ct

, Ind

Finite element method

inh

r th

g fa

di

nsit

e p

nic

vale

ni

re

he

other process parameters.

rts by mminatetter qumponese shes worklar cupdrawitial dir

localized necking initiates (i.e., when the existing state of strain

theSwiftus instingcking

extended the MK model by assuming the groove at an angle to

Contents lists available at ScienceDirect

.e

Finite Elements in An

Finite Elements in Analysis and Design 47 (2011) 11041117product experimentally using the grid marking method and thenE-mail address: pmd@iitk.ac.in (P.M. Dixit).becomes unstable) as the sheet is formed into the product. This the major stress axis to obtain a necking criterion when theprincipal strains are either tensile or compressive.

In practice, the FLD of a material is employed in predicting theformability of a product by estimating the critical strains in the

0168-874X/$ - see front matter & 2011 Elsevier B.V. All rights reserved.

doi:10.1016/j.nel.2011.04.003

Corresponding author. Tel.: 91 512 2597094; fax: 91 512 2597408.product is predicted using the forming limit diagram. The forminglimit of a sheet metal is dened to be a state at which the

criterion when both the principal strains are positive. (This isnormally called the MK model.) Hutchinson and Neale [7]non-circular deep drawing processes, box drawing is the most basicprocess (Fig. 2). In box drawing process, the metal ow rates aredifferent in the straight walls than in the box corner region. Thisresults in an uneven material distribution around the box wall.

Traditionally, the formability of a fracture-free sheet metal

the strain state is less than the induced increment inequivalent stress, corresponds to diffuse necking. Whereas[4] and Hill [5] assumed that the sheet metal is homogeneonature, Marciniak and Kuczynski [6] assumed a pre-exigroove, perpendicular to the major stress axis, to obtain a nethe radial direction. The stress condition in the wall and bottomregions, however, is different. The stress elds exhibited innon-circular deep drawing processes are more complicated asthe mechanics of deformation differs very appreciably from thatobserved in the circular cup drawing process. Among the

proposed by Swift [4], Hill [5] and Marciniak and Kuczynski [6] forthe plane stress conditions. Hills [5] criterion assumed thatlocalized necking takes place along the direction of zero rate ofextension. Swifts [4] criterion, which assumed that necking takesplace whenever the yield stress increment caused by a change in1. Introduction

Manufacturing of sheet metal pais a cost effective process since it eliand welding operations giving a beFurther, it enables production of coAmong the manufacturing of thedrawing is an extensively used presthe deep drawing processes, circubasic process (Fig. 1). In circular cupthe cup is compressed in the tangen& 2011 Elsevier B.V. All rights reserved.

eans of press workings expensive machiningality nished product.nts at a very high rate.et metal parts, deeping process [1]. Amongdrawing is the most

ng, the ange region ofection and stretched in

limit is conventionally represented as a curve in the 2D strainspace of major and minor strains. This curve is called the forminglimit diagram (FLD). Initial attempt to measure the principalstrains at necking was made by Keeler and Backhofen [2] bystretching sheets over hemispherical punches and recording thestrains at the discontinuous increase by a grid marking technique.From this experiment, one can construct only the right side ofFLD. This technique of strain measurement was extended to otherexperiments by Goodwin [3] so as to obtain the left side of theFLD. Early attempts to construct theoretical FLDs were based onthe necking criteria, in terms of the major and minor strains,Numerical analysis of damage for predideep drawing

Ravindra K. Saxena, P.M. Dixit

Department of Mechanical Engineering, Indian Institute of Technology, Kanpur 208016

a r t i c l e i n f o

Article history:

Received 29 October 2010

Received in revised form

7 April 2011

Accepted 7 April 2011

Keywords:

Deep drawing

Ductile fracture

Continuum damage mechanics

Elasto-plastic analysis

a b s t r a c t

The sheet metal may have

imperfections grow unde

fracture is often a limitin

fracture allows a prior mo

nancial savings. The inte

This paper deals with th

continuum damage mecha

the principle of strain equi

Lemaitre. An in-house 3D

height at which the fractu

study of the maximum cup

journal homepage: wwwion of fracture initiation in

ia

erent voids/imperfections present because of preprocessing. These voids/

e applied load resulting into nal fracture. The occurrence of ductile

ctor in metal forming processes. Prediction of the initiation of ductile

cation of the process which can result in a defect-free nal product with

y of voids is often represented by introducing a variable called damage.

rediction of fracture initiation in deep drawn cup using Lemaitres

s model. The damage is incorporated in the constitutive equation through

nce. The damage is evaluated using the damage growth law proposed by

te element formulation is employed for the analysis. The maximum cup

initiates is determined using the critical damage criterion. A parametric

ight is carried out to investigate the inuence of material, geometric and

lsevier.com/locate/finel

alysis and Design

comparing these critical strains with the FLD. However, therehave been some attempts to use the FLD of a material to predictlocalized necking in a deep drawn product using the niteelement method (FEM). Evangelista et al. [8] proposed a mod-ication in the MK model [6] (in the form of an inhomogeneityfactor) to construct the FLD of SAE 1010 steel using FEM. Then,this FLD was employed to predict the maximum cup height atlocalized necking in circular and square cups using FEM with shellelements. However, there is no experimental validation of thepredicted maximum cup height at localized necking. Chen et al.[9] determined the FLD of a magnesium alloy AZ31 using theexperimental method of Keeler and Backhofen [2]. Since themagnesium alloy has a poor formability at room temperature,the FLD is constructed at elevated temperatures. Then, they usedthis FLD and the PAM-STAMP nite element software (with four-noded shell elements) to predict the localized necking in squarecups of AZ31 material. They obtained the maximum cup height atwhich the simulated values of the major and minor strains liebelow the FLD. They observed a good agreement between thepredicted maximum cup height with the experimental results upto moderate temperatures. They also conducted the parametricstudies of the maximum cup height with respect to the threeparameters: the punch prole radius, the die corner radius andthe forming temperature.

The FLD of a material depends on the strain path. As a result,for a single material, there are different FLDs for different paths.To overcome the drawback of the strain path dependence of theFLD, Arrieux et al. [10] and Stoughton [11] proposed an alternatestress-based FLD which is path-independent in the stress space.However, it is difcult to construct the stress-based FLDs experi-mentally due to practical difculties involved in measuring theprincipal stresses on a deformed sheet. As a result, normally thestress-based FLDs are constructed theoretically. Arrieux et al. [12]used the MK model [6,13] to construct the FLD of mild steelin the stress space, instead of in the strain space, by employingthe fourth order RungeKutta method. They employed this FLD(which they called as the forming limit stress curve or FLSC inshort) to predict the occurrence of localized necking in deepdrawing of a square cup using FEM with triangular at elements.They found that the necking region is in the cup corner above thepunch radius region. Further, the location of this region matcheswith the location of the strain localization region determined bythem experimentally.

Instead of using the FLD in the strain or stress space

rppt0

Dd

Punch

Z

rdp

Dp

X

Die

Blankholder

X

thick

R.K. Saxena, P.M. Dixit / Finite Elements in Analysis and Design 47 (2011) 11041117 1105Db

Fig. 1. Schematic diagram of circular cup drawing process (Dbblank diameter,t0blank thickness, Dppunch diameter, Dddie opening diameter, rpppunchprole radius, rdpdie prole radius).

rdc

rpc

Y

Fig. 2. Schematic diagram of square cup drawing process (Sbblank size, t0blank

rpcpunch corner radius, rdcdie corner radius).(constructed experimentally or numerically) for predicting theformability of a product, an alternative approach is to use anappropriate ductile fracture criterion. Quite a few semi-empiricalcriteria have been employed for predicting the fracture initiationin metal forming processes. Takuda et al. [14] used four differentsemi-empirical ductile fracture criteria (Cockroft and Latham [15],Brozzo et al. [16], Clift et al. [17], Oyane et al. [18]) in niteelement simulation of circular cup drawing process for twoaluminum alloys (A1100 and A2024) to predict the maximumcup height without fracture. They also conducted experimentsand observed that the fracture location predicted by these fourcriteria matches with the experimental location of fracture. Theyfurther observed that, whereas the fracture occurs in the cup wallregion near the cup bottom in A1100 material, in A2024 material,the fracture location is either near the cup bottom or mid-waybetween the ange and the cup bottom. About the latter fracturelocation, Takuda et al. [14] observed that the fracture is notpreceded by localized necking and, hence, may not be predictableby the FLD. The main limitation of using any of the semi-empiricalductile fracture criteria for the prediction of fracture initiation isthat they are based on an inadequate knowledge of the physics ofthe ductile fracture process in metals.

It is well-known that ductile fracture in metals occurs mainlydue to void nucleation, growth and nally coalescence into amicro-crack. Three broad approaches have emerged that try to

rpp

Sd

Sp

Sb

rdp

t0

Punch

Z

Blankholder

Die

ness, Sppunch size, Sddie size, rpppunch prole radius, rdpdie prole radius,

dam

R.K. Saxena, P.M. Dixit / Finite Elements in Analysis and Design 47 (2011) 110411171106leads to the linear damage growth law (i.e., the damage depend-ing linearly on the equivalent plastic strain) that does not accountfor the void nucleation. Dhar et al. [32] proposed a damage[24grostic potential is not well-established in the literature. Lemaitre26] proposed a simple expression for the second part whichplath respect to the thermodynamic force (corresponding to theage) gives the damage growth law. The second part of thewipredict ductile fracture initiation on the basis of the phenomenaof void nucleation, growth and coalescence:

porous plasticity model of Gurson [19,20], combination of void nucleation, growth and coalescence mod-

els [2123], continuum damage mechanics model of Lemaitre [2426].

In porous plasticity model, the material with voids is idealizedas a porous plastic material. Thus, the constitutive equation isderived from the plastic potential of porous material. Based onBergs [19] model of dilatational plasticity, Gurson [20] proposeda plastic potential for porous plastic materials. Later Tvergaard[27,28] modied this plastic potential to account for the voidinteraction. In this model, the rate of change of void volumefraction _f is considered as the sum of the void nucleation rate_f nucleation and the void growth rate

_f growth. The void nucleation rateis assumed to depend on the equivalent plastic strain rate, whichis called as the cracking model. The void growth rate is related tothe hydrostatic part of the plastic strain rate tensor. In thisapproach, fracture initiation is characterized by the critical valueof void volume fraction (fc) which should be determined from asuitable void coalescence criterion, but is usually obtainedexperimentally [29]. Doege et al. [30] used the Gursons model[20] and the ABAQUS code (with eight-noded brick elements) tostudy the void volume distribution in circular cup drawing of X5CrNi 1810 stainless steel.

Thomason [23] combined the results of de-cohesion model ofGoods and Brown [21] on void nucleation, those of Rice and Tracy[22] on void growth (a single spherical void in uniform stress andstrain rate elds) and his own on void coalescence (i.e., on plasticinstability of the inter-void matrix) to arrive at a fracture criterionin the form of a graph of fracture strain versus the hydrostaticpart of stress. Thus, in this model, the effects of void nucleationand growth are incorporated not in the constitutive equation butin the fracture criterion itself. A limitation of this approach is thatwhile integrating the void growth equations of Rice and Tracy[22], Thomason [23] assumed that the principal directions of thestrain rate tensor remain xed in the direction throughout thedeformation path. This, in general, is not valid for the nitedeformation and/or rotation.

Kachanov [31] described the deterioration of the materialstrength due to voids by introducing a scalar variable D calleddamage, which is dened as the surface density of void tracesin any plane of volume element. Based on this idea, and usingthe theory of continuum thermodynamics, Lemaitre [2426]proposed a continuum damage mechanics (CDM) model. In thismodel, the plastic potential of a damaged material consists of twoparts: (i) rst part corresponding to the yielding and hardeningand (ii) second part corresponding to the damage. The rst part isusually obtained from an appropriate yield function using theprincipal of strain equivalence. This principal states that thedeformation behavior of a damaged material can be describedby the same constitutive equation as that of the virgin material ifthe Cauchy stress is replaced by the effective stress, which isthe Cauchy stress divided by the factor (1D). The derivative ofthe rst part with respect to the stress gives the constitutiveequation (i.e., the ow rule). The derivative of the second partwth law based on the experimental results of Leroy et al.[33] on the measurements of area void fractions at different strainlevels in tension test. This accounts for both the void nucleation aswell as growth. In continuum damage mechanics approach,fracture initiation is characterized by the critical value of damage(Dc) which is determined either by a void coalescence criterion orexperiments.

Quite a few researchers have used the CDM model of Lemaitre[2426] with or without his damage growth law to predict ductilefracture in deep drawing processes. Elgueta [34] used the damagegrowth law of Lemaitre [24,25] and the in-house nite elementcode (with degenerate shell elements) to simulate the drawing ofthe central sector of a steel kerosene stove. The critical value ofdamage (Dc) was determined experimentally using the micro-hardness measurements in tension test. He evaluated the damagedistribution at the end of the drawing process both in single as wellas two-stage drawing operations. He observed that the maximumdamage in the two-stage drawing was less than Dc but in the single-stage, it reaches the level of Dc. The maximum damage occurs in thewall region close to the bottom. His experimental results on thesingle-stage drawing show that the experimental location of fracturematches with the predicted location of the region where D reachesthe critical value Dc. Khelifa et al. [35] used a modied version of theLemaitres damage growth law [24] and the ABAQUS/explicit code(with eight-noded brick elements) to simulate the drawing of aanged circular cup of aluminum alloy AL5754. They used Hills [36]anisotropic yield function to obtain the constitutive relation of thematerial. However, the damage is assumed to be isotropic as in theLemaitres model. They obtained the damage distributions at variouslevels of the punch displacement. They did not determine the criticalvalue of the damage. Instead, the damage is allowed to reach themaximum value of 1. The maximum value occurs in the wall regionclose to the bottom. Their experimental results show that theexperimental location of fracture matches with the predictedlocation of the region where D reaches the value of 1. Fan et al.[37] also used the Lemaitres damage growth law [25] and theABAQUS/explicit code (with eight-noded brick elements) to simulatethe drawing of a square cup of mild steel. They obtained the damagedistributions at various levels of the punch displacement. Theyfound that the maximum damage occurs in the cup corner regioncloser to the ange. They also conducted experiments and observedthat the experimental fracture location matches with the predictedlocation of the region where D reaches the critical value. They alsoconducted the parametric study of damage growth (during thepunch travel) with respect to the blank-holder force and frictioncoefcient. They observed that the damage increases with theblank-holder force and friction. The values of the blank-holder forceused by them are quite small. As a result, the wrinkling also occurs.

The above literature shows that even though the ductile fractureprediction in circular and square cup drawing processes has beenattempted using various methods, there is a need to study the effectsof various geometric and material parameters on the maximum cupheight that can be drawnwithout fracture. This is the objective of thepresent work. Since the use of FLD or semi-empirical ductile fracturecriteria has certain limitations as mentioned earlier, and the pre-dicted fracture location using the Lemaitres CDM model (along withhis damage growth law) matches with the experimental results, thelatter is used for this purpose. Updated Lagrangian formulation isemployed to develop the incremental nite element equations usingeight-noded brick elements. Incremental logarithmic strain measureis used. The incremental stress is made objective by evaluating it in aframe rotating with the material particle. The material is assumed tobe elasto-plastic strain hardening yielding according to the vonMisesyield criterion. The strain hardening behavior is modeled by a powerlaw. The damage is incorporated in the constitutive equation throughthe principal of strain equivalence. Further, the damage growth law

proposed by Lemaitre [24,25] is used. The critical value of damage

an evolution law for micro-voids which incorporates both the

tY ts2eqtRv

t 23

R.K. Saxena, P.M. Dixit / Finite Elements in Analysis and Design 47 (2011) 11041117 11072E1 D

tRv 2

31n312n

tsmtseq

!24

where E is the Youngs modulus and n is the Poissons ratio,

tsm 1 tsii 5potential of a damaged material is given by [2426]

jftsij , tR; tDfdtY ; tD 2

where f and fd are the plastic potentials associated with yieldingand damage, respectively, and tsij is the Cauchy stress tensor attime t. The quantities tY and tR are the dissipative parts of thethermodynamic forces at time t corresponding, respectively, tothe rates of the damage variable tD and the hardening variable tp.The expression for tY is given by [2426]2.1.1. Incremental stressstrain relation for damaged material

When the temperature change is not signicant, the plasticremental damage at time t by tDD.

formincvoid nucleation as well as the growth and a condition for micro-crack initiation based either on a

coalescence model or experimental observations.

As stated earlier, in this work, the continuum damage mechanics(CDM) model proposed by Lemaitre [2426] is employed for theprediction of fracture initiation. In this model, the behavior ofa material containing micro-voids is described by introducingan additional variable called the damage variable. If the materialbehavior is isotropic, then this variable is scalar [2426]. For isotropicmaterial, the damage at a point (at a time) is dened as the area voidfraction in a plane at that point at that time:

D limDA-0

DAvDA

1

where DA is the innitesimal area around that point and DAv is thearea of void traces contained in DA. Since, this is an incremental

ulation, the damage at time t is denoted by tD and the(available in the literature) is used for predicting the fractureinitiation. Body forces are neglected and due to small accelerationsinertial forces are not included. Modied NewtonRaphson iterativetechnique is used to solve the non-linear incremental nite elementequations. The code is rst validated by comparing the predictedpunch variation and thickness strain distribution with the experi-mental results available in the literature.

2. Formulation

2.1. Continuum damage mechanics

As stated in the introduction, a ductile fracture occurs mainlydue to micro-void nucleation, growth and nally coalescence into amicro-crack. The void growth also affects the constitutive relation ofthe material. A realistic model for the prediction of ductile fracturemust include the following three things:

the effect of micro-voids on the constitutive relations of thematerial,3is the hydrostatic part of tsij (called the mean stress) and

tseq 3

2ts0ij ts

0ij

r6

is the equivalent stress at time t. Here, ts0ij is the deviatoric part oftsij . The quantity tsm=tseq is called the triaxiality. For the case ofstrain hardening, tp is identied as the equivalent plastic straintepLeq and is given by

tp tepLeq X

tDepLeq , tDe

pLeq

2

3 tDepLij tDe

pLij

r7

Here, the sum is to be carried over all the increments up to time tand tDe

pLij is the plastic part of the incremental logarithmic strain

(dened in Section 2.3) in an earlier increment. Note that, tepLeq isthe work conjugate variable corresponding to tR. Therefore, for f,often tepLeq is chosen as one of the independent variables ratherthan tR. Thus, f is considered as a function of tsij , tepLeq and tD.

The principle of strain equivalence states that the deformationbehavior of a damaged material can be represented by the sameconstitutive relation as that of the virgin material (i.e., materialfree from defects) if the Cauchy stress tsij is replaced by theeffective Cauchy stress tsij [2426]:

tsij tsij

1 tD 8

Using this principle, the plastic potential f for a material yieldingaccording to the von Mises criterion is given by

ftsij ,tepLeq ; tD tseqtsy 9

where tseq is the equivalent stress corresponding to tsij denedby a relation similar to Eq. (6) and sy is the variable yield stress ofthe material. The dependence of sy on tepLeq (i.e., the hardeningcurve of the material) is assumed to be given by a power law:

tsy HtepLeq sy0KtepLeq n 10

where sy0 is the initial yield stress and K and n are the harden-ing parameters. Using the associated ow rule where the plasticpotential is given by Eq. (9), employing the additivity of elasticand plastic parts of the incremental logarithmic strain and assum-ing the elastic behavior to be linear, the incremental elasto-plasticstressstrain relationship for isotropic material becomes [38,39]

tDsij Z tDtt

tCEPijkldtDepLkl 11

where the fourth order elasto-plastic constitutive tensor tCEPijkl is

given by

tCEPijkl 2m dikdjln

12n dijdkl9mts0ij

ts0kl23m tH

0ts2eq

8:

9>=>; tFtfDuge 22

Here, the vector tfDuge contains the nodal values of the incrementaldisplacement components at the elemental nodes and the matrixtF contains the shape functions that are known functions of thecoordinates. Substitution of Eq. (22) into the virtual work expression(20) and assembly over all the elements leads to the followingalgebraic equation:

tKtfDug tff g tDtfFg 23Here, tfDug is called the global (incremental) displacement vectorand tK, tff g and tDtfFg, respectively, denote the global coefcientmatrix (at time t), global internal force vector (at time t) and globalexternal force vector (at time tDt), the expressions for which aregiven in any standard text on non-linear FEM [45]. Since

tDtfFg tfFg tfDFg and tfFg tff g 24Eq. (23) can be written as

tKtfDug tfDFg 25The solution of Eq. (25) represents only an approximate

solution to the governing equations, because of the linearizationand approximation involved in arriving at expression (20). Tominimize the error of the approximate solution, the modiedNewtonRaphson algorithm [45] is used. Here, the equation

tKtfDugi tDtfRgi1

for i 1,2, . . . 26where

tDtfRgi1 tDtfFg tDtff gi1 for i 2,3, . . .

tDtfRg0 tfDFg 27is solved till the (global) unbalanced force vector tDtfRgibecomes sufciently small. After solving Eq. (26), the net incre-mental displacement vector

tfDug Xi

tfDugi 28

is used to compute rst the incremental logarithmic strain tensor

using Eqs. (15) and (16) and then the incremental stress tensor

to be rigid. Due to the symmetry in geometry and loading, only

appears that for ith contact node, the sign of contact force haschanged, the node is declared to be free and if jth non-contact nodepenetrates the punch/die/blank-holder prole, the node is projectedon the prole. Taking into account such a change of contactconditions, the incremental step is repeated. The transition to thenext increment is accomplished only if there is no change of contactstate after such a recalculation. The blank-holder force (Fb) isassumed to be uniformly distributed over the sheet. The totalblank-holder force is applied incrementally for a rst few increments.

3.1. Validation

As stated earlier, the FE code is rst validated for the simulation ofsquare cup drawing. First, the predicted punch load variation andthickness strain distributions are compared with the experimentalresults of Ref. [47]. The geometric dimensions used in the squarecup drawing experiment are blank size110.0 mm, blank thick-ness0.86 mm, punch size40.0 mm, die size42.5 mm, die pro-le radius5mm, punch prole radius5mm, punch cornerradius3.2 mm, die corner radius4.45 mm [47]. The material usedis Aluminum-killed steel [47] with Youngs modulus211.744 GPa,

pL 0:269

R.K. Saxena, P.M. Dixit / Finite Elements in Analysis and Design 47 (2011) 11041117 1109one-quarter of the blank is considered for the analysis. Sliding frictionis assumed at sheet-die interface. The coefcient of friction isassumed to be constant. Sticking friction is assumed at sheet-punchinterface, which is a fair assumption considering the large compres-sive force generated at the sheet-punch interface. As the deformationprogresses in the sheet, the position of the nodes and the relatedboundary conditions get affected. Therefore, the boundary conditionsare updated in each increment. Flags are assigned to the nodesaccording to their position, to designate the change in the boundaryconditions.

The algorithm for the contact problem assumes that thetooling geometry is represented by a set of straight line segmentsand circular/elliptical arcs. Such a representation of toolinggeometry makes it possible to employ simple formulae to controlwhether a node crosses the tooling surface (straight line segmentor arc). The nodes, if at any particular increment, penetrate thepunch, die or blank-holder proles, are checked for penetrationinto these proles and are given specied displacement to projectthem onto the punch/die/blank-holder prole. The whole incre-ment is repeated by considering these nodes at the specied/projected points on the prole.

At the beginning of each incremental step, the contact state andthe displacements eld obtained from the previous step are used toinitiate the described iterative procedure. The converged results ofthe iterative procedure at a given contact state are checked tousing Eq. (11). The integration in Eq. (11) is carried out usingthe Euler forward integration scheme. Finally, the Cauchy stresstensor at time tDt is updated using the procedure described inRef. [40].

2.4.1. Evaluation of damage

At each convergent equilibrium state (i.e., at time t), theCauchy stress tensor tsij is decomposed into the hydrostatic(or mean) part tsm and the deviatoric part

tsij0. From the

deviatoric part, rst the equivalent stress tseq is evaluated usingEq. (6) and then the triaxiality (tsm=tseq) is calculated. From thetriaxiality, the quantity tRn is evaluated using Eq. (4). Then, at theend of increment, the incremental logarithmic strain is calculated.As its elastic part is small, the whole of the incremental logarith-mic strain is considered as its plastic part. From the plastic part,the equivalent plastic strain increment tDe

pLeq is calculated using

Eq. (7). From tRn and tDepLeq , the incremental damage tDD is

evaluated using the damage growth law given by Eq. (14).Finally, the damage is updated as

tDtD tD tDD 29The fracture initiation is checked using the condition tDtD Dc.If this condition is satised for at least one nite element, theanalysis is stopped.

3. Results and discussion

An in-house 3D FE code based on the formulation of Section 2is used. The code is rst validated for the simulation of square andcircular cup drawings by comparing the predicted punch loadvariation and thickness strain distribution with the experimentalresults available in the literature. Since the damage data is notavailable for these materials, the validation study is carried outwithout incorporating the damage. Then, the code is used for theprediction of fracture initiation in square and circular cup drawing.

The geometry of the problem for circular cup drawing fromcircular blank is shown in Fig. 1 and for square cup drawing fromsquare blank in Fig. 2. The punch, die and blank-holder are assumeddetermine whether the contact state needs to be changed. If itPoissons ratio0.291, stressstrain curve: sy 172574:5eeq(stress in MPa). The other process parameters are (i) blank-holderforce Fb20.0 kN and (ii) friction coefcient m 0:04 [47]. Fig. 3shows the comparison of the predicted punch load variation (withpunch displacement) with the experimental data [47]. There seemsto be a reasonably good agreement between the two.

Figs. 4 and 5 show the thickness strain distributions in Alumi-num-killed steel sheet along the X- (or Y-) direction and the diagonaldirection of the square cup. For comparison purpose, the experi-mental results [47] are also shown in the gures. Figs. 4 and 5 showthat the thinning of the sheet is most severe in the clearance space.Further, the magnitude of the thickness strain is maximum alongthe diagonal direction compared to the transverse direction. There-fore, the potential fracture initiation site in square cup is along thediagonal direction near the die throat region where the thinning ismaximum. The trends of the thickness strain distributions areconsistent with the experimental results of Ref. [47].

Next, the FE code is validated for the simulation of circular cupdrawing. The predicted punch load variation is compared with theexperimental result of Ref. [48]. The geometric dimensions usedin the circular cup drawing experiment are blank diameter110 mm, blank thickness0.74 mm, punch diameter50 mm,

Fig. 3. The variation of punch load with punch displacement in square cup

drawing.

R.K. Saxena, P.M. Dixit / Finite Elements in Analysis and Design 47 (2011) 110411171110die opening diameter52.5 mm, die prole radius8 mm, punchprole radius8 mm [48]. The material used is Aluminum-killedsteel [48] with Youngs modulus200 GPa, Poissons ratio0.3,yield strength (sy0 162MPa, stressstrain curve: sy 5620:00941epLeq0:266 (stress in MPa). The other process parametersare (i) blank-holder force Fb20.0 kN and (ii) friction coefcientm 0:17 [48]. Fig. 6 shows the comparison of the predicted punchload variation (with punch displacement) with the experimentaldata [48]. There seems to be a reasonably good agreement betweenthe two.

3.2. Typical results

First, the FE code is used for the analysis of damage up tofracture initiation in square cup drawing. The geometric dimen-sions used for the analysis are given in Table 1. The material usedfor the analysis is forming steel [34], whose properties are givenin Table 2. The analysis is performed with a total blank-holderforce of Fb15.0 kN to avoid wrinkling. The coefcient of fric-tion used is m 0:05. The number of nite elements used for

Fig. 5. The variation of thickness strain along diagonal direction in square cupdrawing.

Fig. 4. The variation of thickness strain along transverse direction in square cupdrawing.the analysis is 1390. The size of the element is 1.5 mm along thesymmetry direction of the blank. There are two elements alongthe thickness direction.

The growth of damage with punch displacement is shown inFigs. 710. In these gures, the values of D/Dc are plotted onthe deformed congurations, where unity represents the criticaldamage i.e., DDc. It is observed that, rst, the damage initiates atthe cup corner near the cup bottom because the initial plasticdeformation takes place in this region (Fig. 7). With a furtherpunch displacement, the damage zone grows in size at the cupbottom and it also initiates at the region in contact with the diethroat. The zone of maximum damage shifts from the corners atthe cup bottom to the corners at the die throat with a furtherincrease in the punch displacement (Figs. 8 and 9). Ultimately, thetwo damage zones merge in the region between the cup bottomand the contact with the die throat. The damage reaches the

Fig. 6. The variation of punch load with punch displacement in circular cupdrawing.

Table 1Geometric dimensions.

Circular cup Square cup

Blank diameter 140 mm Blank size 124 mm

Blank thickness 1 mm Blank thickness 1 mm

Punch diameter 85 mm Punch size 70 mm

Die opening diameter 88 mm Die size 74 mm

Die prole radius 8 mm Die prole radius 5 mm

Punch prole radius 7 mm Punch prole radius 8 mm

Punch corner radius 10 mm

Die corner radius 12 mm

Table 2Material properties of forming steel and AA6060-T5.

Forming steel AA6060-T5

Youngs modulus (E) 210 GPa 70.8 GPa

Poissons ratio (n) 0.3 0.3Initial yield stress (sy0) 276 MPa 108 MPaHardening coefcient (K) 421 MPa 231 MPa

Hardening exponent (n) 0.8 0.1807

Damage parameters

eD 0.04 0.06eC 0.56 0.11Dc 0.32 0.8

fracture initiation is in good agreement with the experimentalresults on steel about the fracture location [37] (Fig. 11) as well as

R.K. Saxena, P.M. Dixit / Finite Elements in Analysis and Design 47 (2011) 11041117 1111Element No. 157 critical value at a punch stroke of 17.01 mm at the cup cornersnear the die throat, for the element number 157, at a depth of9.5 mm from the die surface (Fig. 10). The predicted location of

the zone of strain localization [12]. Further, the predicted zone ofmaximum damage is consistent with the zone of maximumdamage of Ref. [37] and the zone of localized necking obtainedusing the stress-based FLD [12]. It is observed that, at the atbottom of the cup and at the ange portion of the cup corners, thetotal damage value is very small at the initiation of fracture.Fig. 10 shows that the maximum cup height that can be achievedwithout fracture is 17.01 mm.

Fig. 12 shows the damage growth with punch displacement fora representative element (element number 157) in which thedamage reaches the critical value (Fig. 10). It shows that thedamage initiates at the punch displacement of 8.4 mm andincreases till the punch displacement of 11.01 mm. After this,the damage growth is small up to a punch displacement of12.7 mm. Again, the damage starts growing up to a punchdisplacement of 14.41 mm except for a small region of constantvalue at 13.1 mm. The damage growth is again small from apunch displacement of 14.4116.31 mm. Finally, it reaches thecritical value at the punch displacement of 17.01 mm.

Since the damage depends on the triaxiality (i.e., the ratio ofthe mean stress to the equivalent stress) and the equivalentplastic strain (through Eqs. (4) and (14)), to understand thebehavior of Fig. 12, the graphs of the variations of triaxialityand equivalent plastic strain with punch displacement are

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

Fig. 7. Damage distribution at a punch displacement of 2.6 mm.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Element No. 157

Fig. 8. Damage distribution at a punch displacement of 6.01 mm.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Element No. 157

Fig. 9. Damage distribution at a punch displacement of 13.6 mm.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Element No. 157

Fig. 10. Damage distribution at a punch displacement of 17.01 mm.Fig. 11. Experimental result for sheet fracture in square cup.

Fig. 12. The variation of damage with punch displacement for element number

157.

constructed for the representative element (element number157). They are shown in Figs. 13 and 14, respectively. Initially,the element is in the ange region and therefore, the tensile stressin the radial direction is more than the sum of the compressivestresses in the circumferential and thickness directions. Thus, thetriaxiality is positive. However, it starts decreasing and continuesto decrease up to the punch displacement of 8.31 mm, exceptfor a small increase at the punch displacement of 5.81 mm.It decreases because the circumferential compressive stressincreases as the element gets drawn in the ange region andmoves toward the die throat. It starts increasing and becomespositive at a punch displacement of 9.81 mm. It continues toincrease up to the punch displacement of 14.41 mm except for asmall decrease at the punch displacement of 10.21 mm. Thisincrease in the triaxiality of the element is due to the stretchingof the sheet between the die and punch. From this point onward,it again decreases up to the fracture initiation. It becomesnegative at the punch displacement of 16.31 mm. It decreasesagain because the circumferential compressive stress increases asthe element moves into the die cavity away from the die throat.

On the other hand, the equivalent plastic strain in the represen-tative element continuously increases with punch displacement upto the fracture initiation (Fig. 14). This is because the element getsplastically deformed continuously as it rst gets drawn in the angeregion up to the die throat and then gets stretched after it enters thedie cavity. The equivalent plastic strain reaches the threshold valueeD 0:04 (i.e., the value at which the damage initiates) at the punchdisplacement of 8.4 mm. This explains why the damage is zerotill the punch displacement of 8.4 mm in Fig. 12. In spite of thecontinuous increase in the equivalent plastic strain, the damagegrowth is small between the punch displacement of 11.01 and12.7 mm because the triaxiality is very small. From the punchdisplacement of 14.41 to 16.31 mm, the damage growth is againsmall because of the decrease in the triaxiality. Beyond the punchdisplacement of 16.31 mm, even though the triaxiality continues todecrease, its magnitude increases. Since, the damage growthdepends on the square of the triaxiality, it starts increasing beyondthe punch displacement of 16.31 mm.

Next, the code is used for the analysis of damage up to fractureinitiation in circular cup drawing. The geometric dimensions usedfor the analysis are given in Table 1. The material used for theanalysis is forming steel [34] whose material properties are givenin Table 2. The analysis is performed with a total blank-holderforce of Fb18.0 kN to avoid wrinkling. The coefcient of frictionused is m 0:05. The number of nite elements used for theanalysis is 1390. The growth of damage with punch displacementis shown in Figs. 1518. Fig. 15 shows that the damage in circularcup drawing initiates at the cup bottom radius region and theange region (near the die throat) because the initial plasticdeformation takes place in these regions. With a further punchdisplacement, the damage initiates in the wall region and growsat the cup bottom radius region. After the punch displacement of

Fig. 13. The variation of triaxiality (ratio of the mean stress to the equivalentstress) with punch displacement for element number 157.

R.K. Saxena, P.M. Dixit / Finite Elements in Analysis and Design 47 (2011) 110411171112Fig. 14. The variation of equivalent plastic strain with punch displacement for

element number 157.00.050.10.150.20.250.30.350.40.450.5

Fig. 15. Damage distribution at a punch displacement of 3.41 mm.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7Fig. 16. Damage distribution at a punch displacement of 13.01 mm.

forming steel [34] and aluminum alloy (AA6060-T5) [49], whosematerial properties are given in Table 2. The results of theparametric study are expressed in terms of the normalizedmaximum cup height d/b where d is the maximum cup heightwithout fracture (i.e., the cup height when DDc) and b is thesheet-diameter for circular cup and sheet-size for square cup.

3.3.1. Effect of sheet thickness

The variation of maximum cup height (that can be achievedwithout fracture) is studied for the following sheet thicknesses inthis section: t0.86, 0.90, 1.0, and 1.1 mm. The analysis is carriedout with the total blank-holder force of Fb15.0 kN for square cupand Fb18.0 kN for circular cup drawing. The tooling geometryremains the same. The material is forming steel. Figs. 19 and 20show the variations of the normalized maximum cup height d/b(without fracture) with respect to the normalized sheet thicknesst/b for square and circular cup drawing processes, respectively. Itis observed that the maximum cup height (without fracture)increases with the sheet thickness for both the cups. This isbecause a thicker sheet is expected to fracture at a higher level of

R.K. Saxena, P.M. Dixit / Finite Elements in Analysis and Design 47 (2011) 11041117 11130

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Fig. 17. Damage distribution at a punch displacement of 17.01 mm.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

113.0 mm, all the nodes under the punch prole radius regioncome into contact with the punch and there is no further plasticdeformation in the cup bottom radius region. Therefore, the zoneof maximum damage shifts from the cup bottom radius region tothe wall region with a further increase in the punch displacement(Figs. 16 and 17). Finally, the damage reaches the critical value atthe punch stroke of 20.01 mm in the cup wall (Fig. 18). Thepredicted location of fracture initiation is in agreement withthe experimental result of material A2024 from Ref. [14]. Asmentioned in Ref. [14], this fracture is not preceded by localizednecking and hence may not be predictable by FLD.

It is further observed that at the at bottom of the cup and atthe ange region, the total damage value is very small at theinitiation of fracture. Fig. 18 shows that the maximum cup heightthat can be achieved without fracture is 20.01 mm.

For circular cup also, one can analyze the damage growth in arepresentative element (in which the damage reaches the criticalvalue) using the procedure similar to that used for the square cup.

In circular cup, the value of equivalent plastic strain at fractureinitiation is observed to be 0.1805 which is much less than thecorresponding value of 0.3063 in square cup (Fig. 14). This showsthat the fracture initiation in square cup is inuenced mostly bythe severe plastic deformation (occurring in the corner regions)than by the triaxiality.

3.3. Parametric study

The modied FE code is used for carrying out the parametricstudy of the maximum cup height that can be achieved withoutfracture for various sets of process parameters. The sheet size/diameter and the coefcient of friction are kept constant in theparametric study. The materials used for the parametric study are

Fig. 18. Damage distribution at a punch displacement of 20.01 mm.

Fig. 19. Effect of sheet thickness on the maximum cup height for square cup(material: forming steel).Fig. 20. Effect of sheet thickness on the maximum cup height for circular cup

(material: forming steel).

deformation. However, the increase in the maximum cup heightis more signicant for circular cup. It is observed that, if only oneelement is used in the thickness direction, the expected trend isnot observed (i.e., the maximum cup height decreases with thesheet thickness).

3.3.2. Effect of die prole radius

The die prole radius affects the maximum cup height that canbe achieved without fracture. For square cup, this effect is studiedfor the following values of the die prole radii: rdp3.0, 4.0, 5.0,and 7.0 mm. The analysis is carried out with the total blank-holder force of Fb15.0 kN. The analysis for circular cup is carriedout for the following values of the die prole radii; rdp6.0, 8.0,9.0, and 10.0 mm. Further, the total blank-holder force isFb18.0 kN. For both the cases, the material is forming steel.The sheet thickness remains the same (1.0 mm). The otherdimensions of the tooling geometry also remain the same.Figs. 21 and 22 show the variations of the normalized maximumcup height d/b (without fracture) with respect to the normalizeddie prole radius rdp/b for square and circular cup drawingprocesses, respectively. For both the cups, the maximum cup

height (without fracture) increases with an increase in the dieprole radius. The material ow from the die contact point to thepunch contact point becomes more smooth with an increase inthe die prole radius. Therefore, the stretching in the cup wallreduces and a larger punch displacement is required for thedamage to reach the critical value when the die prole radius isincreased. For square cup, however, the increase in the maximumcup height is insignicant at higher values of the die proleradius. The material ow in the corners of square cup is con-strained. Because of this, the improvement in the smoothness ofthe material ow is less signicant at higher values of the dieprole radius.

3.3.3. Effect of punch prole radius

The punch prole radius also has an important inuence onthe fracture initiation in deep drawing. The analysis for squarecup is performed for the following values of the punch proleradii: rpp6.0, 8.0, 9.0, and 10.0, and with the total blank-holderforce of Fb15.0 kN. The analysis for circular cup is carried out forthe following values of the punch prole radii: rpp6.0, 8.0, 9.0,

Fig. 21. Effect of die prole radius on the maximum cup height for square cup(material: forming steel).

R.K. Saxena, P.M. Dixit / Finite Elements in Analysis and Design 47 (2011) 110411171114Fig. 22. Effect of die prole radius on the maximum cup height for circular cup

(material: forming steel).and 10.0 mm. Further, the total blank-holder force is Fb18.0 kN.For both the cases, the material is forming steel. The sheetthickness remains the same (1.0 mm). The other dimensions ofthe tooling geometry also remain the same. Figs. 23 and 24 showthe variations of the normalized maximum cup height d/b (with-out fracture) with respect to the normalized punch prole radiusrpp/b for square and circular cup drawing processes, respectively.For both the cups, the maximum cup height (without fracture)increases with an increase in the punch prole radius. When thepunch prole radius is increased, then also the material ow fromthe die contact point to the punch contact point becomes moresmooth thereby reducing the stretching in the cup wall. Becauseof this, a larger punch displacement is needed for the damage toreach the critical value. This result is consistent with the observa-tion of Ref. [9] that the higher punch prole radius enhances theformability of square cup drawing. The observation of Ref. [9] isbased on the use of the FLD in the strain space as the formabilitycriterion.

3.3.4. Effect of blank-holder force

The change in the maximum cup height (without fracture) isstudied for different values of the blank-holder force: Fb16.0,18.0, 20.0, and 22.0 kN. The sheet thickness (1.0 mm) and the

Fig. 23. Effect of punch prole radius on the maximum cup height for square cup

(material: forming steel).

the maximum cup height (without fracture).

(material: forming steel).

R.K. Saxena, P.M. Dixit / Finite Elements in Analysis and Design 47 (2011) 11041117 11153.3.5. Effect of material properties

The material properties also affect the maximum cup heightthat can be achieved without fracture. This effect is studied in twostages. First, for the forming steel, this effect is analyzed byvarying the initial yield stress sy0 and the hardening parametersK and n. In the second stage, this effect is studied for two differentmaterials (the forming steel and the aluminum alloy AA6060-T5)for which the Dc (the critical damage value) is also differenttooling geometry remain the same. The material used is formingsteel. It is found that the blank-holder force has no effect on themaximum cup height (without fracture). The blank-holder forcedoes affect the triaxiality in the ange region. However, when thecritical element is in the ange region, the equivalent plasticstrain is less than the threshold value eD (Figs. 7 and 14). Thus,the damage in the critical element does not initiate till it entersthe die cavity. Therefore, the blank-holder force does not affect

Fig. 24. Effect of punch prole radius on the maximum cup height for circular cup(material: forming steel).besides sy0, K and n.In the rst stage, the effect of material properties is studied by

rst keeping the hardening parameters K and n constant andvarying only the initial yield stress: sy0 200, 276, 350, and450 MPa. The analysis is carried out with the total blank-holderforce of Fb18.0 kN for circular cup and Fb15.0 kN for squarecup drawing. The sheet thickness (1.0 mm) and the toolinggeometry remain the same. Figs. 25 and 26 show the variationsof the normalized maximum cup height d/b (without fracture)with respect to the normalized yield stress sy0=276 (276 MPa isthe initial yield stress of the forming steel as mentioned inTable 2) for square and circular cup drawing processes, respec-tively. It is observed that the maximum cup height (withoutfracture) increases with an increase in the initial yield stress. Anincrease in the initial yield stress delays the occurrence of plasticdeformation, thereby increasing the maximum cup height (with-out fracture). However, the effect of variation in sy0 is not verysignicant in square cup drawing.

Next, the hardening parameters K and n are varied one at atime, by keeping sy0 and the other hardening parameter xed. Itis observed that an increase in the hardening coefcient (K)increases the maximum cup height (without fracture) in circularcup drawing, whereas, in square cup drawing it has no inuenceon the maximum cup height (without fracture). Further, it isFig. 25. Effect of initial yield stress on the maximum cup height for square cupobserved that an increase in the hardening exponent (n) reducesthe maximum cup height (without fracture) in both the cups. Thisis because the triaxiality at fracture initiation increases with anincrease in n. However, the effect of n is not very signicant insquare cup drawing. In general, the plastic deformation is inu-enced mostly by the geometric parameters whereas the triaxialityis affected largely by the material properties. As stated earlier,compared to circular cup, the fracture initiation in square cup isinuenced mostly by the severe plastic deformation (occurring inthe corner regions) than by the triaxiality. This explains why thematerial properties do not have much effect on the maximum cupheight (without fracture) in square cup.

The parameters used in the second stage of the parametricstudy on material properties are as follows. The sheet thicknessesfor the forming steel are t0.86, 0.90, 1.0, and 1.1 mm and for thealuminum alloy are t0.9, 1.0, 1.1, and 1.2 mm. The total blank-holder force is Fb15.0 kN for square cup and Fb18.0 kN forcircular cup for the forming steel and Fb22.0 kN for square cupand Fb28.0 kN for circular cup for the aluminum alloy. Thishas been done to avoid wrinkling. Since, the blank-holder forcedoes not affect the maximum cup height, this difference in the

Fig. 26. Effect of initial yield stress on the maximum cup height for circular cup(material: forming steel).

blank-holder force does not affect the objective of the parametricstudy, i.e., to study the effect of only the material properties. Thetooling geometry remains the same. Figs. 27 and 28 show thevariations of the normalized maximum cup height d=b (withoutfracture) with respect to the normalized sheet thickness t/b forsquare and circular cup drawing, respectively. It is observed thatfor square cup, the maximum cup height (without fracture) ismore for the aluminum alloy whereas for circular cup, it is morefor the forming steel.

The reasons for these trends can be explained as follows. Asper Table 2, all the parameters sy0, K and n are higher for theforming steel than for the aluminum alloy AA6060-T5. Themaximum cup height (without fracture) increases with sy0and K and decreases with n. The net effect seems to be that themaximum cup height (without fracture) is more for the formingsteel in circular cup (Fig. 28).

However, the trend in square cup is different (Fig. 27). Notethat the above observations about the dependence of themaximum cup height (without fracture) on sy0, K and n arebased on the parametric study in which the critical damage valueDc is kept constant. However, the maximum cup height (withoutfracture) is the cup height at which the damage reaches Dc.

At microscopic level, ductile fracture occurs rst due to thenucoafralendamcomsen

an

This work is a part of the research work under QIP scheme

R.K. Saxena, P.M. Dixit / Finite Elements in Analysis and Design 47 (2011) 110411171116Fig. 27. Effect of material properties on the maximum cup height for square cup.Fig. 28. Effect of material properties on the maximum cup height for circular cup.sanctioned by AICTE, INDIA.

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Numerical analysis of damage for prediction of fracture initiation in deep drawingIntroductionFormulationContinuum damage mechanicsIncremental stress-strain relation for damaged material

Damage growth lawGoverning equationsFinite element formulationEvaluation of damage

Results and discussionValidationTypical resultsParametric studyEffect of sheet thicknessEffect of die profile radiusEffect of punch profile radiusEffect of blank-holder forceEffect of material properties

ConclusionsAcknowledgmentReferences