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Journal of Materials Processing Technology 212 (2012) 15731590

Contents lists available at SciVerse ScienceDirect

Journal ofMaterials Processing Technology

journal homepage: www.elsevier .com/ locate / jmatprotec

Mechanics offracture in single point incremental forming

Rajiv Malhotra, Liang Xue, Ted Belytschko, Jian Cao

Department of Mechanical Engineering, Northwestern University, Evanston, IL, USA

a r t i c l e i n f o

Article history:

Received 8 September 2011

Received in revised form 1 February 2012

Accepted 29 February 2012

Available online 12 March 2012

Keywords:

SPIF

Fracture

Material instability

a b s t r a c t

Single point incremental forming (SPIF) is a sheet metal forming technique which has gained considerable

interest in the research community due to its enhanced formability, greater process flexibility and reduced

forming forces. However, a significant impediment in the industrial adoption ofthis process is the accurate

prediction of fracture during the forming process. This work uses a recently developed fracture modelcombined with finite element analyses to predict the occurrence offracture in SPIF oftwo shapes, a cone

and a funnel. Experiments are performed to validate predictions from FEA in terms of forming forces,

thinning and fracture depths. In addition to showing excellent predictions, the primary deformation

mechanism in SPIF is compared to that in conventional forming process with a larger geometry-specific

punch, using the deformation history obtained from FEA. It is found that both through-the-thickness shear

and local bending ofthe sheet around the tool play a role in fracture in the SPIF process. Additionally, it is

shown that in-spite ofhigher shear in SPIF, which should have a retarding effect on damage accumulation,

high local bending of the sheet around the SPIF tool causes greater damage accumulation in SPIF than

in conventional forming. Analysis of material instability shows that the higher rate of damage causes

earlier growth of material instability in SPIF. A new theory, named the noodle theory, is proposed to

show that the local nature ofdeformation is primarily responsible for increased formability observed in

SPIF, in-spite ofgreater damage accumulation as compared to conventional forming.

2012 Elsevier B.V. All rights reserved.

1. Introduction

In single point incremental forming (SPIF) a peripherally

clampedsheetis imparteda desired shape bymovinga singlehemi-

spherical ended tool along a desired profile so as to locally deform

the sheet along this path (Fig. 1). The sum total of these local defor-

mationsgives thesheet itsfinal shape. Significant advantages of this

process over conventional forming include greater formability, low

forming forces and generic tooling configuration. One of the major

research problems of considerable interest to the sheetmetal form-

ing community is the accurate prediction of fracture in SPIF. This

is important because an underestimation of the fracture depth will

result in a loss of theadvantage of enhanced formability of the pro-

cess and an overestimation will cause component failure duringthe forming process itself. Furthermore, a better physical under-

standing of the mechanisms of deformation and fracture in SPIF

is of great importance since this can aid the choice of appropriate

process parameters for the process and can lead to modifications

of the process to further enhance the achievable formability.

Early work in SPIF (Jeswiet et al., 2005) indicated that the maxi-

mum formable wall angle could be a good indicator for material

Corresponding author. Tel.: +1 847 467 1032.E-mail address:[email protected] (J. Cao).

formability in SPIF. More recently, Hussain et al. (2007) formed

axisymmetric funnel shapes in which the profiles of the compo-

nents were arcs of different radii of curvature and showed that

the maximum formable wall angle depended on the radius of the

curvature of the funnel components profile. This indicates that

formabilityin SPIF depends on a combination of theglobal shape of

the component and the process parameters, and therefore essen-

tially on the deformation mechanics of the process. Filice et al.

(2006) explored the possibility of detecting fracture in real time

based on the trend of the forming force. Szekeres et al. (2007)

showed that while the force trend methodology for detecting fail-

ure can be used for a cone shape, it does not work for a pyramid

shape. This observation again highlights the fact that occurrence

of failure in SPIF depends significantly on the process mechanicsitself.

Attempts have also been made to use the concept of conven-

tional forming limit curves (FLCs) to characterize formability limits

in SPIF. Filice et al. (2002) demonstrated that the failure strain in

SPIF significantly exceeds that in conventional forming. Hussain

et al.(2009) derived empiricalforming limit diagrams(FLDs) which

used the reduction in cross sectional area at tensile failure as a

means of predicting failure in SPIF. However, they stated that their

empirically derived FLD depended on the process mechanics itself

and was only valid within the range of the tool diameters, incre-

mental depths and feed rates used in their work. At the same time

0924-0136/$ seefrontmatter 2012 Elsevier B.V. All rights reserved.

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1574 R. Malhotra et al. / Journal of MaterialsProcessing Technology212 (2012) 15731590

Fig. 1. Schematic of SPIF (dotted lines show motion of tool in theprofile view).

Emmens and van den Boogaard (2009) showed that FLCs have cer-

tain drawbacks when it comes to predicting failure in SPIF. It is

known that FLCs are not valid when there is bending and through-

the-thickness shear,bothof whichare significant in SPIF. As a result

modifications of conventional FLCs by incorporating the effects of

changing strain paths (Yao and Cao, 2002) and the effects of large

normal contact pressures (Smith et al., 2003) would still not be

able to predict failure in SPIF accurately. Therefore, the predictionof formability in SPIF with conventional FLCs might notbe feasible.

Numerical investigations usingFEA havealso been conducted to

investigate the deformation force and mechanisms in SPIF. Henrard

et al. (2007) modeled the contact between the tool and the sheet

using a moving spherical tool method, in which a dynamic explicit

time integration scheme was used instead of the usual penalty

based contact algorithm. The main improvement was that a bet-

ter force prediction was obtained using their methodology even

though computational time was reduced by using a larger ele-

ment size. Cerro et al. (2006) simulated SPIF of a pyramid with a

75 wall angle with shell elements and obtained a 5% differencebetween the maximum values of the measured and calculated tool

z forces. However, no attempt was made to predict fracture. van

Bael et al. (2007) extended a Marciniak-Kuczyisnki analysis (M-K analysis) to predict localized necking and fracture in SPIF. They

showed that while the forming limit predictions were higher than

that for monotonic loading, their models still underestimated the

forming limits obtained experimentallyin SPIF.This was attributed

to the fact that the input for the M-K model was obtained at a

pre-determined location through the thickness of the sheet which

meant that interaction between different layers of the sheet was

not considered. Huang et al. (2008) used Oyanes fracture criterion,

an empirical fracture criterion, to predict failure during forming of

a conical cup using SPIF. The model was found to capture forming

limits in SPIF reasonably well, however, the predictions of forming

forces were not satisfactory. Silva et al. (2009) extended a mem-

brane analysis of SPIF to incorporate a damage model in which

damage accumulation depended on hydrostatic stress and showedthat such an approach could be used to predict fracture strains in

SPIF. Malhotraet al. (2010a)investigatedthe use of various material

models and element types to simulate SPIF using FEA and showed

that a fracture model considering triaxility and shear as presented

in Xue (2007a,b) can predict forming forces and fracture occur-

rence muchbetter thanother common materialmodels. The results

were promising and have led to the further investigations on the

mechanicsof fracture in SPIF, whichwillbe presentedin this paper.

While predicting forming force and failure limits in SPIF is

important, a more interesting challenge is to understand why SPIF

results in a much higher formability compared to the conventional

forming process. Emmens et al. (2009) proposed that while bend-

ing, shear, cyclic straining and hydrostatic stress are the dominant

deformation mechanisms in SPIF, pinpointing which factors are

primarily responsible for failure is difficult. Jackson and Allwood

(2009) showed experimentally that deformation in SPIF consists

primarily of stretching perpendicular to the toolpath and through-

the-thickness shear perpendicular to and along the directionof the

toolpath. They also showed that shear increased with the depth of

deformation, was greater on the inner side of the sheet and was

greater along the direction of toolpath motion than in the direc-

tion perpendicular to the toolpath motion. They observed that asthe component was formed the structure became stiffer and the

deformation transitioned from a more widely distributed area to

being concentrated into a local area around the tool contact zone.

In addition, they mentioned that unusual choice of material, sheet

thickness and wall angle used in the present experiments probably

caused strains thathave somedifferences to moretypicalISF exper-

iments which use thinner sheets and steeper wall angles. Allwood

et al. (2007) also showed experimentally that significant through-

the-thickness shear is present in SPIF, by tracing thehistory of a pin

inserted perpendicularly into the blank during deformation of the

blank. A valuable workwith reference to the mechanisms responsi-

blefor increasedformabilityin SPIF was performed by Allwood and

Shouler (2009) in which the M-K analysis was extended so that all

six components of the stress tensor were non-zero. This representsthe typical state of deformation in SPIF. This work provided sig-

nificant circumstantial evidence that through-the-thickness shear

might play a significant role in fracture in SPIF.

The current workgoes beyond the previous discussions focusing

on whether hydrostatic pressure or through-the-thickness shear

contributes to the significant increase of forming limit in SPIF.

Instead, we uncover the unique role thatmateriallocalizationplays

in SPIF. The approach is to use a fracture model in FEA to analyze

the mechanics of deformation in the SPIF of a 70 wall angle coneand a funnel shape. The experimental setup is shown in Section

2. The material model is briefly described in Section 3, in which

the fracture envelope is expressedin the stress space and is a func-

tion of thehydrostatic pressure andthe deviatoric stress state (Xue,

2007a). More recently, it has been shown (Xue, 2007b) that onlystress triaxiality effects cannot explain the phenomenon of frac-

tureinshearandthematerialmodelusedinthisworkcombinesthe

effects of plastic strain, hydrostatic pressure and shear on fracture.

Therefore, this model is ideal for examining the combined effects

of stretching along the component wall and local bending around

the tool (indicated by the hydrostatic pressure and plastic strain)

and through-the-thickness shear (indicated by the Lode angle).

Note that stretching, bending and shear are among the deforma-

tion mechanisms said to very dominant in SPIF (Emmens et al.,

2009). Corresponding experiments are performed to compare the

forming force history and fracture location predictions from FEA.

The mechanics of SPIF are then analyzed in-depth in Section 4 by

examining the deformation history through the thickness of the

sheet at four locations along the formed component wall and the

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R.Malhotra et al. / Journal of Materials Processing Technology212 (2012) 15731590 1575

Fig. 2. Profiles of (a)70 cone (b)funnel formedas part of experiments.

primary deformation mechanisms which affect failure in SPIF are

found. Contributions from hydrostatic pressure and through-the-

thickness shear to fracture are quantified. In Section 5 additional

simulations are performed with a reduced tool-sheet friction coef-

ficient to examine the effect that through-the-thickness shear has

on formability in SPIF.Furthermore, to better understand thefailure

mechanism, simulations are also performed using the same mate-

rial model for forming of the 70cone with a conventional forming

process using a larger punch. The deformation mechanisms andthe occurrence of material localization in the larger punch case are

compared to those in the SPIF case in Section 6. Motivated by the

results from these two cases,a newnoodle theoryis proposed here

to explain the essence of enhanced formability seen in SPIF. Finally,

inferences on the nature of fracture in SPIF are discussed in Section

7 followed by conclusions in Section 8.

2. Experimental setup of forming a cone and a funnel using

SPIF

Two shapes, a 70 wall angle cone and a variable angle funnel(Fig. 2) were formed using 1 mm thick AL5052 sheet, with a tool

diameter of 9.5 mm and feed rate of 150 mm/min. The blank size

available for forming inside the clamp was 80mm80mm squarefor the cone and an 80mm diameter circular area for the funnel

case. No tool rotation was allowed and PTFE based grease was used

as lubricant at the toolsheet interface. The incremental depths

used for the cone and the funnel were 1.0 mm and 0.5 mm respec-

tively. A spiral toolpath was used in which the tool followed the

three dimensional profile of the shape to be formed while moving

simultaneously in thex,y andzdirections (Malhotra et al., 2010b).

The z forces on the tool throughout the forming process were

measured using a Kistler dynamometer model 9255A mounted

below the fixture and the tool tip depth at which fracture occurred

wasalso recorded. Eachexperimentwas performedthree timesand

the final fracture depth for a component was taken as the average

of the observed fracture depths from the three experiments.

3. Fracturemodel and its implementation in FEA

This section describes the material model used and its imple-

mentation in FEA to predict fracture in incremental forming. The

predictions from FEA are compared to measurements obtained

from experiments.

3.1. Fracture model

The present work uses the damage plasticity model proposed

by Xue (2007a) in which two independent internal variables, i.e.,

the so-called damage variable and the plastic strain, are used to

describe the material status. The constitutive relationship consists

of a damage coupled yield function, the evolution laws for the

plastic strain (p) and a reference fracture strain (f). In particu-

lar, we chose the stress-based fracture envelope for its simplicity.

The damage coupled yield function is

= eq w(D)M 0 (1)

where eq is the equivalent stress, M is the undamaged matrixstress which is a functionof theplastic strainto include strainhard-

ening, andw(D) is a weakening factorused to describe the material

deterioration. The weakening factorw is related nonlinearly to thedamage variableD byw(D) = 1D, where is a material constant.The weakening factor w is treated as a scalar for isotropic dam-

age models like the present one. To model matrix resistance the

Swift-type hardening relationship is adopted as show in Equation

(2)

M= y0

1 + p0

n(2)

where y0 is the initial yield stress, 0 is the pre-strain, p is theplastic strain andn is the hardening exponent. The associated flow

rule is enforced.Accumulation of damageis modeled as a nonlinear

function of theplastic strain(p) anda reference fracture strain(f)and is expressed in the rate form as

D = m

pf

m1 pf

(3)

wherem is a material constant. In the finite element model an ele-

ment completely loses its load carrying capacity and is removed

when the value of the damage variableD reaches 1.0. When all the

elements through the thickness of the sheet are removed then a

crack is said to have occurred at that location. The reference frac-

ture strain (f) is first expressed as a stress envelope Mf, whichtakes the form of a modified Tresca type of pyramid (Xue, 2009) as

follows,

Mf=

f01 + kpp

3

2cos L(4)

where f0 is a reference fracture stress, kp is a material constantrelated to pressure sensitivity,p=(kk/3) is the hydrostatic pres-sure and L is the Lode angle, via which the deviatoric componentof the stress state is incorporated into the reference fracture strain.

An example of this reference fracturestrain in thespace of theprin-

cipal strains (1, 2and 3) and the hydrostatic stressp is shown inFig. 3.

Fig. 3. Illustrative example of the reference fracture strain envelope used in this

work.

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The reference fracture strainfused in Equation (3) is obtained

from the inverse of the matrix stress as:

f(p, L) = 0

f0y0

(1/n)1+ kpp

32cos L

(1/n) 1

(5)

Therefore, damage accumulation in this material model

depends on the plastic strain (p) as well as on the current state ofhydrostatic pressure and deviatoric stress state, via the reference

fracture strain (f). Xue (2010) showed that this type of I1-J2-J3constitutive model can predict both the onset of ductile fracture

and material instability at the same time. Note that in this material

model there are four material constants which cannot be obtained

from tensile tests and therefore, a calibration procedure with dif-

ferent mean stress and deviatoric stress states should be employed.

These parameters aref0, kpwhich quantify the reference fracturestrain (Equation (4)), and,mwhichare related to howthe damageand material softeningevolves (Equations (1) and (3) respectively).

The material constantf0 is thematrix stress at fracture under zeromean stress condition, i.e., tensile condition. The parameter kp is

responsible for the contribution of hydrostatic pressure towards

the evolution off. The material constant m signifies the rate at

which damage accumulates in an element and therefore how fast

the element is removed. The inclusion of a nonlinear damage evo-lution law takes into consideration the common observation that

damage accumulation rate increases as the plastic strain increases.

The value of dictates how the material loses its capacity to carry

load as damage accumulates. Note that sincem influences the evo-

lution of the damage (Equation (3)) and influences how thisdamage affects the weakening of the material (Equation (1)), both

these material constants are together responsible for the softening

behavior of the material as damage develops.

In addition to the prediction of damage accumulation and

fracture, the occurrence of diffused and localized necking during

deformation is predictedanalytically usingthis materialmodel.Dif-

fusedneckingis predictedusing a three dimensional generalization

of Consideres maximum force criterion to a maximum power cri-

terion, as derived in Xue (2010). The closed form expression usedto predict the onset of diffused necking is shown in Equation (6),

h

M+ wDD

w cos

L +

6

(6)

where wD =w/D, D =D/p, h=eq/p and L = Lode angle.The equal sign in Equation (6) denotes the onset of diffused neck-

ing. The effectof damageaccumulation andweakening on material

instability is included in this expression via the term (wDD/w).

Note that higherratesof weakeningor damageacceleratethe onset

of diffused necking. The occurrence of localizednecking, i.e.,of local

shear bands post diffused necking, is also predictedin a Hadamard-

Hill sense by examining the positive definiteness of the acoustic

tensor at the current stress state, as shown in Xue and Belytschko

(2010). The form of the acoustic tensorAis as follows

A = h : Cedp: h (7)

where Cedpis the elastic-damage-plasticity tangent matrix andh is

thedirection in which instability might develop. Thetangentmatrix

Cedpis expressed as

Cedp= wC0 +

2G0h+ 3G0

wDDw

2wG0r r (8)

where r is the deviatoric stress direction, =1/3kkI+2/3eqr, G0is the shear modulus of the material, C0 is the isotropic Hookean

matrix for the undamaged material. C0 is expressed as shown in

Equation (9).

C0=

2G0II

+ K0 2

3

G0 I I (9)

Table 1

FEM model details.

Simulation case Tool speed

(mm/s)

Target time

increment for

mass scaling (s)

Simulation

time (CPU h)

SPIF: cone 3000 1.5e06 78SPIF: funnel 1500 1.2e06 115Conventional forming of cone 3000 1.5e06 13

Table 2

Material properties calibrated from tensile tests.

(kg/m3) E(GPa) y0 (MPa) n 0

2680 68.6 0.3 117 0.22 0.0045

whereK0 isthe bulk modulus ofthe material,I and IIare thesecond-

order and fourth order identity tensors respectively. Equation (10)

shows the conditionin whichthe material is stable in a Hadamard-

Hill sense, where g is a vector denoting the direction of particle

velocity. The material has lost stability at a given state when, for

some directionh and velocity g, the left hand side of Equation (10)

is lesser than zero.

g: A: g 0 (10)

3.2. Implementation of the material-model in FEA

The above described material model was implemented using a

user subroutine in LS-DYNA. A schematic of the FEA model used

for simulating SPIF is shown in Fig. 4a. The blank material was

discretized using eight reduced integration linear brick elements

through the thickness of the sheet. The blank was meshed so that

the region to be formed had a radial mesh with a maximum in-

plane element size of 0.50mm in the radial direction and 0.35mm

in the circumferential direction (Fig. 4b). The tools and the top and

bottom clamps were discretized using planar shell elements with

an element size of 0.20mm for the tool and 2.5 mm for the clamps.

Furthermore, the tool speed was artificially increased and massscaling was used to speed up the simulation. The friction coeffi-

cient at thetoolsheet interfacewas specifiedas 0.15 (Eyckensetal.,

2010). The Belytschko-Tsay hourglass formulation was usedto con-

trol hourglassing. All simulations were performed on a workstation

with four processors at 3.66 GHz speed. Details on tool speed, mass

scaling and simulation time are shown in Table 1.

The material parameters E, y0, n and 0 were obtained fromuniaxial tensile tests and the values are shown in Table 2. The

four material constants f0, kp, m and were calibrated manu-ally by matching the tool z forces from simulation with the same

obtained from experiments for the case of the 70 cone (Table 3).The methodology used to find the values for these materials con-

stant is as follows.

The values ofkp and f0 control the initial slope of the forcecurve.A highervalueofkpandf0results in a greater slope and viceversa. The values of the parametersm and are usually between 2.0and 3.0 for most metals (Xue, 2007a,b). If the value ofm is too high

then the occurrence of fracture is delayed and the softeningpart of

the force curve is higher than that from experiments. In this case,

the value ofm is decreased to increase the rate of damage accumu-

lation so that fracture occurs at the correct depth. After this, if the

Table 3

Reference fracture strain parameters calibrated using trial simulations with 70

cone.

f0 (MPa) kp(MPa1) m

490 0.0001 1.80 2.5

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Fig. 4. (a) Schematic of the FEA model used to simulate SPIF. (b) Top view ofthe meshusedto discretize the blank.

initial slope of the force curveand thefracture depthare foundto be

matching well but excessive weakening is observed then the value

of needs to be increased. In this work three trial simulation runs

were performed to calibrate these material parameters (Table 3).To show the quality of the damage model calibration the forming

forces in thezdirection obtained from experiments and simulation

for the 70 cone are compared in Fig. 5a.

3.3. Verification of the material calibration results

The calibrated FEA predicted that the cone fractured at a tool

tip depth of 16.1mm in experiments as compared to a tool tip

depth of 14.8mm predicted by FEA. Furthermore, the maximum

thinning just before fracture was 64% from experiment and 63%

from simulation. The same model was then used to predict form-

ingforce andfracture depthfor the funnelcase(Fig.2b). Inthis case,

theexperimentallymeasured fracturedepth fromexperiments was

15.2mm and that predicted by FEA was 14.5mm. The toolzforces

from experiments and simulation for the funnel are compared in

Fig. 5b. Thethinning beforefracture wasmeasured to be 65.8%from

experiments and 63.64% from simulation.

The forces on the tool relate to the state of stress in the mate-

rial, maximum thinning relates to the strain experienced by the

material and the fracture height relates to how well the damage

evolution function (Equation (3)) incorporates the physical effects

that govern fracture in SPIF. It can be seen that the toolzforce pre-

diction, the maximum thinning and the fracture depth prediction

from FEA all agree quite well with those from experiments.

3.4. FEMmodel of the conventional deep drawing process

To analyze the difference between SPIF and the conventional

deep drawing process, the forming of the 70 cone with a largerpunch instead of with a SPIF tool was also simulated. A schematicof the FEA model used for this simulation is shown in Fig. 6a. The

cornerradius of thepunchwas the same as the SPIF tool radius, i.e.,

4.75mm, and the material properties and mesh size of the blank

as well as contact properties were the same as those in SPIF. In the

SPIF simulation, the outer periphery of the blank was constrained

so that the blank was completely fixed in the xy plane, i.e., there

was no material draw in. The top and bottom clamps were used

to constrain motion of the unformed region of the blank in the z

direction. For the conventional forming simulation, the boundary

of the blank was not constrained, i.e., draw in was allowed. The

blank holder and the die were only used to prevent motion of the

unformed region in the zdirection. The punch was displacement

controlled such that if the SPIF tool tip was at a depth z at time

t then the flat face of the punch was also at the same depth zat

time t. The coefficient of friction between the fixture and the blank

was specified as 0.15, the same as in the case of SPIF. The punch

movement in the negativezdirectionwas set such that theflat face

of the punch was at the samezdepth as the tip of the SPIF tool at

any point during the simulation. The predicted fracture depth for

this larger punch forming case was 13.5 mm, i.e., the formed depth

in SPIF was greater than that in the punch forming case. The plastic

strain just before fracture was 1.43 for the conventional forming

case, as compared to a strain of 1.83 for SPIF. Also in the punch

Fig. 5. Comparison of toolzforcesbetween FEA and experiments for (a)70cone (b)funnel.

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Fig. 6. (a)Schematic of theFEA model (b) occurrence of crack predicted by FEA, forthe punch forming case.

forming case fracture occurred as a continuous crack all along the

circumference of the component (Fig. 6b).

A comparison of the hourglass control energy, kinetic energy

and internal energy of deformation for SPIF and conventional form-

ing simulations showed that the kinetic energy and hourglass

control energy were less than 4% of the internal energy of defor-

mation. Therefore, the effects of hourglass control, mass scaling

and artificially speeding up the tool on the simulation results was

negligible.

4. Deformation analysis of incremental forming

The predictions in terms of forming force, fracture depth and

thickness reduction obtained from the FEM model embedded with

the fracture model provide a good foundation for further analyz-

ing the fundamental deformation mechanics in SPIF. The question

to be answered is, which factors contribute the most to the occur-

rence of fracture in SPIF? This section analyses the key indicators

that can be obtained from the FEA to answer this question. These

indicators are: the damage variable (D) in Section 4.1, plastic strain

(p) in Section 4.2, hydrostatic pressure (p) in Section 4.3, through-

the-thickness shear (13, 23) in Section 4.4 and fracture strain (f)in Section 4.5. The combined effect of these indicators is analyzed

in Section 4.6, to pinpoint the primary deformation mechanisms

affecting fracture in SPIF. All the analysis starts with an examina-

tion of the deformation history at four sections A, B, C and D along

theprofile of thedeformedshapesin thesimulation, section D being

closest to the axis of symmetry of the formed component (Fig. 7a).

At each of these sections the four elements through the thickness

of the sheet are labeled 1 to 8, where 1 is the element on the

inner side of the sheet which is in contact with the tool and 8 is

the element on the outer side of the sheet (Fig. 7b). This work con-

centrateson examining the deformationof elements1 and 8 ateach

of the four sections A, B, C and D. The key deformation indicators

mentioned above are plotted versus thezdisplacement of the tool

tip.

4.1. Damage evolution indicatedby damage index (D)

The damage plasticity model uses a damage variable D to sig-

nify the accumulation of damage and loss of the materials ability

to take stresses during deformation. An element is removed when

the damage variable D reaches a value of 1.0 and a crack is said to

occur at a location where all elements through the thickness of the

sheethave beenremoved.Since we are interested in the occurrence

of fracture therefore, in this sub-section, the evolution of damage

variable, D, in SPIF will be examined first. Fig. 8a and b show the

evolutions of the damage variableD from section A to section D for

the cone and the funnel, respectively.

Itcanbeseenthatinboththeconeandthefunnelcases,thedam-

agevariable evolves fasterfor element 8, i.e., onthe outer side of the

sheet,thanfor element 1 whichis onthe inner side of the sheet.As a

result, in FEA the element on the outer side of the sheet is removed

first. This phenomenon was also confirmed by a visual examina-

tion of the FEA results. Physically this implies that in SPIF the crack

initiates on the outer side of the sheet and propagates inwards.

Note that in Equation (3) the damage variableD is directly propor-

tional to plastic strainp andinversely proportionalto the referencefracture strain f. Therefore, to investigate the reasons behind the

trends for damage evolution shown in Fig.8, the evolutionofpandf at sections A, B, C, D shall be examined next.

4.2. Evolution of plastic strain (p)

Fig. 9 shows the evolution of plastic strain (p) for elements 1and 4 at sections A, B, C and D for the cone and the funnel cases.

For both the cone and the funnel cases, the plastic strain at any

section is greater for the outer side of the sheet (element 8) than

it is for the inner side of the sheet (element 1). This is because of

local stretching and bending of the sheet around the tool which

causes the element on the outer side of the sheet (element 8) tostretch more as compared to the element on the inner side (ele-

ment 1). This results in a higher plastic strain on the outer side of

the sheet. Since, damage accumulation is directly proportional to

plastic strain (Equation (3)), a straightforward conclusion is thatby

itself, localbending ofthe sheetaround thetoolwill attempt to increase

damage accumulation on the outer side of the sheet as compared to

the inner side of the sheet.

However, attributing the damage behavior observed in Fig. 8

entirely to this phenomenon would be premature. As shown in

Equation (3) it is not only the plastic strain p but also the ref-erence fracture strain f that plays a role in determining damage

accumulation. The following sections will explore this facet further

by examining in greater detail the factors that affect f, i.e., hydro-static pressure in Section 4.3 and through-the-thickness shear in

Section 4.4.

4.3. Hydrostatic pressure (p)

The evolution of hydrostatic pressure (p) in forming the cone

and the funnel is shown in Fig. 10. The hydrostatic pressure, p, is

positive on the outer side of the sheet (i.e., element 8) andnegative

on the inner side of the sheet (i.e., element 1). This is because once

the tool has passed over a certain region of the sheet, that region of

the sheetundergoes local springback. This results in the hydrostatic

pressure on the outer side becoming positive and on the inner side

becoming negative. This effect is seen more clearly by examining

thehydrostaticpressure contours in a certain regionof theblankat

consecutive timesteps during the simulation, as shown in Fig. 11. In

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Fig. 7. (a) Schematic of sectionsA, B, C and D along thecomponent wallat which thedeformation history from FEAis examined.(b) Nomenclatureof elements through the

thicknessof thesheet(contours of damage variable D shown).

Fig. 8. Evolution of damage variable D atsections A,B, C,andD for the (a) 70cone (b) funnel shape.

Fig. 9. Evolution of plastic strain pat sectionsA, B,C, and D for the (a) 70cone (b) funnel shape.

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Fig. 10. Evolution ofhydrostatic pressure at sections A, B, C, and D for the (a) 70cone, (b) funnelshape.

the contours shown in Fig. 11a thehighlighted regionof the blank is

in contact with thetool. Fig. 11b shows thesameregion of theblank

after the tool has passed over this region and local springback has

occurred. It canbe seen that thelocal springback results in negative

hydrostatic pressure on the inner side of the sheet and positive

hydrostatic pressure on the outer side of the sheet.

In terms of the effect of hydrostatic pressure (p) on f(Equa-tion (5)), the trends shown in Fig. 11 mean that the pressure term

(1 + kpp)1/n in Equation (5) will be greater than 1.0 for element 8

and lesser than 1.0 for element 1.

4.4. Through-the-thickness shear (zx, zy)

Since the Lode angle L, in Equation (5) denotes the shear com-ponentof thestressit isalsoworthwhileto look atthe shearstresses

in SPIF. Along the direction of the tool motion (i.e., along the hoop

Fig. 11. Contours of hydrostatic pressure on thedeformedblank at simulation time of (a) 2.3885s (b) 2.3942s.

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Fig. 12. Evolution of through-the-thickness shear along the tool motion direction (zx) atsections A,B, C,D for the (a) 70cone (b) funnel shape.

direction of the component) the tool drags the material along with

it, causing a through-the-thickness shearzx. Fig. 12 shows thatzxincreases from sections A to D and is higher on the inner side of the

sheet, i.e., for element 1 than for element 8.

Since a spiral toolpathis being used,the toolcontinuously moves

down in thezdirection while moving in thex andy directions and

therefore drags material in a direction perpendicular to the tool-

path as well. Fig. 13 shows the evolution of through-the-thicknessshear in a directionperpendicularto thetool motion, i.e.,zy. Again,shear is greater on the inner side of the sheet as compared to

the outer side of the sheet. Note that there are three important

trends associated with shear, i.e., (1) greater shear on the inner

side of the sheet, (2) increase in shear along the toolpath motion

with deformation depth, and (3) shear along the toolpath direction

being greater than shear in a direction perpendicular to the tool-

path motion. All these trends have been shown in the past work of

Jackson and Allwood (2009). The effects of the observed trends of

hydrostatic pressure and through-the-thickness shear on fwill bediscussed below.

4.5. Reference fracture strain (f)

In the present material model f depends on the hydrostatic

pressure p and the on the shear, via the Lode angle L as shownin Equation (5). The product term [(1 +kpp) (

3/2 cosL)]

(1/n), in

Equation (5), signifies the combined effect of the hydrostatic pres-

sure andthe Lode angleof the current stress state onf. To examinethe individual effects ofp and L, this product term is split up into apressure term (1+ kpp)

(1/n) and a Lode angle term (

3/2 cosL)(1/n).

Theevolutionofthesefactorswithplasticstrainisindividuallyplot-

ted(Figs. 14and15 respectively),for elements 1 and4, at section D,

i.e., where the through-the-thickness crack first begins. Note here

that a higher value of the product term implies a higherfwhich inturn means that damage accumulation is retarded, and vice versa.

It can be observed that for both the cone and the funnel the

pressure term is lesser than 1.0 for element 1 and greater than 1.0

for element 8. This is to be expected since the hydrostatic pressure

is negative for element 1 and positive for element 8 in both cases

(Fig. 10). Therefore, if the Lode angle terms for elements 1 and 8

are comparable, the product term for element 8 should be higher

thanthat for element 1. This should in turncause anincrease inthe

correspondingffor element 8. However, Figs. 14and 15 show thatfor both the cone and the funnel the modification term is actually

lower ontheoutersideof the sheet (element8) thanit ison the sidein contact with the tool (element 1). This is because in-spite of the

increasing pressure term theLode angle term is so dominant that it

reduces the modification term more significantly on the outer side

of the sheet.

Since the Lode angle is representative of the deviatoric com-

ponent of the stress, this difference in Lode angle terms can

be attributed to the difference in through-the-thickness shears

between the outer and inner sides of the sheet (Figs. 12 and 13).

Therefore, it is the through-the-thickness shear and not the hydro-

static pressure which dominates evolution of the reference fracture

strain fin SPIF. Furthermore, since fis inversely proportional to thedamage variable D, taken by itself, higherf on the inner side of thesheetwill cause retardation in damageaccumulationon the inner side

of the sheet as compared to the outer side.Thereforeit can be said that highershearon theinnerside of the

sheet is a deformation mechanism which will try to reduce dam-

age accumulation on the inner side of the sheet as compared to

the outer side. This also correlates well to past work by (Allwood

et al., 2007) which proposed that greater shear can enhance

formability.

4.6. Combined effect of local bending and through-the-thickness

shear

The analysis performed till now has shown two deformation

mechanisms, which drive damage accumulation in SPIF. These

mechanisms are as follows:

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Fig. 13. Evolution of through-the-thickness shear perpendicular to the tool motion direction (zy) atsections A,B, C,andD for the (a) 70cone (b) funnelshape.

Fig. 14. Comparison of modification factor, pressure factor and Lode angle terms for 70cone.

Fig. 15. Comparison of modification factor, pressure factor andLode angle terms forthe funnelshape.

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Fig. 16. Evolution of (a)damagevariable:D (b) plastic strain: p, at sectionsA, B,C and D for the punch forming case.

the onset of diffused necking occurs earlier in SPIF than that in

the conventional forming case. Consequently, the occurrence of

localized necking is also earlier than in the punch forming case.

If material localization is used as an indicator of fracture then one

would expect that since localized necking begins earlier in SPIF

the fracture depth in SPIF would also be lower. However, this is

not the case. The reasons for this will now be discussed.

A very significant difference between SPIF and the punch form-

ing cases arises after localized necking begins. The z depth at

fractureinSPIFis14.81mm,i.e.,about6.4mmofthealreadyformed

material is in a state of localized necking before fracture finally

occurs (Fig. 21c). This is more than twice that in the punch form-

ing case. This is because, in SPIF, after the tool deforms a certain

region and causes localized necking it moves on and deforms new

material. As a result the amountof deformation experienced by the

previously unstable region is lesser than it would be with a more

global deformation as in the punch case. So the shear bands do

not grow as fast as expected. Therefore, local deformation in SPIF

is responsible for the existence of a large localized necking region

before fracture finally occurs. The effect of the previously formed

material in SPIF undergoing localizednecking without goingall the

way to fracture is that this previously formed region is able to take

up some ofthe deformationcaused insubsequentpasses ofthe tool.

As a result the component can be formed to a greater zdepth and

a greater plastic strain without fracture using SPIF as compared

to the punch forming case. This effect is also seen by examining

the plastic strain contours for both cases which show that in the

punch case (Fig. 20c) the plastic strain becomes concentrated very

quickly into the shear bands in the localized necking region before

fracture. However, in SPIF (Fig. 21c) the plastic strain is distributed

Fig. 17. Evolution of through-the-thickness shear along (a) hoop direction (zx) (b) component wall (zy),at sectionsA, B,C and D for the punch forming case.

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Fig. 18. Comparison of modification factor, pressure factor and Lode angle terms for 70cone formedwith (a) SPIF (b) large punch.

over a much larger localized necking region and has greater mag-

nitude before fracture, as compared to the punch case. Therefore,

the component wall has a more uniform strain distribution in SPIF.

This phenomenon can be better understood by a simple analogy,

i.e., the so-called noodle theory which is as follows. Consider a

single string of wheat noodles that is held at one end and then

needs to be stretched as much as possible.

One obvious strategy is to start pulling at the free end of the

string (Fig. 22a). As a result, at some location on the string the

material will begin to localize, as shown in red vertical stripes in

Fig. 22b, a strain concentration will develop and eventually fracture

will occur (Fig. 22c). This is similar to what happens in conventional

forming. An alternatestrategywould be to stretch by smaller incre-

ments allalong thestring. One would startat a pointa littlebit awayfrom the fixed end of the string(section AAin Fig. 22d) and stretchby a small increment (says), while moving the location at which

the deformation is applied by small regular increments (say c)towards the free end of the string(Fig. 22e), i.e., from section AAto

DD. In this case, material instability would begin much earlier ascompared to the previous strategy (Fig. 22d). However, ifs is lowenough the localized material would not go all the way to fracture.

So, after some time, when the string is being stretched at section

DD (Fig. 22e) some of the deformation would be taken up by thepreviously localized region, i.e., at sections AA, BB and CC. As aresult, the strain would get distributed more uniformly along the

entire length of the string. With the right combination ofcands at each section the string could be stretched to a greater lengthwithout breaking (Fig. 22f). This is very similar to what happens in

SPIF.

It might be thought that if the previous neck is taking up some

of the deformation in subsequent tool passes then it should grow.

The question might arise that if this is the case then why fracturein SPIF does not occur at the originally formed local shear band,

instead of at the contact zone around the tool tip. The ability of

a local shear band to share some of the subsequent deformation

without going to fracture depends not only on the extent of the

Fig. 19. Ratio (p/f) atsections A,B, C,andD for 70cone formedusing (a) SPIF (b) large punch.

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Fig. 20. Contours of localization flag,zdepth and plastic strain at (a) at onset of diffused necking (b) at onset of localized necking (c) just beforefracture, forpunch forming

ofthe 70cone.

deformation but also on the location of application of the defor-

mation. Note that in the schematic representation of the noodle

theory shown in Fig. 22(ef), for the SPIF strategy, the localized

neck growth is more in the region near the actual section of load

application. This is because as the distance of the neck from the

actual point of load application increases the ability of this neck to

share some of the deformation reduces. As a result, after the onset

of localizednecking, it is alwaysthe neck closest to thecontact zone

thatgrows to fracture.Essentially, the portion of the necking region

that is responsible for sharing most of the subsequent deformation

also moves along with the location of load application.

Further supporting evidence for the noodle theory is provided

by the plots of plastic strain shown in Figs. 9a and 16b. As men-

tioned earlier, these plots show that the plastic strain increases

gradually and very regularly along the profile of the SPIF cone,

i.e., from sections A to D, indicating a more uniform distribution

Fig. 21. Contours of localization flag,zdepth and plastic strain at (a) at onset of diffused necking (b) at onset of localized necking (c) just before fracture, for SPIF of the 70

cone.

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Fig. 22. (a) Stretching the string at the free end (b) material localization at a single location on thestring. (c) Fracture at location of material localization. (d) Stretching the

string by s at location cfrom the free end (e)continuous material localization along length of thestring (f)elongation to a greater length without fracture.

of plastic strain along the component wall. This also means that

even after the tool has moved on the previously formed region

is actually taking up some deformation in the subsequent tool

passes. On the other hand for the punch forming case, the plastic

strain is concentrated at section D. In fact, at section D after a

certain point (at which point this region is in a state of localized

necking) the plastic strain rate increases dramatically indicating a

rapid concentration of strain in this region after localized necking.The existence of a larger localized region in SPIF is further sup-

ported by theobservationof materiallocalizationall along theouter

surface of the formed SPIF components (Fig. 23). The components

shown are funnels formed with an incremental depth of 0.5 mm.

The z depth between the localized bands was measured using a

depthgaugeto be approximately 0.5mm. Furthermore, it was visu-

ally observed during the forming process that these shear bands

initiated and grew in the regions where the tool was currently in

contact with the sheet. Fracture always occurred at a previously

generated shear band closest to the current position of the tool.

6. Discussion

The goal of this section is interpret the results obtained in thiswork in an attempt to correlate it to work done in the past and

to examine the implications of the observations made, on the SPIF

process.

6.1. Deformation mechanisms in SPIF

Experimental work in the past (Jackson and Allwood, 2009) has

shown that in SPIF shear along thetoolpathincreases with depth of

deformation, is greater along the direction of the toolpath motion

than perpendicular to it and is greater on the inner side of the

sheet than on the outer side. This is supported by the current work

(Figs. 12 and 13). Furthermore, it is shown that the outer side of

the sheet is subjected to greater plastic strain due to local bending

of the sheet around the tool (Fig. 9). The lower shear and greater

plastic strain on the outer side of the sheet cause greater damage

accumulation on the outer side of the sheet (Fig. 8). This causes the

crack in SPIF to begin on the outer side of the sheet and propagate

inwards. Therefore, when considering failure in SPIF the combined

effect of both local bending and shear must be accounted for.

6.2. Influence of shear on formability in SPIF as compared to

conventional forming

It has been proposed in the past (Allwood et al., 2007; Allwood

and Shouler, 2009) that increased shear in SPIF could be the rea-

son for increased formability in SPIF as compared to conventional

forming. This work partially supports this theory by showing that

increased shear does increase f in SPIF (Fig. 18) which shouldcause lesser damage accumulation in SPIF. At the same time, as

compared to conventional forming, the increase in damage accu-

mulation caused by local bending of the sheetaround the SPIF tool

overwhelms the reduction in damage accumulation due to higher

shear (Fig. 19). Consequently, damage accumulation is faster in SPIF

thanin conventional forming (Figs. 8a and 16a). Therefore, attribut-

ing the increased formability in SPIF as compared to conventional

forming, solely to shear, might not be a complete explanation. Thisraises the question of why formability is higher in SPIF than con-

ventional forming.

6.3. The Noodle theory of failure in SPIF

Martins et al. (2008) proposed that failure in SPIF occurs by uni-

form thinning without evidence of localized necking. This work

shows what appears to be localized necking on the outer side of

the components formed with SPIF (Fig. 23). At the same time it is

important to note that the occurrence of localizedneckingis a fairly

subjective phenomenon as far as experimental observations are

concerned. It is very difficult to decide whether the occurrence of

what might appearto be a neck is purelya geometricphenomenon,

purely a material deformation affect or a combination of both of

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Fig. 23. Regions along the outer surface of SPIF components indicating material localization.

these. However, their claim that in SPIF a neck does not grow com-

pletely to failure upon initiation is supported by the current work

as well as shown in Section 5.3.

Emmens et al. (2009) examined formability in SPIF in terms of

the suppression or retardation of necking. They noted that whilelocalization of material in SPIF is inevitable, the increase in forma-

bility as compared to conventional forming can be explained in

terms of mechanisms which reduce stress at the location of the

originated neck to a level below that required for further growthof

theneck. Thiswork shows thatmaterial localizationis verymuch an

essential characteristic of SPIF due to the local nature of deforma-

tion in the process (Figs. 20 and 21). This happens because in-spite

of higher shear the accumulation of plastic strain, and therefore

of damage, is much higher very early on during the deformation in

SPIF. This shouldcause earlier onset of materialinstabilityand frac-

ture in SPIF as compared to conventional forming. While material

instability does begin early on, actual fracture occurs much later in

SPIF than in conventional forming.

A new noodle theory is proposed that explains the increasedformability in SPIF as compared to conventional forming in-spite

of the fact that damage accumulation and onset of material insta-

bility is faster in SPIF. The theory goes farther than taking material

localization as an indicator of the occurrence of fracture by ana-

lyzing what happens after material localization. It is shown that

the inherently local nature of deformation in SPIF allows the gen-

eration of a larger region of unstable, but not fractured, material

before actual failure occurs (Fig. 21). It is proposed that it is the

ability of this region, in essence, to share some of the deforma-

tion in the subsequent passes of the tool which is the root cause for

increased formability in SPIF (Fig. 22). Thefact that the crack occurs

around thetoolcontactzone instead of at the first originated neck is

explainedby thefactthatthe extentto whicha neck growsdepends

on not only the extent of the subsequent deformation but also onthe location of the application of this deformation.

This theory might also explain the reason for the inability to

accurately predict failure in SPIF using conventional FLCs. Conven-

tional FLCs predict the occurrence of material localization. Using

FLCs to predict failure in conventional sheet forming (Filice et al.,

2002; Hussain et al., 2009) makes the assumption that the transi-

tion from material localization to fracture is so fast that keeping

a margin of safety from the occurrence of material localization is

enough to prevent fracture. This assumption is true for conven-

tional processes, as is shownin thesimulationof the punchforming

case, where material localization is quickly followed by fracture

(Fig. 20). However, for SPIF the transition frommateriallocalization

to actualfracture is much slowerin SPIF than in conventionalform-

ingdueto the local nature of deformationin the process. This is why

FLCs, which predict material localization, are unable to accurately

predict fracture in SPIF.

A practical concern that arises from the observation of a larger

localized necking region in SPIF is that parts formed by SPIF will

be severely damaged even before going into actual operation. It isimportant to note that any material has internal voids which are

subjected to damage under even a small tensile stress. The damage

evolution in general plays a more important role in final fracture

compared to the damage initiation. Therefore, simply taking the

initiation of localized necking at one material point as an indica-

tor for damage is premature. Additionally, another relevant work

(Agrawal et al.,submitted for publication), a studyon a channel part

has shown thatSPIF formed parts have a significantlylonger fatigue

life compared to parts obtained from conventional machining or

bending processes.

6.4. Effect of process parameters on fracture in SPIF

Process parameters such as incremental depth, tool size, tool

rotation, feed rate and friction at the toolsheet interface affect

both local bending and shear in SPIF. Therefore, they also control

damage accumulation and the occurrence of localized and diffused

necking which subsequently controls the final fracture depth. In

addition to predicting fracture and explaining higher formability

in SPIF as compared to conventional forming, this link between

operational parameters and the occurrence of fracture provides a

powerful means to qualitatively predict the effect of operational

parameters on fracture in SPIF.

For example, plastic strain evolution is one factor that affects

damage accumulation. The phenomenon of higher plastic strain

on the outer side is induced by local bending of the sheet and is

therefore qualitatively inherent to SPIF. A change in the incremen-tal depth and the tool size, might reduce the rate at which this

plastic strain and therefore damage accumulates. This will cause a

delayin onsetof diffused neckingand subsequentlocalized necking

which would in turn lead to a greater fracture depth. This might be

a possible explanation for the well documented observation that a

reduction in incremental depth or tool size results in an increased

fracture depth.

Whether shear or local bending dominates fracture in an SPIF

operation is a question that can only be answered subjectively

with reference to another SPIF operationwith different operational

parameters. The answer depends on which operational parameters

dominantly affect plastic strain and shear strain and by howmuch,

andon which parameters aredifferent between the two operations

being compared.

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Smith, L.M., Averill, R.C., Lucas, J.P., Stoughton, T.B., Matin, P.H., 2003. Influence oftransverse normal stress on sheet metal formability. Journal of Plasticity 19,15671583.

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