Materials and Design 68 (2015) 134–145

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Materials and Design

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Analysis of forming limits based on a new ductile damage criterion inSt14 steel sheets

http://dx.doi.org/10.1016/j.matdes.2014.12.0290261-3069/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +86 29 88474117; fax: +86 29 88492642.E-mail address: fuguolx@nwpu.edu.cn (F. Li).

Xinkai Ma, Fuguo Li ⇑, Jinghui Li, Qianru Wang, Zhangwei Yuan, Yong FangState Key Laboratory of Solidification Processing, School of Materials Science and Engineering, Northwestern Polytechnical University, Xi’an 710072, China

a r t i c l e i n f o

Article history:Received 28 September 2014Accepted 18 December 2014Available online 27 December 2014

Keywords:Ductile damage criterionForming limit diagramNumerical simulationNakazima’s test

a b s t r a c t

Analytical and numerical analyses of forming limit in sheet metal hydraulic bulging test under combinedinternal pressure and independent axial feeding are researched in this paper. To predict the initiation offracture, a new normalized ductile damage criterion based on strain hardening exponent, stress triaxialityand strain lode parameters are adopted. Eventually, elastic modulus is chosen as a characterizationparameter to measure the ductile damage during the process of plastic deformation of the material. Inaddition, the explicit expressions of elastoplastic constitutive equations related to this damage criterionare compiled and implemented in ABAQUS/CAE software as the user subroutines. After compared againstthe analytical and the experimental forming limit diagrams (FLDs), the simulative FLDs demonstrate to bereasonable so that this model can be extended to predict a wide range of sheet metals’ forming processes.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Sheet metal forming is widely used for producing various struc-tural components, especially in automotive and aeronautic indus-tries [1]. Forming limit diagrams (FLDs) are commonly used inevaluating the workability of sheet metals and diagnosing produc-tion problems in the forming processes. The formability analysiswas required to determine whether the amount of deformationexceeds the forming limit at any point of the formed part. Theformability limit of sheet metal and tube are usually determinedby the initiation of localized necking that precedes fracture [2,3].

Marciniak and Kuczynski [4] presented the most well-knownmethod (e.g., the M–K theory) to calculate sheet metal forminglimits. They assumed that an initial inhomogeneity in the thicknessof the material was existent, and they assessed plastic instabilityphenomenon using two-zone model. In literature, there are manyresearches (e.g. [5–8]), using the M–K method to obtain the FLDs.

Yoshida et al. [9] simulated the hemispherical punch stretchingusing the elasto-plastic three-dimensional finite element program‘‘ROBUST’’. They predicted the limited cup height and rupture loca-tion for mild steels, high-strength steels, austenitic stainless steels,and aluminum alloy using FE calculation results and FLD curve pro-posed by Storen and Rice [10]. They found that the calculatedresults are in well accordance with the experimental results.Therefore, numerical analysis is proposed for the forming limits.

Some researchers have presented a methodology to predict theforming limit stress diagram (FLSD) and to reexamine the effect ofstrain path or through-thickness normal stress on it [6,11]. Themethodology is based on the M–K model. To calculate sheet metallimiting strains and stresses, a numerical approach applying themodified Newton–Raphson with globally convergence methodhas been used. The evaluation of the theoretical results has beenperformed by using the published experimental data for ST12low carbon steel alloy.

Hashemi et al. [12] investigated the influence of sheet thicknesson sheet metal forming limits. Some investigations indicate that itis worthwhile to consider the effect of thickness on the forminglimit diagrams, while others suggest that it is of less importance.His results strongly support the conclusion that the absolute valueof the thickness has no influence on forming limits of St14 steelsheet. Besides he [13]also established an extended strain-basedFLD based on equivalent plastic strains and materials flow direc-tion at the end of forming, which showed much less strain pathdependent than the conventional FLD and more easier to use andinterpretation than FLSD. The extended strain-based FLD wasverified by some available published experimental [14–17].

Drucker [18] and later Hill [19] introduced a general conditionfor non-bifurcation which was based on the positiveness of thesecond-order work. This primarily diffuse necking criterion pro-vides a lower bound and appears to be too conservative for theaccurate prediction of FLDs in practical applications [20].

Ozturk and Lee [21] pointed out that all the common criteriacould not be used to determine the FLD of sheet metal alone. A

Nomenclature

A cross-sectional area of the gauge section, mm2

A0 initial cross-sectional area of the gauge section, mm2

AD cross-sectional area after deformation damage, mm2

Aj corresponding cross-sectional area to rj, mm2

D damage variableR damage variableDC the critical value of ductile damageE Young’s modulus, MPaE0 initial Young’s modulus of virgin material, MPaED Young’s modulus after deformation damage, MPaK monotonic strength coefficient, MPan monotonic strain-hardening exponentG shear modulus, MPaLn natural logarithm

exp powers of eF fracture toughness, MPa m1/2

hp applied force load

Greek letterr;rs rb;rj;rf ;r�f true stress, yield strength, engineering ulti-

mate strength, true ultimate strength, fracturestrength, engineering fracture strength, MPa

e; es eb; ef ; e�f true strain, corresponding strain to rs rb;rf ;r�fm Poisson’s ratioRr stress triaxialityle rod parameters of strain

X. Ma et al. / Materials and Design 68 (2015) 134–145 135

new methodology was proposed to estimate the ductile fracturecriterion for the strain localization process. The results are in goodagreement with the experimental and analytical results.

Experimental determination of a FLD is very time consumingand requires special equipment, which may not always be avail-able to most researchers. Because of that, many researchers havebeen developing analytical and numerical models as an alternativeto eliminate tremendous amount of experimental work that wouldbe of considerable benefit [22]. While most of their work concen-trated on the application of ductile fracture criteria and the soft-ware of finite element analysis, few work was accomplished bythe union of damage criteria and the software of finite elementanalysis, let alone compile these damage criteria into finite ele-ment software’ subroutine.

For room temperature sheet forming, as the first trial of thiskind, forming conditions were limited to simple tension test andregular/modified hemispherical dome stretching tests (which arethe tools to measure the forming limit criterion) as well as the cir-cular cup drawing test, all at room temperature, in which the plas-tic deformation induced heat effect was ignored for simplicity [23].

In this paper, elastic modulus, as a characterization parameter,which will decrease in the process of materials’ plastic deformationaccording to the principle of continuum damage mechanics, is con-tained in the new normalized ductile damage criterion to measurethe ductile damage. The new normalized ductile damage criterioncompiled into ABAQUS/CAE’ user subroutine for the computationof the two critical principal strain is developed, which makes usjudge the initiation of crack and acquire the strain-based FLD eas-ily. Compared with other methods which were remained to deter-mine the criteria after the end of calculation in finite elementsoftware about the prediction of FLD, these methods could achievea effect that the calculating would not be stopped until the fracturewas found so as to make the calculating of finite element softwaremore precise and efficient. Besides these methods under the condi-tion of Nakazima’s test are selected as in numerical simulation topredict the formability of St14 steel sheet which is widely usedin sheet metal forming processes of automotive industry. Thereforeit is very significant to evaluate the formability of St14 via the FLDutilizing the new normalized ductile damage criterion.

Fig. 1. A schematic diagram of stress–strain curve of metal materials subjecteddamage deformation.

2. Theory basis of deformation damage and ductile fracturecriterion

2.1. The constitutive equation of materials

The intrinsic material property can be reflected by thestress–strain relation of plastic damage material under large

plastic deformation, which is reasonably characterized by a consti-tutive equation.

r ¼ Kðe0 þ eÞn ð1Þ

When the initial strain e0 ¼ 0, the constitutive equation isdescribed by a common power function model

re ¼ Eee; for r 6 rs

rp ¼ Kenp; for r P rs

(ð2Þ

And Fig. 1 shows a schematic diagram of stress–strain relation ofnormal metal materials.

Since the materials’ regular volume is invariant during the pro-cess of large plastic deformation, its expression is as follows

A0L0 ¼ AL ð3Þ

By evaluating the logarithm of Eq. (3), we can get Eq. (4)

lnA0

A¼ ln

LL0¼ e; A ¼ A0e�e ð4Þ

At the uniform deformation stage, the expression of applied load Fis as follows

F ¼ rA ð5Þ

By evaluating the differential of Eqs. (4) and (5), we can get the fol-lowing equations.

dA ¼ �A0e�ede ð6Þ

dF ¼ Adrþ rdA ð7Þ

136 X. Ma et al. / Materials and Design 68 (2015) 134–145

When applied load F reaches the maximum, Eqs. (4) and (6) aresubstituted into Eq. (7):

A0e�edrþ rð�A0e�edeÞjr¼rj¼ 0 ð8Þ

It can be seen from the last equations that when materials aresubjected to the uniaxial tensile strength limit, r = dr/de, whichmeans e = n.

Similarly, we can get the equation of the actual uniaxial tensilestrength limit.

rj ¼Fmax

Aj¼ Fmax

A0

A0

Aj¼ rbeej ð9Þ

And the nominal uniaxial tensile strength limit is obtained asfollows.

rb ¼ ðKnnÞe�ej ¼ Kne

� �nð10Þ

For metal materials, whose stress–strain curves conform theform of power function during cold deformation, the fracture stressis based on the theory of continuum damage mechanics and micro-scopic damage mechanics

rf ¼ rjepn2 ¼ r

pn2þnb ð11Þ

where P denotes the material parameters related to damage defor-mation. P = 1 during homogeneous deformation and P = 9(1 � m)/4during heterogeneous deformation.

Substituting strength factor K for nominal fracture stress rb, wecan get next two equations

rf ¼ kðnffiffiffiepÞpn ð12Þ

Fracture strain ef ¼ npep2 ð13Þ

Therefore, the total strain is composed of elastic strain and plasticstrain

e ¼ ee þ ep ¼ r=Eþ ðr=KÞ1n � e0

h ið14Þ

Usually e0 = 0 and the incremental formulas of constitutive equationare as follows

Drnþ1x ¼ Drn

0 þ 2GðDenx � Den

0Þ;Dsnþ1xy ¼ GDcn

xy

Drnþ1y ¼ Drn

0 þ 2GðDeny � Den

0Þ;Dsnþ1xy ¼ GDcn

xy

Drnþ1z ¼ Drn

0 þ 2GðDenz � Den

0Þ;Dsnþ1xy ¼ GDcn

xy

9>>=>>; ð15Þ

Fig. 2. The process of calculation and judgment in sheet’s plastic deformation.

2.2. The damage description of materials’ forming

Plastic deformation is a process of damage accumulation. Andcrack will not occur until the value of damage variable reaches a crit-ical value. Elasticity modulus will decrease as plastic deformationincrease due to the accumulation of damage in material. These accu-mulation of damage led to the decrease of effective bearing area aswell as the apparent elasticity modulus ED in deformable materials.Apparent elasticity modulus ED is conducted to predict damage per-formance for metal materials, which can be expressed as follows.

ED ¼ E0e�pe ð16Þ

When the initial strain e0 ¼ 0, damage variable based on elastic-ity modulus is exhibited as follows

D ¼ 1� ED

E0¼ 1� e�pe ¼ A0 � AD

A0ð17Þ

In view of loading histories (accumulated plastic strain), stresscondition parameters (stress triaxiality) and deformation pattern(strain lode parameter), damage variable R is proposed to describe

the degree of ductile damage and fracture during the forming pro-cess of metal materials. Moreover, the unified damage function R isin demand to describe various material damages and to remarkdamage as 0 when virgin and as 1 when failure. [24], which canbe expressed as follows.

R ¼ Di

DCð18Þ

where DC is the critical value of ductile damage, DC in the uniaxialtension stress state (stress triaxiality Rr = 1/3 and strain lodeparameter le = �1) can be obtained by:

DC ¼ f ðRr;leÞð1� e�pef Þ ð19Þ

where ef is fracture strain, ef ¼ npep2. P = 1 and P = 9(1 � m)/4, which

represent true fracture strain and nominal fracture strain respec-tively. The value of P is related to the strain-hardening exponentin materials’ plastic deformation; f ðRr;leÞ is the function of stressstate parameters (Rr and le), which can be expressed as:

FðRr;leÞ ¼ 0:558 sinhð1:5RrÞ � 0:008le coshð1:5RrÞ ð20Þ

It is convenient to calculate the process of the finite elementsanalysis that adopting the incremental formulas as the expressionof damage accumulation.

DDi ¼ f iðRr;leÞpe�pei Dei; i ¼ 1;2 . . . k ð21Þ

Di ¼ Di�1 þ DDi ð22Þ

Combining the above equations, when Di 6 0, the value ofEq. (22) is equal to zero. Combining Eq. (18), the damage variablevalue R can be acquired. When R = 0, it means that no damage inmaterials or original damage has been repaired. When 0 < R < 1,it means that damage exists in materials while plastic deformationcould be continued. When R = 1, it means that materials reach thelimit of deformation damage or the ductile fracture begins.

All the above equations were compiled in the User-definedMaterial Mechanical Behavior of Abaqus so that it is convenientfor us to apply this damage criterion to other materials.

2.3. The analytical forming limit diagram

In the process of sheet metals forming, namely the process fromthe dispersion instability to concentration instability and fracture,a closed link exists among stress–strain, yield criterion, strengthenlaw and instability law. The degree of sheet metal forming sub-jected to different load states is determined by its yield criterionand strengthen law. What is more, instability criterion means todetermine a condition that when stress or strain meet thiscondition, we regard this strain as the ultimate strain under that

Fig. 4. The relation between b and le.

X. Ma et al. / Materials and Design 68 (2015) 134–145 137

condition. This theory can be applied to predict the limits of sheetmetal forming. Detailed process is represented as shown in Fig. 2.

Swift’s diffused necking criterion [25] for thin sheets and Hill’slocalized necking criterion [26] associated with Hill’s non-qua-dratic yield function are used to construct the FLC for the bi-axialtensile strain zone and tensile-compressive strain zone, respec-tively [16]. Throughout the analysis of plastic instability, the fol-lowing are assumed:

(1) The stress state of the planar sheet is r1 P r2 P 0;r3 ¼ 0.(2) The load state can also be simplified to simple load that is

a ¼ r2r1; b ¼ e2

e1. And for the isotropic material under this load

condition, b ¼ 2a�12�a .

(3) The hardening curve of sheet accords with power exponen-tial function, that is �r ¼ k�en

The condition of localized instability deduced by Hill is asfollows

dr1

r1¼ dr2

r2¼ �de3 ð23Þ

Based on the instable condition of Hill’s theory and Misses yieldcriterion, the ultimate strain is obtained as follow [23],

When 0 6 a 6 0:5 e1 ¼2� a1þ a

n e2 ¼2a� 11þ a

n ð24Þ

When 0 6 a 6 0:5 e1 ¼2ð2� aÞð1� aþ a2Þð1þ aÞð4� 7aþ 4a2Þ n

e2 ¼2ð2a� 1Þð1� aþ a2Þð1þ aÞð4� 7aþ4a2Þ n ð25Þ

The analytical forming limit diagram in Fig. 3 shows the effect ofthe value of n (the strain-hardening exponent) of the sheet materi-als on the forming limit diagrams using Hill’s non-quadratic yieldfunction. It is apparent that the value of n has a significant influenceon the forming limit diagram. For materials with large n value willsuffer a larger plastic deformation before necking occurs. The form-ing limit diagram will rise as the value of n increases.

Fig. 4 shows the effect of the value of b (principal strain ratio) ofthe sheet material on the deformation properties. In this figure, le(lode parameter of strain) is chosen to describe the deformationproperties, which contain four special conditions, including A –balanced biaxial tension (b = 1), O – plane strain (b = 0), B – uniaxialtension (b = �1/2) and C – pure shear (b = �1). What is more, seg-ment-BOA represents the change of b under the strain state of ten-sion–tension. In a similar way, segment-CBO represents the change

Fig. 3. Effects of the n value on the FLD with Hill’s and Swift’s theory.

of b under the strain state of tension–compression, segment-CDOrepresents the change of b under the strain state of compres-sion–tension and segment-DOE represents the change of b underthe strain state of compression–compression. All these stress statesof steel sheet during the process of plastic deformation can befound in Fig. 4

3. Experimental procedures

3.1. Material

The St14 steel sheets investigated in this paper is a low-carbondrawing quality steel stock with thickness of 0.8 mm, 1.0 mm and1.2 mm, which has been extensively used in the sheet metal form-ing industry. And nominal chemical composition of the material isgiven in Table 1.

3.2. Uniaxial tensile test

A Zwick/Z electronic tensile machine was applied to accomplishthe tensile test at room temperature according to ISO6892-09 [27]and ISO10275:2007 [28]. Specimens oriented 0�, 45� and 90� to therolling direction together with thicknesses of 0.8 mm,1.0 mm and1.2 mm were cut for later use. The specimen gauge length was50 mm and a non-contact extensometer was used to measure theengineering strain. A cross-head speed of 2 mm/min was usedand at least three tests performed for each specimen orientation.To describe the tensile behavior, Eq. (2) were fitted to the truestress–strain (r–e) data for each specimen direction tested [29].And during homogeneous deformation (before necking), the truestress and strain were converted from the engineering stress andstrain

rT ¼ rEð1þ eEÞeT ¼ lnð1þ eEÞ

ð26Þ

According to the uniaxial tensile data, the relevant result isshown in Tables 2–4. Furthermore the true stress–strain (r–e)curves can be acquire based on Eq. (26).

Table 1Nominal chemical composition of st14 steel sheets (wt.%).

Element Fe C Mn P S

Content Balance 0.08 0.40 0.03 0.03

Table 2Mechanical properties of St14 with 0.8 mm thickness.

E (GPa) l r0.2 (MPa) rb (MPa) n K (MPa) q (g mm�3)

210 0.3 165.34 287.57 0.23 507.52 7.85

Table 3Mechanical properties of St14 with 1.0 mm thickness.

E (GPa) l r0.2 (MPa) rb (MPa) n K (MPa) q (g mm�3)

210 0.3 169.73 299.29 0.23 524.58 7.85

Table 4Mechanical properties of St14 with 1.2 mm thickness.

E (GPa) l r0.2 (MPa) rb (MPa) n K (MPa) q (g mm�3)

210 0.3 180.92 306.33 0.23 523.75 7.85

Fig. 5. The out-of-plane test setup [22].

138 X. Ma et al. / Materials and Design 68 (2015) 134–145

3.3. Hemispherical dome stretching test

For purpose of measuring the formability of the sheets, the out-of-plane Nakazima test was performed, which was clampedbetween circular die rings, over a hemispherical punch as shown

Fig. 6. Different specimen geometry of the

in Fig. 5 [22]. The diameter of the hemispherical punch and thedie cavity were 100 mm and 105 mm, respectively. The externaldiameter of blank holder was 120 mm. This test requires the useof specimens with different geometries and lubrication conditionto generate all possible strain states and stress, which are exhibitedin Fig. 6 [30,31]. The specimens with width of 20 mm, 40 mm,60 mm and 80 mm had an hourglass shape which was requiredon narrow blanks to prevent fracture from occurring at the blankholder. Besides the first four specimens of hourglass shape weredeformed with PVC (polyvinyl chloride polymer) film lubricant ofthe same thickness, while other four square specimen weredeforming with PVC film lubricant of thickness from 0.04 mm to0.06 mm between specimen and punch.

A 600 kN hydraulic press punch was employed to fulfill therequired force. The specimens were deforming until the first crackwas visually observed in the sheet. To calculate the ultimatestrains, circular grids with a diameter of 2.25 mm were markedon the sheets using electro-chemical etching equipment. Typically,the circular grid was closest to necking or fracture zone and hadthe largest deformation, which was measured. Experimentalresults were averaged in. The engineering strains were obtainedfrom Eqs. (27) and (28) respectively and then transformed to truestrains by Eq. (26):

e1 ¼d1 � d0

d0� 100 ð27Þ

e2 ¼d2 � d0

d0� 100 ð28Þ

Deformed specimens were designed according to ISO 12004-2:2008 [32], and each specimen repeated three times to avoid arti-ficial error and represented one specific strain path on the FLD. Antypical deformed specimens s with 40 mm width is showed inFig. 7.

3.4. The experimental forming limit diagram

A total number of three FLDs with different thickness weredetermined and the ultimate strains were recorded to acquire

hemispherical dome stretching test.

Fig. 7. The deformed test sample for sheets of 40 mm width.

X. Ma et al. / Materials and Design 68 (2015) 134–145 139

the experimental FLDs. Fig. 8 shows the FLDs of different sheetwith different thickness It is obvious that all the date points areevenly distributed around the fitted curves. However, in the threeFLDs, the date points of minor strain between �0.3 and 0 werefewer than other interval. The emergence of this problem may berelated to the bending effects resulted from curvature of the punch.It is desirable that this missing date point does not affect the wholevariation rule of FLDs acquired in Section 2.3. Fig. 9 shows that thefitted curves of the experimental FLDs have a similar trend asexpected with analytical FLDs in Fig. 3 and other materials[2,20,33].

Fig. 8. The experimental FLDs with different thickness

It should be pointed out that the right side of FLD is obtained bychanging the friction conditions between punch and sheet. Whilethe left side of FLD is acquired by changing the width of sheetmetal [34]. Those points closing to plane strain condition at theright side of FLD are difficult to acquire because friction coefficientsbetween surfaces employed in the Nakazima’s test are not yetknown quantitatively.

From these figures we could conclude that the forming limitsdid not alter significantly with variations of thickness so as tonot change the microstructure of the material. Similar conclusioncould be found in Hashemi et al. [12].

: (a) t = 0.8 mm, (b) t = 1.0 mm and (c) t = 1.2 mm.

Fig. 9. Fitted curves of the experimental FLD with different thickness.

140 X. Ma et al. / Materials and Design 68 (2015) 134–145

4. Finite element simulation

4.1. Finite element analysis mode

All the specimens in Fig. 6 were simulated using commerciallyavailable finite element code ABAQUS/Explicit. A factor of 1000was set for mass scaling to reduce computation times. The wholefinite element modeling of hemispherical dome stretching testshould be based on the actual hemispherical dome stretching test.While one quarter of the sheet was modeled because of the sym-metry boundary condition to reduce the work of calculation, asshown in Fig. 10. The dies and sheet were assumed as analytic rigidand deformable surfaces, respectively. It is suggested that the min-imum length of element should be higher than the shell thickness(the thickness of test specimens were 0.8 mm, 1.0 mm and1.2 mm). For example, based on the mesh sensitivity study, ele-ment size with 2 mm was selected as optimum element size forall finite element simulations when the thickness of test specimenwas 0.8 mm [22]. In the process of numerical simulations, all theanalyses were realized using an explicit finite element approach.Besides in simulation of the stretch drawing and the hydraulicbulge test, three-dimensional eight-node solid elements were used[35]. According to the experiment process, it requires different

Fig. 10. Finite element modeling of hemispherical dome stretching test.

specimen geometries and lubrication condition, such as frictioncoefficient l = 0.1, 0.12, 0.15, 0.18 and 0.2, to acquire as morestress states as possible.

4.2. Simulation and calculation

The material was modeled as elastic–plastic where the plastic-ity were modeled as anisotropic and isotropic, while the elasticityconsidered to be isotropic. What is more, the constitutive relation,yield criteria and ductile damage criteria of this material werecompiled in vectorized user material subroutine (VUMAT) of Aba-qus. And the VUMAT offered 23 variables, for example SD17 (dam-age value R) and SD23 (equivalent strain), which all could be usedas a damage variable to represent the damage degree of materials’deformation. It should be noted that the damage variable can becharacterized by different variables which is related to materials’deformation. And it is interchangeable between different variables,which are utilized to describe the same damage process. In order tofacilitate the comparative analysis of the result of the same dam-age state described by different variables and make the damagevariable comparable, normalization process is usually requiredfor damage variables. After this process of normalization, the valueof damage variables become one, which can be seen in Eq. (18).

To avoid pretension force, the blank holder and lower diemoved 0.5 mm up and down, respectively. And the analysis wasperformed in three steps to ensure the accuracy of numerical sim-ulation. In the first step, the lower die was fixed and the blankholder moved down to contact sheet. In the second step, the sheetwas fixed and the force was applied to the sheet from the blankholder. In the third step, the hemispherical punch moved downalong the YSYMM axis until the expectant displacements wasattained. In other words, the finite element analysis of Abaqus willnot stop before the damage variable R reach the critical value-one.At the same time, the element which was closed to the highest crit-ical value-one at this increment was considered to be ‘‘the firstfractured element’’. And the element number, increment number,major and minor principal strain values of this first fractured ele-ment were recorded for the plotting of the simulative FLDs.

For this materials of St14 steel sheets, the critical value of duc-tile damage DC in Eq. (19) can be obtained after the calculation of n,q, Rr, le. The result is 0.1374.

As an additional effort of this study, some specimen geometries,which are not occured in ISO 12004-2:2008 [32], are also simu-lated in this paper in order to generate more possible strain andstress states for the simulative FLDs.

For the sake of verifying this finite element model, the numer-ical simulation results are in contrast with the experimentalresults, and a typical specimen is choose to study from numbersof specimens. Fig. 11 shows the numerical simulative deformedspecimens of St14 steel sheets of 60 mm width with differentthickness, which are associated with the experimental deformedspecimens. There is no significant difference among the simulativedeformed specimens with different thickness because of the insen-sitivity of thickness for the St14 steel sheets, which was similar tothe trend of experimental test and also verified in Ref. [12]. As wedescribed in last section, the crack will initiate wherever the valueof ductile damage R (SD17) reaches one. Therefore the crack initi-ation and propagation of the specimens occurred around the centerof the specimen for both hourglass and square shape, this phenom-enon also had been found by Panich et al. [5] and Liu et al. [36].What is more, in order to get a clear acquaintance for the shapeof crack in St14 steel sheets, the contour plot of the numerical sim-ulative deformed specimen of 60 mm width and 1 mm thickness isexhibited in Fig. 12, which shows the contour plot of SD23 (equiv-alent strain). The analysis results of finite element simulation andthe actual cracking are consistent closely.

(a)

(b)

(c)

Fig. 11. The deformed shapes and damage degree of test specimens of 60 mm width with different thickness: (a) t = 0.8 mm, (b) t = 1.0 mm and (c) t = 1.2 mm.

X. Ma et al. / Materials and Design 68 (2015) 134–145 141

4.3. The simulative forming limit diagrams

After the finite element analyze of above simulation and calcu-lation, the simulative forming limit diagram was acquired by uti-lizing the new normalized ductile damage criterion. Due to the

additional specimens simulated in the finite element analysis soft-ware of ABAQUS, the number of the simulative points is greaterthan the number of experimental points.

From the simulative FLDs showed in Fig. 13, we can see that theductile damage criteria successfully predicted the FLDs of St14

Fig. 12. The equivalent strain contour plot of the deformed specimen with 1.0 mm thickness.

Fig. 13. The simulative FLDs with different thickness: (a) t = 0.8 mm, (b) t = 1.0 mm and (c) t = 1.2 mm.

142 X. Ma et al. / Materials and Design 68 (2015) 134–145

steel sheets. What is more, the fitted curves of these simulativeFLDs are also exhibited in Fig. 14. It should be noted that the ruleof these simulative FLDs are in good agrement with the experimen-tal points.

At the right side of the simulative FLDs, as the result of manydifferent friction coefficients applied in the process of numerical

simulation, it is desirable to have more stress paths close to theplane strain condition [20].

The fitted curves of the simulative FLDs acquired by the use ofthe Origin software are shown in Fig. 14. In the region of planestrain, these simulative FLDs are also in good accordance with eachother. The fitted curve with 1.2 mm thickness is higher than the

Fig. 14. Fitted curves of simulative FLDs with different thickness.

X. Ma et al. / Materials and Design 68 (2015) 134–145 143

others in biaxial tension region, while it is a little lower than othersin uniaxial tension region. This problem will be discussed in nextsection.

5. Results and discussion

5.1. Comparative analysis of different forming limit diagram

The FLDs obtained by theoretical calculation, experiment andnumerical simulation are thoroughly explained in its own sections.

Fig. 15. Examples of predicted FLDs compared with experimental

These diagrams include three experimental FLDs with differentthickness, three simulative FLDs with different thickness and onetheoretical FLD, which are showed in Fig. 15(a–c).

Compared with experimental and simulative FLDs, the theoret-ical or analytical FLDs based on Hill’s localized necking criterion inthe right side and Swift’s diffused necking criterion in the left sideare too conservative to predict the actual FLDs exactly. Chung et al.[37] found this phenomenon in DP600 steel sheets, which was alsoobserved in the similar pervious work by Stoughton and Zhu [38].The discrepancy may be mainly incurred by the assumption ofSwift’s criterion that the loading procedure is simultaneous sta-tionary loads, which is rarely observed experimentally. MoreoverHill’s localized necking criterion is restricted to negative minorstrain [20] and ignore the strain rate sensitivity of steel sheets [37].

From the comparisons of experimental FLDs and simulative pre-dictions in Fig. 15(a–c), both of the results have a similar trend. Thesimulative results show a little lower than the experimental resultsin the FLDs with different thickness. This difference betweenexperimental and simulative results may be ascribed to the artifi-cial error, model simplification or the accuracy of numerical simu-lation model [36].

At the same time, it should be noted that the reason why thesimulative FLDs is a little larger than the analytical FLDs. First,the new normalized ductile damage criterion about the largedeformation of materials in VUMAT considers the factors of stressstate, strain state and non-uniform necking during the tensiledeformation of materials, which are more closer to the actual thanthe factors considered in analytical solution. Besides, above

FLD points: (a) t = 0.8 mm, (b) t = 1.0 mm and (c) t = 1.2 mm.

144 X. Ma et al. / Materials and Design 68 (2015) 134–145

variables are always dynamic changes during the process ofmaterials’ large deformation, so the simulation calculation ismore scientific and accurate than analytical calculation. Thereforesimulative FLDs have a wider range of application, while theapplication of analytical FLDs are restricted due to its boundednessof assumption about stress and strain state.

5.2. The character and improvement of the simulative FLD

In the biaxial tension region, the error between experimentaland simulative results is larger than other regions. In other words,the accuracy of the right simulative FLD is a little lower than leftregions. This difference may be attributed to the lack of enoughspecimens in the biaxial tension region or the improper parame-ters, which is essential to the calculation of finite element analysis.The accuracy of simulation is determined by the VUMAT based onthe ductile damage criteria and planar anisotropic constitutivemodel. And this new ductile damage criteria is also used in Heet al. [24], who successfully applied this theory to predict the effec-tive elastic modulus by the repeated loading–unloading tensiletests. The simulation results by the use of VUMAT including planaranisotropic constitutive model show that the forming process canbe precisely simulated [39–41]. The simulative prediction will besuccessful only if the correct parameters, boundary conditionsand VUMAT were set up for the finite element model accurately.

Furthermore, the majority of criteria used to predict FLDimplied special modeling restrictions, (i.e. isotropic, pre-strain,work hardening, lubrication or damage). So the finite elementmodel based on the ductile damage criteria should be modifiedto adapt different conditions.

6. Conclusions

In this study, a new normalized ductile damage criterion is putforward and compiled in user subroutine of ABAQUS, which is usedin the process of numerical simulation. And simulative results areverified by the hemispherical dome stretching test of St14 sheetmetals, meanwhile the criterion based on continuum damagemechanics and constitutive equations have been demonstrated ofhigh reliability.

The new normalized ductile damage criterion takes intoaccount the factors of stress state, strain state, non-uniform neck-ing during materials’ tensile deformation, which are more closer tothe actual than the factors in analytical solution. Besides, the ana-lytical forming limit diagrams using Swift’s diffused and Hill’slocalized necking criteria are proved to be too conservative forthe accurate prediction of FLDs in practical applications.

From the hemispherical dome stretching test and its numericalsimulation, it can be seen that the forming limits did not alter sig-nificantly with variations of thickness so as to not change themicrostructure of the material. And different kind of lubricantshould be employed in the experiments in order to acquire morestress paths close to the plane strain condition.

Now most work about the prediction of FLD remained to deter-mine the criteria after the end of calculation in finite element soft-ware, while the application of ductile damage criteria as a failuresubroutine written in user defined subroutine could achieve thatthe calculating would not be stopped until the fracture was found,which made the calculating of finite element software more pre-cise and efficient.

The predicted FLDs using the user defined subroutine are ingood agreement with the experimental forming limits. Thereforethe new normalized ductile damage criterion compiled in VUMATis recommended to predict FLDs and should be verified in othermaterials in the future work.

Acknowledgments

The authors are very grateful for the support received from theNational Natural Science Foundation of China (Grant No.51275414), the Aeronautical Science Foundation of China (GrantNo. 2011ZE53059) and the Graduate Starting Seed Fund of North-western Polytechnic University (Grant No. Z2014007).

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