Int J Mater Form (2011) 4:129–140DOI 10.1007/s12289-010-1013-8

THEMATIC ISSUE: TWENTE

Simulation of stretch forming with intermediate heattreatments of aircraft skinsA physically based modeling approach

Srihari Kurukuri · Alexis Miroux · Harm Wisselink ·Ton van den Boogaard

Received: 11 May 2010 / Accepted: 29 November 2010 / Published online: 23 December 2010© The Author(s) 2010. This article is published with open access at Springerlink.com

Abstract In the aerospace industry stretch forming isoften used to produce skin parts. During stretch form-ing a sheet is clamped at two sides and stretched overa die, such that the sheet gets the shape of the die.However for complex shapes it is necessary to useexpensive intermediate heat-treatments in between, inorder to avoid Lüders lines and still achieve large defor-mations. To optimize this process FEM simulations areperformed. The accuracy of finite element analysis de-pends largely on the material models that describe thework hardening during stretching and residual stressesand work hardening reduction during heat treatmentsdue to recovery and particle coarsening. In this paper,a physically based material modeling approach usedto simulate the stretch forming with intermediate heattreatments and its predictive capabilities is verified.The work hardening effect during stretching is calcu-lated using the dislocation density based Nes model

S. Kurukuri (B) · A. Miroux · H. WisselinkMaterials Innovation Institute (M2i), P.O. Box 5008,2600 GA Delft, The Netherlandse-mail: s.kurukuri@m2i.nl, s.kurukuri@gmail.com

S. Kurukuri · H. Wisselink · T. van den BoogaardFaculty of Engineering Technology, University of Twente,P.O. Box 217, 7500 AE Enschede, The Netherlands

H. Wisselinke-mail: h.wisselink@m2i.nl

T. van den Boogaarde-mail: a.h.vandenboogaard@utwente.nl

A. MirouxFaculty of 3mE, Delft University of Technology,P.O. Box 5025, 2600 GA Delft, The Netherlandse-mail: a.miroux@m2i.nl

and the particle coarsening and static recovery effectsare modeled with simple expressions based on physicalobservations. For comparison the simulations are alsoperformed with a phenomenological approach of workhardening using a power law. The Vegter yield func-tion is used to account for the anisotropic and biaxialbehavior of the aluminum sheet. A leading edge skinpart, made of AA 2024 has been chosen for the study.The strains in the part have been measured and areused for validation of the simulations. From the usedFEM model and the experimental results, satisfactoryresults are obtained for the simulation of stretching ofaircraft skins with intermediate heat treatments and itis concluded that the physics based material modelinggives better results.

Keywords Stretch forming · Heat treatment · Materialmodel · Aluminum sheet · FEM simulation

Introduction

Stretch forming is a process in which a desired shapeis obtained by stretching a sheet blank or extrusionsections over a tool surface while applying a tensilestress. It is a manufacturing process well suited to thefabrication of parts which are convex in nature, withrelatively low levels of bending [1]. This often makesstretch forming the process of choice in the aircraftindustry for manufacturing of large open shapes suchas wing leading edges and engine cowlings [2]. Thebasic principles of the stretch forming process of theleading edge of an aircraft wing is shown in Fig. 1.An advantage of this process is that only one die isneeded. The undeformed sheet is clamped with two

130 Int J Mater Form (2011) 4:129–140

Gripper1 Gripper2

Trajectory1 Trajectory2Die

Sheet

Fig. 1 Principle of stretch forming

CNC controllable grippers which determine the trajec-tory of deformation of the clamped sheet relative tothe die.

The stretch forming of the leading edge can be di-vided into four steps. During the first step, the sheetis draped around the die without any contact with thedie by moving the grippers towards each other. In thesecond step, the grippers are moved down until thesheet touches the die. During the third stage, the sheetis stretched plastically around the die until the sheetand die are completely in contact. In the last stretchingstep, extra strains are induced in the sheet by movingthe grippers vertically downwards in order to minimizethe effect of spring back [3].

The most commonly used material for aircraft skinparts is the heat treatable aluminum alloy AA 2024.In the fabrication of significant doubly curved surfacesusing heat treatable aluminum alloy sheet, large straingradients are induced across the part. That can leadto surface defects like Lüder lines and orange peel,causing rejection of the components. In order to avoidthese failures and still achieve large deformations, amulti stage forming approach is normally employed. Inthis procedure several forming steps with intermediateannealing treatments are employed to return the sheetto a more ductile condition [2, 4].

Traditionally, the forming steps with inter-stage an-nealing and the die shape are defined using productionexperience and are improved by trial and error. Thisis a costly and time consuming way that may lead tosub-optimal solutions. Models of the stretch formingprocess are useful to achieve such an optimal processcontrol. The advantage of models is that they can beused before the tools are made, avoiding lengthy andexpensive trial and error runs for modifications of thetools and for reducing the number of expensive inter-mediate annealing steps. “Thorough” models, such asmodels based on the finite element method are neededto gain fundamental knowledge of the stretch formingprocess [5, 6]. Hence the goal of the finite elementmodel is to facilitate the optimization of the number

and possibly the duration of each intermediate anneal-ing step and therefore of the whole process. However,material models used in the finite element analysis ofstretch forming of aircraft skin parts with intermediateheat treatments are still mainly based on phenomeno-logical laws [6]. Hence, further improvement requiresthe use of a set of thermo-mechanical material modelsthat explicitly consider the effect of microstructure onplastic deformation, the static recovery and particlecoarsening due to precipitation during intermediateheat treatment [7]. These physical mechanisms shouldbe modeled up to a level that is required for macro-scopic process simulations.

Stretch forming characterization tests for AA 2024

The stretch forming process used for producing theaircraft skin part consists of several deformation stepswith intermediate annealing treatments at 340 ◦C. Theannealing treatments reduce the strain hardening of theprevious step. The material properties of AA 2024 arewell known for several tempering conditions, amongstothers the high strength conditions (T3, T4, T8 andtheir derivatives), and also the full-annealed condition[8]. However, the properties after intermediate an-nealing for AA 2024 are unknown. Only a schematicgraph of the influence of several ageing and overageingconditions are given by [8]. This graph shows that thestrength decreases due to overageing while the strainto fracture only increases upon strong overageing.

A number of tensile tests were performed at Na-tional Aerospace Laboratory (NLR) by Stork Fokkerand NLR, in order to understand the effect of in-termediate annealing steps on stretching. Tests wereperformed on AA 2024 Clad in T3 condition with athickness of 1.6 mm. Each sample was first annealed ata temperature of 340 ◦C for different annealing times,which is a common treatment to relieve stress. Afterthis treatment the samples were given several strainsteps with each strain step followed by an intermediateheat treatment. All the samples considered were loadedin the rolling direction with the strains applied in oneor two steps (2 %, 4 %, 6 %, and 2 × 2.5 %). Completedetails of these experimental results were published in[2, 9]. These results are used for validating and com-paring the implemented material models in this work.For brevity, the important findings are described here.In [2], it was found that, the first annealing treatmentapplied on T3 material causes a large decrease of themechanical properties. The ultimate tensile strengthdecreases from 440 MPa to 290 MPa, yield strengthdrops from 340 MPa to 170 MPa and fracture strain

Int J Mater Form (2011) 4:129–140 131

0

75

150

225

300

375

450

0 0.02 0.04 0.06 0.08

true

str

ess

(MPa

)

true strain (−)

T3 (as received)T3 + HT

T3 + HT +2, 4% ε + HT T3 + HT + 2.5% ε + HT + 2.5% ε + HT

Fig. 2 Experimental stress–strain curves with intermediate heattreatments (HT: heat treatment at 340 ◦C for 20 min.)

decreases from 17 % to 10 %. An annealing of 20 min.at 340 ◦C is sufficient to significantly annihilate the de-formation structure and recover the mechanical prop-erties, as well as during the first annealing step thanafter up to 6 % stretching. Consecutive strain steps andintermediate anneals slightly decrease the tensile andyield strengths. There is no influence of the materialbatch. A representative set of true stress–strain curvesof these experiments are presented in Fig. 2.

The initial material was in T3 condition i.e. it wassolutionized, cold deformed and aged at room tem-perature. Due to the natural ageing period, very smallprecipitates and/or GP zones are present. During thesuccessive annealing at 340 ◦C several phenomena canhappen: recovery, precipitation/coarsening of the parti-cles. The microstructure evolution during stretch form-ing with intermediate annealing of AA 2024 was in-vestigated by Teyssier and Miroux [7]. They focusedmainly on understanding the relationship between theevolution of the microstructure, effect of chemical com-position and the changing mechanical properties duringthe stretch forming with intermediate heat treatments.

Electrical resistivity measurements were performedto follow precipitation. The observed sharp decreaseof resistivity during the first annealing on T3 condi-tion (initial material) indicates a decrease of the solutecontent accompanying the nucleation and/or growth ofprecipitates. Optical and scanning electron microscopywere used to measure the grain size and the size anddensity of precipitates. From the micrographs, no ev-idence of recrystallization was found as there was nonucleation or growth of new grains [7]. Recovery andprecipitation kinetics were determined from the hard-ness measurements. X-ray diffraction analysis (XRD)was used to identify the different types of precipitatespresent during heat treatments.

The decrease of mechanical properties during thefirst annealing can be explained by the overageing, i.e.coarsening of precipitates, that follows the initial fastprecipitation. During the subsequent annealing treat-ments, no visible precipitate evolution was observedand only coarsening is assumed to continue playing arole.

Recovery may happen during the first annealingand more surely during the following annealing stepsto remove the work hardening introduced by stretchforming passes.

Work hardening model during stretching

To describe the deformation due to stretching, the mi-crostructural based Nes work hardening model is used,in which the evolution of microstructure is defined bythree internal state variables [10–12]. The Nes modelis based on a statistical analysis of athermal storageand dynamic recovery of network dislocations. The Nesmodel directly takes into account the chemical com-position such as the solute concentrations, grain size,volume fraction and size of the precipitates. Thus, theNes model is quite capable to describe the changingmaterial behavior during the ageing of heat treatablealloys. Extensive presentations of the work harden-ing model are given elsewhere [10–12], and only abrief overview of the analytical expressions constitutingthe core elements of the model are presented in thefollowing.

Microstructure evolution

In order to provide a realistic description of the mi-crostructure evolution the problems of athermal stor-age of dislocations and their annihilation by dynamicrecovery need to be solved. In the present multi pa-rameter microstructural description this implies thatthe evolution equations must be properly defined andsolved. On the basis of the treatments of [10–12], theseevolution equations can be written in compact form asfollows

dρi

dγ= 1

(1 + f (q2

b − 1))

2

b Leff− ρivρ

γ̇(1)

dδ

dγ= −2δ2ρi

κ0ϕ

SL2

Leff+ bvδ

γ̇(2)

dϕ

dγ= g (ρi, δ, ϕ) (3)

132 Int J Mater Form (2011) 4:129–140

Equations 1 and 2 represent the combined effect ofathermal storage of dislocations (first term on the right-hand side in these equations) and their dynamic recov-ery (second term).

Athermal storage of the dislocations In the storageterms, Leff is the effective slip length, the average radiusthe dislocation loops expand before being stored. Itis determined by interactions between the mobile andthe stored dislocations in the dislocation forest. If thereare no other barriers than the dislocations stored inthe Frank network this length scales with the spacingbetween the stored dislocations:

Lρ = C√ρ

(4)

Here C is a statistical parameter inversely proportionalto the chance for a dislocation to become stored inthe Frank network. Which depends on the solute con-tent. However, in most commercial aluminum alloys,other barriers will be present like grain boundaries andprecipitates. These barriers will also impede mobiledislocations and affect the athermal storage rate. Thepresence of grain boundaries and particles reduces theslip length and are taken into account by the followingrelation[

1

Leff

]2

=[

1

Lρ

]2

+[

1

Lp

]2

+[

1

LD

]2

(5)

with

Lp = r/2 fr

κ3, LD = D

κ2

Here κ2 and κ3 are constants accounting for geometricalshapes and the measuring methods applied and areexpected to be of the order of unity. Lp and LD rep-resent the slip length contributions due to precipitatesand grain size (D) respectively. In Eq. 2, the first termrepresents the athermal storage in the newly formedsubgrain boundaries and S is a cell/subboundary stor-age parameter, for more details see [11, 12].

Dynamic recovery The last terms in Eqs. 1 and 2represent, respectively, the dynamic recovery effectson the dislocation density inside the cell ρi, and thesubgrain size, δ. In other words, the dynamic recoveryis incorporated by analyzing the stability of the cellinterior dislocations ρi in terms of the recovery in Franknetwork and the subboundary structure in terms ofsubgrain growth.

In Eq. 1, the second term in principle defines the ratecontrolling annihilation frequency of the dislocations

inside the subgrain. For aluminum alloys, the annihi-lation frequency can be written in the following form:

vρ = ρiblρξρ BρνD exp

(−U∗

kT

)2 sinh

(VρGb

√ρi

kT

)(6)

where lρ ≈ bc−1sc ω−1, U∗ = Us + Us, and Vρ ∝ 2ξρb 3.

Here, lρ is an activation length, csc is the solute con-centration at the dislocation core, ω is a constant (≈ 1).Us is the activation energy for diffusion of solute atomsand Us is the interaction energy between the soluteatoms and the dislocation. For a detailed analysis ofdynamic recovery of subgrain interior dislocations, see[12, 13].

The last term in Eq. 2 contains a velocity term cor-responding to the growth rate for a given subgrain size.However, the subgrain growth phenomenon can occuronly when the deformation takes place at elevatedtemperatures. Thus it can be neglected in this work, asthe deformation takes place at room temperature. Fordetails see [11, 12].

The misorientation aspect When dislocations are con-sumed by the sharpening of old boundaries they arestored in a way that does not contribute to the hard-ening of the material. To handle this aspect in theNes model, the following empirical equation based onphysical observations is used:

dϕ

dγ=

[fII

b KII

ϕLeff+ (1 − fII) 0.02

][

1 −(

ϕ

ϕIV

)3]

(7)

Here fII must be equal to unity during small strainregime and vanish at larger strains. For more details,see [11, 12].

The flow stress

The flow stress, τ , at a known microstructure is com-monly defined in terms of a thermal component, τt,and an athermal stress component τa, so that τ = τt +τa. The thermal component τt is due to the rate andtemperature dependent interactions with short rangeobstacles and the athermal component characterizesthe rate and temperature independent interactions dueto long range barriers.

The thermal stress To calculate the thermal stress τt,it is assumed that the mobile dislocation density ρm isproportional to the total stored dislocation density ρ,i.e. ρm ∝ mρ. From the Orowan equation:

γ̇ = ρmbv (8)

Int J Mater Form (2011) 4:129–140 133

The average speed of the mobile dislocations, v, maygenerally be expressed as:

v = la BtνD exp

(− Ut

kT

)2 sinh

(Vtτt

kT

)(9)

where k is the Boltzmann constant, νD is the Debyefrequency, Vt is an activation volume and Ut is an acti-vation energy, Bt is a constant and T is the temperature.If the solute content is sufficiently large the averagedislocation speed is controlled by the dragging of ele-ments in solid solution. Then the relaxation distance,la = bcet−1

sc /ωt, is the distance traveled by a climbing jogin between thermal activation and subsequent pinningof a solute atom. For alloys, Ut is defined as summationof activation energy for solute diffusion Us and inter-action energy between solute and dislocations Us.The activation volume is given by the spacing of soluteatoms along the dislocation core, Vt = b 2ls = b 3ωtc

−etsc .

Where ls scales with the spacing of the solute atomsalong the dislocation core, ωt is a parameter whichneeds to be determined from experimental flow curvesand the exponent et is expected to have a size in therange 0.5 to 1.

The athermal stress The athermal component, τa isthe stress contribution due to precipitate particles (τp)and stored dislocations (τd) and are considered as longrange contributions.

τa = τp + τd (10)

In age hardening aluminum alloys, the strengtheningeffect mainly comes from precipitation hardening dueto shearing and/or bypassing of particles by disloca-tions. Shearing is possible only if the interface betweenparticle and matrix is coherent i.e. continuity of thecrystal structure but not necessarily the same inter-atomic distances. Bypass or Orowan mechanism occursif the interface between particle and matrix is incoher-ent [14]. Precipitates can also indirectly contribute tothe strengthening of alloys by influencing the storageof dislocations during microstructure evolution.

In this work, the contribution due to precipitates forthe strengthening is described according to [15]. Forthis calculation the particles are assumed to be sphericaland of equal size (i.e. monodispersive), equal to themean size. This means that the constituent equationswere derived on the basis of the classical analytical solu-tions for the dislocation-particle interaction. Accordingto Myhr et al. [15] the stress contribution due to a

population of precipitates with a radius r and volumefraction fr is given by:

τp = 2βGbr

(3 fr

2π

)1/2

min

[(rrc

)3/2

, 1

]

(11)

rc is the critical radius deciding between shearing andbypassing mechanisms. In the above equations, G isthe shear modulus of the aluminum matrix and β is aconstant close to 0.5 . The precipitate size and volumefractions (input parameters) are determined by fittingthe proportional limit calculated with the model ofMyhr et al. [15] to experimental measurements for thedifferent stretching and consecutive heat treatmentspresented in Fig. 2.

The unified relation for the stress contribution (τd),due to stored dislocations with statistical distribution ofsubgrain sizes in different stages of work hardening isgiven by:

τd = α1Gb[�1

(qc

δ√

ρi

)√ρi + �2

(qc

δ√

ρi

)qc

δ

]

+ α̂2Gb[�2 (0)

1

δ+ 1

D

] (12)

with

α̂2 = fscαsc2 + (1 − fsc)α2 (13)

where

αsc2 = α1 f (qb − 1)qc (14)

and

fsc =1 − �1

[qc

δ√

ρi

]

1 − �1 (1)(15)

Here the functions �1 and �2 are given as follows

�1 (x) =∫ x

0 x3f(x)dx∫ ∞

0 x3f(x)dx= 1 − exp (−5x)

7∑

n=0

(5x)n

n! (16)

and

�2 (x) =∫ ∞

x x2f(x)dx∫ ∞

0 x3f(x)dx= 5

7exp (−5x)

6∑

n=0

(5x)n

n! (17)

with the subgrain size distribution given by f(x) =55x4

24 exp(−5x) and x =(

qcδ√

ρi

).

In the model, it was assumed that the initial materialin T3 condition is fully recovered after the first heattreatment at 340 ◦C for 20 min. and the correspondingdislocation density ρ0 was chosen to be 1011 m−2. Soluteconcentrations come from the mass balance.

134 Int J Mater Form (2011) 4:129–140

0

75

150

225

300

375

450

0 0.02 0.04 0.06 0.08

true

str

ess

(MPa

)

true strain (−)

T3 (as received)

T3 + HT

T3 + HT + 2% ε + HT

Nes modelexperiment

Fig. 3 True stress–strain curves with intermediate heattreatments—experiments and model (HT: heat treatment at340 ◦C for 20 min.)

The sensitivity study of the Nes model in respect tothe model parameters shows that only two parametershave a significant influence on the stress–strain relation:

– qc controls the work hardening (storage of disloca-tion and subgrain formation and misorientation)

– ξρ controls the dynamic recovery.

Both parameters have roughly similar effect on thestress–strain curve so an infinite number of combi-nations of these parameters can fit the experimentalstress-strain curves. However, each combination gives adifferent result for the dislocation density and subgrainevolution. The qc and ξρ parameters are obtained byfitting the stress–strain curves on the material after firstheat treatment on initial material in T3 condition at

Table 1 Parameters of the Nes model for AA 2024 (T3 + HT)

νD 1.0 · 1013s−1 b 2.86 ·10−10mκ 1.3 · 10−23J/K qc 18.0C 40.0 f 0.1α1 0.3 α2 3.0κ0 3.5 m−1 Us 1.5 · 105J/molUsd 1.3 · 105J/mol Ut 1.5 · 105J/molκ2 2.0 κ3 0.7ϕc 0.3489 Bt 57.0ωt 20.0 et 0.54Bρ 0.23 ξρ 45.0ωs 1.0 eρ 0.67Bδ 100.0 ξδ 20.0eδ 0.65 ρ0 1.0 ·1011m−1

SIV 0.15 qIV 2.0ϕIV 0.052 rad m 0.8r 13.4 ·10−9m fr 0.013β 0.36 rc 5.0 ·10−9mc 0.0042

340 ◦C for 20 min. giving the most realistic dislocationdensity and subgrain size. The fit of the strain–stresscurve is not perfect for all strains, which is clearly visiblein Fig. 3. All the other parameters have negligibleinfluence on work hardening when they are variedwithin their range of possible values. Default valuesare used based on literature and the selected and fittedparameters of the Nes model are presented in Table 1.With these parameters, prediction of the stress–straincurves for the other conditions are reasonably good(Fig. 3).

Material model during heat treatment

To describe the particle coarsening and static recoveryduring intermediate heat treatments, simple equationsbased on physical observations are used.

Isothermal ageing model

During age hardening, Guinier Preston (GP) zonesnucleate and grow from the quenched solid solutionand deplete it. During the nucleation and growth phase,both the radius of the precipitates and their vol-ume fraction are variable. When the solubility limitis reached, precipitates coarsen by competitive growthand in the model the radius of the precipitates isconsidered as a variable and the volume fraction ofthe precipitates becomes constant. It was found in theexperiments of Teyssier and Miroux [7] that nucleationand growth of GP zones is observed mainly duringthe first 2 min. of initial heat treatment at 340 ◦C for20 min. on T3 condition and no or very little nucle-ation of GP zones were observed in the consecutiveheat treatments. Hence, ageing is controlled by thecoarsening kinetics and the particle coarsening effectsare considered by a relatively simple semi-empiricalexpression. Coarsening is described here by the growthof the mean particle radius r of assumed spherical shapewith time t. No attempt has been made at present toinclude changes in particle shape or in the distributionof particle size during coarsening, though it is acknowl-edged that these can be important effects. According toRylands et al. [16], the size evolution in each annealingsequence is:

r = (rn

i + kt)1/n (18)

with

k = k0

Texp

(− Q

RT

)(19)

Int J Mater Form (2011) 4:129–140 135

Table 2 Volume fraction and size of the particles calculated fordifferent stretching and consecutive heat treatments

Condition Vol. frac (%) Size (m)

T3 1.3 5.0 · 10−9

T3+HT 1.3 13.4 · 10−9

T3+HT+2 to 6% stretch+HT 1.3 16.6 · 10−9

where Q is the activation energy for volume diffu-sion of Cu atoms between particles which is equalto 136 kJ/mol, R is the universal gas constant, t isthe process annealing time and the bulk diffusioncoefficient n = 3 is considered in the simulations. k0 isthen obtained by fitting the predicted particle radiuspresented in Table 2 and equal to 4.5 · 10−13K·m3/s.

Static recovery model

The term “recovery” refers to the restoration of mater-ial properties back to their original or undeformed stateafter deformation and prior to recrystallization [17].Static recovery is specifically attributed to the processthat occurs after deformation whereas dynamic recov-ery takes place during the course of deformation. It iswell known that recovery of most metals is primarilydue to changes in the dislocation structure [18]. Dislo-cation recovery is not a simple microstructural change,but is constituted by a series of events: dislocationtangling immediately after deformation, dislocation cellformation, annihilation of dislocations within cells, sub-grain formation and subgrain growth [13, 18].

In this work, an empirical approach based onphysical observations of recovery kinetics based onHumphreys type 2 model [18] is used. According toHumphreys, the total dislocation density evolution dueto recovery kinetics is given by:

ρ =[√

ρ0 + (√ρd − √

ρ0)(1 + (q − 1)Bt)

(− 1

q−1

)]2

(20)

where ρ0 is the total dislocation density of fully softmaterial, ρd is the dislocation density of deformedconfiguration just before annealing and t is the anneal-ing time. The recovery kinetics parameters B and qare material and temperature dependent and fitted forthe temperature of interest. Usually the exponent q isfound to be in the range of 3 to 5; and q = 4 is chosenin this work.

When deforming the material after annealing, it isexpected that the deformation of the recovered struc-ture starts with stage II. Therefore the in-cell dislo-cation density during recovery is related to the total

dislocation density using the same scaling relation asduring stage II:

ρi = ρ

f(q2

b − 1) + 1

(21)

The fully soft dislocation density is taken as ρ0 =1011m−2 According to the experimental results, 20 min.annealing at 340 ◦C is supposed to be enough to obtaina fully recovered microstructure. The parameter B isthen chosen in such a way that ρi gets close to ρ0 after20 min. annealing at 340 ◦C. It is found that after a sharpdecrease, ρi decreases only slowly with time, whateverthe value of B. A value of B = 0.1s−1 is chosen asa compromise between the value of ρi after 20 min.annealing at 340 ◦C and the rate of decrease of ρi at thebeginning of annealing.

The yield criterion

To describe the anisotropy and biaxial behavior ofthe sheet, the Vegter yield function [20] is used. TheVegter yield function is one of the most accurate yieldfunctions for aluminum alloy sheets defined in planestress condition. The Vegter yield locus is constructedin principal stress space and uses Bezier interpolationto connect the measured yield stresses in equi-biaxial,plane strain tension, uniaxial tensile and pure sheartests as shown in Fig. 4. This leads to 4 stress pointsin the region where σ1 > σ2. For the situations whereσ1 < σ2 can be determined if the sample is rotated by90 ◦. The compressive part of the yield locus is usuallynot measured from the experiments, but is defined byassuming that the initial yield stress in compression isequal to that in tension.

σ 1

σ 2σ eb

σ ps

σ un

shσ

Fig. 4 Basic stress points required to construct the Vegter yieldlocus

136 Int J Mater Form (2011) 4:129–140

Table 3 Measured stress and strain ratios for the AA 2024-Osheet (Vegter yield locus data)

Angle with RD 0◦ 45◦ 90◦

(σ1/σRD1 )un 1.0 0.98 0.99

fps 1.14 1.1 1.09fsh 0.576 0.579 0.577fbi 0.96R-value 0.52 0.67 0.52(ε2/ε1)bi 1.0635

Anisotropy of the sheet is captured by using the flowstresses of the experiments under different angles withrespect to the rolling direction. The flow stresses onintermediate angles are defined by a harmonic interpo-lation function.

In the implementation of this model for an orthog-onal material, experiments in 3 directions (0◦, 45◦ and90◦) are required to determine the 14 material parame-ters. The parameters for the investigated AA 2024 alloyare given in Table 3.

Stretch forming of aircraft skin experiments

A number of sheets have been stretched using manuallydetermined trajectory of the grippers. The dimensionsof the sheet are 1130 ×1920 ×3.5 mm. Five intermediateheat treatments are used. To measure the strain inthe final part a grid of dots with known dimensionswas applied to the undeformed sheet. Due to the largedimensions of the sheet only part of the sheet which isknown to deform most was measured. The final part isshown in Fig. 5.

The PHASTTM strain measurement system [21] hasbeen used to measure the strains after stretch forming.This system is based on 3D image processing. Digitalphotos are taken from different positions of the product

Fig. 5 Skin parts after stretch forming

Fig. 6 Parts with recognized grid points and beacons

and some beacons. The strains are calculated fromthe recognized grid points and beacons (Fig. 6) usingphotogrammetry.

A correction in the thickness calculation must beconsidered as it was calculated from the strains mea-sured on the outer plane of the sheet instead of themidplane. In this case the thickness is corrected up to0.1 mm, as was confirmed by direct thickness measure-ments at the edges.

Finite element simulations

The initial shape of the sheet and tools is shown inFig. 7. As the geometry of the die is symmetric andtherefore the geometry of the final product is also sym-metric only half of the sheet is modeled. The part of thesheet that is clamped between the gripper jaws is notmodeled, as ideal clamping is assumed. The prescribeddisplacement boundary conditions are applied to the

Fig. 7 Initial Sinusoidal shape of sheet and tools

Int J Mater Form (2011) 4:129–140 137

remaining edges of the sheet in the gripper jaws. Thecheeks (part of the grippers which are making contactwith the product) are modeled as cylinders. Both cheeknodes and prescribed sheet nodes are given the sameprescribed displacements.

It can be seen in Fig. 7 that the initial shape of thesheet is curved, which agrees with the industrial prac-tice as the sheet is slightly bent before clamping it intothe machine. The initial curvature is modeled by takingthe initial shape of the sheet equal to a shallow cosine.This shape is equal to the shape of the buckle mode ofa compressed sheet. In this example the edges of thesheet are moved only 15 mm inwards. It is assumed thatthis bend causes no stresses and plastic deformationsin the sheet. By modeling this initial cosine shape itis ensured that the sheet deforms in the correct up-ward direction, when the sheet is draped around thedie in the first part of the stretch process. This alsoavoids numerical instabilities during the finite elementsimulation.

The cheeks and die are assumed to be rigid bodies,i.e. they are undeformable with respect to the sheet. Apenalty method is used to describe the contact betweenthe rigid tools and the sheet. The friction between thesheet and rigid tools (die and cheeks) is described withCoulomb’s friction model with a friction coefficient of0.1 .

The sheet has been meshed with 2810 triangularelements with a refinement close to the symmetry line.The size of the element edges ranges from 20 to 40 mm.The used discrete Kirchhoff shell elements [19], takemembrane as well as bending stiffness into account.

The used trajectory of the grippers is shown in Fig. 8.It is a piecewise linear approximation of the trajectoryused for the experiments. The rotations of the grippers

800

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0 100 200 300 400 500 600 700 800 900

z-co

ord

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circleInitial shape

TrajectoryHeat treatment

Fig. 8 Trajectory of the grippers used in the simulation

have been neglected, as they are relatively small. Thepositions at which an intermediate heat treatment isgiven to the sheet are marked.

Two different hardening models: a phenomenolog-ical power law model and the physically based Nesmodel are used for the simulations. The power law pa-rameters are determined from the experimental stress–strain curves presented in Fig. 2.

In the complete stretch forming process on a saddleshaped skin part, the sheet was annealed five times. Inthe simulation the annealing step is included by usingthe simple models for particle coarsening and staticrecovery presented in the previous section. During theheat treatment, the new values for the precipitate sizeand total dislocation density for a given annealing tem-perature and time are calculated. Further deformationsteps are continued using the physically based Nes workhardening model with the calculated values of precip-itate size and dislocation density at every integration

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knes

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knes

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(b) without heat treatment

Fig. 9 Thickness along symmetry line with and without heattreatments (HT)

138 Int J Mater Form (2011) 4:129–140

point of the DKT elements. In the case of the powerlaw model, the annealing step is included only phenom-enologically by resetting the (equivalent) plastic strainto zero, thus deleting complete work hardening in allintegration points.

In Fig. 9 the thickness along the symmetry line ispresented after all stretching phases with and withoutheat treatments for the Nes and Nadai models. Withheat treatments, the deformations are stable and theyare almost completely kinematically determined. Thephysically based Nes model gives better thickness pre-diction, particularly at the critical region of the partas shown in Fig. 10, than the phenomenological Nadaimodel (see Fig. 9a). The die force–gripper displacementdiagram in Fig. 11a shows a large drop in the die forcebecause the stresses are set to zero after the heat treat-ment. For the Nes model, the force does not completelydrop to zero because the work hardening effect is notcompletely deleted and residual stresses are left in thematerial.

Figure 11b shows that without intermediate heattreatments steps, the die force is higher as there is noremoval of work hardening. Correspondingly withoutheat treatments, more thinning is observed in the crit-ical region of the part shown in Fig. 9b. This thinningis more severe with the Nes model. Apart from theheat treatments, a drop in vertical force on the die, forexample for the gripper displacement between 500 mm

2.493

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3.383

Fig. 10 The predicted thickness in the final product with heattreatments using Nes model

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Fig. 11 Vertical force on die with and without heat treatments(HT)

to 600 mm corresponds to loss of contact between thedie and the sheet.

In Fig. 12, the effect of different approaches for heattreatments are studied. In this study, simulations areperformed with a physics based approach for heat treat-ments by considering the particle coarsening and staticrecovery and a phenomenological approach i.e. assum-ing the work hardening effect is completely removedby resetting plastic strains to zero. Simulations are alsoperformed without intermediate heat treatments. In allthe three situations, the work hardening effect is de-scribed using the physically based Nes model. From Fig.12a, it is clearly seen that the force–displacement curveis more stable with the physically based approach forheat treatments than the phenomenological approachas it shows jiggles in the force–displacement curve.With the phenomenological approach, after every in-termediate heat treatment the vertical force on the die

Int J Mater Form (2011) 4:129–140 139

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No heat treatment

(a) Vertical force on die

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HT: phenomenological approachNo heat treatment

(b) Thickness along symmetry line

Fig. 12 Influence of heat treatment on predicted vertical forceon die and thickness along symmetry line

almost drops to zero as the stresses are completelyremoved. However, this is not completely realistic.From the thickness along the symmetry line plot shownin Fig. 12b it follows that with the phenomenologicalapproach the thinning effect in the critical region isunderestimated and the physically based approach ismuch better. Without heat treatments too much thin-ning effect is observed.

Conclusions

The main objective of this work was to develop anduse a suitable numerical model for the finite elementanalysis of the stretch forming of aircraft skin parts withintermediate heat treatments. An important part of thenumerical model is the modeling of the material behav-ior. Both the hardening behavior during the stretching

process, and the reduction of work hardening behaviorduring the heat treatment process are considered. Twohardening models are compared: a phenomenologicalNadai hardening model and a physically based modelaccording to Nes. The reduction in work hardeningduring heat treatment is modeled by considering theunderlying physics such as the overageing model forparticle coarsening and the static recovery model. Aphenomenological approach is also used for compar-ison by deleting all work hardening effect from theprevious stretching phases.

From the presented simulations, it can be concludedthat the physics based material modeling gives betterresults. The predicted thickness distribution especiallyin the critical region of the part using the Nes model forwork hardening and the overageing and static recoverymodels for intermediate annealing gives better agree-ment with the experiments.

In this work the material models are compared onlywith uniaxial tensile experiments and the predictedthickness distribution are used for the validation. Inorder to have more confidence on the implementedmaterial models, further validation of FEM results suchas the vertical force on die and the strain distributionwould be useful. There is also room for improvementin the material modeling as in the current models,the effect of particle coarsening and static recoveryare treated separately. In reality, the effect of particlecoarsening and recovery on the reduction of work hard-ening during heat treatments is coupled.

Acknowledgements This research was carried out under theproject number MC1.02106 in the framework of the ResearchProgram of Materials innovation institute M2i (http://www.m2i.nl), the former Netherlands Institute for Metals Research.

Open Access This article is distributed under the terms of theCreative Commons Attribution Noncommercial License whichpermits any noncommercial use, distribution, and reproductionin any medium, provided the original author(s) and source arecredited.

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