Computers and Structures 231 (2020) 106222

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Computers and Structures

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Numerical implementation of modified Chaboche kinematic hardeningmodel for multiaxial ratcheting

https://doi.org/10.1016/j.compstruc.2020.1062220045-7949/� 2020 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: hylee@sogang.ac.kr (H. Lee).

Jungmoo Han, Karuppasamy Pandian Marimuthu, Sungyong Koo, Hyungyil Lee ⇑Sogang University, Department of Mechanical Engineering, Seoul 04107, Republic of Korea

a r t i c l e i n f o

Article history:Received 8 September 2019Accepted 29 January 2020

Keywords:Modified Chaboche modelMultiaxial cyclic plasticityNumerical implementationFinite element analysisImplicit radial return method

a b s t r a c t

For simulating multiaxial ratcheting behavior, the modified Chaboche kinematic hardening model wasnumerically implemented by using the framework of a small-strain elastic-plastic theory. Unlike earlymodels, this improved multiaxial model is difficult to implement using finite element methods owingto its complicated constitutive relations, such as radial evanescence terms and the fourth hardening rulewith a threshold. We present an effective procedure for numerical implementation using Voigt notationsand the implicit radial return method with Newton-Raphson iterations. All the equations of constitutenumerical integration and consistent tangent operator (CTO) are simply solved using matrix operations.The integration algorithm is validated by using both numerical examples and analytical solutions. TheCTO is verified by additional stress calculations. The model detects variations in the cyclic indentationresponse with changes in a multiaxial-dependent parameter. The numerical implementation allowssimulations of both biaxial and general multiaxial ratcheting behaviors.

� 2020 Elsevier Ltd. All rights reserved.

1. Introduction

In metallic materials, ratcheting is a phenomenon in whichplastic strain accumulates after yielding owing to repeated stressor thermal deformation. As a result, ratcheting produces excessiveplastic deformation, which is one of the major causes for loss ofstructural integrity [1–5]. Therefore, the mechanical componentsoperating under cyclic loading environments should be designedafter determining the ratcheting deformation of these components.Investigations regarding ratcheting behaviors of the materials usedin gears, bearings, rails, turbine disks, pressure vessels and atomicreactors were performed for safety and fatigue life improvements[6–11]. It was suggested that ratcheting can be analyzed effectivelyby using kinematic hardening models rather than isotropic harden-ing models [12].

Based on the linear kinematic hardening models of Prager [13]and Ziegler [14], Armstrong and Frederick [15] initially proposed anonlinear kinematic hardening model (AF model) by adding adynamic recovery term to the linear kinematic hardening model.Later, Chaboche [16,17] and Ohno and Wang [18] improved theAF model by introducing the concept of multiple backstress com-ponents and proposed the Chaboche model and Ohno-Wangmodel, respectively. These two types of constitutive models are

most commonly used for uniaxial ratcheting simulations. For mul-tiaxial [19–26] and time-dependent (viscoplastic) [27–31] ratchet-ing simulations, improved kinematic hardening models have beendeveloped by modifying the Chaboche or Ohno-Wang model;detailed reviews are provided by Chaboche and his coworkers[32,33]. Madrigal et al. [34] showed that biaxial ratcheting undercombined axial-torsional loading can be analyzed via simplenumerical computation. However, the cyclic loads acting on actualengineering structures and components are often multiaxial.Therefore, an advanced finite element method (FEM) is necessaryalong with kinematic hardening models for practical applicationsand extensive multiaxial ratcheting analysis.

Several researchers performed numerical implementations ofkinematic hardening models using the FEM. Kobayashi and Ohno[35], Kang [36], and Zhu et al. [37,38] implemented improved kine-matic hardening models based on the Abdel Karim-Ohno model[22], which is one of the Ohno-Wang-type models. Kulling andWippler [39] and De Anglis [40] also implemented the early Cha-boche model and its viscoplastic version, but these are not suitablefor multiaxial ratcheting analysis. Although both Ohno-Wang- andChaboche-type models have advantages of obtaining the modelparameters from uniaxial experiments [23,41,42], when comparedto Ohno-Wang-type models, the Chaboche-type models for multi-axial ratcheting have not been intensively investigated; moreover,their numerical implementation is not discussed in any literature.Analytical solutions of Chaboche-type models exist for uniaxial

Nomenclature

nþ1ð�Þ current state variable ð�Þnð�Þ prior state variable ð�ÞDð�Þ increment of ð�Þdð�Þ differential of ð�ÞE;v Young’s modulus, Poisson’s ratiol shear modulusr stressrtr trial stressro yield strengthrm mean stressra amplitude stressrh circumferential stressrhc magnitude of circumferential stressee elastic strainep plastic straine total strain (= ee + ep)ez axial strainezc amplitude of cyclic axial strainehp circumferential plastic strainp equivalent plastic strain

s deviatoric stressstr deviatoric part of trial stressa total backstress on deviatoric planeam mth backstress�a4 threshold level for 4th backstressCm rate of yield surface movement of mth backstress com-

ponentcm relaxation rate of yield surface movement of mth back-

stress componentk multiaxial dependent parameterq relative stress ðs� aÞqtr trial relative stress ðstr � aÞDe elastic tangent operatorDep elastoplastic tangent operatorw �1 in case of each tension and compressionn unit normal to yield surface on deviatoric planeP indentation loadh indentation depthho initial indentation depth at first loadingN number of cycles

2 J. Han et al. / Computers and Structures 231 (2020) 106222

conditions, which makes it possible to effectively determine themodel parameters through optimization [43–45]. For multiaxialratcheting prediction, Bari and Hassan [23] modified the Chabochekinematic hardening model. Numerical implementation of themodified Chaboche model with finite element (FE) software canbe useful for performing complex ratcheting simulations.

Therefore, in this study, the modified Chaboche kinematic hard-ening model is numerically implemented by developing user sub-routines i.e. UMAT (user-defined material model) in Abaqus/Standard [46]. The constitutive equations in terms of the fourth-order tensor are developed in a reduced form by using Voigt nota-tions and the backward Euler scheme. The numerical integrationalgorithm, comprising the radial return method, computes threeequations for stress calculations with the Newton iterationmethod. The numerical algorithm is then verified by comparingthe numerical results with (i) analytical solutions of the constitu-tive equation and (ii) the results of another study [23] on biaxialratcheting simulation. To validate the consistent tangent operator(CTO), additional stress calculations are conducted using the CTOand then compared with the stress values computed by numericalintegration. Finally, cyclic indentation FE analysis is performedalong with uniaxial and biaxial numerical examples using theimplemented model; this model can be regarded as a generalmodel for simulatingmultiaxial cyclic responses of actual engineer-ing parts.

2. Constitutive model (Bari and Hassan, 2002)

Bari and Hassan [23] developed an improved kinematic harden-ing model (BH model) for analyzing multiaxial cyclic plasticity.Similar to typical kinematic hardening models, the BH model alsoconsists of three basic concepts. They are the (i) yield criterion,(ii) flow rule, and (iii) hardening rule (Eqs. (1)–(3)), which repre-sent the combined Chaboche [17] and Burlet-Cailletaud [47]models.

Von-Mises yield criterion

F s� að Þ ¼ f s� að Þ � ro ¼ 0; f s� að Þ

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi32

s� að Þ : s� að Þr

ð1Þ

Flow rule

dep ¼ffiffiffi32

rdpn; n ¼

ffiffiffi23

r@F s� að Þ

@r¼

ffiffiffi23

r@f s� að Þ

@r

¼ffiffiffi32

rs� að Þro

ð2Þ

Hardening rule

dam ¼ 23Cmdep � cm kam þ 1� kð Þ am : nð Þnf gdp; for m

¼ 1;2;3

da4 ¼23C4dep ; f a4ð Þ< �a4; f a4ð Þ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi32a4 :a4

q23C4dep�c4 ka4þ 1�kð Þ a4 :nð Þnf g 1� �a4

f a4ð Þ

� �dp ; f a4ð ÞP �a4

8><>:

ð3Þ

da ¼ da1 þ da2 þ da3 þ da4 total backstressð Þ

In a multiaxial ratcheting simulation, the multiaxial dependentparameter k improves both the over-prediction by Chaboche’s ruleand underestimation by Burlet and Cailletaud’s rule by weightingeach model. Parameter k affects only the multiaxial ratchetingresponse and not the uniaxial response. For the uniaxial loadingcondition, the backstress evolution (Eq. (3)) becomes the same asthe initial Chaboche hardening rule [17]. The analytical solutionfor the uniaxial case is described as follows:

am ¼ am;o � 23w

Cm

cm

� �� �exp �wcm ep � epo

� �þ 23w

Cm

cm

� �;

m ¼ 1;2;3 ð4Þ

a4¼a4;o� 2

3 wC4c4þ�a4

� �h iexp �wc4 ep�epoð Þ½ �þ2

3 wC4c4þ�a4

� �; a4 > 2

3�a4

a4;oþ 23C4 ep�epoð Þ; a4j j6 2

3�a4

a4;o� 23 wC4

c4��a4

� �h iexp �wc4 ep�epoð Þ½ �þ2

3 wC4c4��a4

� �; a4 <�2

3�a4

8>>><>>>:

ð5Þ

Here, w takes values equal to + 1 and �1 for tension and compres-sion, respectively. The above equations are used to verify the

p (%)0 2 4 6 8 10 12 14 16

(M

Pa)

-200

-100

0

100

200

300

400cycle : 20 40 60 80 100

m, a = 50, 250) MPa

Fig. 1. Total stress-plastic strain curve for 100 cycles for (rm , raÞ = (50, 250) MPa.

Fig. 2. Evolution of three backstresses ða1�3Þ for (rm , raÞ = (50, 250) MPa.

J. Han et al. / Computers and Structures 231 (2020) 106222 3

numerical integration algorithm for stress calculations, discussed inSection 4.1.

2.1. Uniaxial ratcheting simulation under various stress conditions

By using the analytical solutions (Eqs. (4) and (5)), uniaxialratcheting simulations are conducted under various cyclic stressconditions. In this paper, the values of material model parametersused by Bari and Hassan [23] are utilized for numerical examplesas listed in Table 1. The seven analysis conditions are designed as(rm, ra) = (50, 250), (50, 275), (50, 300), (100, 250), (150, 250),(50, 130), (50, 131) MPa to consider the effects of rm and ra onthe uniaxial ratcheting response. For the case of (rm, ra) = (50,250) MPa, the ratcheting phenomenon is simulated as the plasticstrain accumulates with the open hysteresis loops of r-ep as shownin Fig. 1. The evolutions of the three backstresses (a1�3Þ for the firstcycle are shown in Fig. 2, whereas the hardening behavior andtransition of the fourth backstress ða4Þ between the linear and non-linear rule under uniaxial cyclic loading are shown in Fig. 3. Itshould be emphasized that unlike the other backstresses, thefourth hardening rule requires transitions at the threshold bound-aries (Fig. 3). As a result, it causes numerical difficulties.

Expressing the plastic strain according to the number of stresscycles N is a common way to investigate the ratcheting phenom-ena. The effect of rm on the uniaxial ratcheting response is evalu-ated by comparing the three cyclic stress conditions ofðrm;raÞ = (50, 250), (50, 275), (50, 300) MPa. Similarly, the effectof ra is analyzed with the conditions of ðrm;raÞ = (50, 250), (100,250), (150, 250) MPa. As a result, Fig. 4 confirms that the rate ofthe accumulation of plastic strain (slope of ep-N curve) increasesas rm or ra increases. These findings are in good agreement withactual material behaviors i:e. the ratcheting strain increases withincreases in rm and ra [48]. The results of ratcheting simulationsunder ra = 130 and 131 MPa with the same rm = 50 MPa are com-pared in Fig. 5. Since the yield stress is ro = 130 MPa, the conditionfor the occurrence of ratcheting (slope – 0) is clearly observed;the ratcheting strain does not occur if ra 6 ro (= 130 MPa) in uni-axial condition. Since the uniaxial ratcheting occurs for asymmet-ric cyclic stress condition ðrm – 0), it should be emphasized thatthe sign of increment of ep may not always match that of rm, how-ever the sign can be affected by the loading history [49].

3. Numerical implementation

3.1. Discretization of constitutive relations

The constitutive relations are discretized by using the backwardEuler scheme and then numerically integrated based on the radialreturn method which is the most popular implicit implementationmethod for various types of kinematic hardening models [35–40].Although Kobayashi and Ohno [35] proposed a nonlinear scalarequation for numerical implementation with a general form ofkinematic hardening rule, the equation cannot be utilized to imple-ment the BH model due to the radial evanescence term of Eq. (3).Therefore, the numerical implementation of the BH model requiresthe inverse of the fourth-order tensor for integration algorithm andCTO and thus high computational cost increases. This studydevelops the constitutive relations in a reduced order by using

Table 1Reference values of model parameters [23].

E (MPa) v ro (MPa)181 � 103 0.3 130

c2 C3 (MPa) c3400 3 � 103 11

Voigt notation to solve the tensor equations using only matrixoperations.

In the Voigt form, the stress and strain are expressed as

rf g ¼ ½r11 r22 r33 r12 r23 r31�T ; T means transpose ð6Þ

ef g ¼ ½e11 e22 e33 2e12 2e23 2e31�T ð7ÞIt is also beneficial to understand that Abaqus [46] stores the stressand strain components in the above form (Eqs. (6) and (7)) toreduce the dimensionality of problems in practical FE analysis[50]. Hence, all the numerical equations formulated with Voigtnotation can be directly coded into the user material subroutineUMAT without any further processing. The constitutive relationsare discretized to yield the following equations (Eqs. (8)–(23)):

nþ1F ¼ nþ1f nþ1q � ro ¼ 0 ; yield criterion

nþ1f nþ1q ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

32

nþ1qf gT E1½ � nþ1qf gq

; nþ1q � ¼ nþ1s

�� nþ1a �

ð8ÞHere, the left upper subscript n +1 denotes the current incrementstate and {nþ1q} represents the relative stress from the backstressto the current deviatoric stress.

C1 (MPa) c1 C2 (MPa)413 � 103 20 � 103 22 � 103

C4 (MPa) c4 �a4 (MPa)103 � 103 5 � 103 34.5

Fig. 3. Evolution and transition behavior of fourth backstress a4 for (rm , raÞ = (50, 250) MPa.

(a) (b) N

0 20 40 60 80 100 120

p (%)

0

20

40

60

80

100

120a = 250 MPa

N0 20 40 60 80 100 120

0

20

40

60

80

100

120m = 50 MPa

Fig. 4. Effects of various cyclic stress conditions on ep �N data for various (a) rm and (b) ra .

N0 20 40 60 80 100 120

p (%)

0.00

0.02

0.04

0.06

0.08m = 50 MPa, o = 130 MPa

130 MPa (no ratcheting)

Fig. 5. ep �N curves for different ra = 130 and 131 MPa with ro = 130 MPa.

4 J. Han et al. / Computers and Structures 231 (2020) 106222

nþ1ep � ¼ nepf g þ Dnþ1ep

� ð9Þ

nþ1n � ¼

ffiffiffi32

rnþ1q �ro

ð10Þ

Dnþ1ep � ¼

ffiffiffi32

rDnþ1p E1½ � nþ1n

� ¼ 32ro

Dnþ1p E1½ � nþ1q � ð11Þ

The subscript n represents the prior increment state. The trial stressassumes that there is no plastic strain increment (Eq. (12)); truestress is calculated by subtracting the plastic corrector from the trialstress. This process is called the radial return method, as shown inFig. 6.

nþ1rtr � ¼ nrf g þ Dnþ1rtr �¼ nrf g þ De½ � Dnþ1e

�; ðtrial stressÞ ð12Þ

nþ1r � ¼ nþ1rtr

�� De½ � Dnþ1ep

�; ðsubtract plastic correctorÞ ð13Þ

Dnþ1r � ¼ De½ � Dnþ1e

�� Dnþ1ep � � ð14Þ

Here, [De] is the elastic stiffness matrix defined as

De½ � ¼

kþ 2l k k

k kþ 2l k

k k kþ 2ll

ll

2666666664

3777777775;

k ¼ Ev1þ vð Þ 1� 2vð Þ ; l ¼ E

2 1þ vð Þ

ð15Þ

The deviatoric parts of Eqs. (12) and (13) are written as

nþ1str � ¼ nsf g þ De½ � Id½ � Dnþ1e

� ð16Þ

nþ1s � ¼ nþ1str

�� De½ � Dnþ1ep �

¼ nsf g þ De½ � Id½ � Dnþ1e �� Dnþ1ep

� �¼ nsf g þ 2l E2½ � Id½ � Dnþ1e

�� Dnþ1ep � � ð17Þ

Here l is the shear modulus. Using Eq. (16), the trial relative stressis defined as

(a) (b)

1

2 3

deviatoricplanena

nσ

1 trn σ

1n σ1n n

1 1 1n n p n e

E

1n

n 1n

n

1 tr 1n n nE

1n p

(elastic predictor)

1n pE(plastic corrector)

o 1n e

Fig. 6. Implicit radial return method in case of (a) uniaxial and (b) general hardening.

J. Han et al. / Computers and Structures 231 (2020) 106222 5

nþ1qtr � ¼ nþ1str �� naf g ð18Þ

The yield condition of the trial stress is determined by substitutingthe above Eq. (18) into Eq. (8) to obtain

nþ1Ftr ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi32

nþ1qtrf gT E1½ � nþ1qtrf gr

� ro ¼ 0 ð19Þ

The total backstress is defined as

nþ1a � ¼ naf g þ Dnþ1a

� ð20Þ

Dnþ1a � ¼

X4m¼1

Dnþ1am � ð21Þ

The increment of each backstress {Dnþ1am} in Eq. (21) is formulatedusing Eq. (3) as follows:

Dnþ1am �¼2

3Cm E2½ � Dnþ1ep

��cm

nþ1am �

kþ 1�kð Þ nþ1am �T

E1½ � nþ1n �

nþ1n �n o

Dnþ1p ð22Þ

Dnþ1a4 �¼2

3C4 E2½ � Dnþ1ep

��c4

nþ1a4 �

kþ 1�kð Þ nþ1a4 �T

E1½ � nþ1n �

nþ1n �n o

g nþ1a4

Dnþ1p

ð23Þ

Here, gðnþ1a4Þ represents the threshold term of the fourth back-stress, and is defined as

g nþ1a4 ¼ 0 ; f nþ1a4

6 �a4 threshold levelð Þ

1� �a4f nþ1a4ð Þ ; f nþ1a4

> �a4

8<: ð24Þ

Based on the above discretized relations (Eqs. (6)–(24)), theimplicit integration algorithm and CTO are formulated. In thisstudy, the brackets {�} and [�] denote the first and second-ordertensors, respectively. [E1] and [E2] are auxiliary matrices maintain-ing the consistency of the constitutive equations in a reduced form,and [Id] is the deviatoric operator matrix. These matrices aredefined as follows:

E1½ � ¼

11

12

22

2666666664

3777777775;

E2½ � ¼

11

11=2

1=21=2

2666666664

3777777775;

Id½ � ¼ 13

2 �1 �1�1 2 �1�1 �1 2

33

3

2666666664

3777777775

ð25Þ

3.2. Formulation of implicit integration algorithm

In this section, the derivations of formulas for numerical inte-gration of the constitutive equations are described in detail to cal-culate three backstress components{nþ1a1;2;3} and three separateiteration functions for deriving the fourth backstress {nþ1a4}, rela-tive stress {nþ1q} and increment of the equivalent plastic strainðDnþ1pÞ. The three backstress components (m = 1, 2, 3) are implic-itly integrated using Eqs. (10), (11) and (22).

nþ1am �¼ namf gþCm

nþ1q �ro

Dnþ1p

�cm k nþ1am �þ3 1�kð Þ

2r2o

nþ1q �

nþ1q �T

E1½ � nþ1am �� �

Dnþ1p

ð26Þ

6 J. Han et al. / Computers and Structures 231 (2020) 106222

Rearranging {nþ1am} on the right side gives

nþ1am � ¼ Lm½ ��1 namf g þ Cm

ro

nþ1q �

Dnþ1p� �

ð27Þ

Here, the square matrix [Lm] is written as

Lm½ � ¼ I½ � þ cm k I½ � þ 3 1� kð Þ2r2

o

nþ1q �

nþ1q �T

E1½ �� �

Dnþ1p ð28Þ

where [I] the is identity matrix. Similarly, using Eq. (23), the fourthbackstress {nþ1a4} is determined to be

nþ1a4 � ¼ na4f g þ C4

Dnþ1pro

nþ1q �� L4½ � nþ1a4

�g nþ1a4 ð29Þ

L4½ � ¼ c4 k I½ � þ 3 1� kð Þ2r2

o

nþ1q �

nþ1q �T

E1½ �� �

Dnþ1p ð30Þ

The three backstress components {nþ1a1;2;3} are directly solved usingEqs. (27) and (28). However, whereas the fourth backstress compo-nent {nþ1a4} could not be directly solved. Therefore, it is calculatedby using the iteration scheme as the threshold term gðnþ1a4Þmade itimpossible to write {nþ1a4} as a single term. Therefore, in this study,the Newton iteration function for computing {nþ1a4} is formulatedconsidering two cases: (i) outside (nonlinear) and (ii) within (linear)the threshold level �a4. Using Eqs. (29) and (30), the iteration func-tion for {nþ1a4} is derived as

nþ1Xnonlin � ¼ nþ1a4

�þ L4½ � nþ1a4 �

1� �a4f nþ1a4ð Þ

� �� na4f g

� C4Dnþ1pro

nþ1q � ð31Þ

@ nþ1Xnonlin �@ nþ1a4f g ¼ I½ � þ L4½ � þ �a4

f nþ1a4ð Þ L4½ �nþ1a4 �

nþ1a4 �T E1½ �

nþ1a4f gT E1½ � nþ1a4f g� I½ �

" #ð32Þ

nþ1a iþ1ð Þ4

n o¼ nþ1a ið Þ

4

n o�

@ nþ1X ið Þnonlin

n o@ nþ1a4f g

24

35

�1

nþ1X ið Þnonlin

n o;

f nþ1a ið Þ4

� �> �a4 ð33Þ

Eq. (33) is the Newton iteration function for {nþ1a4} in case ofoutside the threshold. Within threshold level, the iteration func-tion is given by

nþ1Xlin � ¼ nþ1a4

�� na4f g � C4Dnþ1pro

nþ1q �

;@ nþ1Xlin �@ nþ1a4f g

¼ I½ � ð34Þ

nþ1a iþ1ð Þ4

n o¼ nþ1a ið Þ

4

n o� nþ1X ið Þ

lin

n o; f nþ1a ið Þ

4

� �6 �a4 ð35Þ

Eqs. (31)–(35) are used for computing the fourth backstresscomponent {nþ1a4}. Unlike in the formula for the early Chabochemodel [35], the relative stress {nþ1q} cannot be directly solvedwithout using the iteration method as it cannot be separated fromthe radial evanescence term of Eqs. (27)–(30). By subtracting Eq.(20) from Eq. (17) and using Eqs. (27)–(30)(11), (27) and (29), com-ponent {nþ1q} is obtained as follows:

nþ1q �¼ nþ1s

�� nþ1a �

¼ nþ1str ��3l

ro

nþ1q �

Dnþ1p�X3m¼1

Lm½ ��1 namf gþCm

ro

nþ1q �

Dnþ1p� �

� na4f g�C4

ro

nþ1q �

Dnþ1pþ L4½ � nþ1a4 �

g nþ1a4 ð36Þ

Similarly, for {nþ1a4}, the Newton iteration function for {nþ1q} is for-mulated as follows:

nþ1U �¼ 1þ3l

roDnþ1p

� �I½ �þDnþ1p

ro

X3m¼1

Cm Lm½ ��1þDnþ1pro

C4 I½ �" #

nþ1q �

� nþ1str �þX3

m¼1

Lm½ ��1 namf gþ na4f g

� L4½ � nþ1a4 �

g nþ1a4

ð37Þ

@ nþ1U �@ nþ1qf g ¼ 1þ 3l

roDnþ1p

� �I½ � þ Dnþ1p

ro

X3m¼1

Cm Lm½ ��1 þ Dnþ1pro

C4 I½ �" #

� 3 1� kð Þ2r3

oDnþ1p� �2X3

m¼1

Cmcm Qm½ �

� 3 1� kð Þ2r2

oDnþ1p

X3m¼1

cm Am½ �

� 3 1� kð Þ2r2

oDnþ1pc4 A4½ �g nþ1a4

ð38Þ

nþ1q iþ1ð Þ � ¼ nþ1q ið Þ �� @ nþ1U ið Þn o@ nþ1qf g

24

35

�1

nþ1U ið Þn o

ð39Þ

Here, the square matrices [Qm], [Am], and [A4] are abbreviatedparameter expressions defined as

Qm½ �¼ Lm½ ��1 E1½ � Lm½ ��1 nþ1q �

nþ1q �Tþ nþ1q

�nþ1q �T

Lm½ ��T E1½ �h i

Am½ �

¼ Lm½ ��1 E1½ � Lm½ ��1 namf g nþ1q �Tþ nþ1q

�namf gT Lm½ ��T E1½ �

h iA4½ �

¼ E1½ � nþ1a4 �

nþ1q �Tþ nþ1q

�nþ1a4 �T

E1½ �ð40Þ

Finally, the Newton iteration function for Dnþ1p is derived bysubstituting {nþ1q} from Eq. (36) into the yield function of Eq. (8)as follows:

nþ1W ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi32

nþ1qf gT E1½ � nþ1qf gr

� ro ¼ 0 ð41Þ

@nþ1W

@Dnþ1p¼

3 nþ1q �T E1½ � @

nþ1qf g@Dnþ1p

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi32

nþ1qf gT E1½ � nþ1qf gq ð42Þ

Here, {nþ1q} is from Eq. (36), and its partial derivative with respectto Dnþ1p is written as

@ nþ1q �@Dnþ1p

¼ �3lro

nþ1q ��X3

m¼1

@ Lm½ ��1

@Dnþ1pnamf g

� 1ro

X3m¼1

Cm Lm½ ��1 nþ1q �

� Dnþ1pro

X3m¼1

Cm@ Lm½ ��1

@Dnþ1pnþ1q �� 1

roC4

nþ1q �

þ @ L4½ �@Dnþ1p

nþ1a4 �

g nþ1a4 ð43Þ

Each partial derivative of [Lm]�1 and [L4] in the above equation iswritten as

(plastic)

Yesyield check

given n , n+1

from Abaqus

call state variables of nth

increment with ROTSIG

F (n+1qtr) > 0Eq. (19)

n+1qtr = n+1str naEq. (18)

No

n+1 e = n e + n+1

n+1 p = n p

n+1 = n+1 tr

n+1am= namn+1p = np

(elastic)

UMAT

update state variables, stress and consistenttangent operator at n+1th increment

No

Yes

compute n+1a4(i+1)

Eq. (33)

compute n+1p(i+1) Eq. (45)

compute n+1q(i+1) Eq. (39)

compute n+1a4(i+1)

Eq. (35)

convergence all ?

Yes

No

i = i+1

1 ( 1) 44 1 ( 1)

4

1n in i

ag af a

1 ( )4 0n ig a

i = 1

1 ( )4 4

n if a a

n+1p(1) = np /100 or 1000, n+1q(1) = nq

n+1a4(1) = na4,

1 (1) 44 1 (1)

4

1nn

ag af a

(initial condition)

Fig. 7. Flow chart for UMAT subroutine.

J. Han et al. / Computers and Structures 231 (2020) 106222 7

@ Lm½ ��1

@Dnþ1p¼�cm k Lm½ ��1 Lm½ ��1þ3 1�kð Þ

2r2o

Lm½ ��1 nþ1q �

nþ1q �T

E1½ � Lm½ ��1� �

@ L4½ �@Dnþ1p

¼c4 k I½ �þ3 1�kð Þ2r2

o

nþ1q �

nþ1q �T

E1½ �� �

ð44ÞThen, using Eqs. (41) and (42), the Newton iteration function forDnþ1p is defined as

Dnþ1p iþ1ð Þ ¼ Dnþ1p ið Þ � @nþ1W ið Þ

@Dnþ1p

!�1

nþ1W ið Þ� �

ð45Þ

Note that each iteration function for the three variables {nþ1a4},{nþ1q}, and Dnþ1p should be calculated by using the updated valuesduring every iteration. Consequently, the stress {nþ 1r} isobtained by substituting each converged variable into Eqs. (10)and (11) and using Eq. (14) as follows:

nþ1r � ¼ nrf g þ Dnþ1r

�¼ nrf g þ De½ � Dnþ1e

�� 32Dnþ1p E1½ �

nþ1q �ro

� �ð46Þ

3.3. Consistent tangent operator (CTO)

As the CTO is closely related to the rate of convergence inthe implicit FEM, it should be derived to ensure the algorith-mic consistency with the integration process [51]. Althoughthe CTO formulation is based on the scheme proposed byKobayashi and Ohno [35], the transition of fourth backstressnþ1a4 is considered by the threshold term g ðnþ1a4Þ (Eq. (24)).The CTO is defined as the derivative of the stress increment{Dnþ1r} for the total strain increment {Dnþ1e} at the (n+1)th

increment, as follows:

Dep½ � ¼ dDnþ1r �dDnþ1ef g ð47Þ

Differentiating each discretized relation of Eqs. (10), (11), (14) and(17) yields the following Eqs. (48)–(51):

dnþ1n � ¼

ffiffiffi32

rdnþ1s �� dnþ1a

�ro

ð48Þ

plastic strain (%)0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

stre

ss (M

Pa)

-300

-200

-100

0

100

200

300

400

o

o

1.5a

Fig. 8. Relation of total stress and backstress under uniaxial condition.

8 J. Han et al. / Computers and Structures 231 (2020) 106222

dnþ1ep � ¼ dDnþ1ep

�¼

ffiffiffi32

rdDnþ1p E1½ � nþ1n

�þ Dnþ1p E1½ � dnþ1n � � ð49Þ

dnþ1r � ¼ dDnþ1r

� ¼ De½ � dDnþ1e �� dDnþ1ep

� �¼ De½ � dDnþ1e

�� 2l E2½ � dDnþ1ep � ð50Þ

dnþ1s � ¼ dDnþ1s

� ¼ 2l E2½ � Id½ � dDnþ1e �� dDnþ1ep

� � ð51ÞFrom Eqs. (8) and (10), the following relation is obtained.

nþ1n �T

E1½ � nþ1n � ¼ 1 ! nþ1n

�TE1½ � dnþ1n � ¼ 0 ð52Þ

Multiplying {nþ1n}T forward on both sides of Eq. (49) and using Eq.(52) gives

dnþ1p ¼ dDnþ1p ¼ffiffiffi23

rnþ1n �T

dDnþ1ep � ð53Þ

Substituting Eq. (53) into the differential of Eq. (20) and using Eqs.(21)–(23) gives

dnþ1a �¼ dDnþ1a

�¼23

X4m¼1

Cm E2½ � dDnþ1ep �

�ffiffiffi23

r X3m¼1

cmknþ1am �

nþ1n �T

dDnþ1ep �

�ffiffiffi23

r X3m¼1

cm 1�kð Þ nþ1am �T

E1½ � nþ1n �

nþ1n �

nþ1n �T

dDnþ1ep �

�ffiffiffi23

rc4k

nþ1a4 �

nþ1n �T

dDnþ1ep �

g nþ1a4

�ffiffiffi23

rc4 1�kð Þ nþ1a4

�TE1½ � nþ1n �

nþ1n �

nþ1n �T

dDnþ1ep �

g nþ1a4

ð54ÞSubstituting Eqs. (48) and (53) into Eq. (49) provides

dDnþ1ep � ¼ E1½ � nþ1n

�nþ1n �T

dDnþ1ep �

þ 32

E1½ � dnþ1s �� dnþ1a

�ro

Dnþ1p ð55Þ

Substituting Eqs. (51) and (54) into Eq. (55) provides

dDnþ1ep �¼ E1½ � nþ1n

�nþ1n �T

dDnþ1ep �

þ3lro

Dnþ1p Id½ � dDnþ1e �� dDnþ1ep

� �

� 1ro

Dnþ1pX4m¼1

Cm dDnþ1ep �

þffiffiffi32

rkro

Dnþ1pX3m¼1

cm E1½ � nþ1am �

nþ1n �T

dDnþ1ep �

þffiffiffi32

r1�kð Þro

Dnþ1pX3m¼1

cm E1½ � nþ1am �T

E1½ � nþ1n �

nþ1n �

nþ1n �T

dDnþ1ep �

þffiffiffi32

rkro

Dnþ1pc4 E1½ � nþ1a4 �

nþ1n �T

dDnþ1ep �

g nþ1a4

þffiffiffi32

r1�kð Þro

Dnþ1pc4 E1½ � nþ1a4 �T

E1½ � nþ1n �

nþ1n �

nþ1n �T

dDnþ1ep �

g nþ1a4 ð56Þ

Separating the differentiation of the total and plastic strain incre-ments gives

2l Id½ � dDnþ1en o

¼ H½ � dDnþ1epn o

dDnþ1epn o

¼2l H½ ��1 Id½ � dDnþ1en o

ð57Þ

with

H½ � ¼ 2ro

3Dnþ1pI½ � � E1½ � nþ1n

�nþ1n �Th i

þ 2l I½ �

þ 23

X4m¼1

Cm I½ � �ffiffiffi23

rkX3m¼1

cm E1½ � nþ1am �

nþ1n �T

�ffiffiffi23

r1� kð Þ

X3m¼1

cmnþ1am �T

E1½ � nþ1n �

E1½ � nþ1n �

nþ1n �T

�ffiffiffi23

rkc4 E1½ � nþ1a4

�nþ1n �T

g nþ1a4

�ffiffiffi23

r1� kð Þc4 nþ1a4

�TE1½ � nþ1n �

E1½ � nþ1n �

nþ1n �T

g nþ1a4

ð58ÞSubstituting Eq. (57) into Eq. (50) gives

dDnþ1r � ¼ De½ � � 4l2 E2½ � H½ ��1 Id½ �

h idDnþ1e � ð59Þ

Finally, the CTO is obtained in the following form:

Dep½ � ¼ dDnþ1r �dDnþ1ef g ¼ De½ � � 4l2 E2½ � H½ ��1 Id½ � ð60Þ

3.4. Structure of user subroutine

The integration equations and CTO were included in the UMATsubroutine code, which was written using Fortran 77. The overallcomputational algorithm for the UMAT with Abaqus/Standard[46] is described in Fig. 7, where the utility routine ROTSIG of Aba-qus [46] is used for considering the rotated coordinate system ofthe integration point. In particular, in the integration algorithm,for the accurate calculation of the fourth backstress component{nþ1a4}, the transition of the fourth hardening rule at the thresholdlevel, which is the mentioned issue in Section 2, is considered dur-ing the iteration process.

4. Validation of numerical implementation

4.1. Verification of integration algorithm

The integration algorithm discussed in Section 3.2 was verifiedby comparing the numerical results from the implemented FEmodel with the analytical solution of the BH model for uniaxialconditions (Eqs. (4) and (5)). The model parameter values used inall the verification processes are listed in Table 1, in which the val-

x

y

z

x

y

z

2pets1pets

Fig. 9. Single-element FE model for uniaxial cyclic loading.

plastic strain0.000 0.005 0.010 0.015

stres

s (M

Pa)

-300

-200

-100

0

100

200

300

400

UMAT FEAanalytical solution

Fig. 10. Comparison of r� ep curves obtained from FEM and analytical solutionsunder uniaxial cyclic loading.

z

(a)t

c

z

tzc

(b)

Fig. 11. (a) 1/8 FE model of thin-walled tube subject to (b) biaxial loading history.

N0 10 20 30 40

p (%

)

0

1

2

3

4

5 k = 1

0.5

0.1

0.010

= c

z

tzc

Fig. 12. Effect of multiaxial dependent parameter k on biaxial ratcheting.

J. Han et al. / Computers and Structures 231 (2020) 106222 9

ues for multiaxial variable k are not listed. From the backstressevolution am Eqs. (4) and (5), the uniaxial stress is calculated as

r ¼ 32

X4m¼1

am þwro ð61Þ

where the factor 3/2 accounts for the deviatoric plane of the back-stress. The relation of the total stress and backstress under uniaxial

condition is described in Fig. 8. The uniaxial FE model is composedof a single continuum three-dimensional solid element (C3D8) withload-controlled cyclic loading conditions as described in Fig. 9. Boththe numerical and the analytical solutions produced exactly thesame stress-plastic strain curves (r� ep), as shown in Fig. 10. As khas no effect on the uniaxial ratcheting response, additional verifi-cation is performed by using the analysis results of Bari and Hassan[23] regarding biaxial ratcheting. Biaxial ratcheting analysis withthe BH model was performed using a 1/8 symmetric FE model ofa thin-walled tube, which consisted of C3D8 elements. By maintain-ing a constant internal pressure in the model, the cyclic axial strainez is varied as shown in Fig. 11. Repetitive ez results in circumferen-tial ratcheting strain ehp [52]. Fig. 12 shows the effect of multiaxialdependent parameter k on biaxial ratcheting, which is consistentwith the tendency of the BH model for biaxial ratcheting [23]. Cir-cumferential ratcheting does not occur for k = 0, whereas ehpbecomes larger as k increases.

4.2. Verification of CTO

In Abaqus/Standard [46] with the UMAT subroutine, the currentstress at every integration point is computed by using the numer-ical integration algorithm, in which the given stress of prior statenr and total strain increment Dnþ1e are used, as described inFig. 7. To verify the CTO, an additional stress rJ is computed using

the CTO from the given Dnþ1e and then compared with the stresscalculated from the integration algorithm described in Section 3.2.rJ is calculated as follows:

2 mm

6 mm 4 mm

10 mm

10 mm

10 mm x

y

z

2pets1pets

2 mm

4 mm

10 mm

10 mm

10 mm x

y

z

6 mm

Fig. 13. Single-element FE model with large deformation for verification of CTO.

increment #.0 500 1000 1500 2000 2500 3000

norm

al st

ress

(M

Pa)

-200

-100

0

100

200

integration algorithmconsistent tangent modulus

k = 0.5

increment #.0 500 1000 1500 2000 2500 3000

norm

al st

ress

(M

Pa)

-200

-100

0

100

200

integration algorithmconsistent tangent modulus

increment #.0 500 1000 1500 2000 2500 3000

norm

al st

ress

(M

Pa)

-200

-100

0

100

200

increment #.0 500 1000 1500 2000 2500 3000

shea

r stre

ss

(MPa

)

-200

-100

0

100

200

increment #.0 500 1000 1500 2000 2500 3000

shea

r stre

ss

(MPa

)

-150

-100

-50

0

50

100

150

increment #.0 500 1000 1500 2000 2500 3000

shea

r stre

ss

(MPa

)

-150

-100

-50

0

50

100

150

Fig. 14. Validation of consistent tangent modulus of UMAT subroutine for BH model.

10 J. Han et al. / Computers and Structures 231 (2020) 106222

h (mm)0.00 0.05 0.10 0.15 0.20

P (N

)

0

200

400

600

800

1000isotropic hardening with Table 2 BH kinematic hardeningwith k = 0.5 and Table 1 740 N

Fig. 16. P � h curves for isotropic and BH kinematic hardening FE model.

Fig. 15. FE model for cyclic spherical indentation test (Ei = 1000 GPa, v i = 0.07).

h (mm)0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

P (N

)

0

50

100

150

200

250k = 0.5 with Table 1Pmax = 190 N N = 1 200

h (mm)0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

P (N

)

0

100

200

300

400

500k = 0.5 with Table 1Pmax = 400 N N = 1 200

P (N

)

200

400

600

800

1000

N = 1 10 25 50 100 200

k = 0.5 with Table 1Pmax = 740 N

Table 2Model parameters for isotropic hardening.

E (MPa) v ro (MPa) n

181 � 103 0.3 130 4.2

J. Han et al. / Computers and Structures 231 (2020) 106222 11

Dnþ1rJ � ¼ Dep½ � Dnþ1e

�nþ1rJ � ¼

Xnþ1

i¼1

DirJ �

: stress at nþ 1ð Þth increment ð62Þ

h (mm)0.0 0.1 0.2 0.3 0.4

0

Fig. 18. P � h curves for various Pmax = 190, 400, 740 N with k = 0.5.

where [Dep] is calculated based on Eq. (47). To consider all the stresscomponents while verifying the CTO, displacement-controlled sin-gle element FE analysis is performed; the displacement conditionsin an oblique direction are shown in Fig. 13. All the stress compo-

h (mm)0.00 0.05 0.10 0.15 0.20

P (N

)

0

200

400

600

800

1000

400 N

740 N

190 N

E, v, o, n from Table 2

(a)

Fig. 17. (a) P � h curves and (b) h� N data for three d

nents obtained using the CTO are the same as those obtained fromthe integration algorithm at every increment during analysis(Fig. 14). Therefore, it can be concluded that the elastoplastic tan-gent operator is consistent with the numerical integration process.

N0 50 100 150 200

h (m

m)

0.00

0.05

0.10

0.15

0.20

0.25Pmax = 740 N

400 N

190 N

(b)

ifferent loads using isotropic hardening FE model.

Table 3Variation of initial indentation depth ho at Pmax (= 190, 400, 740 N) for various k.

Pmax (N) ho (mm) max. gap (%)

k = 1.0 k = 0.9 k = 0.7 k = 0.5 k = 0.3 k = 0.1

190 0.0492 0.0493 0.0495 0.0498 0.0502 0.0507 3.0400 0.1004 0.1007 0.1013 0.1020 0.1026 0.1037 3.2740 0.1984 0.1986 0.1990 0.2000 0.2013 0.2019 1.7

N0 50 100 150 200

h/h o

1.0

1.1

1.2

1.3

1.4

1.5Pmax = 190 N

N180 190 200 210 220

h (m

m)

1.32

1.33

1.34

1.35

1.36

1.37

1.38k = 1.0

0.9

0.7

0.5

0.30.1

cut-off

N0 50 100 150 200

h/h o

1.0

1.1

1.2

1.3

1.4

1.5

1.6Pmax = 400 N

N180 190 200 210 220

h (m

m)

1.40

1.45

1.50

1.55k = 1.0

0.9

0.7

0.5

0.3

0.1cut-off

N0 50 100 150 200

h/h o

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4Pmax = 740 N

N180 190 200 210 220

h (m

m)

1.6

1.7

1.8

1.9

2.0

2.1

2.2

k = 1.00.90.70.5

0.3

0.1

cut-off

Fig. 19. h/ho � N curves for various Pmax = 190, 400, 740 N with various k.

12 J. Han et al. / Computers and Structures 231 (2020) 106222

4.3. Numerical examples

The indentation test is one of the modern methods for evaluat-ing mechanical properties [53–55], especially for contact problemswhere contact and friction forces play important roles. Most of theengineering structures and parts such as gears, bearings, and railsare subject to severe and repeated contact loads. Cyclic indentationtests have been utilized to analyze the cyclic plastic behavior ofmetallic materials [56–58]. In this study, the cyclic spherical inden-tation test is simulated by using both isotropic hardening andmodified Chaboche kinematic hardening (i.e. implemented BHmodel) models to validate the numerical implementation. By con-

sidering the axial symmetry of the load and the geometry, anaxisymmetric FE model was created using four-node continuumaxisymmetric elements (CAX4). The FE model for the sphericalindenter and specimens comprised 16,600 nodes and 16,000 ele-ments (Fig. 15). The spherical indenter was 1 mm in diameter (D= 1 mm) and reflected the mechanical properties of diamond(Young’s modulus Ei = 1000 GPa and Poisson’s ratio v i = 0.07).The friction coefficient between the indenter and specimen wasset as 0.1. For the model parameters listed in Tables 1 and 2, sim-ulations are performed with the maximum indentation loads Pmax

(= 190, 400, 740 N) and corresponding initial indentation depthho=D (= 0.05, 0.1, 0.2), which are the typical range of depths in

J. Han et al. / Computers and Structures 231 (2020) 106222 13

spherical indentation [54,55]. The cyclic loading conditionsincluded 200 repetitions with each maximum load ðPmaxÞ with fullunloading.

4.3.1. Isotropic hardening modelKoo et al. [12] demonstrated that the uniaxial ratcheting behav-

ior cannot be simulated by using isotropic hardening model as itleads to a purely elastic behavior under uniaxial cyclic loads. How-ever, FE analyses of cyclic indentation with isotropic hardeningmodel are performed prior to analyses with the BH model. For iso-tropic elastic-plastic material model, a stress-strain relation inpiecewise power law form [59] is

eteo

¼rro

for r 6 ro

rro

� �nfor r > ro

8<: ð63Þ

The isotropic hardening parameters (Table 2) are determined byfitting the initial monotonic load-depth ðP � hÞ curve with thatbased on the BHmodel (Fig. 16). For the isotropic hardening model,immediate elastic shakedown behavior (the cyclic response istotally elastic [60]) is observed as shown in Fig. 17. Therefore,the isotropic hardening model cannot describe the ratcheting phe-nomenon induced by the cyclic indentation for the consideredthree loading levels.

4.3.2. Implemented BH kinematic hardening modelThe model parameters for the BH model are the same as those

listed in Table 1 with k = 0.1–1.0 to investigate the effect of multi-axial parameter k on the ratcheting response during cyclic indenta-tion. Unlike the isotropic hardening model, the indentation depthaccumulates in the form of open hysteresis loop for Pmax = 190,400, 740 N as shown in Fig. 18. It is noted that changing the k valuedoes not affect monotonic loading response; the initial indentationdepth ðhoÞ at the first Pmax is nearly the same for all k values with amaximum error of 3.2 % (Table 3). For various values of k, the ratioof indentation depth to the initial depth (h/hoÞ per cycle N are plot-ted as shown in Fig. 19. Before the cut-off point, which variesaccording to Pmax, the effect of k on the h/ho � N curve is not signif-icant After the cut-off point, the deviation between h/ho distribu-tions increases with increasing k. This suggests that k can beeffectively adjusted to simulate the indentation ratcheting behav-ior while maintaining a similar response of the monotonic (noratcheting) domain for different k. The cyclic indentation modeldemonstrates potential for the practical applicability of the imple-mented model for engineering problems involving multiaxial cyc-lic loads, which is rather challenging to be solved withconventional methods such as isotropic and early kinematic hard-ening models.

5. Summary and conclusion

Numerical implementation of the modified Chaboche kinematichardening model using the FEM was studied as the commerciallyavailable FE codes are based on early theories regarding cyclic plas-ticity and not suitable for numerical analysis of multiaxial ratchet-ing [61]. A user material subroutine for Abaqus was written inFortran and then validated with various FE models. The reducedforms of constitutive equations were developed by using Voigtnotations and the backward Euler scheme to simply solve the for-mulated equations of the integration algorithm and that of the CTOvia matrix operations; the fourth-order tensor terms were reducedto the second-order tensor terms.

The numerical integration algorithm was verified by comparingFE results with the (i) analytical solutions for uniaxial cyclic load-ing conditions and (ii) results of Bari and Hassan [23] for biaxial

ratcheting. Similarly, the stress components calculated using theCTO were compared with the results of the numerical integrationalgorithm at each increment to confirm that the elastoplastic tan-gent operator is consistent with the integration process. As anumerical example, cyclic spherical indentation was simulated toexplore the ratcheting under multiaxial cyclic loadings and theeffect of multiaxial parameter k on indentation ratcheting wasinvestigated. It was further demonstrated that the isotropic hard-ening cannot be applied to simulate the multiaxial ratchetingresponse. Accordingly, we concluded that the proposed numericalimplementation allows simulations of both biaxial and generalmultiaxial ratcheting. Further, the proposed numerical implemen-tation process can be effectively utilized in numerical applicationsof similar advanced kinematic hardening models that incorporatetemperature dependency, viscoplasticity and isotropic-combinedhardening to solve engineering problems involving complex cyclicloads.

Acknowledgment

This work was supported by the National Research Foundationof Korea (NRF) grant funded by the Korea government (MSIT)(NRF-2017R1A2B3009706).

References

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