradiation effect is related to many factors such as irradiation dose [2], temperature [3], microstructure
and defects [4,5]. Formation and evolution of microstructure have a
more substantial impact among all factors [6].
Crystal plasticity theory for metal reveals that macroscopic
deformation of crystals is closely related to dislocation slip. Taylor

[7] formulated a model for the relationship between texture and
mechanics. A mathematical description of crystal plastic deformation and kinematics was derived rigorously by Hill and Rice [8].
Asaro [9] and Peirce [10] further developed and improved the
plastic constitutive theory of crystals. Combining crystal plasticity
theory and finite element method can be used to introduce

microstructure defects to the study of the mechanical response of
crystal materials [11]. A. Ma [12] established a constitutive model
based on dislocation density for face-centered cubic(FCC) crystals
and implemented the model under crystal plasticity finite element
framework.
There are some studies on the application of crystal plasticity
theory to the research of irradiation effect of materials. Vincent [13]
carried out the modeling of RPV steel brittle fracture using crystal
plasticity computations on polycrystalline aggregates, and the
modeling attempted to predict the brittle fracture by calculating
and discussing the largest values of the stresses. Fabien Onimus [14]
provided a polycrystalline modeling to study the mechanical
behavior of neutron irradiated zirconium alloys, and the model
described the effects of the dislocation channeling mechanism on
the mechanical behavior of irradiated zirconium alloys. Xiao [15]
studied the strain softening in BCC iron induced by irradiation
used crystal plasticity, and it was also indicated that the flow stress
increases under neutron irradiation. Generally, crystal plasticity is
an effective approach to study the mechanical properties and
microstructure evolution induced by irradiation for metal

* Corresponding author.
E-mail address: (J. Nie).
/>1738-5733/© 2018 Korean Nuclear Society, Published by Elsevier Korea LLC. This is an open access article under the CC BY-NC-ND license ( />licenses/by-nc-nd/4.0/).



502

J. Nie et al. / Nuclear Engineering and Technology 51 (2019) 501e509

materials.
In this paper, a body-centered cubic(BCC) crystal plasticity
model comprising irradiation effect and crystal plasticity damage
based on dislocation evolution is proposed and numerically
implemented under finite element framework. Influences of
irradiation and temperature on the mechanical properties of
Chinese A508-3 steel which has a BCC structure are studied based
on the model. Furthermore, crystal plasticity damage is introduced to the model, and the damage behavior of the material is
also simulated. The simulation results agree well with the
experimental results.
2. BCC crystal plasticity model based on microscopic
dislocation mechanism

2.1. Kinematical theory of crystals

F ¼ F * $F P

(1)

Where F * and F P are the elastic part and plastic part of the deformation gradient, respectively.
Velocity gradient tensor is related to the deformation gradient,
and can be decomposed into elastic part and plastic part as:



L ¼ F $F À1 ẳ L* ỵ LP

(2)



L* ẳ F * $F *1

(3)
n
X

aẳ1



ga s*a 5m*a

(4)

vectors s*a and m*a are the slip direction and normal to the slip
plane of the slip system a in the intermediate configuration,


respectively. ga is the slipping rate of the slip system a.
2.2. Basic crystal plasticity model formulation
2.2.1. Constitutive relation
Following the elastic constitutive relation proposed by Hill and
Rice, the relation describing the distortion and rotation is given by
V


s Ki ¼ C : D
V

n
X

ma ẳ

(6)


1 *a
s 5m*a ỵ m*a 5s*a
2

(7)

Ã

This section is a simple summary of the kinematical theory for
the mechanics of crystals, following the work of Taylor [7], Hill [8],
Rice [8] and Asaro [9]. In order to facilitate the analysis, crystal
deformation is assumed to be divided into two processes in crystal
plasticity theory. Firstly, the crystal translates from the original
reference configuration to the intermediate configuration via
dislocation slip; then the crystal undergoes lattice distortion and
torsion and reaches the current configuration. The two processes
respectively correspond to plastic deformation and elastic deformation of the crystal. Deformation gradient of the crystal is
decomposed as:




2.2.2. Dislocation slip formulation and flow rules
In the dislocation motion theory, it is argued that plastic
deformation of the material is accomplished via the slipping of
dislocations on the slip plane. The dislocation slip is assumed to
obey Schmid's law [16]. When the resolved shear stress on the slip
plane exceeds the corresponding slip resistance, the dislocations
start to slip resulting in the plastic deformation in the crystal. 48
potential slip systems should be considered in the BCC crystal
which is 12 in the FCC crystal.
In the process of finite deformation, Schmid resolved shear
stress is given by

ta ¼ sKi : ma

BCC crystal plasticity model based on dislocation evolution will
be described in the following chapter.

LP ¼ F P $F PÀ1 ẳ

deformation rate.

Ã

ẵC : ma ỵ ua $sKi sKi $ua Šga

The slipping rate ga is generally expressed as the following
thermal activation form [17].


8 
a
>
>
>g ¼ 0 ;
<

jta j < g a
!p #q )

"
(
a
a


>
a ¼ ga exp À Q0 1 À jt j À g
>
>
g
:
a
0
_
kT

t


sgnðta Þ ; jta j > g a
(8)

_a

_a

t ¼ t0

G
G0

(9)

where Q0 denotes the activation energy of sliding without extern
force, p and q are flow rules related parameters, G and G0 are the
elastic shear modulus at current temperature T and 0K, respec_a

tively, t is the maximum of slip resistance without thermal activation energy, g a is the slip resistance caused by dislocations in the
material.
2.2.3. Hardening based on dislocations
Dislocation mechanism of crystal plastic deformation proposed by Orowan [18] has been generally acknowledged. With the
dislocation density increasing, interactions among dislocations
are stronger and the slip resistance increases. A hardening formula based on dislocation density according to Taylor's hardening
law [7] is raised so as to represent the hardening behavior of the
crystal:

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u
N h


i
u X
Aab rbM ỵ rbI

ga ẳ Gbtqr

(10)

bẳ1

where b is the magnitude of the Burgers vector, qr is a statistical
parameter denoting the deviation between real distribution and
hypothetical regular distribution of dislocations, Aaa (no sum) and
Aab ðasbÞ are self and latent hardening coefficients respectively
and denote the contribution of each slip system to the slip resistance of the current slip system, rbM is the mobile dislocation den-

(5)

a¼1

Where s Ki denotes the Jaumann derivative of the Kirchoff stress
tensor sKi in the original configuration, C is the stiffness tensor, D is
the deformation rate tensor, u is the spin rate and m is the

sity and rbI is the immobile dislocation density of slip system b.
The evolution of dislocations in a BCC crystal mainly has the
following patterns: multiplication and annihilation of dislocations
[19,20], mobile dislocations being trapped as immobile dislocations
[21], dynamic recovery of immobile dislocations [22]. The evolution

formulas of mobile and immobile dislocations are integrated and


J. Nie et al. / Nuclear Engineering and Technology 51 (2019) 501e509

given by


raM ¼


raI ¼



Table 1
Parameters of the elastic modulus(GPa).

 
kmul 2Rc a
1  
À
rM À a ga 
bld
b
bl

(11)

  

 
1
a ga 
À
k
r
dyn I 

bla

(12)



2.3. Evolution of dislocation loops induced by irradiation
A lot of researches have shown that collisions between incident
particles and atoms of the crystal lattice induce irradiation defects
during the irradiation. For BCC crystals, dislocation loops are the
primary defect structure. Reactions between dislocation loops and
mobile dislocations inhibit the motion of dislocations and result in
the irradiation hardening. A term with respect to dislocation loops
is introduced into Eq. (10) to reflect the contribution of irradiation
to slip resistance. The hardening formula coupling with irradiation
effect is

(13)

where Ni denotes the dislocation loop density of slip system a, di is
the mean size of dislocation loops in slip system a, qi is the hardening strength of dislocation loops. Deo [23] indicated that the
density and mean size of dislocation loops at the initial stage of

defect accumulation are respectively proportional to the square
root of the irradiation dose on the basis of the experiments.

N ai ¼ A,dpa2

(14)

dai ¼ B,dpa2

(15)

1

A and B are material constants, the scope of application is
0.0001~10dpa.
It is widely considered that the interactions between dislocation
loops and dislocations raise the evolution of dislocation loops while
the real process remains unknown. Similar to the evolution of
dislocations, results of the interactions between dislocation loops
and dislocations can be described by the following patterns:
annihilation of dislocation loops, translation from dislocation loops
to mobile dislocations, mobile dislocations cut by dislocation loops.
Patra [24] proposed a phenomenological model to describe the
annihilation of dislocation loops. In order to make the model more
universal, we change the exponential constant in the origin formula
to a variable. The final annihilation formula of dislocation loops is
given as:

Ác À a Á1Àc a
Ra À

a
rM
N_ i;ann dai ¼ i N ai dai
jg_ j
b

C12

C44

G0

G

236

134

119

87.6

82.534

g_ 0 ðsÀ1 Þ

p

q


b
t 0 ðMPaÞ

107

0.47

1.1

390

Q0 ð10À19 JÞ
Tð CÞ
unirr
irr

À100
1.85
2.41

20
2.15
2.81

288
3.98
4.71

equivalent to increasing the multiplication rate of mobile dislocations, the cutting process between dislocation loops and mobile
dislocations decreases the density of mobile dislocations. Both of

the processes are fitted by adjusting kmul in Eq. (11). Due to the
introduction of dislocation loops, the ld and la have changed into
new forms:


,v
u N
uX b
t
ld z1
rM ỵ Nbi dbi

(16)

where Rai is the critical size of the annihilation of dislocation loops,
c is used as the annihilate index reflecting the influence caused by
dislocation loop density and mobile dislocation density on the
annihilation rate of dislocation loops.
The translation from dislocation loops to mobile dislocations is

(17)

b¼1

1

b¼1

1


C11

Table 2
Parameters of the plastic flow law.

where kmul is the multiplication coefficient of mobile dislocations,
ld is the mean length of mobile dislocation fragments,
ffi
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
PN
a
b
ld z1=
bÀ1 rM , Rc denotes the critical size of the annihilation, l
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
is the mean free path of the trapping process, la ẳ 1=br raM ỵ raI ,
kdyn is the dynamic recovery coefficient of immobile dislocations.

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u
N h

i
u X
a
Aab rbM ỵ rbI
ỵ qi N ai dai
g ẳ Gbtqr


503

1

1

ẳ
ỵ
ẳ
la lar lai

q

q

br raM ỵ raI ỵ bi Nai dai

(18)

Up to now the model has been completely described. Moreover,
the BCC crystal plasticity model based on dislocation evolution
coupling with irradiation effect is implemented into ABAQUS as an
interface of user material (UMAT) [25] for numerically
computation.
3. Material parameters and modeling
The RPV material used in this study is Chinese A508-3 steel. The
material properties were characterized via tensile testing carried
out by Lin Yun [26] on the unirradiated and irradiated conditions.
Parameter selection mainly refers to the tests. It is assumed that 48
slip systems in A508-3 steel have the same initial value of critical

resolved slip resistance, 390MPa [27]. The dislocation density of
steel up to 107/mm2 and increases as the deformation increasing.
Assuming an uniform initial mobile and immobile dislocation
density for each slip system, 2 Â 107/mm2. The elastic constants can
be obtained from Refs. [28,29]. Irradiation damage is estimated at
0.1 dpa [30]. Ashby [31] suggested the activation energy for dislocation glide Q0 to be of the order of 0:5Gb3 for irradiated materials.
The optimized values of the material parameters are obtained by
minimizing the difference between experimental results and the
macroscopic response of the polycrystalline Finite Element computations. The optimum set of parameters is given in Tables 1e4.
Voronoi [32] method is generally used to construct a
Table 3
Parameters of the hardening law.
qr

qi

bðnmÞ

kdyn

Aaa

0.06

1.0

0.248

275


1.0

Rc ðnmÞ

br

Aab

kmul
unirr

irr

1.5

0.074

0.2

0.0735

0.0955


504

J. Nie et al. / Nuclear Engineering and Technology 51 (2019) 501e509

Table 4
Parameters of the irradiation effect.


bi

Að1013 mmÀ3 Þ

Bð10À6 mmÞ

c

0.1

5

3:7

0.8

Fig. 3. Simulation stress-strain curves of the models with 3 sets of grid numbers.

Fig. 1. The 3D voronoi model consists of 40 grains with random distributed crystlline
orientations.

polycrystalline model. A 3D voronoi polycrystalline model is shown
in Fig. 1. To be consistent with the actual grain size, the model is
assigned a length of 5mm for each side, containing 40 grains. The
distribution of the grain orientations is random and periodic
boundary conditions are set for all directions. It is a quasi-tension,
and the constant stretch rate is 0.2 mm=min which is the same as
the experimental condition [21]. The element type is C3D10.
The validity of the model is analyzed from the perspective of

grain orientations and grid numbers. Owing to the randomness of
grain orientations, the stress and strain of the polycrystal are
calculated by volume averaging the values at all integral points of
each grain. According to Ref [33], 5 sets of grain orientations are
tested, including [111], [110] and 3 sets of random orientations. The
simulation results agree with theoretical analysis [34] that the
uniaxial tensile stress-strain curves of the models with random

Fig. 2. Simulation stress-strain curves of the models with 5 sets of grain orientations.

orientations are between the stress-strain curves of the models
with orientation [111] and orientation [110]. The stress-strain
curves of the models with random orientations are very close and
have the same trend (Fig. 2), indicating that 40 grains can better
reflect the tensile property of the material. In order to be more
representative, the model with the second set of random orientations is used.
The models are divided into 10589, 48123 and 71897 grids
separately. As the number of the grids increases, the stress is
basically unchanged (Fig. 3), which proves that the results are
convergent and the model can represent the real stress level of the
structure. For the purpose of saving computing time, the model is
divided into 48123 grids.
The comparison of the stress-strain curves between experimental and simulation results at À100  C, 20  C, and 288  C are
plotted in Fig. 4 and Fig. 5.

4. Crystal plasticity damage
Generally, metal materials exhibit better ductility and can support a large plastic deformation before fracture. In the process of

Fig. 4. Experimental and simulation stress-strain curves for samples at different
temperatures before irradiation.



J. Nie et al. / Nuclear Engineering and Technology 51 (2019) 501e509

505

Table 6
Parameters of the reference stress B0 .
B0 ðMPaÞ
Tð CÞ

À100

20

288

unirr

1420

1300

1165

irr

1650

1320


1230

initiation term of the damage corresponds to a specific plastic strain
threshold, below which no damage by microcraking occurs. The
initiation term of the damage is described by cumulative slip strain
and the model is validated in Ref. [37]. The final damage evolution
law is

"

# ( )
b m
s
f ẳ Hg gsth ị bI5I þ ð1 À bÞ I
:
B0
·

Fig. 5. Experimental and simulation stress-strain curves for samples at different
temperatures after irradiation.

Table 5
Parameters of the initial value of cumulative slip strain at the beginning of damage
gsth .

gsth
Tð CÞ

À100


20

unirr

0.4

0.3

0.3

irr

0.4

0.25

0.32

288

ð4Þ

(21)

〈〉 in Eq. (20) or (21) is the McCauley symbol. When the value in
parentheses is greater than 0, the value is the same; instead, the
value is 0. The guidelines deems that only the material being
stretched may appear damage. HðÞ is the Heaviside function, controlling that the damage occurs when the cumulative slip strain in
all slip systems reaches the threshold.

Effective stress and damage factor is introduced into the UMAT.
Coupled with the damage, parameters selection and the calculation
results are listed in Tables 5 and 6. The values b ¼ 0:5; n ¼ 20 are
adopted to fit the experimental data. The simulation results with
damage model are shown in Fig. 6 and Fig. 7 and agree well with
the experimental results of breaking process of the tensile test.

5. Results and discussion
the growth of plastic strain, the damage begins to occur and
accumulate, resulting in the degradation of mechanical properties.
The material will lose the carrying capacity when the damage
reaches a certain extent.
Lemaitre [35] put forward the principle of strain equivalence
and concluded that the constitutive model of the material with
damage could be derived from the constitutive relationship of the
material without damage by simply replacing the stress with the
effective stress. The definition of effective stress tensor is

b ¼ ðI À fÞÀn $ s $ðI À fÞÀn ¼
s

3
X

s

s

bin
bi

b i i5 n
s

(19)

The stress and strain contours at room temperature are shown
in Fig. 8 and Fig. 9. Fig. 8 is the Mises stress contour and Fig. 9 is the
maximum principal strain contour. Both figures show obvious
inhomogeneous distribution. The values of stress and strain are
larger around the grain boundary than interior of the grain, which
indicate that stress concentration and non-uniform large deformation occur at the grain boundary. Actually, the grain boundary
will absorb the point defects induced by irradiation and reduce the
defect density inside the grain. In this study, the grain boundary is
treated as pure geometric surface, the process of the absorption is

i¼1

where I is the unit tensor, f is the damage factor tensor, n is a
b i and n
b
b si are the ith eigenvector of the stress s
material constant, s
respectively.
In order to study the damage behavior of materials in the case of
large deformation, anisotropic damage model improved by Lu Feng
[36]. Defining a damage factor tensor f, then the rate of damage is
Ã

"


4ị

f ẳ bI5I ỵ 1 bị I

#
:

3 ( )m
X
b
si
b si 5 n
b si
n
B
0
iẳ1

(20)

4ị

where I is four-order identity tensor, b and B0 are material parameters, b ¼ 1 is corresponding to the isotropic damage and b ¼ 0
is corresponding to the complete anisotropic damage.
Ref [35] has explained the introduction of the initiation term of
the damage, and it described that before the microcracks are
initiated, they must nucleate by the accumulation of microstresses
accompanying incompatibilities of microstrains or by the accumulation of dislocations in metals. For the pure tension case, the

Fig. 6. Experimental and simulation stress-strain curves comprising damage for

samples at different temperatures before irradiation.


506

J. Nie et al. / Nuclear Engineering and Technology 51 (2019) 501e509

Fig. 7. Experimental and simulation stress-strain curves comprising damage for
samples at different temperatures after irradiation.

Fig. 10. The error bars associated with the experimental data compared with numerical results before irradiation.

Fig. 8. The contour of Mises stress at room temperature after irradiation.

Fig. 9. The contour of the maximum principle strain at room temperature after irradiation.


J. Nie et al. / Nuclear Engineering and Technology 51 (2019) 501e509

507

Table 7
Experimental and simulation results of the yield stress before and after irradiation at different temperatures.
Yield Stress(MPa)

Relative error(%)

T
ð CÞ


Unirr(exp/sim)

Irr(exp/sim)

Incre-Mental (%)

unirr

irr

À100
À40
20
100
200
288

555/547
N/447
409/417
N/408
N/405
389/395

638/655
N/575
517/534
N/510
N/475
441/472


14.9
28.7
26.4
25.1
17.3
13.3

1.4

2.7

1.9

3.3

1.5

7.0

Table 8
Experimental and simulation results of the tensile strength before and after irradiation at different temperatures.
Tensile Strength(MPa)

Relative error(%)

T
ð CÞ

Unirr (exp/sim)


Irr (exp/sim)

Incre-mental (%)

unirr

irr

À100
À40
20
100
200
288

708/691
N/642
573/607
N/599
N/597
556/584

766/768
N/696
639/635
N/625
N/625
605/612


8.1
8.4
11.5
4.3
4.7
8.8

2.4

0.3

5.9

0.6

5.0

1.1

ignored though it will cause the defects to converge at the grain
boundary and enhance the resistance to the motion of dislocations.
The calculation results of stress-strain curve (Figs. 6e7) agree
well with the experimental results and the validity of the model is
verified. The error bars associated with the experimental data [26]
before irradiation are plotted in Fig. 10. The results of yield strength
are within the margin of error. The error bars of tensile strengths
are small so the computed results are not in the error range, the
actual deviation is not large. The error values are in Table 7 and
Table 8.
To analysis the trends of mechanical properties, additional

temperature conditions are calculated and the data of critical points
are extracted into Tables 7 and 8.
The variations of the yield stress and tensile strength with
temperature are shown in Fig. 11 and Fig. 12. The results indicate
that temperature has a great influence on the mechanical properties of the material. The effect of thermal activation on the
dislocation motion is enhanced with the increasing of temperature, which reduces the resistance of dislocation slip. Therefore,

the increasing temperature entails decrease of the yield strength
and tensile strength of Chinese A508-3 steel. Temperature will
also influence the interactions between dislocation loops and
dislocations and the irradiation hardening effect is also affected.
At room temperature, the yield stress and tensile strength after
irradiation increase more significantly, which are 26.4% and 11.5%
respectively. That means the irradiation effect is more obvious at
room temperature. On the whole, the irradiation hardening effect
increases first and then decreases from low temperature to high
temperature.
Irradiation damage can be reflected by parameters gsth and B0 .
gsth is the slip strain when the damage of the material occurs.
Reference stress B0 controls the rate and degree of the damage.
According to the fitted values of gsth and B0 , the two parameters
also show a temperature dependence. The influence of irradiation
on the damage is more pronounced at room temperature for that
the damage occurs much earlier after irradiation at a cumulative
slip strain of 0.25. The influence is inhibited and the values of gsth
are larger at low temperature and high temperature.

Fig. 11. The variations of the yield stress and tensile strength with temperature before
and after irradiation.


Fig. 12. The incremental of the yield stress and tensile strength after irradiation with
temperature.


508

J. Nie et al. / Nuclear Engineering and Technology 51 (2019) 501e509

Dta and Dsij are written in the function of Dεij .

6. Conclusion
In this study, a crystal plasticity model based on dislocation
evolution for BCC crystals is constructed and the model is coupled
with the irradiation effect via introducing irradiation defects evolution. The model is numerically implemented and then the mechanical properties of Chinese A508-3 steel with irradiation at
different temperatures are simulated. The conclusions have been
presented as followed:
$ The model can better describe the irradiation hardening of the
material in a certain temperature range.
$ The model considering damage evolution can better describe
the degradation of mechanical properties of RPV steel with
irradiation and unirradiation.
$ The model can reflect the variations of the mechanical properties and damage behavior with temperature.
Future work will focus on developing a more detailed classification of irradiation defects and a numerical simulation method to
capture the mechanical properties of the material. Then the model
can easily predict the mechanical properties of other BCC crystal
materials after irradiation with a higher accuracy.
Acknowledgements
The support of the National Natural Science Foundation of China
under Grant No. 11202114, Beijing Higher Education Young Elite
Teacher Project under Grant No. YETP0156 and National Science

and Technology Major Project of China, Chnia, Grant No.
2017ZX06902012 are gratefully acknowledged.
Appendix
Incremental form is beneficial to the implementation of subroutines. The tangent modulus method for rate dependent solid
developed by Peirce [38] is used in the subroutine. It is assumed
that the increment of ga within the time increment Dt is defined as

Dga ẳ ga t ỵ Dtị ga tị

(22)

the linear form is



Dga ẳ Dt 1 qịg_ at ỵ qg_ atỵDt



(23)

q is the integral parameter whose value between 0 and 1. Seeing
that g_ a is a function of ta and g a , we can substitute the Taylor
expansion of g_ atỵDt into Eq. (22) and get

Dga ẳ Dt g_ a þ q
vg_ a
vta

vg_ a a

vg_ a
Dt þ q a Dga
a
vt
vg

!

_a
and vvgga can be obtained according to Eq. (8)

(24)

i

h

2

Dta ¼ Lijkl makl ỵ uaik sjk ỵ uajk sik $4Dij
Dsij ẳ Lijkl Dεkl À sij Dεkk À

3
X b
b
mij Dg 5

(27)

b


i
Xh
Lijkl makl þ uaik sjk þ uajk sik Dga
a

(28)

Dga is written in the funtion of dislocation density.

Dga ẳ



N
!
Gbị2 X
ab kmul 2Rc ra À k
a Dgb 
A
q
r


r
dyn I
2ga
bld
b M


(29)

b¼1

References
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J. Nie et al. / Nuclear Engineering and Technology 51 (2019) 501e509
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