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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

@n¼ﬃﬃﬃ

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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