1-s2.0-S0045794908001703-main

Computers and Structures 87 (2009) 128–139

Contents lists available at ScienceDirect

Computers and Structures

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

Estimation of forming limit diagrams by the use of the finite element methodand Monte Carlo simulation

Ø. Fyllingen a,*, O.S. Hopperstad a, O.-G. Lademo a,b, M. Langseth a

a Structural Impact Laboratory (SIMLab, Centre for Research-based Innovation), Faculty of Engineering Science and Technology, Norwegian University of Science andTechnology, 7491 Trondheim, Norwayb SINTEF Materials and Chemistry, Department of Applied Mechanics and Corrosion, Richard Birkelands vei 2, 7465 Trondheim, Norway

a r t i c l e i n f o a b s t r a c t

Article history:Received 1 November 2007Accepted 22 July 2008Available online 7 September 2008

Keywords:FormabilityFinite elementsProbabilityShellsAnisotropic material

0045-7949/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.compstruc.2008.07.002

* Corresponding author. Tel.: +47 73 59 47 00; fax:E-mail address: [email protected] (Ø. Fylling

A stochastic finite element-based approach for forming limit calculations of sheets is proposed and eval-uated. Material inhomogeneities are represented by spatial thickness variations of the sheet which aremodelled by the use of random fields. The effects of changing the smoothness, wavelengths, amplitudeand anisotropy of the field realisations on the forming limit diagram are investigated. Further, the effectsof the patch size and plastic anisotropy on the forming limit diagram are studied. The assumed thicknessvariations result in a quite wide scatter band, and changes of the characteristics of the thickness fieldresult in changes of the shape and variance of the forming limit diagram.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Sheet metal forming operations are widely used in severalindustries. One of the main limitations of such processes is thedevelopment of a localised neck. As the strains increase up to acritical limit the sheet necks and eventually fail in localisation.Combinations of all critical major and minor strains give rise to aforming limit diagram (FLD). A substantial amount of experimentaland theoretical work has been done in order to determine forminglimit diagrams.

Theoretical calculations for the limiting strains for the diffusenecking condition have been derived by Swift [1], assuming thatplastic instability occurs at a maximum load for proportional load-ing. Hill [2] first described localised necking in thin sheets along adirection of zero extension. Hill’s criterion does not allow for local-ised necking for the material subjected to biaxial stretching. In or-der to predict the FLD for biaxial stretching of the material withsmooth yield surfaces and associated plastic flow, inhomogeneitieshave to be introduced. The first analytical model, which could pre-dict localised necking in biaxial stretching, was introduced byMarciniak and Kuczynski [3]. By increasing the strains of a plate,with a groove of infinite length perpendicular to the largest princi-pal stress and strain, localisation initiates in the groove above acritical strain state. The geometrical defect represents a hypothe-sised initial local weakness. By the use of the finite element meth-

ll rights reserved.

+47 73 59 47 01.en).

od, other geometrical defects may be introduced. Bate and Wilson[4] examined sheets with circular regions of reduced thickness andlater Neglo et al. [5] studied elliptical-shaped imperfections.

The material parameters controlling the plastic deformationmay be considered at different scales. At macroscale, the constitu-tive parameters are derived from the overall material behaviourssuch as strain hardening, strain-rate hardening and anisotropy,and at microscale the structural parameters are physically deter-mined from an understanding of the different mechanisms in-volved in plastic deformation: roughness, grain size, texture,dislocations microstructure, inclusions content, internal damageetc. The two scales are strongly related, but still further investiga-tion is needed in order to obtain the parameter on macroscaleaccurately from the parameters on microscale. In the present worka macroscale approach is considered.

Marciniak and Kuczynski [3] showed that the predicted forminglimit was strongly influenced by the strain hardening and normalanisotropy as well as the size of the inhomogeneity. The Marci-niak–Kuczynski approach (M–K approach) has been widely usedto predict the forming limits of sheets subjected to biaxial stretch-ing. Barlat et al. [6] and Barlat and Jalinier [7] relate the reducedthickness in the groove to the distribution of voids in the material.Barata da Rocha et al. [8] used the Marciniak–Kuczynski approachto predict FLDs of anisotropic sheets subjected to linear and non-linear loading. They showed that for an in-plane anisotropic mate-rial, the critical orientation of the groove may not be perpendicularto the largest principal strain. Further on, the shape of the FLD isstrongly influenced by plastic anisotropy. Several other recent

Ø. Fyllingen et al. / Computers and Structures 87 (2009) 128–139 129

studies using the M–K approach with various constitutive equa-tions and for the proportional and non-proportional strain pathshave been performed (e.g. [9–17]). A review of the theoreticalmodels of the strain-based FLD and their relevance to the stress-based FLD has been performed by Stoughton and Zhu [18]. Further,Banabic et al. [19] have made a comparison between differentmodelling approaches to predict the FLD for sheet metal forming.

Variation in the critical strains has been observed when per-forming several stretching experiments under similar conditions.Van Minh et al. [20] studied such variations for four different gradesof mild steel, and developed a probabilistic model of limit strains insheet metals [21]. They claim that ‘‘the variation in forming limits ismuch greater than that due to the experimental error” and it is pro-posed that this scatter reflects an intrinsic property of the materialwhich is important in determining material formability. Two recentstudies considering the statistical variation of the forming limit dia-gram have been performed by Janssens et al. [22] and Strano andColosimo [23]. Banabic et al. [24] have developed a Marciniak–Kuczynski approach to predict the forming limit band. They deter-mine the forming limit band by taking into account the variation ofthe parameters of the yield locus and the hardening rule. Narasim-han et al. [25] used Monte Carlo simulation and finite elementmethods to predict the scatter band in forming limit strains. Theyconsidered a patch of material with random thickness distribution.A similar approach has been proposed by Lademo et al. [26]. Here, asquare patch of material subjected to linear strain paths is simu-lated the by use of the finite element method. The thicknesses atthe nodes are independent and normal distributed. As the strain in-creases, localisation is detected when the negative strain incrementin thickness strain in one element is much larger than the averagenegative strain increment in the thickness direction of the wholepatch. Lademo et al. [27] used the method to predict FLDs for differ-ent anisotropic aluminium alloys. The shapes of the predicted FLDsare similar to the experimental FLDs.

A further development of the approach developed by Lademoet al. [26] is investigated in this study. A square patch of material,modelled by the use of the finite element method, is subjected to aset of proportional strain paths within the right-hand side (i.e. firstquadrant) of the FLD. The material inhomogeneities are repre-sented by spatial thickness variations. Hence, the weaker zonesare thinner and the stronger zones are thicker. The thickness vari-ations are represented by the use of isotropic Gaussian randomfields with a Matérn covariance function. By changing the threeparameters of the Matérn covariance function, the smoothness,wavelengths and amplitude of the field realisations vary. If thematerial contains defects with large aspect ratios orientated in a

1

2

aba

tb ta

a

Fig. 1. (a) Marciniak and Kuczynski two z

certain direction, these may be represented by anisotropic fields.Different realisations lead to different forming limits. By the useof Monte Carlo analysis, a stochastic forming limit diagram is con-structed. The effects of changing the parameters of the covariancefunction and the degree of anisotropy in the inhomogeneity fieldare studied. The adopted material model accounts for plasticanisotropy and a comparison between predictions obtained for iso-tropic and anisotropic materials is carried out.

The advantage of FEM-based models compared to analyticalmodels is that physical inhomogeneities of any kind and generalconstitutive models, loading modes and loading paths can bestraightforwardly represented. Further, the proposed model is di-rectly applicable to prediction of plastic instability in componentsand structures subjected to forming operations or crash events.

2. Model

As emphasized in the introduction, Marciniak and Kuczynski [3]introduced the first analytical model which was able to predict theforming limits for biaxially stretched material. They assumed thatthe inhomogeneities in the material, such as impurities, varyingtexture, different size and orientation of the grains, can be repre-sented by an equivalent geometrical inhomogeneity. The shapeof the inhomogeneity is a groove oriented perpendicular to the axisof the largest principal stress and strain, see Fig. 1a. The ratio be-tween the thickness outside the groove ta and inside the groovetb is defined as the inhomogeneity factor f = tb/ta. Region a is sub-jected to a biaxial stress state with a constant ratio of strain incre-ments qa = Dea2/Dea1. As the deformation progresses, the negativestrain increment in the thickness direction is larger inside thegroove than outside the groove in order to satisfy the force equilib-rium across the groove. The limiting strains are achieved whenlocalisation takes place, which may be determined numericallywhen the ratio b = Deb3/Dea3 exceeds a critical limit. By repeatingthe procedure along several linear strain paths, an FLD may beconstructed.

Lademo et al. [26] adopt the ideas introduced by Marciniak andKuczynski [3] to the finite element method environment. They con-sider a square patch of material, as depicted in Fig. 1b discretizedinto a grid of nel � nel square-shaped membrane elements. The ini-tial width of the patch is w0, and the initial thickness is described bythe field t0(x,y). It is assumed that the plate is thin such that theassumption of plane stress can be used. The advantage of a finiteelement model is that there are few restrictions with respect tothe thickness field. The patch is subjected to proportional straining.In order to obtain proportional straining, the patch is constrained at

w0

ux

uy

w0

1 2

34

x

y

b

one model, (b) finite element model.

sin(θ)ε

cos(θ)ε

θ

εx

ε

εy

εtot

t

ε

ttot

a b

Fig. 2. (a) Linear strain path, (b) loading curve.

130 Ø. Fyllingen et al. / Computers and Structures 87 (2009) 128–139

sides 1–2 and 1–4 and subjected to displacement ux and uy at sides2–3 and 3–4, respectively. The relations between the displacementsand strains are

ex ¼ cosðhÞe ¼ ln 1þ ux

w0

� �; ey ¼ sinðhÞe ¼ ln 1þ uy

w0

� �ð1Þ

where e ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie2

x þ e2y

qand h is the angle between the x-axis and the

direction of straining, see Fig. 2a. By rewriting Eq. (1), the displace-ments for a linear strain path may be found as

ux ¼ w0ðexpðcosðhÞeÞ � 1Þ; uy ¼ w0ðexpðsinðhÞeÞ � 1Þ ð2Þ

The simulations are performed by the explicit solver in the FE-codeLS-DYNA. In order to run the quasi-static analysis in the explicit sol-ver quite fast, the following straining function is used:

e ¼ etot 1� cospt

2ttot

� �� �ð3Þ

where t is the current time, ttot is the termination time and etot is thestrain e at termination. A plot of the function is demonstrated inFig. 2b. The displacements as function of time are found by insertingEq. (3) into Eq. (2).

During the simulation of biaxial stretching, the strain gradientswithin the sample will grow and eventually produce large localisedstrains in weak regions corresponding to the formation of a local-ised neck. In order to predict localisation a failure criterion isneeded. The negative strain increment in the thickness directionis much larger within the neck than outside it when localisationoccurs. The average strain increment in the thickness directionfor the whole patch, DeX

z , can be found numerically by

DeXz ¼

1N

XN

i¼1

Dez;i ð4Þ

where Dez,i is the strain increment in the thickness direction for ele-ment number i and N is the total number of elements in the domainX, here chosen as the whole patch of elements. It has been assumedthat the area of each element is approximately equal before locali-sation occurs. The ratio between the element strain increment Dez,i

for element i and the average strain increment DeXz in the thickness

direction is called bi and defined by

bi ¼Dez;i

DeXz

ð5Þ

Localised necking is assumed to occur at a time tcr if any bi exceeds acritical value bcr at tcr and during nit consecutive time steps. Byinserting the time at failure tcr into Eq. (3) and making use of Eq.(1) the forming limit strains of the patch may be found for the ac-

tual path. These limit strains include both the elastic and plasticparts of the strain components. If the procedure is repeated for sev-eral different h, an FLD can be constructed. The failure criterion hasbeen implemented into a user-defined material model in the FE-code LS-DYNA by Berstad et al. [28]. In the material model the yieldcondition proposed by Barlat et al. [29] (Yld2000) and later by Aretz[30] (Yld2003) has been adopted:

f ðr̂;RÞ ¼ 12ðjr̂01j

m þ jr̂02jm þ jr̂001 � r̂002j

m� �1

m

� ðr0 þ RÞ 6 0 ð6Þ

where

r̂01r̂02

� �¼ a8r̂x þ a1r̂y

2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2r̂x � a3r̂y

2

� �2

þ a24r̂2

xy

s

r̂001r̂002

� �¼ r̂x þ r̂y

2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia5r̂x � a6r̂y

2

� �2

þ a27r̂2

xy

s ð7Þ

Here ðr̂x; r̂xy; r̂yÞ are the stress-components referred to the princi-pal axes of anisotropy, (a1,a2, . . . ,a8,m) are material constants forplastic anisotropy and shape of the yield surface, and r0 is the yieldstress. This yield function has been developed for anisotropic sheetmetals, and in particular aluminium alloy sheets. The isotropicstrain hardening variable R is defined by

Rð�eÞ ¼X2

i¼1

Q ið1� expð�Ci�eÞÞ ð8Þ

where �e is the equivalent plastic strain and (Q1,C1,Q2,C2) are workhardening parameters. The associated flow rule is assumed. A de-tailed description of the material model is given by Reyes et al. [31].

The width of the localisation band that develops in sheet metalsis typical in the order of the thickness. For shell elements the widthof the localisation band is independent of the thickness and is typ-ically equal to the width of the elements. Hence, the localisation isvery mesh dependent. In order to avoid the mesh dependency insoftening materials a non-local approach was originally proposedby Pijaudier-Cabot and Bazant [32]. In non-local failure theories,the failure criterion depends on the material within a radius ofinfluence which surrounds the integration point. Wang et al. [33]applied a non-local approach with respect to the thinning, wherethe strain increment in the thickness direction in one element iscalculated as a weighted average of the thickness increments ofelements within a radius r from the centre of the correspondingelement. This radius is typically of the order of half the thicknessof the plate. For the mentioned material model, this is done byuse of the *MAT_NONLOCAL card in LS-DYNA. The thickness

Ø. Fyllingen et al. / Computers and Structures 87 (2009) 128–139 131

increments of the elements within a radius r will be equallyweighted here.

Lademo et al. [26] assumes that the thicknesses of the (nel + 1)2

nodes are independent and normal distributed with mean value, l,and variance, s2. They found that typical values of the coefficient ofvariation CoVt = s/l are in the range 0.001 6 CoVt 6 0.005. A weak-ness with this method is that the thickness variation will dependon the number of nodes. Hence, a mesh refinement will lead to adifferent field. In this study, this problem is avoided by describingthe characteristics of the thickness field independently of the finiteelement mesh.

3. Representation of inhomogeneities and Monte Carlosimulation

Van Minh et al. [21] states that ‘‘. . . if non-uniformities exist inthe sheet, it may be expected that these will differ in severity frompoint to point and may be described in terms of a population hav-ing some statistically described distribution. Non-uniformitiescould arise from many different sources such as voids, inclusions,second phase particles and grain boundary effects as well as geo-metric factors, surface defects and local variations in plastic prop-erties.” The void volume fraction of dual phase steel has beenanalysed by Barlat and Jalinier [7], who state that ‘‘The microscop-ical observations carried out for a large number of sites showimportant differences in the volume fraction of voids at the scaleof a few centimetres. In these different places, the size of the cav-ities is the same, only their number is different”. From the mea-surements they found an equivalent groove-shaped defect andused the Marciniak–Kuczynski model to predict the FLD.

On the basis of the abovementioned statements, it seems likethe defects can be expected to vary spatially. In this study it isinvestigated how different randomly varying inhomogeneities willeffect the variation and shape of FLDs. A randomly varying field iscalled a random field or stochastic process in statistics. By analys-ing detailed measurements of inhomogeneities it could be possibleto find the characteristics of such fields. A method for the calcula-tion of the FLDs is proposed, and it is assumed that the inhomoge-neities can be represented by random fields. The effect on how theFLD changes by varying the characteristics of the random field isinvestigated. The thickness field1 T0(x,y) is partitioned into a meanterm l(x,y) and a residual term Z(x,y):

T0ðx; yÞ ¼ lðx; yÞ þ Zðx; yÞ ð9Þ

The mean term is assumed to correspond to global variations, whilethe residual term captures the local behaviour. For the patches ana-lysed in this study, the mean term is constant, l(x,y) = l and theresidual term is assumed to be a stationary zero mean Gaussianrandom field. From a theoretical point of view, the Gaussian distri-bution is not appropriate to represent thicknesses, since it is possi-ble to have samples with values from minus infinity to infinity. Inorder to avoid values far away from the mean the distributionmay be truncated. However, in this study the distribution was nottruncated since the probability of having samples far away fromthe mean is extremely low.

There exist several covariance functions for common paramet-ric isotropic models. Two of the commonly used two-parametercovariance functions are the Exponential and Gaussian covariancefunctions. Both functions are special cases of the Matérn covari-ance function. It was chosen to use the Matérn covariance functionsince it is quite adaptable

1 Notation: Capital letter is used to denote a random variable or field, while smallletter is used for their realisations.

CðhÞ ¼s2

2m�1CðmÞ ðh=RÞmKmðh=RÞ; h > 0

s2; otherwise

(ð10Þ

where Km and C are the modified Bessel function of order one andthe Gamma function, respectively. s2 is called the sill or varianceparameter, m is the smoothness parameter and R is referred to asthe range parameter. Because the covariance function reaches itsultimate value asymptotically, it is common to define an effectiverange. A common definition of the effective range is the distanceh0 at which the correlation, q(h) = C(h)/s2, has dropped to 0.05[35]. For a given set of h0 and m, the range parameter R may be foundnumerically. In Fig. 3a the Matérn correlation function is shown fordifferent values of m, while h0 = 20.0 mm. The effect of changing h0 isdepicted in Fig. 3b, while m = 5.0.

In order to generate anisotropic fields, coordinate transforma-tion is applied. A direction of maximum range and an anisotropyratio / has to be chosen. / is defined as the ratio between maxi-mum and minimum ranges [36]. The coordinates are first trans-formed to the isotropic space, simulation is done and then thecoordinates are back-transformed.

There exist several codes for the generation of random fields. Oneof these is the code R [37] with the package geoR [36]. In this studythe function grf(�) in geoR is used. This function generates simulationsof Gaussian random fields for the given covariance parameters.

Now, some samples of T0(x,y) will be given. It was chosen to usea mean thickness l equal to 2.0 mm. Lademo et al. [27] found thata coefficient of variation about 0.005 worked well on aluminiumalloy. Hence, in the demonstration the sill s2 will be equal to0.012 mm2. The field is generated on a grid of 41 � 41 nodesequally spaced on (x,y) 2 [(0.0 mm,100.0 mm), (0.0 mm,100.0mm)]. In Fig. 4, the effective range is set equal to 20 mm, whilethe smoothness parameter is 2.0 in (a) and 10.0 in (b). In Fig. 5,the smoothness parameter is kept constant at 5.0, while the effec-tive range is 10 mm in (a) and 30 mm in (b). These figures demon-strate that the effect of increasing the effective range h0 is longerwavelengths and the increase of the smoothness m leads tosmoother samples.

The simulations of the FLDs are done by the use of Monte Carlosimulation. In Monte Carlo analysis, n samples of the input vari-ables are randomly drawn from the given probability density func-tions. The n simulations are run and statistical properties of theoutput variables may be estimated. Here, the random input is thethickness field, which is represented by the use of a random fieldwith defined characteristics. The output variables that are to becharacterised are the strains at the time of localisation. It is desir-able to estimate an interval, with a given level of confidence, atwhich a certain percentage of all observations are expected to fallwithin. This is called a tolerance interval. If the output variablesseem to follow a known distribution, as the normal distribution,such an interval may be constructed quite easily. The size of theinterval decreases as the number n increases. An example is givenassuming that a variable x is normal distributed with unknownmean and variance. The estimated mean and variance are �x ands2, respectively. Then a tolerance interval is given byð�x� ks; �xþ ksÞ, where k is determined so that one can assert with100(1 � c)% confidence that the given limits contain at least theproportion 1 � a of the measurements [38]. The parameter c isthe probability of selecting a random sample that will producean interval containing less than the proportion 1 � a of the mea-surements. Some values of k for different n is listed in Table 1,where 1 � c = 0.99 and 1 � a = 0.99. From the table, it may be ob-served that k decreases quite rapidly as n increases for low valuesof n, while k decreases quite slowly for large values of n. If the var-iable in concern seems to follow any other known distribution,similar tables might be constructed.

0 5 10 15 20

0.2

0.4

0.6

0.8

1.0

h [mm]

ρ(h)

v = 2.0v = 5.0v =10.0

0 5 10 15 20 25 30

0.0

0.2

0.4

0.6

0.8

1.0

h [mm]

ρ(h)

h0 = 10 mmh0 = 20 mmh0 = 30 mm

a b

Fig. 3. Plot of correlation function: (a) varying smoothness; (b) varying effective range.

x

0

50

100

y

0

50

100

t

1.98

2.00

2.02

x

0

50

100

y

0

50

100

t

1.98

2.00

2.02

a b

Fig. 4. Samples of T0(x,y): (a) h0 = 20 mm, m = 2.0, (b) h0 = 20 mm, m = 10.0.

x

0

50

100

y

0

50

100

t

1.98

2.00

2.02

x

0

50

100

y

0

50

100

t

1.98

2.00

2.02

a b

Fig. 5. Samples of T0(x,y): (a) h0 = 10 mm, m = 5.0, (b) h0 = 30 mm, m = 5.0.

Table 1n and k (found in [37])

n k

2 242.320 4.16150 3.339100 3.0961000 2.7181 2.576

132 Ø. Fyllingen et al. / Computers and Structures 87 (2009) 128–139

4. Numerical study

Simulations of the model described in the previous sections areperformed here. First an evaluation is done in order to find a proper

element type, element size, loading time and bcr. Then a central de-sign is defined and the effects of varying the random field param-eters around this design are investigated. These numerical studiesare performed assuming isotropic material properties. On this ba-sis, the effects of material anisotropy are studied. Finally, the effectof the size of the patch is investigated by dividing a patch into foursmaller patches and by performing FLD calculations on each ofthem. The resulting FLDs from the smaller patches are comparedto the FLD for the initial patch.

The parameters described in the following are common in mostof the simulations. It has been chosen to investigate a patch withw0 equal to 100 mm and mean thickness, l, of 2.0 mm. The mate-rial is the aluminium alloy AA7108. For this material the Young’smodulus, Poisson’s ratio and density are approximately70000 MPa, 0.33 and 2700 kg/m3, respectively. Lademo et al. [39]

Ø. Fyllingen et al. / Computers and Structures 87 (2009) 128–139 133

found the parameters for the hardening rule (Eq. (8)) as presentedin Table 2. In the initial numerical studies it is assumed that the al-loy is isotropic with respect to strength and plastic flow and hencethe ai with i from 1 to 8 in Eq. (7) are all equal to 1. According toLogan and Hosford [40], the exponent m in Eq. (6) is typically 8for face centred cubic (f.c.c.) materials.

4.1. Evaluation of the finite element model

Since Monte Carlo simulation is going to be performed, it isimportant to reduce the FE simulation time as much as possiblewithout reducing the quality of the results substantially. Someimportant factors that are influencing the simulation time of an ex-plicit solver are the element type, the element size and the loadingtime.

There are two aspects related to discretization. The discretiza-tion into finite elements, which is essentially necessary from theviewpoint of local stress and strain gradients, and another discret-ization of stochastic fields related to the field properties. The thick-ness of the elements in LS-DYNA is given at the nodes. Hence, thethickness field can be discretized by giving node i at (xi,yi), thethickness t0(xi,yi). By refining the mesh, the description of the thick-ness field is improved. The other aspect concerns the behaviour ofthe structure. By refining the mesh, the element solution shouldconverge towards an exact solution. The size of the finite elementsshould be such that both the stochastic field and the stress andstrain gradients are well represented.

Table 2Material parameters for the isotropic hardening rule [38]

r0 [N/mm2] Q1 [N/mm2] C1 Q2 [N/mm2] C2

87.0 11.4 724.4 133.6 15.5

x

0

50

100

y

0

50

100

t1.98

2.00

2.02

x

0

50

t

1.98

2.00

2.02

a b

c

Fig. 6. Plot of t0(x,y): (a) 160 � 160 elements, (

The time at localisation is detected when bcr is exceeded in anyof the elements during nit consecutive time steps. Because of thepropagating stress waves, bcr has to be exceeded over several timesteps in order to detect localisation when it really occurs. In theprevious studies a bcr equal to 2.0 and nit equal to 250 seemed towork well. These values are used in the evaluation of element size,element type and loading time.

First an evaluation of the element size is done. The covariancefunctions which are going to be used in the next sections are thesame as plotted in Fig. 3. The covariance function withh0 = 10.0 mm and m = 5.0 leads to the most oscillating field realisa-tions and hence needs the finest mesh for a proper field represen-tation. A realisation of a thickness field with this covariancefunction is shown in Fig. 6a. The field has been generated on a reg-ular grid of 161 � 161 nodes. On coarser meshes the field values atthe element nodes may be found by bilinear interpolation of thethickness data. In Fig. 6b and c, the field realisation has been rep-resented on a grid of 81 � 81 and 41 � 41 nodes, respectively. Sim-ulations of FLDs for these three meshes are done along four linearstrain paths, with h equal to 0�, 15�, 30� and 45�, respectively.

The Belytschko–Tsay membrane element (El 5) is used and etot/ttot is set equal to 50/s. The simulations are done both with andwithout the non-local thickness increment averaging method(nlt) described in Section 2. The radius r of the thickness strainaveraging was set to 1.0 mm. The resulting strains at the time oflocalisation, ecr, for the different meshes are shown in Table 3,where the three first rows of results are obtained without nlt, whilethe three last rows of results are found by using the nlt. Theapproximate simulation time for the three meshes, starting withthe coarsest mesh was about 3 min, 23 min and 180 min, respec-tively. The ecr does not seem to converge when refining themesh without nlt, while with nlt the values are quite similar forthe three different meshes. Since the model with nlt results inlocalisation bands of a certain size, and hence is less sensitive to

x

0

50

100

y

0

50

100

t

1.98

2.00

2.02

100

y

0

50

100

b) 80 � 80 elements, (c) 40 � 40 elements.

Table 3Forming limit strains ecr for different meshes

nlt Mesh h

0� 15� 30� 45�

Without 40 � 40 0.119 0.154 0.202 0.24680 � 80 0.118 0.149 0.191 0.234160 � 160 0.115 0.144 0.185 0.229

With 40 � 40 0.119 0.154 0.202 0.24680 � 80 0.123 0.159 0.209 0.247160 � 160 0.122 0.157 0.204 0.247

Table 4Levels of the covariance parameters

Level h0 [mm] m s/l

1 10 2 0.00252 20 5 0.0053 30 10 0.01

134 Ø. Fyllingen et al. / Computers and Structures 87 (2009) 128–139

mesh refinement, it is used in the following. The resulting FLDwith a mesh of 40 � 40 elements does not deviate substantiallyfrom the FLD found with a mesh of 160 � 160 elements. Thus, amodel of 40 � 40 elements is used in the following. With a meshof 40 � 40 elements, the non-local thickness increment averagingmethod is not active, since the element size is larger than twicethe radius r.

Now a comparison between two different element types isdone, namely, the Belytschko–Tsay membrane element (El 5) andthe fully integrated Belytschko–Tsay membrane element (El 8).The FLDs predicted by the use of the two elements are almost iden-tical, see Fig. 7a. The solution time for El 5 was about 3 min, whileit was about 4 min for the fully integrated B–T element. Hence, El 5is preferred.

The effect of changing etot/ttot is shown in Fig. 7b. The simulationtime for etot/ttot equal to 25/s, 50/s and 100/s are 6 min, 3 min and1.5 min, respectively. The FLDs for 25/s and 50/s are almost identi-cal while at 100/s the FLD deviates. Hence, a etot/ttot equal to 50/sseems to be adequate. Finally, the effect of changing bcr is demon-strated in Fig. 7c. The effect of increasing bcr leads to an increasedforming limit. In the following bcr will be equal to 2.

4.2. Variation of field characteristics

In this section the effect of changing the parameters of thecovariance function is investigated. The parameters that are con-sidered are the effective range, h0, the smoothness parameter, m,and the coefficient of variation, s/l. By the use of Monte Carlo sim-ulation, a stochastic FLD may be constructed for each combinationof the covariance parameters. The procedure is as follows: n sam-ples of the random field are generated and mapped onto n patches.For each patch the forming limits along l linear strain paths are cal-culated. Hence, the total number of simulations is n � l. Along eachof the l linear strain paths, the mean and standard deviation of ecr

are estimated. All the results are plotted in a diagram called a sto-chastic forming limit diagram. In this diagram the result of eachsimulation is given as a dot, while the sample mean of ecr is indi-

0.00 0.05 0.10 0.15 0.20

0.0

00

.05

0.1

00

.15

0.2

0

εy

ε x

element 5

element 8

0.00 0.05 0

0.0

00

.05

0.1

00

.15

0.2

0

ε x

a b

Fig. 7. FLDs: (a) comparison between element types 5 and 8, (b) compariso

cated by a dotted line and the sample mean of ecr ± k sample stan-dard deviations of ecr is indicated by two lines.

A central design has been chosen as level 2 in Table 4. The effectof each parameter around the central design is found by alteringthe parameter in consideration to levels 1 and 3, and keeping therest of the parameters fixed on the values at level 2. The total num-ber of combinations is 7 including the central design. Even thoughthe density of ecr for each direction might deviate from the normaldensity, Table 1 is used as an indication on how many samples thatare necessary. For 100 samples it is expected on a 99% level of con-fidence that 99% of all possible outcomes are contained within thesample mean ±3.096 sample standard deviations, see Table 1. Byincreasing the number of samples to 1000, the value of k is reducedby 12%, while the number of samples is increased by 900%. Sincethe model is based on many assumptions and simplifications itshould be sufficient with a sample of size 100. Four linear strainpaths are considered: h = {0�,15�,30�,45�}. Then each FLD will con-sist of 4 � 100 = 400 simulations.

The effects of changing h0, m and s/l are presented in Figs. 8–10,respectively. The number k was set to 3 for all the FLDs presented.From these figures it can be observed that most of the points arecontained within the limits of ±3 sample standard deviations fromthe mean. The central design is called 2_2_2 and is depicted in allthese figures. The lower level (1) is shown to the left in the figures,while the upper level (3) is shown to the right in the figures. In Ta-ble 5, the sample mean and standard deviation (sd) of ecr arepresented.

In general, the mean and standard deviation of ecr increase as hincreases. From Fig. 8 and Table 5 it may be observed that an in-crease in h0 leads to an increase in the standard deviation of ecr,while the mean of ecr remains almost unchanged. Further on, theeffect of changing m on both the mean and standard deviation ofecr is very small according to Fig. 9 and Table 5. Finally, fromFig. 10 and Table 5, it may be observed that the effect of increasings/l reduces the mean of ecr, especially for large h, while the stan-dard deviation of ecr does not change a lot. It is important to notethat the observed effects are only valid within the chosen parame-ter range. Smaller values of m and h0 may lead to different effectsand a finer mesh is needed in order to describe the resulting fieldrealisations.

.10 0.15 0.20

εy

εtot

ttot

= 25/s

εtot

ttot

= 50/s

εtot

ttot

= 100/s

0.00 0.05 0.10 0.15 0.20

0.0

00

.05

0.1

00

.15

0.2

0

εy

ε x

β = 2

β = 4

β = 8

c

n of different strain rates: 25/s, 50/s and 100/s, (c) effect of varying bcr.

0.00 0.05 0.10 0.15 0.20 0.25

0.0

00

.05

0.1

00

.15

0.2

00

.25

1_2_2

εy

ε x

0.00 0.05 0.10 0.15 0.20 0.25

0.0

00

.05

0.1

00

.15

0.2

00

.25

2_2_2

εy

ε x

0.00 0.05 0.10 0.15 0.20 0.25

0.0

00

.05

0.1

00

.15

0.2

00

.25

3_2_2

εy

ε x

a b c

Fig. 8. FLDs with m = 5 and s/l = 0.005: (a) h0 = 10 mm, (b) h0 = 20 mm, (c) h0 = 30 mm.

0.00 0.05 0.10 0.15 0.20 0.25

0.0

00

.05

0.1

00

.15

0.2

00

.25

2_1_2

εy

ε x

0.00 0.05 0.10 0.15 0.20 0.25

0.0

00

.05

0.1

00

.15

0.2

00

.25

2_2_2

εy

ε x

0.00 0.05 0.10 0.15 0.20 0.25

0.0

00

.05

0.1

00

.15

0.2

00

.25

2_3_2

εy

ε x

a b c

Fig. 9. FLDs with s/l = 0.005 and h0 = 20 mm: (a) m = 2, (b) m = 5, (c) m = 10.

0.00 0.05 0.10 0.15 0.20 0.25

0.00

0.05

0.10

0.15

0.20

0.25

2_2_1

εy

ε x

0.00 0.05 0.10 0.15 0.20 0.25

0.00

0.05

0.10

0.15

0.20

0.25

2_2_2

εy

ε x

0.00 0.05 0.10 0.15 0.20 0.25

0.00

0.05

0.10

0.15

0.20

0.25

2_2_3

εy

ε x

a b c

Fig. 10. FLDs with m = 5 and h0 = 20 mm: (a) s/l = 0.0025, (b) s/l = 0.005, (c) s/l = 0.01.

Table 5Mean and standard deviation of ecr

Name 0� 15� 30� 45�

Mean sd Mean sd Mean sd Mean sd

2_2_2 0.118 0.003 0.147 0.007 0.186 0.011 0.240 0.0131_2_2 0.118 0.002 0.147 0.006 0.187 0.010 0.244 0.0113_2_2 0.120 0.004 0.150 0.010 0.192 0.016 0.242 0.0152_1_2 0.118 0.003 0.147 0.007 0.186 0.011 0.241 0.0132_3_2 0.119 0.003 0.149 0.007 0.189 0.011 0.241 0.0132_2_1 0.127 0.002 0.169 0.008 0.224 0.012 0.299 0.0142_2_3 0.108 0.004 0.126 0.007 0.152 0.010 0.185 0.012

Ø. Fyllingen et al. / Computers and Structures 87 (2009) 128–139 135

Since the same patches were used for all the strain paths, thecorrelation between the critical strains at the different paths may

be calculated. In Table 6 the correlation matrix for the central de-sign is presented. The critical strains at 0�, 15� and 30� are highlypositively correlated, while the critical strains at 45� are less corre-lated to the critical strains at the other directions. The pattern wasquite similar for the other six combinations. The high positive cor-relation between the critical strains at 0�, 15� and 30� indicatesthat the patches which have high (low) forming limits in one direc-tion compared to the mean value are likely to have high (low)forming limits compared to their respective mean values in theother two directions.

Pictures of the predicted localisation bands for the three firstsamples of the central design are depicted in Fig. 11. Each samplecorresponds to one row and the x- and y-axes are orientated as inFig. 1b. For plane strain and at 15� a localised neck perpendicular to

Table 6Correlation coefficients between the critical strains at different paths for 2_2_2(central design)

Variables 0� 15� 30� 45�

0� 1.00 0.90 0.80 0.3615� 1.00 0.96 0.4130� 1.00 0.4245� 1.00

Fig. 11. Pictures of the localisation bands for the first three patches of 2_2_2.

Table 8Material parameters for anisotropy and shape of yield surface [38]

a1 a2 a3 a4 a5 a6 a7 a8 m

0.847 0.998 0.893 1.358 0.884 1.092 1.200 1.137 8

Table 7Estimated mean and standard deviation of ecr

Variable 0� 15� 30� 45� 60� 75� 90�

Mean 0.113 0.135 0.168 0.230 0.217 0.165 0.126sd 0.003 0.006 0.009 0.015 0.015 0.009 0.003

136 Ø. Fyllingen et al. / Computers and Structures 87 (2009) 128–139

the x-axis develops. At 30� the localisation initiates in the sameneck, but in addition necks perpendicular to the first neck devel-ops. The initiation of necks at 45� seems to be more complex andthis may be the cause for the lower correlation between the criticalstrains at this direction and the rest. The orientation of the meshcould be important for the development of the localisation. Itwould have been interesting to analyse the development of strainlocalisation by the use of irregular meshes of triangular or tetrahe-dral elements. However, this was not done since the main objectivewith this study is to investigate when the localisation occurs andnot on the development of the localisation.

A comparison between classical M–K theory and the presentedFLD is performed next. Typically, values used for the inhomogenityfactor f found in the literature are in the range 0.98–0.995. InFig. 12a FLDs found by the use of M–K analysis are demonstrated.In the analysis, localisation was detected when the ratio betweenthe thickness strain increment within the groove and the thickness

0.00 0.05 0.10 0.15 0.20 0.25

0.0

00

.05

0.1

00

.15

0.2

00

.25

εy

ε x

f = 0.980

f = 0.985

f = 0.990

f = 0.995

a

Fig. 12. (a) M–K analysis, (b) FLD res

strain increment outside the groove exceeded 2.0. For the M–Kanalysis the plastic limit strains are plotted, while the total limitstrains are plotted for the FE-predictions. Since the elastic strainsare quite small compared to the plastic strains the plots are com-pared directly. The mean of most of the FLDs found by the use ofrandom fields are within the range of FLDs produced by M–K anal-ysis with f ranging from 0.985 to 0.995. An exception is the mean of2_2_1, which is above the M–K FLDs.

Now, the effect of an anisotropic field is investigated. For theisotropic fields it was assumed that the FLDs are symmetric abouta line directed along h = 45�. For an anisotropic field this is not nec-essarily valid and hence the following strain paths are considered:h = {0�,15�,30�,45�,60�,75�,90�}. This results in 7 � 100 = 700 sim-ulations for each FLD. The chosen parameters for the covariancefunction are the same as for 1_2_2, and the anisotropy ratio / isequal to 3. The direction of maximum range is parallel to the y-axis, which results in longer wavelengths in the y-direction thanin the x-direction. An anisotropic field may represent, for instance,directional-dependent inhomogeneity arising from the grainmorphology.

In Fig. 12b the resulting FLD is presented. It may be observedthat the FLD is not symmetric about a line aligned at h = 45�. In Ta-ble 7 the estimated mean and standard deviation of ecr are pre-sented. Especially, large differences are observed whencomparing the estimated mean and standard deviation of ecr ob-tained at 30� and 60�.

4.3. Material anisotropy

Until now, it has been assumed that the AA7108 alloy is isotro-pic with respect to the strength and plastic flow. This material isindeed quite anisotropic. Lademo et al. [39] found the materialparameters in Eq. (7) as presented in Table 8. Contour plots of

0.00 0.05 0.10 0.15 0.20 0.25

0.0

00

.05

0.1

00

.15

0.2

00

.25

anisotropic

εy

ε x

b

ulting from an anisotropic field.

Ø. Fyllingen et al. / Computers and Structures 87 (2009) 128–139 137

the identified yield surface are depicted in Fig. 14. The yield surfaceused previously is demonstrated in Fig. 13.

For the anisotropic material in consideration, the FLD is not ex-pected to be symmetric about a line aligned at h = 45� even with anisotropic field. Hence, the whole first quadrant of the FLD is consid-ered. It has been chosen to use the same values of parameters of thecovariance function as for the central design (2_2_2) in Section 4.2.The resulting FLD is presented in Fig. 15a. In order to compare thisFLD with an FLD assuming isotropic material, the same simulationswere done, but assuming isotropy. The resulting FLD is presented inFig. 15b.

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

σx

σ0

σ yσ 0

Fig. 13. Identified Yld2003 yield surface assumin

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

σx

σ0

σ yσ 0

Fig. 14. Identified Yld2003 yield surface taking into ac

0.00 0.05 0.10 0.15 0.20 0.25

0.00

0.05

0.10

0.15

0.20

0.25

εy

ε x

a

Fig. 15. (a) FLD for anisotropic materi

The shape of the two FLDs is quite different, and especially largedifferences are observed for h around 45�. Hence, a proper charac-terisation of the yield surface is important.

Experimental and numerical forming limit predictions ofAA7108 have been carried out by Lademo et al. [39]. The materialis referred to as material B in their article. The material experi-enced a low formability in the equibiaxial mode (h = 45�). The esti-mated shape in Fig. 15a is much closer to the experimental FLDthan the estimated shape in Fig. 15b. Other studies on the effectof plastic anisotropy have been done by Barlat and Richmond[39] and Barata da Rocha et al. [8].

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

(σx+ σ

y) 2σ

0

σ xy

σ 0

g isotropy, height between contours = 0.05.

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

(σx+ σ

y) 2σ

0

σ xy

σ 0

count anisotropy, height between contours = 0.05.

0.00 0.05 0.10 0.15 0.20 0.25

0.00

0.05

0.10

0.15

0.20

0.25

εy

ε x

b

al, (b) FLD for isotropic material.

1 2

3 4

0.00 0.05 0.10 0.15 0.20 0.25

0.0

00

.05

0.1

00

.15

0.2

00

.25

main

εy

ε x

0.00 0.05 0.10 0.15 0.20 0.25

0.0

00

.05

0.1

00

.15

0.2

00

.25

sub 1

εy

ε x

a b c

Fig. 16. (a) Patch divided into four smaller patches, (b) FLD for the whole patch, (c) FLD for patch 1.

Table 9Estimated mean and standard deviation of ecr

Part 0� 15� 30� 45�

Mean sd Mean sd Mean sd Mean sd

Main 0.119 0.003 0.149 0.007 0.189 0.012 0.243 0.014Sub 1 0.122 0.004 0.156 0.010 0.204 0.017 0.262 0.019Sub 2 0.121 0.004 0.155 0.010 0.203 0.017 0.262 0.020Sub 3 0.122 0.004 0.158 0.010 0.205 0.016 0.265 0.017Sub 4 0.121 0.004 0.156 0.010 0.203 0.016 0.265 0.020

138 Ø. Fyllingen et al. / Computers and Structures 87 (2009) 128–139

4.4. Effect of the patch size

In the last numerical study the effect of the patch size is inves-tigated. A patch of 100 mm � 100 mm is divided into four smallerpatches of 50 mm � 50 mm, see Fig. 16a. FLD simulations for thelarge patch and the four smaller patches are performed and com-pared. The covariance parameters are the same as for the centraldesign in Section 4.2, and material isotropy is assumed. Hundredsamples of the large patch are generated, which results in4 � 100 smaller patches. The simulations are done along four linearstrain paths, which result in a total of 5 � 100 � 4 = 2000 simula-tions. The results are presented in Fig. 16b and c, and in Table 9.

From Table 9 it can be observed that both the mean and stan-dard deviation (sd) of ecr are higher for the smaller patches thanthe large patch. Hence, for the chosen case, the effect of investigat-ing a smaller patch results in larger mean and variance of ecr. Onlythe FLD of the main part and subpart 1 is presented in Fig. 16. TheFLDs for the other subparts were quite similar. Further on, positivecorrelation of about 0.60 at each of the strain paths was found be-tween ecr for the large patch and the minimum ecr of thesubpatches.

5. Concluding remarks

The assumed thickness variations resulted in a quite wide scat-ter band, and a change of the characteristics of the thickness fieldresulted in a change of the shape and variance of the FLD. An in-crease in the effective range resulted in a larger variance, a changein the smoothness parameter did not affect the FLD much, and anincrease in the sill reduced the critical strain ecr, especially for haround 45�. By imposing an anisotropic field the FLDs were notsymmetric about a line aligned at h = 45�. Further on, the effectof taking into account material anisotropy for the AA7108 alloyled to a quite different shape of the FLD, and hence a proper mate-rial characterisation is very important. The effect of reducing thepatch size gave an increase in both the mean and variance of ecr

for the chosen case. The proposed model is also applicable to the

prediction of plastic instability in components subjected to form-ing operations or crash events.

In order to compare the model to experimental data, it is desir-able to relate the field characteristics to the measured inhomoge-neities. Both variations in geometry and material should beincluded.

Acknowledgements

This work has been carried out with financial support from TheNorwegian University of Science and Technology and the BIP pro-ject ROBDES. The ROBDES project is founded by the Research Coun-cil of Norway, Hydro Aluminium Structures, Volvo Car Corporation,Ford, Scania, SSAB Tunnplåt and Gestamp Hardtech.

References

[1] Swift HW. Plastic instability under plane stress. J Mech Phys Solids1952;1:1–18.

[2] Hill R. On discontinuous plastic states, with special reference to localizednecking in thin sheets. J Mech Phys Solids 1952;1:19–30.

[3] Marciniak Z, Kuczynski K. Limit strains in the processes of stretch-formingsheet metals. Int J Mech Sci 1967;9:609–20.

[4] Bate P, Wilson DV. Strain localisation in biaxially stretched sheets containingcompact defects – II. Int J Mech Sci 1983;26:363–72.

[5] Neglo K, Chater E, Neale KW. Effects of the shape of a geometric defect and ofinteractions between defects on limit strains for biaxially stretched sheets. Int JMech Sci 1987;29:807–20.

[6] Barlat F, Barata Da Rocha A, Jalinier JM. Influence of damage on the plasticinstability of sheet metals under complex strain paths. J Mater Sci1984;19:4133–7.

[7] Barlat F, Jalinier JM. Formability of sheet metal with heterogeneous damage. JMater Sci 1985;20:3385–99.

[8] Barata Da Rocha A, Barlat F, Jalinier JM. Prediction of the forming limitdiagrams of anisotropic sheets in linear and non-linear loading. Mater Sci Eng1984–1985;68:151–64.

[9] Friedman PA, Pan J. Effects of plastic anisotropy and yield criteria on predictionof forming limit curves. Int J Mech Sci 2000;42:29–48.

[10] Cao J, Yao H, Karafillis A, Boyce MC. Prediction of localized thinning in sheetmetal using a general anisotropic yield criterion. Int J Plasticity2000;16:1105–29.

[11] Kuroda M, Tvergaard V. Forming limit diagrams for anisotropic metal sheetswith different yield criteria. Int J Solids Struct 2000;37:5037–59.

[12] Banabic D, Dannenmann E. Prediction of the influence of yield locus on thelimit strains in sheet metals. J Mater Process Technol 2001;109:9–12.

[13] Yao H, Cao J. Prediction of forming limit curves using an anisotropic yieldfunction with prestrain induced backstress. Int J Plasticity 2002;18:1013–38.

[14] Butuc MC, Banabic D, Barata da Rocha A, Gracio JJ, Ferreira Duarte J, Jurco P,et al. The performance of Yld96 and BBC2000 yield functions in forming limitprediction. J Mater Process Technol 2002;125–126:281–6.

[15] Butuc MC, Gracio JJ, Barata da Rocha A. A theoretical study on forming limitdiagrams prediction. J Mater Process Technol 2003;142:714–24.

[16] Wu PD, Jain M, Savoie J, MacEwen SR, Tugcu P, Neale KW. Evaluation ofanisotropic yield functions for aluminium sheets. Int J Plasticity2003;19:121–38.

[17] Aretz H. Numerical analysis of diffuse and localized necking in orthotropicsheet metals. Int J Plasticity 2007;23:798–840.

Ø. Fyllingen et al. / Computers and Structures 87 (2009) 128–139 139

[18] Stoughton TB, Zhu X. Review of theoretical models of the strain-based FLD andtheir relevance to the stress-based FLD. Int J Plasticity 2004;20:1463–86.

[19] Banabic D, Aretz H, Paraianu L, Jurco P. Application of various FLD modellingapproaches. Modell Simulat Mater Sci Eng 2005;13:759–69.

[20] Van Minh H, Sowerby R, Duncan JL. Variability of forming limit curves. Int JMech Sci 1974;16:31–44.

[21] Van Minh H, Sowerby R, Duncan JL. Probabilistic model of limit strains in sheetmetal. Int J Mech Sci 1975;17:339–49.

[22] Janssens K, Lambert F, Vanrostenberghe S, Vermeulen M. Statistical evaluationof the uncertainty of experimentally characterised forming limits of sheetsteel. J Mater Process Technol 2001;112:174–84.

[23] Strano M, Colosimo BM. Logistic regression analysis for experimental determi-nation of forming limit diagrams. Int J Mach Tools Manuf 2006;46:673–82.

[24] Banabic D, Vos M, Paraianu L, Jurco P. Theoretical prediction of the forminglimit band. In: 10th ESAFORM conference on material forming, 2007. p. 368–72.

[25] Narisimhan K, Zhou D, Wagoner RH. Application of the Monte Carlo and finiteelement methods to predict the scatter band in forming limit strains. ScrMetall Mater 1992;26:41–6.

[26] Lademo OG, Berstad T, Hopperstad OS, Pedersen KO. A numerical tool forformability analysis of aluminium alloys. Part I: Theory. Steel Grips2004;2:427–31.

[27] Lademo OG, Berstad T, Hopperstad OS, Pedersen KO. A numerical tool forformability analysis of aluminium alloys. Part II: Experimental validation. SteelGrips 2004;2:433–7.

[28] Berstad T, Lademo OG, Hopperstad OS. Weak and strong texture models in LS-DYNA. SINTEF REPORT STF80MK F05180; 2005.

[29] Barlat F, Brem JC, Yoon JW, Chung K, Dick RE, Lege DJ, et al. Plane stress yieldfunction for aluminium alloy sheets – Part I: Theory. Int J Platicity2003;19:1297–319.

[30] Aretz H. Applications of a new plane stress yield function to orthotropic steeland aluminium sheet metals. Modell Simulat Mater Sci Eng 2004;12:491–509.

[31] Reyes A, Hopperstad OS, Lademo OG, Langseth M. Modeling of texturedaluminium alloys used in bumper system: material tests and characterization.Comput Mater Sci 2006;37:246–68.

[32] Pijaudier-Cabot G, Bazant ZP. Nonlocal damage theory. J Eng Mech ASCE1987;113(10):1512–33.

[33] Wang T, Hopperstad OS, Lademo OG, Larsen PK. Finite element analysis ofwelded beam-to-column joints in aluminium alloy EN AW 6082 T6. FiniteElem Anal Des 2007;44:1–16.

[35] Ribeiro Jr PJ, Diggle PJ. geoR: a package for geostatistical analysis. R-NEWS2001;1:14–8.

[36] R Development Core Team. R. A language and environment for statisticalcomputing. R Foundation for Statistical Computing, Vienna, Austria; 2004.

[37] Walpole RE, Myers RH, Myers SL. Probability and statistics for engineers andscience. Prentice-Hall International; 1998.

[38] Lademo OG, Pedersen KO, Berstad T, Furu T, Hopperstad OS. An experimentaland numerical study on the formability of extruded AlZnMg alloys. Eur J MechA–Solids 2008;27:116–40.

[39] Logan RW, Hosford WF. Upper-bound anisotropic yield locus calculationsassuming h111i-pencil glide. Int J Mech Sci 1980;22:419–30.

[40] Barlat F, Richmond O. Prediction of tricomponent plane stress yield surfacesand associated flow and failure behaviour of strongly textured F.C.C.polycrystalline sheets. Mater Sci Eng 1987;95:15–29.