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A Modified Isotropic–Kinematic Hardening Model to Predict the Defects in Tube Hydroforming Process

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A Modified Isotropic–Kinematic Hardening Model to Predict the Defects in Tube Hydroforming Process

Abstract

Numerical simulations of tube hydroforming process of hollow crankshafts were conducted by using finite element analysis method. Moreover, the modified model involving the integration of isotropic–kinematic hardening model with ductile criteria model was used to more accurately optimize the process parameters such as internal pressure, feed distance and friction coefficient. Subsequently, hydroforming experiments were performed based on the simulation results. The comparison between experimental and simulation results indicated that the prediction of tube deformation, crack and wrinkle was quite accurate for the tube hydroforming process. Finally, hollow crankshafts with high thickness uniformity were obtained and the thickness distribution between numerical and experimental results was well consistent.
A Modified Isotropic–Kinematic Hardening Model
to Predict the Defects in Tube Hydroforming Process
Kai Jin, Qun Guo, Jie Tao, and Xun-zhong Guo
(Submitted September 21, 2016; in revised form June 22, 2017)
Numerical simulations of tube hydroforming process of hollow crankshafts were conducted by using finite
element analysis method. Moreover, the modified model involving the integration of isotropic–kinematic
hardening model with ductile criteria model was used to more accurately optimize the process parameters
such as internal pressure, feed distance and friction coefficient. Subsequently, hydroforming experiments
were performed based on the simulation results. The comparison between experimental and simulation
results indicated that the prediction of tube deformation, crack and wrinkle was quite accurate for the tube
hydroforming process. Finally, hollow crankshafts with high thickness uniformity were obtained and the
thickness distribution between numerical and experimental results was well consistent.
Keywords crack, ductile fracture criteria, isotropic–kinematic
hardening, tube hydroforming, wrinkle
1. Introduction
Tube hydroforming (THF) process has been increasingly
attracting attention in recent years, due to the increasing
demands for lightweight components in automotive, aircraft
and aerospace industries (Ref 1). The THF has become a
popular method for producing complex three-dimensional
structural shapes because of its enormous advantages such as
part consolidation, weight reduction, improved structural
strength and stiffness and reduction in the associated tooling
and materialsÕcost, over the more traditional processes
including stamping, welding, deep drawing and roll forming.
The ultimate purpose of this process is to form a tube of
complex shape with varying cross sections without causing any
defects such as crack and wrinkle. Therefore, extensive
research efforts have been devoted to study and analyze the
process in order to understand the aspects of THF. Different
optimizations have been proposed by researchers to optimize
the loading paths conditions. The loading paths in the THF
process are traditionally determined using trial-and-error pro-
cedures. The THF process further becomes complicated if new
materials and geometries are used; furthermore, the process is
time-consuming and expensive. Therefore, an integrated
approach to the problem involving finite element (FE) analysis,
a failure model and an optimization code is required. Recently,
the numerical simulation method, in particular, the FE method,
has been widely used to predict and estimate the formability of
the THF process (Ref 2,3).
For extremely large plastic deformation under complex
loading conditions where Bauschinger effect exists, isotropic
hardening elastoplastic constitutive models are no longer
suitable. Therefore, kinematic hardening model was taken into
consideration and combined with isotropic hardening model to
construct a new modified constitutive model. Besides, metal
softening effect becomes obvious as the damage accumulation
increases until fracture initiation. Therefore, an appropriate
damage model should also be adopted in the constitutive
model.
Till date, numerous constitutive models have been proposed
to simulate nonlinear behaviors of metal materials and struc-
tures. Frederick and Armstrong (Ref 4) introduced a relaxation
term into the back-stress to describe the metal plasticity. Based
on Armstrong–Frederick model, Chaboche (Ref 5,6) employed
several independent back-stresses with different evolution rules
to simulate different nonlinear behaviors of various materials.
Zaverl and Lee (Ref 7) put forward a combined nonlinear
kinematic hardening (NKH) model with the isotropic hardening
evolution rule, which can well describe the nonlinear behaviors
of many metals. Zang et al. (Ref 8,9) combined ChabocheÕs
model and ZaverlÕs model together and proposed a two-term
kinematic hardening model. Besides, two-surface model with
non-isotropic hardening memory surface was proposed and
termed as the Yoshida–Uemori model (Ref 10). In this study,
ZangÕs model was adopted and modified to describe the metal
behavior of THF process.
However, the model exhibited some limitations when plastic
strain exceeded over 50%, leading to the initiation of the
fracture. Thus, integration of a ductile fracture model into the
constitutive model was required to predict the fracture initia-
tion. A micromechanical model, namely Rice and Tracey
Kai Jin and Qun Guo have contributed equally to this work.
Kai Jin, Institute of Advanced Materials and Forming Technology,
Nanjing University of Aeronautics and Astronautics (NUAA), Nanjing
211100, PeopleÕs Republic of China; Jiangsu Key Laboratory of
Nuclear Energy Equipment Materials Engineering, Nanjing 211100,
PeopleÕs Republic of China; and College of Mechanical and Electrical
Engineering, Nanjing University of Aeronautics and Astronautics
(NUAA), Nanjing 211106, PeopleÕs Republic of China; Qun Guo, Jie
Tao, and Xun-zhong Guo, Institute of Advanced Materials and
Forming Technology, Nanjing University of Aeronautics and
Astronautics (NUAA), Nanjing 211100, Peoples Republic of China;
Jiangsu Key Laboratory of Nuclear Energy Equipment Materials
Engineering, Nanjing 211100, Peoples Republic of China; and
College of Material Science and Technology, Nanjing University of
Aeronautics andAstronautics (NUAA), Nanjing 211100, Peoples
Republic of China. Contact e-mail: guoxunzhong@nuaa.edu.cn.
JMEPEG ASM International
DOI: 10.1007/s11665-017-2937-7 1059-9495/$19.00
Journal of Materials Engineering and Performance
(R&T) model, was employed, which studies the evolution of
spherical voids and governs the voids growth by the stress
triaxiality (Ref 11). Based on the integration of the ductile
fracture model into ZangÕs model, a modified isotropic–
kinematic hardening model was proposed in this study. By
using this model, the metal behavior could be described well
and the defects in hydroforming could be predicted precisely.
2. Constitutive Model
Regarding the ductile fracture criteria, the R&T model was
used to describe the damage accumulation and crack initiation.
The model analyzes the relationship between the radius of a
void and stress triaxiality based on a simplified model of a
spherical void in a remote simple tension strain rate field. It
provides good evaluation of the ductile fracture under a large
variety of stress conditions. The term of critical void growth
index is vcr (Ref 12-16). Assuming an ideal case, the stress
triaxiality is constant in the entire loading history, and the
relationship between fracture strain
efand stress triaxiality g
can be formulated as follows:
ef¼vcr e3
2gðEq 1Þ
It is assumed that no damage accumulates when the stress
triaxiality gis below 1/3 and equivalent strain is below
critical strain e
D
. The damage increment can be expressed as
follows:
_
D¼
_
ep
vcre3
2g¼hRT _
epif
ep>eDand g>1
3
0 Otherwise
(ðEq 2Þ
The scalar D (0 £D£1), which is an internal variable,
was adopted to describe the isotropic damage. Taking the
impact of damage into account, the expression of actual flow
stress rDis given by Eq 3as follows:
rD¼1DðÞrM¼1DðÞr0þk1em
ep

C1
r1
1er1
ep


ðEq 3Þ
where rMis the flow stress of undamaged material, r0is the
initial yield stress, and k,m,C1and c1are material hardening
parameters.
The global work hardening in hydroforming can be
represented by the equation as an isotropic hardening model
if the tube has only tension or compression stress states.
However, the actual stress states often change from tension to
compression or vice versa for the forming of non-symmetrical
components with complex shape. Therefore, kinematic hard-
ening model should be adopted to describe the hardening
behavior under strain path reversal. The Bauschinger effect is
captured by the kinematic motion of yield surface. Combining
isotropic and kinematic hardening together, a plasticity consti-
tutive model integrated with ductile fracture criteria was
proposed. The von Mises yield surface is defined as follows:
U¼frD¼ffiffiffiffiffiffiffiffiffiffiffiffi
3
2n:n
rrD
ep

¼0ðEq 4Þ
n¼SaðEq 5Þ
where nis the stress difference, which is the stress measured
from the center of the yield surface. Sis the deviatoric stress
tensor and ais the back-stress tensor, which determines the
central position of the current yield surface. The elastoplastic
formulation was derived by the associated flow rule and
hypo-elastic assumption. The plastic strain increment is given
by:
dep¼dk@f
@nðEq 6Þ
For the kinematic hardening model, the normal to the von
Mises yield surface can be written as follows:
Q¼@f
@n¼ffiffi
3
2
rn
rD
ðEq 7Þ
Thus, the plastic part of the strain rate is given by a
normality condition and the equivalent value is
dep¼dkQðEq 8Þ
d
ep¼dkðEq 9Þ
With the assumption of a small elastic and large plastic
deformation, strain increment decan be decomposed into
elastic component deeand plastic component dep,
de¼deeþdepðEq 10Þ
For isotropic elasticity, the generalized HookeÕs law states
that Cauchy stress tensor is proportional to elastic strain tensor,
as follows:
dr¼Ce:dee¼Ce:dedep
ðÞ ðEq 11Þ
where Ceis the fourth-order elastic tensor.
To describe the hardening behavior under strain path
reversal, a two-term kinematic hardening model was adopted,
which was proposed by Zang et al. One term a1was adopted
from a nonlinear Chaboche kinematic hardening model which
is used to express the transient behavior. Another term a2was
obtained from a linear Ziegler kinematic hardening model,
which is used to represent the permanent softening. Thus, the
back-stress acan be written as follows:
a¼a1þa2ðEq 12Þ
A nonlinear kinematic motion can be generated for the yield
surface, and the evolution of back-stresses, da1and da2, can be
defined, respectively, as follows:
da1¼C1
nSaðÞd
epc1a1d
epðEq 13Þ
da2¼C2
nSa
ðÞ
d
epðEq 14Þ
where C1,c1and C2are material hardening parameters.
If plastic deformation occurs, the stress state remains on the
yield surface (F = 0) and the plastic compliance factor dk
becomes nonzero. Once dkis obtained, dep,dr,daand drD
could be determined and their corresponding accumulated
values could also be updated.
Journal of Materials Engineering and Performance
3. Experiments
3.1 Uniaxial Tensile and Cyclic Loading Tests
To verify the proposed model, uniaxial tensile and cyclic
loading tests were carried out. The geometry of test specimens
made by SUS304 and the setup of experiments are shown in
Fig. 1. Material properties of SUS304 derived from tensile test
are given in Table 1. Loading paths of tests are displayed in
Fig. 2. Herein, the sample was loaded under uniaxial tension
(Fig. 2a). The sample was loaded under a cycle loading history
(Fig. 2b).
3.2 Tube Hydroforming Test of a Non-symmetrical
Component
A hollow crankshaft is a typical non-symmetrical compo-
nent, which is used in the engine of small airplane or unmanned
aerial vehicles (as shown in Fig. 3). During hydroforming
process, the blank tube is formed to fit the non-symmetrical
cavity of hydroforming die, simultaneously by internal pressure
and axial compressive loads. Thus, the tube suffers complex
loading conditions and large deformation to achieve the desired
shape.
The final geometry of the SUS304 crankshaft is shown in
Fig. 3, which is formed from a tube blank with 300.0 mm
length, 38.0 mm outer diameter and 1.0 mm thickness. The
expansion ratio was 33% in A–A section and 15.4% in B–B
section. The experimental setup is shown in Fig. 4. Molybde-
num disulfide was used as lubricant. After clamping the die,
high-pressure oil was allowed to flow into the tube and two
punches moved inward.
According to the study of Yuan et al. (Ref 17), before
designing the loading paths, the calibration pressure (the
maximum internal pressure) has to be acquired by using Eq 15:
pc¼t
rc
rsðEq 15Þ
where p
c
is the closing force, r
c
is the minimum transition fil-
let radius of the part, tis the thickness of part and r
s
is the
material yield strength. For crankshaft, r
c
= 5.31 mm,
t= 1 mm and r
s
= 245 MPa. Pc was calculated to be
46.89 MPa from Eq 1. Based on the theoretical calculation
results, step-increased loading paths were built, as shown in
Fig. 5.
Figure 6(a) exhibits that a designed geometry of hollow
crankshaft was obtained along loading path 1. However, the
wrinkle occurred at the bulge transition region along loading
path 2 (Fig. 6b) because the deformation caused by pressure
increment was much smaller than that caused by axial feeding
speed. Therefore, expansion was insufficient in circumferential
direction and wrinkle got accumulated in axial direction. In
contrast, if internal pressure was too large such as loading path
3 at the initial yield stage, the thickness of tube decreased
rapidly, and it busted as shown in Fig. 6(c). Clearly, there is a
coordination relationship between internal pressure and axial
feeding speed. The appropriate values could also be obtained
by optimum design.
4. Finite Element Analysis
4.1 Material Property and Model Parameters
Numerical analysis was carried out to validate the correct-
ness and calibrate the parameters of the proposed constitutive
Fig. 1 Setup of uniaxial tensile and cyclic loading tests: (a) test
machine and (b) geometry of test specimen
Table 1 Material property parameters of SUS304
Material PoissonÕs ratio Yield strength, MPa Tensile strength, MPa Density, kg/m
3
Number K, MPa
SUS304 0.28 245 408 7850 0.32 537
Journal of Materials Engineering and Performance
Fig. 2 Loading paths of material property tests: (a) uniaxial tensile and (b) loading test
Fig. 3 Geometry and cross sections of hollow crankshaft
Fig. 4 Experimental setup for hydroforming of hollow crankshaft Fig. 5 Loading paths of hydroforming tests
Journal of Materials Engineering and Performance
model. Using the same conditions as in experiments, hydro-
forming simulations were performed by using the explicit
module of ABAQUS, implemented in user subroutine VUMAT.
Test specimens were axisymmetric; therefore, one-eighth model
of specimen was utilized and symmetry planes were con-
strained as shown in Fig. 7. Nearly 5000 C3D8 elements were
meshed and the region with big deformation was divided by
dense mesh.
Under uniaxial tension, the proposed model could accurately
predict the fracture as shown in Fig. 8(a). Moreover, by
employing NKH model, the calculation of the von Mises stress
is consistent with experimental results in cyclical loading test.
The Bauschinger effect was also predicted accurately as shown
in Fig. 8(b). The parameters of the constitutive models related
to the damage calculation were calibrated by using the
monotonic tensile as shown in Fig. 8(a), and the parameters
related to the kinematic hardening were determined by full
cyclic loading test as shown in Fig. 8(b). All calibrated
parameters for the constitutive model are listed in Table 2.
Fig. 6 Hollow crankshaft formed by different loading paths: (a)
path 1, (b) path 2 and (c) path 3
Fig. 7 FE model for uniaxial tensile and cyclic loading tests (unit mm)
Fig. 8 Comparison between experiments and numerical analysis results: (a) uniaxial tensile test and (b) cyclical loading test
Journal of Materials Engineering and Performance
4.2 Tube Hydroforming Simulation
The hydroforming simulation model was composed of the
preformed part. Figure 9shows the FE model of hydroforming
including upper and lower dies, punch and tube blank for
hydroforming. The punches and die were set as rigid body, and
the tube blank was meshed by hexahedral element. Five layers
were meshed in thickness direction and the total number of
elements was nearly 50000. The feeding speed of punch and
the internal pressure were given (Fig. 5).
5. Results and Discussion
5.1 The Effects of Friction on Tube Hydroforming Results
Figure 10 clearly indicates that an increasing friction
coefficient leads to a gradual decrease in the protrusion height,
while the maximum thinning rate remains constant. At the
friction coefficient of 0.05, the protrusion height was 31.5 mm
and the maximum thinning rate was 36%. When the friction
coefficient exceeded 0.05, the protrusion height was insuffi-
cient. The friction coefficient had insignificant effect on the
maximum thinning rate. Consequently, a friction coefficient of
0.05 was selected as the optimum.
5.2 The Effects of Loading Paths on Forming Results
Different loading paths result in different mechanical
behaviors of material in the THF process. If the pressure is
too small, the axial feed of the materials is provided in time by
the punches, and top of the protrusion is the main deformed
region with wrinkle defects. However, if the internal pressure
on the inner face of the hollow crankshaft is too large, the axial
feed of the materials is not provided in time by the punches,
causing the top of the protrusion to be the main deformed
region with crack defects.
Clearly, the profiles of three cases calculated by FE
analysis are consistent with experimental results. The pre-
dicted wrinkle distribution as shown in Fig. 11(b) is similar to
that obtained from the experiment. The position and direction
of predicted cracks as shown in Fig. 11(c) are also in good
agreement with experimental results. Figure 11 demonstrates
the cutting of the hollow crankshaft along longitudinal
direction, for further comparison of profile and thickness of
bulge region. Sixteen points of the bulge region were selected,
and corresponding thickness and outer diameter coordinates
were measured as shown in Fig. 12. The outer diameter
difference is 5% in path 1, 7% in path 2 and 9% in path 3. The
thicknessdifferenceis6%inpath1,8%inpath2and10%in
path 3. Clearly, the protrusion height and thickness distribu-
tion of the crankshaft depend on the compressive action of
internal pressure, feed distance and friction coefficient. From
the above-mentioned simulation results, the optimized param-
Table 2 Calibrated model parameters of constitutive
models
One-surface cyclic plastic
constitutive model
Rice and Tracey
damage model
r0265.57 MPa vcr 0.25
k359.67 MPa eD0.75
m5.37
C13000
c120
C2200
Fig. 9 Initial setup of hydroforming for finite element simulation
Fig. 10 Friction vs. the protrusion height and thickness thinning
rate
Fig. 11 Profile of hollow crankshaft: (a) loading path 1, (b) loading
path 2 and (c) loading path 3
Journal of Materials Engineering and Performance
eters are as follows: loading path 3 and a friction coefficient of
0.05.
5.3 Comparison of Simulation Results of Proposed Model
and Isotropic Hardening Model
In proposed model, NKH model was adopted. Further-
more, isotropic hardening (ISO) model, which is used mostly
in FE analysis, was also used for the simulation along loading
path 1. Figure 13(a) shows that the outer diameter difference
is bigger than that obtained from NKH model. The thickness
reduces much more compared to the results from NKH model
as shown in Fig. 13(b). This is attributed to the fact that the
stress state is changed from compression to tension in the
transition zone where the tube is bended by fillet and then
expands in the expansion zone. Herein, the change in
maximum principal stress of a selected element can clearly
explain the phenomenon (Fig. 14). When the selected element
entered the transition zone, compression stress was obtained
because the internal pressure pushed it to the fillet. As the
element flowed to the expansion zone, the stress changed from
compression to tension. At this time, the Bauschinger effect
occurred. If only isotropic hardening model was used, the
deformation and stress calculation would not have been
accurate. Therefore, consideration of the kinematic hardening
model was required and it was integrated with the isotropic
hardening model. In our proposed model, mixed isotropic–
kinematic hardening model could reasonably describe the
deformation.
Fig. 13 Comparison of isotropic hardening model and proposed model: (a) outer diameters of crankshaft formed by loading path 1 and (b)
thickness distribution of crankshaft formed by loading path 1
Fig. 12 Comparison of experiments and numerical analysis: (a) outer diameters of crankshaft and (b) thickness distribution of crankshaft
Journal of Materials Engineering and Performance
6. Conclusion
This study proposed a straightforward method to analyze the
plastic deformation under extreme large strain loading. A ductile
fracture criterion (Rice and Tracey model) was also integrated
into the constitutive model and implemented in numerical
analysis. Furthermore, the proposed model was applied to tube
hydroforming of non-symmetrical components. The formability
of tube material was predicted through numerical simulations.
Thus, the following conclusions can be drawn:
1. A modified plastic constitutive model was proposed,
which was based on nonlinear kinematic hardening, and
isotropic hardening and further integrated with a damage
model. The model provided almost the same predicted
crack displacements of the experiments, in each of which
the proposed cyclic fracture models were employed.
Compared to experimental data, the model was verified
to be accurate under various loading paths.
2. Through using the proposed constitutive model in tube
hydroforming process, the deformation and formability of
tube were predicted accurately. Comparing the outer
diameter and thickness of each case, the feasibility of
proposed model was verified. A coordination relationship
was obtained between internal pressure and axial feeding
speed. Moreover, the appropriate values could also be ob-
tained by optimum design.
3. Comparison of numerical analysis results between isotro-
pic hardening model and mixed isotropic–kinematic hard-
ening model indicated that the FE analysis required the
consideration of nonlinear kinematic hardening if a part
was formed by cyclical loading or with a complex geom-
etry such as non-symmetrical shape.
Acknowledgments
The authors greatly acknowledge the financial support from the
Natural Science Foundation of Jiangsu Province (Grant No.
SBK2015022427), the Research Fund of Nanjing University of
Aeronautics and Astronautics (Grant No. YAH17019), the Funda-
mental Research Funds for the Central Universities (Grant Nos.
NJ20150023, NJ20160035 and NJ20160036) and Project Funded
by the Priority Academic Program Development of Jiangsu Higher
Education Institutions.
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Journal of Materials Engineering and Performance
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Tube hydroforming (THF) experiments were performed on TP2 copper tubes under different loading velocities and fluid velocities using a self-developed measurement system to investigate dynamic frictional characteristics in the guiding zone. The results show that the coefficient of friction (COF) dynamically changes during forming experiments and decreases with tube deformation. The average descending rate and amplitude of the COF increase with increasing loading velocity. Microscopically, the micro-protrusions on the tubular surface are flattened, and the surface scratches are finer and more uniform, as the loading velocity increases, resulting in a decrease in COF. At the same external loading velocity, the COF increases with increasing fluid velocity and is also extremely sensitive to it. Moreover, improving and predicting the formability of such tubes by accurately adjusting and controlling fluid velocity in THF is valuable and critical for the future.
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