F. CHEN et al.: VOID-CLOSURE BEHAVIOR AND A NEW VOID-EVOLUTION MODEL FOR VARIOUS STRESS STATES
105–113
VOID-CLOSURE BEHAVIOR AND A NEW VOID-EVOLUTION
MODEL FOR VARIOUS STRESS STATES
NOV MODEL ZAPIRANJA VOTLIN IN RAZVOJ NOVIH
POD VPLIVOM RAZLI^NIH NAPETOSTNIH STANJ
Fei Chen, Xiaodong Zhao, Huiqin Chen
*
, Jinyu Ren
College of Materials Science and Engineering, Taiyuan University of Science and Technology, Taiyuan 030024, P. R. China
Prejem rokopisa – received: 2020-07-07; sprejem za objavo – accepted for publication: 2020-10-05
doi:10.17222/mit.2020.128
Cavity defects including a shrinkage cavity, air hole and porosity are typical defects of an ingot, which are difficult to avoid.
These defects can disappear during deformation at a reasonably high temperature and heat-preservation process so as to ensure
the quality of forgings. However, the current understanding of the mechanism of the void closure is insufficient, and the models
that can accurately predict the evolution of a void are relatively rare. Based on the representative-volume-element model, a theo-
retical analysis and numerical simulation, the influence of the Lode parameter and stress triaxiality on the void closure is ex-
pounded. Besides, based on the above research, a new void-evolution model considering all kinds of macro-stress states is pro-
posed. The model can predict the closure degree of voids in different positions and various macro-stress states during the plastic
deformation of ingots. Comparing the new void-evolution model with other models, numerical simulations and physical-simula-
tion results, the accuracy of the proposed void-evolution model is verified.
Keywords: void closure, representative volume element, stress triaxiality, Lode parameter, void-evolution model
Kavitacijske napake, nastale zaradi kavitacijskega kr~enja, kot so: zra~ni mehur~ki in poroznost, so tipi~ne napake v ingotih, ki
se jim v metalur{ki praksi te`ko izognemo. Te napake lahko popravimo s pomo~jo visokotemperaturne deformacije in s
postopkom termi~ne poprave. S tem lahko zagotovimo izdelavo kvalitetnih odkovkov. Vendar pa je dana{nje razumevanje
mehanizmov zapiranja votlin (majhnih praznin v materialu, npr.: pore in lunkerji) nezadostno in so modeli, s katerimi lahko
natan~no napovemo razvoj votlin, zelo redki. Na osnovi modela volumskega elementa so avtorji razlo`ili mehanizem zapiranja
votlin, na osnovi teoreti~ne analize in numeri~ne simulacije, upo{tevajo~ Lodeov parameter in triosno napetostno stanje. Na
osnovi podane raziskave so avtorji poleg tega predlagali nov model razvoja votlin, z upo{tevanjem vseh vrst makronapetostnih
stanj. Model lahko napove stopnjo zapiranja praznin na razli~nih polo`ajih in razli~na makronapetostna stanja med plasti~no
deformacijo ingotov. Primerjava novega modela razvoja praznin s podobnimi modeli, simulacija in njeni rezultati, so potrdili
natan~nost njegovih napovedi.
Klju~ne besede: zapiranje votlin, reprezentativni volumski element, napetostna triosnost, Lodeov parameter, model razvoja
votlin
1 INTRODUCTION
Cavity defects including a shrinkage cavity, air hole
and porosity are typical defects of ingots, which are dif-
ficult to avoid, especially in large ingots.
1,2
These defects
need to be eliminated using deformation with a reason-
ably high temperature and a heat-preservation process to
restore the continuity of materials and ensure the quality
of forgings.
3,4
Many scholars carried out a lot of research on the be-
havior of the void closure and the establishment of a
void-closure model. Tanaka et al. considered that the
equivalent strain and hydrostatic stress were the impor-
tant factors affecting the void closure, and proposed pa-
rameter Q, which was defined as the integral of the stress
triaxiality to equivalent strain to assess the void clo-
sure.
5,6
Nalawade et al. studied the void-closure behavior
of a large-section as-cast bar during hot rolling and ex-
plored the relationship between parameter Q and the
void closure.
7
Based on the distribution of the compres-
sive-strain field around the void, Chen et al. obtained the
quantitative relationship between the void evolution and
the compressive strain through a formula derivation and
mathematical fitting based on the Gaussian function.
8
X.
X. Zhang et al.
9
developed a computational program to
simulate the void evolution. It could show the change in
the relative volume of voids synchronously and intu-
itively in a simulation of forging of large forgings. N.
Harris et al.
10
analyzed the effects of the far-field equiva-
lent strain, stress triaxiality and material properties on
the void closure by studying the void-closure behavior of
a large billet forging, and finally proposed a new
void-closure model. In order to improve the internal
quality of shaft forgings, D. Tian et al.
11
proposed the
floating-pressure method to eliminate the void in an in-
got. They investigated the changes in the stress, metal
flow and void shape using a numerical simulation, and
expounded the mechanism of the floating-pressure
method to eliminate the void.
11
Park conducted an
in-depth study on the evolution law of void under a
high-stress triaxiality and found that the stress triaxiality
and equivalent strain were important factors affecting the
Materiali in tehnologije / Materials and technology 55 (2021) 1, 105–113 105
UDK 620.1:532.613:620.172.2 ISSN 1580-2949
Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 55(1)105(2021)
*Corresponding author's e-mail:
chen_huiqin@126.com (Huiqin Chen)

void closure. With the increase in the stress triaxiality,
the void-closure mode changed from collapse to contrac-
tion.
12
Based on the energy principle, S. H. Zhang et al.
13
established a new criterion for measuring the closure of
an ellipsoidal void during thick-plate rolling. The closure
behavior of seven kinds of ellipsoidal voids during roll-
ing was simulated with the finite-element method. The
new criterion was compared with the results of the fi-
nite-element method, and its accuracy was verified. L.
Zhang
14
et al. explored the evolution rule of voids with
different shapes, orientations and positions during a
stretching process and obtained the critical forging ratio
for the void closure of a large billet.
In summary, previous researches on the void-closure
behavior required a significant amount of work. How-
ever, most of them focused on the evolution behavior of
a void with a specific position and shape in an ingot un-
der a specific deformation condition. In recent years,
with the development of the finite-element software and
the popularity of the representative volume element
(RVE), many scholars have begun to analyze the evolu-
tion behavior of voids under different stress states based
on the RVE model. But this research is still in its infancy.
Using the RVE model, the evolution behaviors of voids
under complex stress states were analyzed and the char-
acteristic parameters that could quantitatively character-
ize the void-closure behavior in most stress states were
extracted. Based on the characteristic parameters, a new
model of the void evolution was established, which com-
prehensively considers various stress states. A general
method was obtained to determine the void evolution in
ingots in different positions and different stress states.
2 REPRESENTATIVE-VOLUME-ELEMENT
MODEL AND STRESS BOUNDARY
CONDITIONS
2.1 Representative-volume-element model
A RVE model helps us establish the quantitative rela-
tionship between the void evolution and macroscopic
stress states.
15
In recent years, many scholars have used
RVE models to describe the growth and aggregation of
voids under various stress states.
16–18
A RVE model is
shown in Figure 1. The surface of the RVE is subjected
to a uniform macroscopic stress. The boundary condition
of the RVE is the macroscopic stress state at the marked
point on the ingot. No matter how complex the stress
state is, it can finally be expressed by the normal stresses
on three pairs of mutually perpendicular planes with a
zero shear stress. These three pairs of normal stresses are
also called principal stresses. By changing the values of
the three principal stresses around the RVE model, the
stress state around the RVE model can be effectively
controlled. In a three-dimensional stress space, the stress
state of an infinitesimal can be described with stress ten-
sor ij
that has nine stress components ( x
, y
, z
, xy
, yx
,
 yz
, zy
, zx
, xz
). The stress tensor can be decomposed into
the spherical stress tensor and deviatoric stress tensor.
When the coordinate axis and the stress principal axis
overlap, the shear stress is 0 and the three normal
stresses are called the principal stresses, namely  1
,  2
,
 3
. For the convenience of the following discussion, the
principal axes of the stress tensor were required to coin-
cide with the principal axes of the Cartesian coordinate
system. Meanwhile, it was postulated that  1
=  x
,  2
=
 y
, 3
= z
, and 1

2

3
.
2.2 Stress boundary conditions
The shape of the void in an actual ingot is varied, but
in order to obtain the general rule with engineering sig-
nificance to the actual production and reduce the difficul-
ties of analyzing and calculating, the smallest sphere
containing the actual cavity defect can usually be taken
as the shape of the void. This increases the volume of the
void, but the results are more conservative. Therefore, a
spherical void was selected to explore the void-closure
behavior.
In this paper, the DEFORM-3D and RVE model were
used for the numerical simulation to study the void-clo-
sure behavior under different macroscopic stress states.
In order to simplify the calculation, the 1/8 RVE model
was selected for the simulation. Figure 2 shows the 1/8
RVE model which is a regular hexahedron of (5×5×5) mm.
The center of the model contains a spherical void whose
F. CHEN et al.: VOID-CLOSURE BEHAVIOR AND A NEW VOID-EVOLUTION MODEL FOR VARIOUS STRESS STATES
106 Materiali in tehnologije / Materials and technology 55 (2021) 1, 105–113
Figure 1: Representative-volume-element model

radius is 1mm. Besides, Pb was selected as the material.
The constitutive model of Pb is shown in Equation (1).
 =+ 160 69
03
..
.
(1)
where, is the equivalent stress whose unit is MPa and
 is the equivalent strain.
The stress triaxiality is one of the most important pa-
rameters in the establishment of a void-closure model by
many scholars.
5,7,9,10
It is defined as the ratio of hydro-
static stress to equivalent stress. According to its defini-
tion, it can be considered that the stress triaxiality repre-
sents the relative magnitude of hydrostatic stress and
equivalent stress. The Lode parameter was proposed for
the study of the material yield criterion.
19
The definition
of the Lode parameter is (2 2
– 1
– 3
)/(  1
– 3
). After an
in-depth research by many scholars, it was found that the
Lode parameter was also of great significance when de-
scribing the strain type.
16
Moreover, it could represent
the relative amounts of the principal values of the
deviatoric stress tensor.
The stress tensor can be divided into the spherical
stress tensor and deviatoric stress tensor. For a certain
material, when the stress triaxiality and Lode parameter
are determined, the main stress on the outer surface of
the RVE model can be determined, and the stress state of
the RVE model can be determined.
20
In order to explore
the influence of different stress states on the evolution
law of a void, we studied the closure of a void under 25
groups of stress states, with stress triaxiality T being –1,
–0.67, –0.33, 0, 0.33, and Lode parameter μ being –1,
–0.5, 0, 0.5, and 1, respectively. Referring to Chbihi’s
method, the macroscopic stress state around the RVE
model can be determined according to the Lode parame-
ter and the stress triaxiality.
20
These stress states basi-
cally cover the stress states corresponding to the com-
mon plastic deformation processes.
3 RELATIONSHIP BETWEEN THE VOID
CLOSURE AND STRESS STATES
From practical experience and numerical-simulation
results, it can be known that with the occurrence of de-
formation, the void shape changes from a sphere to an el-
lipsoid.
20
Therefore, when the length of the short axis of
a void is zero in the direction of compression, the void
can be considered closed. Therefore, the initial diameter
of the spherical void was defined as d
0
. During deforma-
tion, the length of the short axis of the void in the Z di-
rection (the maximum-compression direction) was de-
fined as r
Z
, and the void shape coefficient in the Z
direction was defined as S
Z
(S
Z
=2 r
Z
/d
0
). When the shape
coefficient in the Z direction is zero, the void is closed.
3.1 Effect of the Lode parameter on the void-closure
behavior
According to the numerical-simulation results, the in-
fluence of the Lode parameter on the void closure can be
obtained. Figure 3a shows the relationship between the
void shape coefficient S
Z
and the macroscopic equivalent
strain E
e
when stress triaxiality T is –0.33 and Lode pa-
F. CHEN et al.: VOID-CLOSURE BEHAVIOR AND A NEW VOID-EVOLUTION MODEL FOR VARIOUS STRESS STATES
Materiali in tehnologije / Materials and technology 55 (2021) 1, 105–113 107
Figure 3: Relationship between S
Z
and E
e
; E
Z
under different Lode
parameters: a) S
Z
– E
e
;b)S
Z
– E
Z
Figure 2: 1/8 RVE model with a void in the center

rameter μ [–1, 1]. It can be seen on Figure 3a that
when Lode parameter μ is 1, the macroscopic equivalent
strain E
e
required for the void closure is minimal. When
the equivalent strain E
e
is constant, the larger the Lode
parameter, the smaller is the void-shape coefficient S
Z
.
The same phenomenon occurs when stress triaxiality
T [–1, 0.33]. It is known that the larger the Lode param-
eter, the smaller is the macroscopic equivalent strain re-
quired for the void closure.
Figure 3b shows the relationship between the
void-shape coefficient S
Z
and the macroscopic compres-
sive strain E
z
when the stress triaxiality T is –0.33 and
the Lode parameter μ [–1, 1]. It can be seen on Figure
3b that when the compressive strain E
z
is constant, the
change in the Lode parameter has little influence on the
change in the void-shape coefficient S
Z
. This effect is
particularly evident when the Lode parameter μ [–0.5,
1]. This is because the equivalent strain is the combined
effect of the strains in the X, Y and Z directions, and it is
numerically larger than the other three principal strains.
The void-shape coefficient S
Z
represents the void-closure
degree in the Z direction, which has no direct relation-
ship with the strains in the other two directions. It can be
found that when the void-shape coefficient S
Z
is selected
to measure the void-closure degree, the maximum
compressive strain E
z
is more suitable as an independent
variable than the equivalent strain E
e
.
3.2 Effect of the stress triaxiality on the void-closure
behavior
Figure 4a shows the relationship between the
void-shape coefficient S
Z
and the macroscopic equivalent
strain E
e
when the Lode parameter is 0 and the stress
triaxiality is different. As is shown in Figure 4a, the
void-shape coefficient S
Z
first decreases rapidly and then
slowly with the increase in the macroscopic equivalent
strain E
e
. When the macroscopic equivalent strain E
e
is
constant, the smaller the stress triaxiality, the smaller is
the void-shape coefficient S
Z
. The void is most easily
closed when the stress triaxiality T is –1. The same phe-
nomenon occurs when the Lode parameter μ [–1, 1].
Figure 4b shows the relationship between the void-
shape coefficient S
Z
and the macroscopic compressive
strain E
Z
when the Lode parameter is 0 and the stress
triaxiality is different. The rule from Figure 4b is consis-
tent with that from Figure 4a.
In short, with a decrease in the stress triaxiality, the
macroscopic equivalent strain required for a complete
closure of a void decreases. That is, the greater the hy-
drostatic-pressure stress, the easier is the void closure.
4 ESTABLISHMENT AND VERIFICATION OF
THE NEW VOID-EVOLUTION MODEL
Generally speaking, there are two methods for assess-
ing the closure of a void. The first one uses the numeri-
cal-simulation software for measuring the void closure,
setting a void into a workpiece, while the other method
uses the void-closure criterion. The advantage of the first
method is that the change in the voids in a workpiece can
be observed visually. However, the disadvantage is that a
simulation can only assess one or a few voids, and the
mesh around the void needs to be partially refined. Dur-
ing the deformation process, the mesh distortion is seri-
ous, the mesh should be redivided many times and the
simulation time is very long. The criterion method re-
quires taking a certain void closure model as the criterion
for predicting the void closure and assessing the void
closure with a certain predicted value obtained with a
calculation. The advantage of the criterion method is that
there is no need to set a void into a workpiece. For a cer-
tain deformation process, the closure degree of a void at
any position in the workpiece can be calculated using the
void-evolution criterion. Moreover, the number of
meshes and simulation time can be greatly reduced and
the simulation period can be shortened.
Since there is a large number of randomly distributed
voids in an ingot, the criterion method is undoubtedly
one of the best methods for analyzing and assessing
whether voids in different positions are closed in a forg-
F. CHEN et al.: VOID-CLOSURE BEHAVIOR AND A NEW VOID-EVOLUTION MODEL FOR VARIOUS STRESS STATES
108 Materiali in tehnologije / Materials and technology 55 (2021) 1, 105–113
Figure 4: Relationship between S
Z
and E
e
; E
Z
under different stress-
triaxiality values: a) S
Z
– E
e
;b)S
Z
– E
Z

ing process. Based on the simulation results of the RVE
model above, a new void-evolution model, taking into
account all kinds of macroscopic stress states, is estab-
lished. It is desirable to obtain a general method of
studying the evolution of voids at arbitrary positions in
an ingot.
4.1 Establishment of the new void-evolution model
As shown in Figures 3 and 4, when the stress
triaxiality T and Lode parameter μ are constant, the vari-
ation of the void-shape coefficient S
Z
with the macro-
scopic compressive strain E
Z
presents a conical relation-
ship. According to the above analysis, when the stress
triaxiality T is constant and the Lode parameter is more
than –0.5, the influence of the Lode parameter on the
void-shape coefficient S
Z
is small. Based on this, the
quadric equation shown as Equation (2) was used in this
work to describe the relationships between the void-
shape coefficient S
Z
, stress triaxiality T and macroscopic
compressive strain E
Z
.
F TE a Eb Ec T E
dTE eT E fT E
zzzz
zzz
(,, )  = ++++
+++
1
22
222 2
(2)
where a, b, c, d, e and f are the correlation coefficients.
When the Lode parameter is more than –0.5, the in-
fluence of the Lode parameter on the void-shape coeffi-
cient S
Z
is small. When the Lode parameter μ=0,the
correlation simulation data is selected for the modeling.
Figure 5 shows the variation relation of the void-shape
coefficient S
Z
with stress triaxiality T and macroscopic
compressive strain E
z
when the Lode parameter μ =0 .
By fitting the data into Figure 5 with the custom func-
tion, Equation (3) can be obtained. The model can be ap-
plied to predict the evolution of the void-shape coeffi-
cient S
Z
under a constant stress triaxiality.
STE E E T E
TE T
zz zz z
z
(, ) . . .
..
=+ + − −
−−
1 157 051 089
034 027
2
22
ET E
zz
−012
22
.
(3)
To increase the universality of the model, we let
Equation (3) take the partial derivative of E
Z
, and Equa-
tion (4) was obtained. It indicates the change rate of the
void-shape coefficient S
Z
when the compressive strain E
Z
is constant and the stress triaxiality T is varied.
∂
∂
STE
E
ETT E
TE
zz
z
zz
z
(, )
.. ..
.
=+ −− −
−−
157 102 089 068
027 0
2
.24
2
TE
z
(4)
Based on the analysis above, the new void-closure
model established in this paper is shown in Equation (5),
which can be used to predict the evolution of the
void-shape coefficient S
Z
when the stress triaxiality T
changes with the deformation. In particular, when E
z
=0,
then S
Z
= 1, and this is the initial state of the void before
deformation.
∂
∂
STE
E
ETT E
T
zz
z
zz
(, )
.. ..
..
=+ −− −
−−
157 102 089 068
027 02
2
4
1
2
TE
S
STE
E
E
z
z
zz
z
z
=+
∂
∂
(, )
d
0
E
z
∫
⎧ ⎨ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ (5)
4.2 Comparison between the new void-evolution model
and the previous models
The establishment of a new model is important, but
its accuracy is also one of the important factors of the
model research. The new void-evolution model was com-
pared with the previous models as follows.
Models 1 and 2: the Gurson model and the G-T model
The famous Gurson model is widely used in damage
mechanics.
15
According to this model, the porosity rate
of change is:
df
ff q
q
T
qf f q
q
T
=
−
⎛ ⎝ ⎜ ⎞ ⎠ ⎟ +−
⎛ ⎝ 31
3
2
21 2
3
2
1
2
3
2
1
2
()s i n h
cosh⎜ ⎞ ⎠ ⎟ d
e
E (6)
Besides, f
n
= f
n–1
+d f. f
0
represents the initial porosity
rate. When f
n
/f
0
approaches 0, the void is closed. When
q
1
= q
2
= q
3
= 1, Equation (6) is the result derived from
the classical Gurson model. Later, Tvergaard and
Needleman modified the model and they believed that
when q
1
= 1.5, q
2
= 1.0 and q
3
= 2.25, the accuracy of the
model is improved.
21
Under this condition, Equation (6)
is the well-known G-T model.
Model 3: the BHS model
Another famous model is the BHS model, which con-
siders the effect of the Norton index of material on the
void closure.
22
This is shown in Equation (7):
F. CHEN et al.: VOID-CLOSURE BEHAVIOR AND A NEW VOID-EVOLUTION MODEL FOR VARIOUS STRESS STATES
Materiali in tehnologije / Materials and technology 55 (2021) 1, 105–113 109
Figure 5: Relationships between S
Z
, E
Z
and T when μ=0

F. CHEN et al.: VOID-CLOSURE BEHAVIOR AND A NEW VOID-EVOLUTION MODEL FOR VARIOUS STRESS STATES
110 Materiali in tehnologije / Materials and technology 55 (2021) 1, 105–113
Figure 6: Geometric models for numerical simulation: a) upsetting, b) plane strain
Figure 7: Comparison graph of calculation results

ln
() (( ) ) V
V
s
n
T
nns
n
dE
E
n
z
0
2
0
3
2
3
2
1
=+
−+ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ∫
 e
(7)
where s is the sign (T), T is the stress triaxiality, (1) =
ln 3 – 2/3, (–1) = 2π/9 3, n is the Norton index of ma-
terial.
Model 4: the Z-C model
Based on the mechanical analysis and numerical cal-
culation, X. X. Zhang obtained an interpolation model of
the void closure by studying the process of a spherical
void evolving into an ellipsoid and, finally, closing into a
gap.
9
According to the numerical results of the model
and void evolution, an evolution model of the void vol-
ume was proposed:
V
V
E
n
T
nn
n
n
0
2
3
2
3
2
15 2
5
=+
−+ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ +
⎡ ⎣ ⎢ ⎢ exp ( )
() ()
sign
me
 ]}
⎧ ⎨ ⎪ ⎩ ⎪ ++++ qT qE qE q
ee 12
2
3
4
4
(8)
Where  m
is the macroscopic hydrostatic stress, T is
the stress triaxiality, n is the Norton index of material
and q
1
, q
2
, q
3
, q
4
are the coefficients associated with the
Norton index of material.
In this part of the research, the void-evolution model
obtained in the previous part was imported into finite-el-
ement software DEFORM-3D through secondary devel-
opment, and two deformation processes including upset-
ting and plane deformation were simulated. Pb was also
used in the simulation. The results of the void-evolution
models obtained by the predecessors and the new
void-evolution model were compared with those ob-
tained with the traditional method of setting a void in a
workpiece.
Geometric models for the numerical simulation of a
set void in a workpiece are shown in Figure 6. The
shear-friction coefficient between the workpiece and the
anvils was set to 0.7. The upset workpiece was a cylinder
and the plane-strain workpiece was a cuboid. Based on
the symmetry of the workpiece, the 1/2 model of the
workpiece was used in the numerical simulation.
By comparing the above four void-evolution models
and the new void-evolution model established in this
work, with the method of a set void in the workpiece, the
final results were obtained as shown in Figure 7. The ab-
scissa of Figure 7 represents the numerical-simulation
results obtained with the method of a set void in the
workpiece, while the ordinate indicates the calculation
result for each model. The closer the data point is to the
black line of slope 1 in the middle, the more accurate is
the model. As shown in Figure 7, the calculation results
for the new void-evolution model are the closest to the
black line. Other models were built under axisymmetric
stress, so errors of some models are relatively large. In
particular, the error is the greatest when the void-closure
degree is predicted under a plane-strain condition.
4.3 Physical-simulation experiment design and model
verification
To further verify the accuracy of the new void-evolu-
tion model, an upsetting test of the cylindrical sample
was carried out. Four different positions of voids were
selected as shown in Figure 8a. Firstly, a  30×30 mm
cylindrical sample was processed along the axis to make
a through-hole of  10 mm. Secondly, a Pb sample was
machined into a cylinder of 10×30 mm using wire-elec-
trode cutting. Thirdly, the cylinder was cut in half radi-
ally. Then, in each of its end, a  4 mm hemispherical
void was made by milling. Finally, they were inserted
into the through-hole of the larger cylinder to simulate
the internal void of the ingot. The samples were pro-
cessed and assembled as shown in Figure 8b. The pro-
cessed cylinder samples were upset on a 100T hydraulic
press and the deformation was about (10, 20, 30 and
40) %, respectively. The deformed cylinders were cut
along the central axis with wire-electrode cutting. The
sections of the samples were highlighted with blue chalk.
Parts of the samples after the cutting are shown in Fig-
ure 9. By measuring the size of the voids with physi-
cal-simulation experiments, the closure degrees of the
voids in these four samples under different deformation
quantities could be obtained.
A cylinder of the same size as the physical-simula-
tion sample without a void was used as the geometric
model of the numerical simulation. The upsetting pro-
cess of the Pb sample was numerically simulated using
F. CHEN et al.: VOID-CLOSURE BEHAVIOR AND A NEW VOID-EVOLUTION MODEL FOR VARIOUS STRESS STATES
Materiali in tehnologije / Materials and technology 55 (2021) 1, 105–113 111
Figure 8: Schematic diagram of a physical-simulation sample: a) schematic diagram of the locations of the voids, b) assembly method for the
sample with void 1#

the new model. The results of the physical-simulation
experiments and the new model were obtained as shown
in Figure 10. In order to verify the reliability of the cri-
terion, the root mean square error (RMSE) was calculated
in accordance with the following formula.
RMSE
n
SS
i
n
=−
=
∑
1
2
1
()
ep
(9)
where S
e
is the experimental data, S
p
is the predicted
data and n is the total number of data points. The results
show that the RMSE error is 0.05. It is clear that the cal-
culation results for the new void-evolution model are in
good agreement with the experimental results, verifying
the accuracy of the new model.
In conclusion, the numerical simulation and experi-
mental results show that the new void-evolution model is
mainly applicable to predict the closure degree of voids
at different positions in an ingot during plastic deforma-
tion. The void-evolution model can be used to predict the
void closure irrespective of whether the ingot undergoes
compressive deformation (such as upsetting) or plane de-
formation (such as plane strain and rolling).
5 CONCLUSIONS
Based on the RVE model, a comprehensive theoreti-
cal analysis and numerical simulation, the influence of
the Lode parameter and stress triaxiality on the void clo-
sure is expounded. In addition, according to the above
research, a new void-evolution model considering vari-
ous macroscopic stress states is proposed, and the fol-
lowing main conclusions are obtained:
1) The Lode parameter and stress triaxiality have im-
portant effects on the void closure. The larger the Lode
parameter, the smaller is the equivalent strain required
for the void closure. When the Lode parameter μ [0, 1],
the difference in the macroscopic compressive strain re-
quired for the void closure is minimal. The smaller the
stress triaxiality, the smaller are the macroscopic equiva-
lent strain and the macroscopic compressive strain re-
quired for a complete closure of a void.
2) When the void-shape coefficient in the direction of
the maximum compressive strain is used to describe the
closure degree of a void, the macroscopic compressive
strain is more suitable as an independent variable than
the macroscopic equivalent strain. Based on this, a new
void-evolution model is established, whose accuracy is
close to the numerical-simulation result for a set void in
a workpiece. The model can predict the void closure un-
der various stress states with high accuracy.
3) The root mean square error (RMSE) of the new
void-evolution model is 0.05 when compared with the
physical-simulation results. The accuracy of the new
model is verified. Using this model, the process design
and optimization of ingot casting can be carried out con-
veniently, and the evolution of voids at any position in an
ingot can be assessed effectively.
Acknowledgment
The work was financially sponsored by the National
Natural Science Foundation of China (No.51575372),
the Start-Up Fund for Scientific Research of the Taiyuan
University of Science and Technology (No.20172011)
F. CHEN et al.: VOID-CLOSURE BEHAVIOR AND A NEW VOID-EVOLUTION MODEL FOR VARIOUS STRESS STATES
112 Materiali in tehnologije / Materials and technology 55 (2021) 1, 105–113
Figure 10: Comparison of the results of the physical-simulation ex-
periments and the new model
Figure 9: Physical-simulation results of upsetting test: a) samples with void 2#, b) samples with void 4#

and the Shanghai Research Center of Engineering Tech-
nology for Large Parts Thermal Manufacturing.
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