Relevance of Incorporating Cavity Shape Change in

Research ArticleRelevance of Incorporating Cavity Shape Change in Modellingthe Ductile Failure of Metals

Mohand Ould Ouali

Laboratory Elaboration and Characterization of Materials and Modelling (LEC2M) University Mouloud MAMMERI of Tizi OuzouBP 17RP 15000 Tizi Ouzou Algeria

Correspondence should be addressed to Mohand Ould Ouali m ouldoualiummtodz

Received 28 February 2018 Revised 29 July 2018 Accepted 28 August 2018 Published 16 October 2018

Academic Editor Stefano de Miranda

Copyright copy 2018 MohandOuldOualiThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The purpose of this paper is to assess the relevance of considering the cavities shape change in the context of physically basedmodelling of the ductile rupture inmetals Two thermomechanical models have been used in this study the GTNmodel developedby Gurson Tvergaard and Needleman for spherical cavities keeping their shape unchanged during loading and the GLD modelproposed by Gologanu Leblond and Devaux for ellipsoidal cavities that can change their shape The GTN and GLD models havebeen extended to take into account material thermal heating due to plastic dissipation These two constitutive laws have beenimplemented into the commercial finite element code AbaqusExplicit in order to simulate the necking of a round bar and thefailure of a sheet deep drawing The results showed the importance of incorporating the shape effects of the cavities for a correctdescription of the material failure

1 Introduction

The ductile fracture of the materials is associated with thedegradation of the mechanical properties under the effect ofthree physical mechanisms nucleation of the cavities theirgrowth under appropriate loading and the coalescence of theneighbouring voids at an advanced deformation stage Sev-eral nucleation criteria have been proposed in the literatureSome of these criterions have been formulated by exploitingthe energetic criterion of germination initially enunciated byGurland and Plateau [1 2] Needleman and Rice proposed anucleation criterion depending on the inclusions rate usinga phenomenological approach [3] Other nucleation criteriahave experimental origins [4] The modelling of void growthhas undergone several important contributions Gurson [5]proposed a kinematically based limit analysis approach ofa hollow sphere with an isotropic rigid-plastic matrix anddeveloped a macroscopic yield criterion depending on theporosity Tvergaard and Needleman extend this model toincorporate cavities interaction and to describe cavity coa-lescence in a phenomenological way [6 7] The new modelis known as GTNmodel Recently Nahshon and Hutchinson[8] proposed an interesting modification of the porosity

evolution law to account material damage under conditionsof low or vanishing triaxiality With the same aim Bai andWierzbick [9] Xue [10] Xue and Wierzbicki [11] Danas andP Ponte Castaneda [12] and Zhou et al [13] studied thedependency of ductile fracture on the Lode angle inmetals byincorporating shear mechanisms effects in the porosity evo-lution law Although these new expressions do not agree withthe classical porosity law derived from homogenization andhaving a physical interpretation these extensions offer goodresults and improve them substantially at low triaxialities seefor instance [14ndash18] All these models have been proposed byconsidering thematrix behaviour governed by vonMises-likemodel Benzerga and Besson [19ndash21] carried out a series ofunit cell calculations for initially spherical voids embeddednow in an orthotropic matrix Cazacu et al [22] assumedthe matrix obeying Trescarsquos criterion when performing limit-analysis of the hollow sphere Gologanu et al [23 24] extendtheGurson analysis to introduce cavity shape effects in the so-called GLDmodelThey proposed a model incorporating theporosity and a new internal variable the shape parameterThemodel has been assessed by Pardoen and Hutchinson [25] incomparison to unit cell prediction with different spheroidalcavities in the case of a hardenable matrix Ould Ouali

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 6454790 9 pageshttpsdoiorg10115520186454790

2 Mathematical Problems in Engineering

b

a

y

z

xΩ

Figure 1 Geometry of the RVE corresponding to theGursonmodel

and Aberkane [26ndash28] studied namely the microstructureevolution during rolling of an A1050 aluminium sheet andintroduced thermomechanical coupling in the GLD modelMadou and Leblondrsquos [29 30] extended the GLD model togeneral ellipsoidal voids having three distinct axes

The numerical simulation of the mechanical behaviourof metals undergoing forming operations has been a greatcontribution to the optimization of manufacturing processesIndeed the study of the material failure during such opera-tionsmust prevent the arising of the internal or surface cracksas well as their propagations before causing the total ruptureof the structure Given the high number of parameterscontrolling the deformation mechanisms under such solici-tations the recourse to modelling is necessary In the presentwork themodelling includes the effects of themicrostructureevolution in the material on the overall response of the partbeing formed However it is generally observed in materialforming processes low triaxiality which can be an obstacleto use the classical physically based models The purpose ofthe study is to assess the relevance of taking into accountthe evolution of cavities shape parameter as proposed inthe GLD model to correctly predict the ductile failure ofmetals Benzerga et al [32] showed that the consideration ofcavity shape change is currently themost appropriate physicaldescription for modelling ductile fracture at low triaxialityTwo applications have been chosen necking of a round barunder tensile loading and the deep drawing of an aluminiumsheet Comparison of the numerical and experimental resultsshows that neglecting cavity shape change leads to poornumerical predictions

2 Mathematical Modelling

21 Voids Growth

211 Gurson and Tvergaard (GT) Model Gurson [5] devel-oped a mechanical model for porous materials by limit anal-ysis of a spherical Representative Volume (RVE) containinga spherical confocal void (Figure 1) Tvergaard [7] proposedan extension of the Gurson plasticity criterion by introducingthree coefficients 1199021 1199022 and 1199023 = 11990221 to account for theinteraction between cavities It also replaced the yield stress1205900 with the flow stress 120590 in order to incorporate the isotropic

work hardening of the matrix The plastic potential is in thefollowing form

Φ = sum21198901199021205902 + 21199021119891 cosh(321199022sum119898120590 ) minus 1 minus (1199021119891)2 = 0 (1)

where sum119890119902 is the macroscopic von Mises equivalent stresssum119898 the macroscopic mean stress and 119891 the porosity of thematerial The matrix hardening was introduced using theequivalence between the macroscopic and the microscopicplastic dissipation as proposed by Gurson [5]

(1 minus 119891) 120590 120576119901 = sum 119901 (2)

where 119901 is the macroscopic plastic strain rate tensor and 120576119901the cumulated plastic strain rate of the matrix

The evolution law of the porosity can be approximatedusing the condition of plastic incompressibility of the matrixIn the case where nucleation of new voids is not taken intoaccount this is given by

119891 = 3 (1 minus 119891) 119901119898 (3)

212 The GLD Model Gologanu et al [23 24] extendedthe Gursonrsquos analysis to a prolate or oblate ellipsoidal RVEcontaining a confocal cavity and subjected to some asym-metric loadings (Figure 2) The authors assumed that thevoids can change their shape under the effect of loadingSuch modelling requires the introduction of a new internalvariable the shape parameter 119878 = ln119882 = ln(119886119887)

The expression of the plastic potential is given by thefollowing [21 25 26]

Φ = 11986212059021003817100381710038171003817100381710038171003817100381710038171003817sum1015840 + 120578sum

ℎ

11988310038171003817100381710038171003817100381710038171003817100381710038172

+ 2119902 (119892 + 1) (119892 + 119891) cosh(120581sumℎ120590 ) minus (119892 + 1)2minus 1199022 (119892 + 119891)2 = 0

(4)

where is the vonMises norm andX a second-order tensordefined by X = (13)(minus119890119909 otimes 119890119909 minus 119890119910 otimes 119890119910 + 2119890119911 otimes 119890119911) sumℎ isthe hydrostatic stress weighted by coefficients depending onthe cavity geometry It governs the voids growth [19] 120581 and gare the coefficients of the GLD related to porosity and shapeparameterThe coefficient q is the equivalent of the Tvergaardcoefficient in the GT model [33]

119902 = tanminus1 [4 (25 minus 119879)] 100381610038161003816100381610038161003816100381610038161003816119887minus1120587100381610038161003816100381610038161003816100381610038161003816 +119887 + 12 (5)

where 119887 = 1 + (0655 minus 175119899 minus 0533 4radic119891)(12 + tanminus1[2(1 minus119878)]120587 minus 00288119890minus108(02+119878))The evolution law of the shape parameter 119878 is obtained by

making some analysis and calibrations with cells calculations[23 24]

119878 = 32ℎ1 (119901119911119911)1015840 + 3ℎ2119901119898 (6)

where ℎ1 and ℎ2 are two parameters depending on 119891 119878 andtriaxiality Τ = sum119898 sum119890119902

Mathematical Problems in Engineering 3

A

a

Bb

cy

z

x

Ω

(a)

B

A

b

a

c

y

z

x

Ω

(b)

Figure 2 Geometry of the RVE corresponding to the GLD model (a) Prolate cavity (b) Oblate cavity

22 Coalescence Stage Prior to coalescence the GT or GLDmodel is used to describe thematerial behaviour [3 31]Thesemodels cannot capture any realistic void volume fractionduring the final coalescence stage So the description ofthe material capacity loss needs to use a coalescence modelIn this study the function 119891lowast proposed by Tvergaard andNeedleman is adopted [7] 119891lowast allow the reproduction ofthe rapid void volumes growth at the material coalescencebeginning from some critical value 119891119888 of 119891

119891lowast = 119891 if 119891 le 119891119888119891119888 + 120575 (119891 minus 119891119888) otherwise (7)

where 120575 = (119891119906 minus 119891119888)(119891119865 minus 119891119888) describe the rapid increase ofthe porosity at the coalescence 119891119888 and 119891119865 are the porosities atthe onset of coalescence and at final failure respectively

In the model the critical porosity can be determinedusing the Thomasonrsquos criterion This criterion has beenformulated with the assumption that the onset of coalescenceoccurs when the limit plastic charge is reached at the mate-rial ligament between voids (see Figure 3) The coalescencecriterion is therefore written in the case of the GTN modelas follows

[01 (119861 minus 119887119886 )2 + 12radic119861119887 ]sdot [1 minus (3radic1205874 1198910)( 1198871198770) exp (119864119911119911)] = 0

(8)

Equation (8) can be reformulated in accordance with thedimensionless variables 120594 = 119887119861 = ((32)119891120582119890minus119878)13 and120582 = 119860119861 = 1205820 exp((32)119864119890119902) used in the GLD model Wewrite the following [19 28]

sum119911119911120590 = (1 minus 1205942) [01 (1 minus 120594120594119882 )2 + 12radic 1120594] (9)

a

b

y

z

x

2A

2B

2C

Ω

Figure 3 Geometry of Thomasonrsquos criterion

23 Thermal Heating due to Plastic Dissipation The GTNand GLD models have been extended to take into accountthe thermal heating due to plastic dissipation This exten-sion is formulated using the hypothesis of a rigid-plasticbehaviour of the matrix according to Gurson assumptionThe thermoplastic coupling is introduced by enforcing thedependence of the flow stress 120590 characterizing the isotropicmatrix hardening on equivalent plastic strain 120576119901 and thetemperature T [27 28] The following hardening law ischosen

120590 = 120590 (120576119901 119879) = 1205900 ( 1205761205760)119899 (1 minus 120603 1198791198790) (10)

where 1205900 is the initial yield stress and 119899 hardening exponent120603 is a material parameter


In the absence of external heat sources the temperatureevolution law is solved with the general heat equation underadiabatic conditions (neglecting the conductivity term)

120588119862 = 120585119891120590 120576119901 (11)

where 120588 is the material density 119862 is the specific heat capacityand 120585119891 = (1 minus 119891 minus 120588120597120595120590120597120576119901) is the expression of the Taylor-Quinney coefficient for porous material

24 The Thermoelasticity Law For the sake of consistencythe following thermoelastic law is introduced [27 28]

Σ = Λ119890 (E minus E119905ℎ) + 120597Λ119890120597119879 (E minus E119905ℎ) (12)

where E119905ℎ = 120572(119879 minus 119879119874)I is the strain tensor due to thermalexpansion and Σ someobjective derivative of the stress tensorI is the second-order unit tensor

25 Numerical Implementation The GTN and GLDmodels have been implemented into the finite elementcode AbaqusExplicit using the Vectorized User MATerial(VUMat) subroutine and used to carry out the numericalsimulations The ldquoelastic predictor-plastic correctionrdquoalgorithm proposed by Aravas [34] is adopted in the localresolution of the constitutive laws GTN and GLDThe detailsof the implementation of these two behaviour laws as wellas the validation in comparison with the predictions ofcell calculations using an explicit technique are given in[26 27 33]

Aravasrsquos algorithm is based on the decomposition of thestress tensor into its hydrostatic 120590119898 and deviator parts 120590119890119902 andthe plastic strain increment tensorΔ120576119901 into its volumetric anddeviator parts

120590 = 1205901198981 + 23120590119890119902n (13)

Δ120576119901 = 23Δ120576p1 + Δ120576119902n (14)

Δ120576p and Δ120576119902 are used by Aravas as primary variables So thesystem is reduced to two non-inear equations

Δ120576p 120597Φ120597120590119890119902 + Δ120576119902120597Φ120597120590119898 = 0 (15)

Φ(120590 119891 120590) = 0 (16)

After the resolution of (15) and (16) the hydrostatic stress Σ119898and the equivalent stress Σ119890119902 are updated

Σ119898 = Σ119890119898 + 119870Δ119864119901Σ119890119902 = Σ119890119890119902 minus 3119866Δ119864119902 (17)

NO

NO

Begin

Φ lt 0

Convergence

Calculation of the trial stressCalculation of the yield condition Φ

Plastic correction

YES

YES

Update the stress tensor

Return

Elastic solutionReturn Compute the corrections cp and cq

Update ΔＪ Δq ΣＪ and ΣＫ

Update ３ f T

Figure 4 Flowchart of the implementation of the GLD model

Then the state variables are actualized according to thefollowing

Δ119891 = (1 minus 119891)Δ119864119901 +AΔ119864119901119897119898Δ119878 = 32ℎ1 (13Δ119864119901 + 119899Δ119864119902) + ℎ2Δ119864119901Δ119879 = 120585120588119862

minusΣ119898Δ119864119901 + Σ119890119902Δ119864119902(1 minus 119891)Δ120590 = (ℏE120590 + ℏ119879 120585120588119888V)

minusΣ119898Δ119864119901 + Σ119890119902Δ119864119902(1 minus 119891)

(18)

where ℏE = 120597120590120597120576119901 and ℏT = 120597120590120597119879 are respectively theplastic and thermal modulus

Details of the implementation procedure are done in theflowchart of Figure 4 and can be found in [19 20 24]

It is important to notice the GLD growth model imple-mentation requires specific treatment due to the ldquolimitsrdquoshapes that the cavities can take during the loading processpenny-shaped crack (119891 997888rarr 0 S997888rarr-infin) ldquosandwichrdquo struc-ture (119891 equiv finite S 997888rarr- infin) spherical (119891 equiv finite S=0) orcylindrical (119891 equiv finite S 997888rarr +infin) shapes or a needle (119891 997888rarr0 S 997888rarr +infin) Specific numerical treatment of the particularexpressions of the plastic potential Φ according to each caseshould be carried out

3 Results and Discussion

In this section we discuss the results obtained with theGTN and GLDmodels extended to take account the thermalheating due to plastic dissipation and to predict the onset ofcoalescence usingThomasonrsquos criterion In order to highlightthe effect of cavity shape change we make use of the same


Table 1 Material parameters used with the GTN and GLD models

1198780 1198910 119891119888 () 119891119865 () 119891119873 () 120576119873 119878119873 119902 parametersGTN - 0 15 25 4 03 01 1199020 = 16GLD 0 0 15 25 4 03 01 1199021 = 15 1199022 = 11

Ｆ0 = 4２0

２0

(a)

BC

A

(b)

Figure 5 Cylindrical bar (a) Initial mesh (b) Deformed mesh at failure

0

02

04

06

08

1

12

14

Nom

inal

Stre

ss

005 01 015 02 025 03 035 04 0450Longitinal Nominal Strain

GLD - VumatGTN - Vumat

(a)

0

02

04

06

08

1

12

14

Nom

inal

Stre

ss

005 01 015 02 025 030Radial Nominal Strain


(b)

Figure 6 Evolution of the nominal stress in function of (a) nominal strain and (b) radius reduction

material parameters for both the GTN and GLDmodels withthe exception of those related to the state variable describingthe cavity shape change the cavity shape parameter is notdescribed in the GTN model Two applications were chosentensile of a round specimen and deep drawing of a sheet

31 Necking of Cylindrical Bar The GTN and GLD modelsare used to simulate the rupture of a round bar subjectedto tensile loading It has been found experimentally that thefailure of the bar occurs according to the so-called ldquocup conerdquomode [7 35] Indeed Thomason observed that the crackappears in the centre zone of the specimen and then spreadshorizontally in the plane perpendicular to the loading axisNear of the outer surface the crack deviates with an angleof 45∘ from the propagation direction forming a cup conefracture surface

The heightradius ratio of the cylindrical bar is takenequal to 11989701198770= 4 Three hundred and twenty (320) axisym-metric elements with reduced integration (CAXR) are used tomesh the quarter of the specimen The initial and deformed

meshes as well as the loading conditions are shown inFigure 5 The constituent material of the specimen is astandard steel with a rigidity 1198641205900 = 500 and Poissonrsquos ratio] = 028 The hardening exponent 119899 = 01

The material is assumed to be initially dense (1198910 = 0) Weuse the same coalescence and nucleation parameters as thoseused by Tvergaard and Needleman when studying necking ofthe cylindrical bar [7] These parameters are done in Table 1

The evolution of the nominal stress 120590119873 = 119865120587119877201205900 versusthe longitudinal nominal strain 120576119871 = 1199061198970 and the radialnominal strain (reduction of radius) 120576119877 = Δ1198771198770 calculatedwith the two models GTN and GLD are represented inFigure 6 During the test the nominal stress 120590119873 increasesto a maximum that corresponds to the onset of neckingthen decreases [7] This diminution of the nominal stress isaccentuated at the end of the test because of the coalescenceof the cavities until the final rupture A difference appearsin nominal stress levels reached by the GTN and the GLDmodels This difference can be explained by the effect ofthe cavity shape change on the resistance of the materials


C

B



0

02

04

06

08

1

12

14

16

18Eq

uiva

lent

Nom

inal

Stre

ss

A

(a)

C

B



0

004

008

012

016

02

024

Poro

sityf

A

(b)

C

B

A


GLD - Vumat

0

03

06

09

12

15

18

21

Shap

e Par

amet

ers S

(c)

Figure 7 Evolution of the stress state and the microstructure at the neck (a) Equivalent nominal stress sum119890119902 1205900 (b) The porosity 119891 (c) Theshape parameter 119878

observed namely in cell calculations [19 24ndash26 33] Theseauthors have related these results to the morphologicalanisotropy generated by the cavities shape that the GLDmodel allows to take into account

In the case of a simulation with the GLD model themaximum force is equal to 119865119866119871119863max = 1344120587119877201205900 whereas thatpredicted by the GTN model is 119865119866119879119873max = 1311120587119877201205900 Thetwo maxima are obtained for the same radial nominal strain120576neck119877 = 00847 and for close longitudinal strains of about120576neck119871 = 004 These main results are reported in Table 2120576fail119871 and 120576fail119877 are respectively the longitudinal and theradial nominal strain of the bar at failure A qualitativeexplanation of this difference was given by Aravas and PonteCastaneda concerning this test using the model proposedby Aravas and Ponte Castaneda [36] In the case of theGLD model the justification can be done regarding theevolution of themicrostructure (the porosity119891 and the shape

Table 2 Representative values of the stress-strain curves corre-sponding to a cylindrical bar under tensile load

120590119873 120576neck119877 120576neck119871 120576fail119877 120576fail119871GTN 1311 00847 00448 0261 0348GLD 1344 00847 00433 0275 0426

parameter 119878) at the neck Figure 7 represents the evolutionof the nominal von Mises stress sum119890119902 1205900 the porosity 119891 andthe shape parameter 119878 versus the nominal radial strain inthree elements (A) (B) and (C) of the bar located at the neck(see Figure 7) It is well-known that the effective nominalstress depends on the net area of the section (without voids)perpendicular to the load at the neck that supports the loadin the case of tensile loading This area decreases when thetotal area of the section of the bar decreases (necking) andor


when the porosity in the plane carrying the load increasesAs previously mentioned the section of the bar for which thedifference between themaximum force predicted by the GLDand GTN models is observed for the same radius reductionΔRR0 = 00847 Consequently the difference in the intensityof the loading force can only be justified by the second pointthe surface occupied by the voids at the neck of the bar

We will study this point in more detail by consideringthematerial microstructure evolution of the bar (Figures 7(b)and 7(c))

(i) Figure 7(b) shows that the porosity predicted by theGTN and GLD models are the same for the threeelements located at the neck of the bar These curvescoincide almost at least until the nominal radial strain120576119877 = 00847 which corresponds to the onset ofnecking

(ii) In Figure 7(c) we can see that the voids tend tolengthen in the direction of loading in the case of theGTN model since the shape parameter S increasesFor the GTN model the cavities keep the sphericalshape It follows that for the same porosity the areaoccupied by the voids in the section at the neck issmaller in the case of the GLD model the cavitiesshape have changed from spherical (initial shape 119878 =0) to elongated voids (119878 gt 0)

So for the same volume fraction of voids the surface ofthe voids in the section of the neck supporting the load issmaller in the case of the GLD model compared to the GTNmodel Indeed during the simulation with the GTN modelthe increase of the porosity is done by the increase of thediameter of the void So the effective surface is smaller thanthat predicted by the GLD model Knowing that the effectivestress is directly related to the applied force the differencebetween the effective surfaces explains the difference betweenthe maximum forces achieved

32 Deep Drawing of a Sheet The deep drawing of a thinsheet of aluminium alloy is studied We are interested in theSwift deep drawing test which is applied to an aluminiumsheet having a thickness of 1mm until failure [31] Thedimensions of the sheet and the tools are shown in Figure 8Due to geometry and applied load symmetries the testis modelled under axisymmetric conditions The mesh ofthe sheet is composed of 640 axisymmetric elements withreduced integration CAXR with 04 layers of elements onthe thickness The friction between the sheet and the toolshas been determined experimentally by the CETIMSenlis(France) and is equal to 120583119891 = 017 [31] The clamping forceapplied by the blanking tool is 119865119904 = 5 119896119873

The material properties of the aluminium alloy sheet areYoungrsquos modulus 119864 = 70119866119875119886 Poissonrsquos ratio ] = 033density 120588 = 27001198701198921198983 and yield strength 1205900 = 114119872119875119886The hardening exponent is 119899 = 0146 In the absence ofexperimental data on the microstructure of the aluminumalloy we have adopted the initial porosity 1198910 = 01 Thislevel of initial porosity is generally observed in such materialFor choosing the initial value of the shape parameter a

R365mm

R176mm

R10

5mm

R40mm

R4mm

1mm

R1695mm

R165mmR5mm

R45mm

Figure 8 Schematization of the deep drawing test initial mesh andboundary conditions

Experiment

1 2 3 4 5 6 7 8 9 10 11 120Displacement u (mm)

0

3

6

9

12

15

18

21

24

27Fo

rce (

kN)


Figure 9 Evolution of the stamping force versus tool displacement[31]

parametric study has been carried out using initial oblatespherical and prolate cavities It is found that a value of 1198780 =138 gives good results in this case

Figure 9 shows a comparison between the experimentalresults obtained by the CETIMSenlis with the numericalpredictions of theGTNandGLDmodelsThe curves describethe evolution of the deep drawing force as a function ofthe punch displacement Good agreement is found betweenexperimental and numerical results in the case of the GLDmodel However a difference appears during the simulationwith the GTN model

Table 3 resumes the main values of the force-displacement curve during deep drawing of the aluminiumsheet maximal force and corresponding displacement In


Failure

(b) (a)

Figure 10 Distribution of the porosity in the sheet at failure and experiment [31]

Table 3 Representative points of the force-displacement curveduring deep drawing of the sheet

Experiment GTN GLDMaximal force (kN) 2592 2231 2566Error () - 1393 100Displacement (mm) 1028 919 1019Error () - 1060 087

the estimation of the error the experimental results aretaken as reference The error made during the simulationwith the GTN model (1393 and 106) is more importantthan that made using the GLD model (1 and 087) Anexplanation of these differences can be found regarding theorthotropic behaviour of the sheet experimentally identifiedby the CETIMSenlis [31] In the GTN model the matrixbehaviour is assumed to be isotropic In the GLD model amorphological anisotropy is introduced by the change of theellipsoidal cavities shape

Figure 10 shows comparison of the stamp shape obtainedexperimentally and that described numerically with the GLDmodel We observe that the two geometries are practicallyidentical and that the rupture appears in the same region ofthe sheet

4 Conclusions

Twophysically based constitutive laws theGTNand theGLDmodels have been implemented in the ABAQUS calculationvia theVUMAT subroutine Bothmodels have been extendedto account for thermal heating due to plastic dissipationunder adiabatic conditions The theoretical and numericalaspects of these formulations are detailedThese models havebeen used for the simulation of a round bar under tensileloading and deep drawing of aluminium sheet From thecomparison of the predictions of these models we can statethat incorporating cavity shape changes is an important stepin the correct modelling of the ductile failure of metals

As a perspective for this work we plan to carry outexperimental tests and unit cell calculations in order toverify the pertinence of the shape parameter evolutionlaw under different stress states Especially in the case ofthermomechanical loading another interesting extension is

to introduce the orthotropic material behaviour existingnamely in thin structures Finally the incorporation of shearmechanisms will also significantly improve the numericalpredictions of ductile fracture arising in some particularengineering problems

Data Availability

Numerical data used to support the findings of this studyare actually restricted by a research convention Requests fordata after publication of this article can be addressed to thecorresponding author for consideration in the context of theconvention

Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The author acknowledges thankfully the General Council ofthe Ardennes (France) for the financial support during thepreparation of part of this work

References

[1] A S Argon J Im and R Safoglu ldquoCavity formation frominclusions in ductile fracturerdquoMetallurgical Transactions A vol6 no 3 pp 825ndash837 1975

[2] J Gurland and J Plateau ldquoMechanism of ductile rupture ofmetals rupture of metals containing inclusionsrdquo Journal of theASM vol 56 pp 442ndash455 1963

[3] A Needleman and J R Rice ldquoLimit to ductility set by plasticflow localizationrdquo in Mechanics of Sheet Metal Forming D PKoistinie and N M Wang Eds pp 237ndash267 Plenum PressNew York NY USA 1978

[4] F M Beremin ldquoCavity formation from inclusions in ductilefracture of A508 steelrdquoMetallurgical Transactions A vol 12 no5 pp 723ndash731 1981

[5] A L Gurson ldquoContinuum theory of ductile rupture by voidnucleation and growth part Imdashyield criteria and flow rulesfor porous ductile mediardquo Journal of Engineering Materials andTechnology vol 99 no 1 pp 2ndash15 1977

[6] J W Hancock and R D Thomson ldquoStrain and stress con-centrations in ductile fracture by void nucleation growth and


coalescencerdquoMaterials Science and Technology vol 1 no 9 pp684ndash690 1985

[7] V Tvergaard and A Needleman ldquoAnalysis of the cup-conefracture in a round tensile barrdquo Acta Metallurgica et Materialiavol 32 no 1 pp 157ndash169 1984

[8] K Nahshon and J W Hutchinson ldquoModification of the GursonModel for shear failurerdquo European Journal of Mechanics -ASolids vol 27 no 1 pp 1ndash17 2008

[9] Y Bai and T Wierzbicki ldquoApplication of extended Mohr-Coulomb criterion to ductile fracturerdquo International Journal ofFracture vol 161 no 1 pp 1ndash20 2010

[10] L Xue ldquoConstitutive modeling of void shearing effect in ductilefracture of porous materialsrdquo Engineering Fracture Mechanicsvol 75 no 11 pp 3343ndash3366 2008

[11] L Xue and T Wierzbicki ldquoDuctile fracture initiation and prop-agation modeling using damage plasticity theoryrdquo EngineeringFracture Mechanics vol 75 no 11 pp 3276ndash3293 2008

[12] K Danas and P Ponte Castaneda ldquoInfluence of the Lodeparameter and the stress triaxiality on the failure of elasto-plastic porous materialsrdquo International Journal of Solids andStructures vol 49 no 11-12 pp 1325ndash1342 2012

[13] J Zhou X Gao J C Sobotka B AWebler and B V CockeramldquoOn the extension of the Gurson-type porous plasticity modelsfor prediction of ductile fracture under shear-dominated condi-tionsrdquo International Journal of Solids and Structures vol 51 no18 pp 3273ndash3291 2014

[14] M Achouri G Germain P Dal Santo and D Saidane ldquoExper-imental characterization and numerical modeling of microme-chanical damage under different stress statesrdquo Materials andCorrosion vol 50 pp 207ndash222 2013

[15] M Achouri G Germain P Dal Santo andD Saidane ldquoNumer-ical integration of an advancedGursonmodel for shear loadingApplication to the blanking processrdquo Computational MaterialsScience vol 72 pp 62ndash67 2013

[16] K L Nielsen J Dahl and V Tvergaard ldquoCollapse and coales-cence of spherical voids subject to intense shearing Studied infull 3Drdquo International Journal of Fracture vol 177 no 2 pp 97ndash108 2012

[17] V Tvergaard ldquoEffect of stress-state and spacing on voids in ashear-fieldrdquo International Journal of Solids and Structures vol49 no 22 pp 3047ndash3054 2012

[18] Z Xue J Faleskog and J W Hutchinson ldquoTension-torsionfracture experiments - Part II Simulations with the extendedGurson model and a ductile fracture criterion based on plasticstrainrdquo International Journal of Solids and Structures vol 50 no25-26 pp 4258ndash4269 2013

[19] A A Benzerga Rupture ductile des toles anisotropes simulationde la propagation longitudinale dans un tube pressurise [PhDthesis] Ecole Nationale Superieure des Mines de paris parisFrance 2000

[20] A A Benzerga ldquoMicromechanics of coalescence in ductilefracturerdquo Journal of the Mechanics and Physics of Solids vol 50no 6 pp 1331ndash1362 2002

[21] A A Benzerga J Besson and A Pineau ldquoAnisotropic ductilefracture Part II Theoryrdquo Acta Materialia vol 52 no 15 pp4639ndash4650 2004

[22] O Cazacu B Revil-Baudard N Chandola and D KondoldquoNew analytical criterion for porous solids with Tresca matrixunder axisymmetric loadingsrdquo International Journal of Solidsand Structures vol 51 no 3-4 pp 861ndash874 2014

[23] P SuquetRecent Extensions of GursonrsquosModel for Porous DuctileMetals CISM Lectures Series Springer New-York NY USA1997

[24] Mihai GologanuEtude de quelques problemes de rupture ductiledes metaux [PhD thesis] Universite de Paris Paris France1997

[25] T Pardoen and J W Hutchinson ldquoExtended model for voidgrowth and coalescencerdquo Journal of the Mechanics and Physicsof Solids vol 48 no 12 pp 2467ndash2512 2000

[26] M Ould Ouali Approche micromecanique de la rupture ductiledans les procedes demise en forme desmateriaux Prise en comptede lrsquoeffet de forme des cavites [PhD thesis] Universite de ReimsChampagne-Ardenne Reims France 2007

[27] M Ould Ouali and M Aberkane ldquoMultiscales modeling ofmicrostructure evolution during asymmetric cold rolling pro-cessrdquo International Journal of Material Forming vol 1 no 1 pp89ndash92 2008

[28] M Ould Ouali and M Aberkane ldquoMicromechanical modelingof the rolling of a A1050P aluminum sheetrdquo InternationalJournal of Material Forming vol 2 no 1 pp 25ndash36 2009

[29] K Madou and J-B Leblond ldquoA Gurson-type criterion forporous ductile solids containing arbitrary ellipsoidal voids -II Determination of yield criterion parametersrdquo Journal of theMechanics and Physics of Solids vol 60 no 5 pp 1037ndash10582012

[30] K Madou J-B Leblond and L Morin ldquoNumerical studies ofporous ductile materials containing arbitrary ellipsoidal voidsndash II Evolution of the length and orientation of the void axesrdquoEuropean Journal of Mechanics - ASolids vol 42 pp 490ndash5072013

[31] M Khelifa M Oudjene and A Khennane ldquoFracture in sheetmetal forming Effect of ductile damage evolutionrdquo Computersamp Structures vol 85 no 3-4 pp 205ndash212 2007

[32] A A Benzerga J-B Leblond A Needleman and V TvergaardldquoDuctile failure modelingrdquo International Journal of Fracturevol 201 no 1 pp 29ndash80 2016

[33] L Siad M Ould Ouali and A Benabbes ldquoComparison ofexplicit and implicit finite element simulations of void growthand coalescence in porous ductile materialsrdquo Materials andCorrosion vol 29 no 2 pp 319ndash329 2008

[34] N Aravas ldquoOn the numerical integration of a class ofpressure-dependent plasticity modelsrdquo International Journal forNumerical Methods in Engineering vol 24 no 7 pp 1395ndash14161987

[35] P F Thomason Ductile fracture of Metals Pergamon PressOxford Uk 1990

[36] N Aravas and P Ponte Castaneda ldquoNumerical methods forporous metals with deformation-induced anisotropyrdquo Com-puter Methods Applied Mechanics and Engineering vol 193 no36-38 pp 3767ndash3805 2004

Hindawiwwwhindawicom Volume 2018

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b

a

y

z

xΩ

Figure 1 Geometry of the RVE corresponding to theGursonmodel

and Aberkane [26ndash28] studied namely the microstructureevolution during rolling of an A1050 aluminium sheet andintroduced thermomechanical coupling in the GLD modelMadou and Leblondrsquos [29 30] extended the GLD model togeneral ellipsoidal voids having three distinct axes

The numerical simulation of the mechanical behaviourof metals undergoing forming operations has been a greatcontribution to the optimization of manufacturing processesIndeed the study of the material failure during such opera-tionsmust prevent the arising of the internal or surface cracksas well as their propagations before causing the total ruptureof the structure Given the high number of parameterscontrolling the deformation mechanisms under such solici-tations the recourse to modelling is necessary In the presentwork themodelling includes the effects of themicrostructureevolution in the material on the overall response of the partbeing formed However it is generally observed in materialforming processes low triaxiality which can be an obstacleto use the classical physically based models The purpose ofthe study is to assess the relevance of taking into accountthe evolution of cavities shape parameter as proposed inthe GLD model to correctly predict the ductile failure ofmetals Benzerga et al [32] showed that the consideration ofcavity shape change is currently themost appropriate physicaldescription for modelling ductile fracture at low triaxialityTwo applications have been chosen necking of a round barunder tensile loading and the deep drawing of an aluminiumsheet Comparison of the numerical and experimental resultsshows that neglecting cavity shape change leads to poornumerical predictions

2 Mathematical Modelling

21 Voids Growth

211 Gurson and Tvergaard (GT) Model Gurson [5] devel-oped a mechanical model for porous materials by limit anal-ysis of a spherical Representative Volume (RVE) containinga spherical confocal void (Figure 1) Tvergaard [7] proposedan extension of the Gurson plasticity criterion by introducingthree coefficients 1199021 1199022 and 1199023 = 11990221 to account for theinteraction between cavities It also replaced the yield stress1205900 with the flow stress 120590 in order to incorporate the isotropic

work hardening of the matrix The plastic potential is in thefollowing form

Φ = sum21198901199021205902 + 21199021119891 cosh(321199022sum119898120590 ) minus 1 minus (1199021119891)2 = 0 (1)

where sum119890119902 is the macroscopic von Mises equivalent stresssum119898 the macroscopic mean stress and 119891 the porosity of thematerial The matrix hardening was introduced using theequivalence between the macroscopic and the microscopicplastic dissipation as proposed by Gurson [5]

(1 minus 119891) 120590 120576119901 = sum 119901 (2)

where 119901 is the macroscopic plastic strain rate tensor and 120576119901the cumulated plastic strain rate of the matrix

The evolution law of the porosity can be approximatedusing the condition of plastic incompressibility of the matrixIn the case where nucleation of new voids is not taken intoaccount this is given by

119891 = 3 (1 minus 119891) 119901119898 (3)

212 The GLD Model Gologanu et al [23 24] extendedthe Gursonrsquos analysis to a prolate or oblate ellipsoidal RVEcontaining a confocal cavity and subjected to some asym-metric loadings (Figure 2) The authors assumed that thevoids can change their shape under the effect of loadingSuch modelling requires the introduction of a new internalvariable the shape parameter 119878 = ln119882 = ln(119886119887)

The expression of the plastic potential is given by thefollowing [21 25 26]

Φ = 11986212059021003817100381710038171003817100381710038171003817100381710038171003817sum1015840 + 120578sum

ℎ

11988310038171003817100381710038171003817100381710038171003817100381710038172

+ 2119902 (119892 + 1) (119892 + 119891) cosh(120581sumℎ120590 ) minus (119892 + 1)2minus 1199022 (119892 + 119891)2 = 0

(4)

where is the vonMises norm andX a second-order tensordefined by X = (13)(minus119890119909 otimes 119890119909 minus 119890119910 otimes 119890119910 + 2119890119911 otimes 119890119911) sumℎ isthe hydrostatic stress weighted by coefficients depending onthe cavity geometry It governs the voids growth [19] 120581 and gare the coefficients of the GLD related to porosity and shapeparameterThe coefficient q is the equivalent of the Tvergaardcoefficient in the GT model [33]

119902 = tanminus1 [4 (25 minus 119879)] 100381610038161003816100381610038161003816100381610038161003816119887minus1120587100381610038161003816100381610038161003816100381610038161003816 +119887 + 12 (5)

where 119887 = 1 + (0655 minus 175119899 minus 0533 4radic119891)(12 + tanminus1[2(1 minus119878)]120587 minus 00288119890minus108(02+119878))The evolution law of the shape parameter 119878 is obtained by

making some analysis and calibrations with cells calculations[23 24]

119878 = 32ℎ1 (119901119911119911)1015840 + 3ℎ2119901119898 (6)

where ℎ1 and ℎ2 are two parameters depending on 119891 119878 andtriaxiality Τ = sum119898 sum119890119902


A

a

Bb

cy

z

x

Ω

(a)

B

A

b

a

c

y

z

x

Ω

(b)







(8)



a

b

y

z

x

2A

2B

2C

Ω



120590 = 120590 (120576119901 119879) = 1205900 ( 1205761205760)119899 (1 minus 120603 1198791198790) (10)




120588119862 = 120585119891120590 120576119901 (11)







120590 = 1205901198981 + 23120590119890119902n (13)

Δ120576119901 = 23Δ120576p1 + Δ120576119902n (14)


Δ120576p 120597Φ120597120590119890119902 + Δ120576119902120597Φ120597120590119898 = 0 (15)

Φ(120590 119891 120590) = 0 (16)


Σ119898 = Σ119890119898 + 119870Δ119864119901Σ119890119902 = Σ119890119890119902 minus 3119866Δ119864119902 (17)

NO

NO

Begin

Φ lt 0

Convergence


Plastic correction

YES

YES


Return



Update ３ f T





minusΣ119898Δ119864119901 + Σ119890119902Δ119864119902(1 minus 119891)

(18)









Ｆ0 = 4２0

２0

(a)

BC

A

(b)


0

02

04

06

08

1

12

14

Nom

inal

Stre

ss



(a)

0

02

04

06

08

1

12

14

Nom

inal

Stre

ss



(b)









C

B



0

02

04

06

08

1

12

14

16

18Eq

uiva

lent

Nom

inal

Stre

ss

A

(a)

C

B



0

004

008

012

016

02

024

Poro

sityf

A

(b)

C

B

A


GLD - Vumat

0

03

06

09

12

15

18

21

Shap

e Par

amet

ers S

(c)















R365mm

R176mm

R10

5mm

R40mm

R4mm

1mm

R1695mm

R165mmR5mm

R45mm


Experiment


0

3

6

9

12

15

18

21

24

27Fo

rce (

kN)







Failure

(b) (a)






4 Conclusions




Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018









A

a

Bb

cy

z

x

Ω

(a)

B

A

b

a

c

y

z

x

Ω

(b)







(8)



a

b

y

z

x

2A

2B

2C

Ω



120590 = 120590 (120576119901 119879) = 1205900 ( 1205761205760)119899 (1 minus 120603 1198791198790) (10)




120588119862 = 120585119891120590 120576119901 (11)







120590 = 1205901198981 + 23120590119890119902n (13)

Δ120576119901 = 23Δ120576p1 + Δ120576119902n (14)


Δ120576p 120597Φ120597120590119890119902 + Δ120576119902120597Φ120597120590119898 = 0 (15)

Φ(120590 119891 120590) = 0 (16)


Σ119898 = Σ119890119898 + 119870Δ119864119901Σ119890119902 = Σ119890119890119902 minus 3119866Δ119864119902 (17)

NO

NO

Begin

Φ lt 0

Convergence


Plastic correction

YES

YES


Return



Update ３ f T





minusΣ119898Δ119864119901 + Σ119890119902Δ119864119902(1 minus 119891)

(18)









Ｆ0 = 4２0

２0

(a)

BC

A

(b)


0

02

04

06

08

1

12

14

Nom

inal

Stre

ss



(a)

0

02

04

06

08

1

12

14

Nom

inal

Stre

ss



(b)









C

B



0

02

04

06

08

1

12

14

16

18Eq

uiva

lent

Nom

inal

Stre

ss

A

(a)

C

B



0

004

008

012

016

02

024

Poro

sityf

A

(b)

C

B

A


GLD - Vumat

0

03

06

09

12

15

18

21

Shap

e Par

amet

ers S

(c)















R365mm

R176mm

R10

5mm

R40mm

R4mm

1mm

R1695mm

R165mmR5mm

R45mm


Experiment


0

3

6

9

12

15

18

21

24

27Fo

rce (

kN)







Failure

(b) (a)






4 Conclusions




Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018










120588119862 = 120585119891120590 120576119901 (11)







120590 = 1205901198981 + 23120590119890119902n (13)

Δ120576119901 = 23Δ120576p1 + Δ120576119902n (14)


Δ120576p 120597Φ120597120590119890119902 + Δ120576119902120597Φ120597120590119898 = 0 (15)

Φ(120590 119891 120590) = 0 (16)


Σ119898 = Σ119890119898 + 119870Δ119864119901Σ119890119902 = Σ119890119890119902 minus 3119866Δ119864119902 (17)

NO

NO

Begin

Φ lt 0

Convergence


Plastic correction

YES

YES


Return



Update ３ f T





minusΣ119898Δ119864119901 + Σ119890119902Δ119864119902(1 minus 119891)

(18)









Ｆ0 = 4２0

２0

(a)

BC

A

(b)


0

02

04

06

08

1

12

14

Nom

inal

Stre

ss



(a)

0

02

04

06

08

1

12

14

Nom

inal

Stre

ss



(b)









C

B



0

02

04

06

08

1

12

14

16

18Eq

uiva

lent

Nom

inal

Stre

ss

A

(a)

C

B



0

004

008

012

016

02

024

Poro

sityf

A

(b)

C

B

A


GLD - Vumat

0

03

06

09

12

15

18

21

Shap

e Par

amet

ers S

(c)















R365mm

R176mm

R10

5mm

R40mm

R4mm

1mm

R1695mm

R165mmR5mm

R45mm


Experiment


0

3

6

9

12

15

18

21

24

27Fo

rce (

kN)







Failure

(b) (a)






4 Conclusions




Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018











Ｆ0 = 4２0

２0

(a)

BC

A

(b)


0

02

04

06

08

1

12

14

Nom

inal

Stre

ss



(a)

0

02

04

06

08

1

12

14

Nom

inal

Stre

ss



(b)









C

B



0

02

04

06

08

1

12

14

16

18Eq

uiva

lent

Nom

inal

Stre

ss

A

(a)

C

B



0

004

008

012

016

02

024

Poro

sityf

A

(b)

C

B

A


GLD - Vumat

0

03

06

09

12

15

18

21

Shap

e Par

amet

ers S

(c)















R365mm

R176mm

R10

5mm

R40mm

R4mm

1mm

R1695mm

R165mmR5mm

R45mm


Experiment


0

3

6

9

12

15

18

21

24

27Fo

rce (

kN)







Failure

(b) (a)






4 Conclusions




Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018









C

B



0

02

04

06

08

1

12

14

16

18Eq

uiva

lent

Nom

inal

Stre

ss

A

(a)

C

B



0

004

008

012

016

02

024

Poro

sityf

A

(b)

C

B

A


GLD - Vumat

0

03

06

09

12

15

18

21

Shap

e Par

amet

ers S

(c)















R365mm

R176mm

R10

5mm

R40mm

R4mm

1mm

R1695mm

R165mmR5mm

R45mm


Experiment


0

3

6

9

12

15

18

21

24

27Fo

rce (

kN)







Failure

(b) (a)






4 Conclusions




Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018
















R365mm

R176mm

R10

5mm

R40mm

R4mm

1mm

R1695mm

R165mmR5mm

R45mm


Experiment


0

3

6

9

12

15

18

21

24

27Fo

rce (

kN)







Failure

(b) (a)






4 Conclusions




Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018









Failure

(b) (a)






4 Conclusions




Data Availability




Acknowledgments


References














































Journal of












Journal of







Volume 2018






Volume 2018















































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








Documents

Relevance of Incorporating Cavity Shape Change in