An analysis of sheet necking

under combined stretching and

bending

N.E. de Kruijf

Master thesis, MT08.43

Supervisors:dr.ir. Ron Peerlingsprof.dr.ir. Marc Geers

Mechanics of Materials,Department of Mechanical Engineering,Eindhoven University of Technology.November, 2008

Abstract

During deformation of structures consisting of metal sheets, like hull structures of ships duringcollision, large curvature bending and stretching can take place. Current finite element calculationsusing conventional shell elements show large dependence on the used mesh size and failure criteria.The ultimate goal of the work reported on here is the development of enhanced shell elementwhich is able predict accurate failure by removing this limitation. In order to understand theinitiation of plastic instability resulting in necking of sheet metal under large curvatures, thedeformation is divided in bending and stretching contributions. The objective of this study istwofold, on the one hand understand the deformation behavior of stretching of sheet metal underlarge curvature, where the order in time plays a crucial role. And on the other the development ofa generalised bifurcation condition for stretching under combined large curvature bending whichcan be incorporated in the enhanced shell element in a later stadium. In finite element packageMSC.Marc Mentat different simulations under the plane strain assumption are performed. Thestretching of sheet metal is simulated where the curvature is increased prior, simultaneous orafterwards. Results show significant dependence of neck initiation on the order in which thecurvature increases. Semi-analytical modelling assuming rigid-plastic isochoric material behaviordelivers a bifurcation condition for stretching under simultaneous curvature increase. The foundcondition is in good agreement with the numerical results.

ii

Contents

Abstract ii

Contents iii

1 Introduction 1

2 Problem statement 32.1 Mesh size issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Considere’s criterion for planar stretching . . . . . . . . . . . . . . . . . . . . . . . 62.4 Combined stretching and bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5 Material model used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Numerical simulations 113.1 Finite element model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.1 Sequential loading; stretch-bending . . . . . . . . . . . . . . . . . . . . . . . 133.2.2 Sequential loading; bend-stretching . . . . . . . . . . . . . . . . . . . . . . . 153.2.3 Simultaneous loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Semi-analytical model 224.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2 Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.3 Tangential force and bending moment . . . . . . . . . . . . . . . . . . . . . . . . . 264.4 Bifurcation condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.5 Force and moment curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.6 Necking diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.7 Material parameter variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.8 Linear vs. nonlinear hardening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.9 Deformation path dependence; an example . . . . . . . . . . . . . . . . . . . . . . . 344.10 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Conclusion and outlook 36

Bibliography 38

A Sensitivity 39A.1 Mesh density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39A.2 Imperfection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

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Chapter 1

Introduction

Schelde Naval Shipbuilding has introduced a new type of sandwich hull structure, with a Y-framecore between the inner and outer hull of the ship. This structure is schematically shown in the topright part of figure 1.1. Large scale experiments have shown the capability of the newly designedstructure to withstand low velocity impact, where the conventional double hull design failed. Crashsafety is improved and outflow of hazardous cargo can be prevented under certain circumstances(Vredeveldt et al., 2004). This proof resulted in an adaptation of tank size regulations for inlandwaterway ships, making larger tank sizes possible. An economical benefit is obtained by the en-largement of the cargo tanks’ capacity and the simultaneous reduction of the number of tanks.

Figure 1.1: Large-scale collision test on the Schelde Y-frame structure (left), a sketch of the cross-sectionof the structure (top right, source: (Rubino et al., 2008)) and a cross-section of the deformed test section(bottom right, source: Schelde Naval Schipbuilding).

The Y-frame structure is designed in such a way that the load is distributed over a large sectionof the hull and thus prevents tearing of the face-sheet. This is due to the relatively compliantweak upper section of the frame (Rubino et al., 2008). In a large scale experiment conducted byTNO, in collaboration with different shipyards including the Schelde Naval Shipbuilding, a shipwith a bulbous bow filled with concrete collided with a test section of the sandwich material withY-frame core. As can be seen on the left in figure 1.1, the Y-frame structure prevailed, even upona second collision on exactly the same section. The conventional structure failed to withstand the

1

impact in an identical test.

At Schelde large scale finite element computations are performed on naval structures such asY-core sandwich hulls. The energy absorption capacity of the structure and failure are of greatinterest in these computations. In most cases large sections of the ship are modeled by finiteelements. To limit computation times, extensive use is made of shell elements. Typically, elementsizes are used by the Schelde in such computations which are fives times the sheet thickness tokeep the computation cost at an acceptable level. At these relatively large element sizes, resultshave been observed to be sensitive to the used failure criteria and mesh density. This problem canbe related to the nature of shell elements (Ehlers et al., 2008), as is shown using an example insection 2.1 of this report. The ultimate goal of the work reported on here, is to remove the aboveproblem by developing an enhanced shell element, which on one hand is cheap to use and on theother predicts the initiation of failure accurately, even if the element size is large with respect tothe sheet thickness.

During deformation of the Y-frame structure the metal sheets of which the beams are constructed,as well as the face sheets, undergo large amounts of stretching and bending, which can be seen inthe left and bottom-right snapshots in figure 1.1. Our first concern is to gain more insight in theresponse of metal sheets under combined stretching and bending. This is done by dividing thedeformation into bending and stretching contributions, where the order in which these componentsare applied in time plays a crucial role.

Localized necking and fracture are the two primary modes of failure anticipated for such de-formations. In this report we focus on necking and disregard the possibility of fracture withoutany prior necking. Classically, in sheet-metal forming, necking criteria are presented by a forminglimit diagram (FLD). However, FLDs, as well as the classical Considere criterion for uniaxial ten-sion (section 2.3) cannot be used in predicting of failure under large curvatures. This due to factthat in this criterion the through thickness stress distribution is assumed to be constant. Thiscondition is violated when imposing large curvatures during or prior to stretching, which leadsto substantial gradients in radial direction. So there is a need for a new criterion to predict theonset of necking under combined stretching and bending. Developing such a criterion is the mainobjective of this report. It can later be used in the efficient shell element formulation for failureas discussed above. Furthermore, the detailed finite element simulations done to study the onsetof necking as well as the post-necking response of the sheet can later be used to calibrate the shellelement against.

The initiation of necking, dependent on the individual contributions, is studied within a finiteelement environment in chapter 3. Analytical modeling using some simplifications is used tosubstantiate and explain the finite element results, which is done in chapter 4.

2

Chapter 2

Problem statement

As discussed briefly in previous chapter, the use of conventional shell elements to predict failureof naval structures has some limitations. In the first section of this chapter this limitation isillustrated using a simple example. Subsequently the strategy which is pursued to remove thislimitation is explained. For planar stretching, Considere’s criterion can predict the stretching limitof metal sheet, as long as the failure is governed by necking, as is further elaborated in section2.3. The remainder of this report is on combined bending and stretching; the approach followedin defining this combined loading is elaborated in section 2.4. Finally the material model usedthroughout this report is detailed.

2.1 Mesh size issues

It is known that conventional shell elements fail to predict the right amount of energy dissipa-tion in large deformations due to mesh dependence. In this section the mesh dependence of shellelements is shown and compared with the behavior of solid elements. In figure 2.1 a schematicrepresentation of a cross-section of a sheet is given. Within MSC.Marc Mentat a model of thissheet of length 2l0 and thickness h0 can be constructed and discretised into either a number ofshell elements or solid elements. Due to the large width, perpendicular to the plane of the sketch,a plane strain condition is assumed in this direction. The mesh dependence is demonstrated belowusing a simple tensile test on this sheet.

ul0

h0

Figure 2.1: Sheet under tensile loading with the corresponding boundary conditions; note that only half ofthe length has been modelled due to symmetry.

Discretising the sheet in figure 2.1 into plane strain elements is fairly straightforward. On the leftside a symmetry plane is used to account for the fact that only half of the length of the sheet hasbeen modeled; nodes on this edge are free to move only in vertical direction except the node inthe center. On the right edge a displacement is imposed in horizontal direction but the nodes arefree to move in vertical direction. Due to these boundary conditions the sheet is free to contractin thickness direction. When discretising the sheet of figure 2.1 using shell elements, one only

3

has to model the middle surface of the sheet, perpendicular to the plane of sketch, and attributethe correct thickness to the shell elements. The displacement boundary conditions are applied tothe two nodes at the ends of the sheet. In the following analysis we use three-dimensional shellelements, whose lateral deformation (perpendicular to the sketch in figure 2.1) is constrained toimpose a planar deformation.

Simulations using the shell elements and a large strain formulation are performed with increas-ing mesh density and the results are compared with the simulation using plane strain elements.Localisation is triggered by an imperfection which is made at the plane of symmetry, on the leftside in figure 2.1. The material is elastoplastic with a power law material hardening. In figure 2.2the response of a sheet discretised by 2560 plane strain elements is compared with that of a sheetdiscretised by ne = 1, 10, 40 and 80 shell elements. The normalised force is shown as a functionof the average axial strain ε = ln(1 + u

l0). The force here and throughout this work is normalised

by dividing it by the initial cross-sectional area A0, which is in fact equal to the thickness of thesheet due to the fact that the width of the sheet is set equal to unity, and the initial yield stressσy0.

0 0.1 0.2 0.30

0.4

0.8

1.2

1.6

ε [−]

FA

0σ

y0

[−]

plane strain elements1 shell element10 shell elements40 shell elements80 shell elements

Figure 2.2: Comparison of the tensile response of a sheet modelled with shell and plane strain elementsin terms of normalised force as a function of the axial strain.

The simulation using plane strain elements first shows an elastic regime which is followed by uni-form plastic deformation up to the point where the maximum force is reached. From thereon aneck is formed at the imperfection, as can be observed in figure 2.3, and the sheet starts to loseits strength by a rapid decrease of the reaction force. Changing the mesh density does not havean appreciable effect on the response (not shown here).

Figure 2.3: Tensile test on a sheet using 2560 plane strain elements where a neck has formed at thesymmetry axis.

In the elastic regime and the first part of the hardening regime the shell element models show thesame behavior as the plane strain model. The mesh dependence of the shell element simulations

4

is apparent beyond the peak force. With increasing mesh density the softening and necking takesplace at lower strains and the predicted amount of dissipated energy is lower.

An important conclusion of the above example is that increasing the mesh density when us-ing shell elements does not necessarily lead to convergence to a unique solution, let alone to thecorrect (post-peak) solution obtained by solid elements. This behavior can be explained by thediscontinuity in thickness which is allowed by conventional shell element formulations. Since onlyone node is used in thickness direction, and only displacement and rotation degrees of freedom areassociated with this node, the thicknesses of adjacent elements are not coupled. As a consequence,upon the initiation of necking, all further deformation localizes in a single element, whereas theother elements unload elastically. This effect is represented schematically figure 2.4.

ne = 3

ne = 9

Figure 2.4: Thickness discontinuity when using shell elements in a tensile test on a sheet.

Upon increasing of the mesh density, i.e. increasing ne, the volume which enters the softeningregime decreases and therefore a stronger softening is observed in the force-strain curve. Thisclearly contrasts with the smooth neck and force-strain response predicted by plane strain el-ements. For this reason conventional shell elements are unreliable in predicting post neckingbehavior of this simple tensile test and also of more complex structures and loadings, even if theelement size is substantially smaller than the length of the neck. Similar problems are to be ex-pected in simulating other failure mechanisms using conventional shell elements.

Continuum-based shell theories have also been developed (Parisch, 1995) which provide eight-nodesolid-like shell elements. These elements have the advantage of thickness distribution continuity.However, to capture neck formation the element size must be considerably smaller than the size ofthe neck, i.e. a fraction of the sheet thickness. Because one generally does not know a priori wherenecking will occur the mesh density must thus be large throughout the model, which makes themcomputationally expensive. Similarly modelling the sheet using solid elements would in principlework, as can be seen in figure 2.3, but is prohibitively expensive in three dimensional modelling.

2.2 Objective

We have seen above that on the one hand there is a need for cheap elements for large scalesimulations and on the other hand neck formation and in a later stadium complete failure must bepredicted. The ultimate goal of our efforts on the subject is the development of an enhanced shellelement which is capable of accurately predicting failure of sheet metal by necking or by othermechanisms, under general loading conditions even if the element size is significantly larger thanthe width. In this report we take a first step towards this goal by developing a criterion for theprediction of solely neck initiation under combined tension and bending. Such a criterion couldlater be incorporated in standard shell elements to detect the onset of necking. When a likelihoodof necking (or some other form of instability) is detected, special shell elements could be inserted(figure 2.5), the response of which has first been calibrated on detailed analyses such as the onesdone in this report.

5

shell elementspecial formulation

Figure 2.5: Example of the use of the enhanced shell element.

2.3 Considere’s criterion for planar stretching

Necking is the outcome of a competition between the hardening of the material on the one handand the geometrical softening on the other. The force is obtained from the multiplication of thecross-sectional area A and the axial stress, i.e. F = Aσ. Geometrical softening results from thedecrease in area A due to the axial stretching whereas the hardening due to plasticity results in anincreasing stress σ. Initially, the hardening is usually stronger than the geometrical softening. Assoon as the softening becomes dominant, however, and the force thus decreases, necking instabilitymay occur.

Consider a sheet of original length l0 and thickness h0 which is stretched to a strain of ε = ln(1+ ulo

)at a constant strain rate ε - cf. the tensile test in section 2.1. This leads to the normalised forcestrain behavior shown in figure 2.6. First the force increases elastically from where the sheet startsto harden uniformly. Slightly beyond ε = 0.2 the peak force is reached, as marked in the figure bya circle, and from hereon the sheet starts to lose its strength.

0 0.1 0.2 0.30

0.4

0.8

1.2

1.6

ε [−]

FA

0σ

y[−

]

Figure 2.6: Normalised force as function of the strain for a stretched sheet.

A simple analysis of this plastic instability , due to Considere (Nadai, 1950), is as follows. Through-out the sheet the force relayed must be constant due to equilibrium, i.e.

∂F (ε)

∂x= 0 (2.1)

6

Initially, an increment in strain ε leads to a uniform increase in force. At a certain stage, howevera state is reached where a second solution ε + ∆ε next to ε is possible at the same force F . Abifurcation point is thus reached, i.e. there is no longer a unique solution and some cross-sectionsof the sheet may follow one equilibrium path, whereas others follow the other. In the limit of ∆ε

going to zero we have

lim∆ε→0

F (ε + ∆ε) − F (∆ε)

∆ε=

∂F

∂ε= 0 (2.2)

which corresponds with the criterion found by Considere. It states that bifurcation into an inho-mogeneous deformation path becomes possible when the peak force is reached. After this pointmore than one solution can be found. One of these paths has a uniform deformation. But nextto it, alternative paths become available in which the deformation is localised in one or multiplenecks. A single neck is the most critical case and therefore this is the response which is usuallyobserved.

Reconsidering the example, because we are also interested in the post-necking behavior an im-perfection is used in this work to trigger the localisation and the equilibrium path, seen in figure2.6, is obtained. Due to this imperfection, after the bifurcation point, no uniform deformationis possible and necking sets in by a more rapid decrease in area where the strains and stresseslocalise.

0 0.1 0.2 0.30

0.2

0.4

0.6

0.8

1

ε [−]

h h0

[−]

hn (thickness inside the neck)hs (thickness outside the neck)

0 0.1 0.2 0.30

0.4

0.8

1.2

ε [−]

η[−

]

Figure 2.7: Area decrease with respect to the imposed strain (left) and the thickness fraction with respectthe strain (right).

The variable η is introduced to characterise the relative thickness strain between the thicknessoutside the neck, hs, and inside the neck, hn, and is defined as

η = log

(

hs

hn

)

(2.3)

In the left diagram of figure 2.7 the decrease of both thicknesses hs and hn is given with respect toε. Both hs and hn first decrease identically to the necking point, from where the area in the neckstarts to decrease more rapidly and the area outside the neck stops decreasing so the localisationtakes place.

7

On the right in figure 2.7 the relative thickness strain η as defined by equation (2.3) is givenas a function of the applied average strain. Comparing the value of strain at which this variablestarts to increase appreciably with the diagram of figure 2.6 shows that the neck initiation is inagreement with the strain at which the peak force occurs.

2.4 Combined stretching and bending

As mentioned in the introduction we are interested in the combined stretching and bending ofsheets. The deformation is divided into bending and stretching contributions which need to bedefined properly. In a segment of sheet which undergoes a certain deformation, a reference surfacecan be placed. This reference surface is depicted on the left in figure 2.8 by the dashed curve andis placed in the sheet’s initial center. Both the curvature and stretch are defined for this surfaceas κref and εref respectively and they are taken to characterise the deformation of the entiresheet. A deformation space can be defined which is spanned by these strain measures, where thecurvature is normalised by half of the initial thickness as h0

2 κref ; note that in a small-strain settingthis quantity would be the bending strain. The deformation space thus obtained is sketched onthe right side in figure 2.8.

r θ

εref , κref

εref [−]

h0

2κref [−]

sequential

sequentialsimultaneous

loading

loading

loading

bend-stretching

stretch-bending

Figure 2.8: Reference strain and curvature definition within a piece of sheet (left) and the deformationspace built from the individual contributions (right).

For example, in a tensile test the deformation path follows the εref -axis. A sequential deformationpath is stretch-bending, where the sheet is stretched followed by bending and this path is depictedin the figure by the dash-dotted line. An other sequential deformation path is bend-stretching,i.e. the sheet is bent to a certain radius and stretched afterwards. This path is depicted in thefigure by the dashed line. The simultaneous loading considered also within this report is a linearcombination of stretching and bending (solid line).

If the simultaneous deformation is imposed by simply applying a displacement and rotation toboth ends of the sheet a problem arises. Applying both at the same time creates unbalanced loadsacting on the sheet. This due to the fact that in our definition the force acting on the sheet hasto follow the curvature. This results in a net force in vertical direction which is not balanced byany other load acting on the sheet. This is visualised by a schematic representation of the forceequilibrium in figure 2.9, in which Fcomp = 2Fv represents the compensation force required tosatisfy equilibrium.

For this reason the bottom of the sheet must be supported at all times to maintain a constant

8

FF FvFv

Fh Fh

Fcomp

Figure 2.9: Unvalance in vertical equilibrium due to a net force in vertical direction.

curvature. A way to achieve this even for evolving curvatures is to use a deformable support. Bycontrolling the shape of this support the curvature of the inner surface can be prescribed at alltimes and care is taken of the vertical compensation force Fcomp. In stretching the sheet is pulledover the support.

In the literature the influence of curvature during stretching has been investigated. Charpentier(1975) observed an increase in stretching limit for smaller punch curvatures during experiments.Charpentier explained this finding by the induced strain gradient through the thickness causedby the bending contribution. The work of Shi and Gerdeen (1991) showed good agreement withthe experimental findings of Charpentier, by theoretically predicted FLDs in punch stretchingbased on the M-K method by introducing a strain gradient term in the constitutive equation.Yoshida et al. (2005) studied fracture limits in sheet stretch bending. On the assumption thatfracture occurs when the force reaches a maximum, a limit wall stretch is given as a function ofthe punch-radius. But in this analysis the focus is on die-corner fracture.

2.5 Material model used

The material data used throughout this report is for grade A 37 steel, a much used material inship building. The local true stress-strain behavior can be represented using a modified power lawhardening relation

σy = σy0 + Kεnp (2.4)

with σy0 the initial yield stress, K the hardening modulus, εp the equivalent plastic strain and n

the strain-hardening exponent. In this work the material is assumed to be isotropic and rate andtemperature effects are neglected. The different material constants and the strain-hardening curvecan be obtained from a fitting procedure using MATLAB curve fitting toolbox on data obtainedfrom the Schelde Naval Shipbuilding (Broekhuijsen, 2003). The curve is given in figure 2.10 andthe parameters in table 2.1.

Table 2.1: Material parameters used in the finite element simulations.

E [GPa] ν [-] σy0 [MPa] K [MPa] n [-]

210 0.33 284 450 0.4

9

0 0.2 0.4 0.6 0.8 1 1.20

0.5

1

1.5

2

2.5

3

εp [−]

σy

σ0

[−]

Figure 2.10: Material model described with a modified power law hardening relation.

10

Chapter 3

Numerical simulations

Using the finite element package MSC.Marc Mentat a finite element model has been constructed.Using this model different deformation paths can be simulated. In this chapter the numericalmethods by which the model has been built are discussed. The deformation of the sheet isprescribed by using boundary conditions which vary in time. Results are shown for both sequentialand simultaneous bending and stretching.

3.1 Finite element model

Within MSC.Marc Mentat half of the sheet is discretised into 80 x 32 = 2560 plane straineight-node quadrilateral elements with reduced integration. A verification of this choice of themesh density can be found in appendix A. The length of the total sheet is five times its thickness.In figure 3.1 a schematic representation of the model is given.

support

control node

u

vϕ

Figure 3.1: Sketch of the undeformed mesh as modeled in MSC.Marc Mentat; note that the number ofelements in the sketch does not correspond with the actual model.

On the left side a symmetry plane is used to account for the fact that only half of the sheet hasbeen modeled; nodes on this edge are free to move only in vertical direction. The right edge isconnected to a rigid body so that nodes on it can slide along the rigid body, but their displacementis otherwise confined by it. The right bottom corner node of the sheet is the control node for therigid body, on which displacements u, v and a rotation φ can be prescribed in time. The momentexerted on the rigid body is defined as the moment around the control node and is therefore rela-tive to the bottom of the sheet. The bottom surface of the sheet rests on a support which consistsof line elements. Each node of this support, except the first, is given a displacement in verticaland horizontal direction so its motion and therefore its shape is fully prescribed. Contact betweenthe support and sheet is modeled using a frictionless touching condition and an analytical, smoothboundary description. Localisation is triggered by reducing the width of the sheet, in the directionperpendicular to the plane of the sketch of figure 3.1, in the first column of elements along the

11

symmetry plane by 1%; also the magnitude of this imperfection is verified in appendix A.

l0

ϕ

u

v1κ

1κ

κ, ε

h0

κref , εref

Figure 3.2: Parametrisation of the problem; note that the rigid body has been removed for clarity.

The variables used to impose the deformation on the sheet are the angle of the rigid body ϕ,the displacements u and v of the right bottom node of the sheet and the displacements of everysingle node on the support. The geometrical parameters are depicted in figure 3.2. The curvatureand stretch of the reference plane in the original center of the sheet, κref and εref respectively,are converted to the those of the bottom surface of the sheet, κ and ε, since this is where thecurvature is actually imposed. The radius of the inner surface 1

κis taken as the radius of the

reference surface 1κref

minus half of the initial thickness of the sheet, h0

2 . Note that this means

the thinning and thickening of the sheet is neglected in prescribing the deformation. Rewritingthis assumption leads to

κ =κref

1 − h0

2 κref

(3.1)

The stretch of the reference plane is defined as the logarithm of its deformed length divided bythe original length l0. The rotation angle ϕ can be expressed in terms of the bottom and referencesurface as

ϕ = l0eεκ = l0e

εref κref (3.2)

Substituting equation (3.1) into (3.2) and rewriting leads to

ε = εref + ln

(

1 − h0

2κref

)

(3.3)

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the displacements u, v can be expressed in terms of the rotation angle ϕ as

u =1

κsin(ϕ) − l0 (3.4)

v =1

κ(cos(ϕ) − 1) (3.5)

The motion of the nodes of the support is determined in the same manner. Relations (3.1)-(3.5)allow one to determine u, v and ϕ as a function of time for a given evolution of εref , κref in time.In all results presented in this chapter the reference surface strain and curvature rates εref andκref have been kept constant to obtain straight lines in the deformation space of figure 2.8.

3.2 Results

In this section results of the finite element simulations for the different deformation paths discussedin chapter 2 are presented. First the results of the two sequential deformation paths are discussedand subsequently those of the simultaneous loading paths. In prescribing the deformation withκref and ε an error is made by assuming the thickness does not decrease.

3.2.1 Sequential loading; stretch-bending

In stretch-bending the sheet is first stretched to a prestretch εpreref from t = 0 to t = 0.5 at a

constant stretch rate εref and subsequently an increase in curvature κref takes place at a constantcurvature rate κref between t = 0.5 and t = 1. The reference strain is kept constant during thecurvature increase. The prestretching is done in such a way that the point of necking is neverreached. As a result, the sheet hardens uniformly and its thickness decreases prior to bending.At t = 0.5 the deformation is switched from stretching to bending so that the curvature startsto increase. In figure 3.3 the deformation of the sheet is shown at t = 0.5 and t = 1. The colorsrepresent the equivalent von Mises stress. At t = 0.5 the sheet has hardened uniformly due to theprestretch and at t = 1 the stresses remain constant along the circumference.

In figure 3.4 the distribution of equivalent plastic strain εp across the thickness of the sheet isgiven for a prestretch ε

preref = 0.2 at t = 0.5, i.e. after the first phase of the deformation, and at

t = 1, i.e. at the end of the deformation. From this figure and from figure 2.10 it can be concludedthat the prestretch leads to uniform hardening in the sheet before the increase in curvature starts.During bending fibers at the top of the sheet are stretched even further and fibers near the bottomare compressed. Note that the neutral surface between these zones does not correspond with theinitial center of the sheet due to the large deformations which occur.

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(a) t = 0.5 (b) t = 1

0 [MPa] 957 [MPa]σVM

Figure 3.3: Deformation in the sequential stretch-bending deformation path with pre-stretch εpre

ref = 0.2.

0 0.1 0.2 0.3 0.40

0.2

0.4

0.6

0.8

1

εp [−]

y0

h0

[−]

t = 0.5 t = 1

Figure 3.4: Equivalent plastic strain as a function of the normalised initial thickness coordinate for aprestretch of ε

pre

ref = 0.2 after prestretching (t = 0.5) and at the end of the deformation (t = 1).

In the diagram on the left in figure 3.5 the normalised moment is given for increasing referencesurface curvature for the second phase of the deformation, i.e. the prestretching has stopped andthe sheet is now bent. The curves in the diagram represent the deformation paths with pre-strainsε

preref of 0, 0.05, 0.1, 0.15 and 0.2. The arrow indicates the direction of increased prestretch ε

preref .

The bending moment is normalised by the second moment of area, which is I = 112 bh3

0, the initialyield stress σy0 and half of the original thickness.

The reduction of the thickness of the sheet during prestretching leads to a lower bending momentof sheets which have undergone a larger amount of prestretch, as can be observed in figure 3.5.Furthermore, as a higher prestretch implies a lower hardening rate (figure 2.10), the slope of thecurves is lower for high ε

preref .

14

0 0.05 0.1 0.15 0.2 0.250

0.5

1

1.5

2

2.5

3

h0

2κref [−]

Mh

0

2Iσ

y0

[−]

εpreref

0 0.2 0.4 0.6 0.8 10.8

0.85

0.9

0.95

1

εpreref

t [−]

h h0

[−]

Figure 3.5: Normalised moment around the inner surface as a function of curvature (left) and the nor-malised thickness as a function of time (right).

In the diagram on the right in figure 3.5 the thickness is given as a function of time. Whereasthe thickness decreases during stretching, it nearly stops decreasing when the curvature starts toincrease. Note that some thinning still occurs during bending, by a maximum of 1.5%. This is aknown effect for large curvature bending (Zhu, 2006).

As a result of the virtually constant cross-section during bending, no necking is expected, be-cause the mechanism needed for necking, geometrical softening, is no longer active. Indeed thefinite element simulations do not show any necking during stretch-bending.

3.2.2 Sequential loading; bend-stretching

In bend-stretching the sheet is first prebent from t = 0 to t = 0.5 to a certain curvature h0

2 κpreref

at a constant curvature rate κref and subsequently the reference stretch εref is increased at aconstant rate εref from t = 0.5 to t = 1, while keeping the curvature constant. This implies thatafter t = 0.5 the curvature of the support is fixed and the sheet is pulled over the support. Infigure 3.6 the deformation observed in the finite element simulations is shown for the two stages ofdeformation. The colors represent the equivalent von Mises stress. The first picture is at t = 0.5,so immediately after bending, and the second at t = 1, where the simulation has ended and thestretching has stopped. At the latter stage, clearly a neck has formed at the symmetry axis. Notethat at the bottom surface the neck remains in contact with the support.

During prebending the cross-sectional area of the sheet does not decrease significantly. How-ever, the material is hardened and the amount of hardening depends on the radial position withinthe sheet, as can be seen in figure 3.7 where the equivalent plastic strain εp is given as function ofthe normalised original thickness y0

h0

for two stages during the deformation. The first curve is fort = 0.5, where the inhomogeneous equivalent plastic strain is caused by the bending. After thispoint the stretching starts and the equivalent plastic strain increases uniformly. The second curveis for t = 0.67, when the peak force is reached.

15

(a) t = 0.5 (b) t = 1

0 [MPa] 957 [MPa]σVM

Figure 3.6: Deformation observed in the sequential bend-stretching for a prebend of h0

2κ

pre

ref = 0.35.

0 0.2 0.4 0.60

0.2

0.4

0.6

0.8

1

εp [−]

y0

h0

[−]

t = 0.50 t = 0.67

Figure 3.7: Equivalent plastic strain distribution across the initial thickness for a prebend of h0

2κ

pre

ref = 0.35

at two stages during the stretch-bending.

On the left in figure 3.8 the normalised force is given with respect to the reference strain for theprebend curvatures h0

2 κpreref of 0.01, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3 and 0.35. The arrow indicates

the direction of increased prebend h0

2 κpreref . In the force-strain curve the peak force is marked by

a circle. Increasing the prebending results in an earlier peak force.

In the diagram on the right in figure 3.8 the normalised thickness is shown as a function ofthe reference strain. After the point the sheet starts necking the thickness, which is taken outsidethe neck, stops decreasing due to the fact the sheet starts to relax outside the neck.

The neck initiation is visualised in figure 3.9, in which η is given as function of the reference strain.

16

0 0.1 0.2 0.30

0.4

0.8

1.2

1.6

εref [−]

FA

0σ

y[−

]

κpreref

0 0.1 0.2

0.8

0.9

1

εref [−]

h h0

[−]

Figure 3.8: Normalised force (left) and normalised thickness (right) as a function of true reference strain.

The occurrence of 1, 10 and 20% thickness difference ∆h between inside and outside the neck isindicated by the diamond, triangle and upside down triangle respectively; these values correspondwith η = 0.01, 0.11 and 0.22. From this figure it can be concluded that with increasing prebendthe neck initiation is promoted. Comparing this trend with that of figure 3.8 (left) shows that itcorresponds with the earlier peak force observed for a higher level of prebend.

0 0.1 0.2 0.30

0.1

0.2

0.3

εref [−]

η[−

]

κpreref

η = 0.01η = 0.11η = 0.22

Figure 3.9: Relative thickness strain as a function of the reference strain for different amounts of prebend-ing κ

preref .

The correlation between peak force and the onset of necking can be made visible when the ob-tained peak force, peak moment and relative thickness strain are given in the deformation space,as has been done in figure 3.10. From this diagram it can be concluded that the peak force is agood predictor for neck initiation in bend-stretching.

17

0 0.1 0.2 0.30

0.1

0.2

0.3

εref [−]

h0 2κ

ref

[−]

strain pathmax(F )max(M)η = 0.01η = 0.11η = 0.22

Figure 3.10: Necking indicators in deformation space for the bend-stretching deformation.

3.2.3 Simultaneous loading

During simultaneous loading, the curvature and reference surface strain are increased linearly dur-ing the time interval from t = 0 to t = 1 at a constant curvature and strain rate κref and εref

respectively. A parameter γ is introduced which indicates the relative amount of bending andwhich is defined as

γ =h0κref

2εref

(3.6)

A small γ implies a small fraction of bending deformation whereas a large γ leads to a bending-dominated deformation path.

In figure 3.11 the deformation and stress observed in the finite element simulation for γ = 0.4is given at two stages. The first state is at t = 0.5 and the second is at t = 1. In the latter a neckhas formed at the symmetry plane. Note that during the simulation the curvature of the supportand thus the curvature of the bottom of the sheet is increased while the sheet is simultaneouslystretched.

In figure 3.12 the equivalent plastic strain distribution across the initial thickness of the sheetis given for three different stages during deformation. The average deformation as well as its slopeincrease monotonically as the sheet is stretched and bent. The equivalent plastic strain is notincreased uniformly, as seen in the bend-stretching, but is dependent on the radius within thesheet due to the simultaneous curvature increase.

18

(a) t = 0.5 (b) t = 1

0 [MPa] 957 [MPa]σVM

Figure 3.11: Deformation for the simultaneous loading using γ = 0.4.

0 0.1 0.2 0.3 0.40

0.2

0.4

0.6

0.8

1

εp [−]

y0

h0

[−]

t = 0.125

t = 0.25 t = 0.5

Figure 3.12: Equivalent plastic strain as a function of the normalised initial thickness coordinate for thesimultaneous loading of γ = 0.4 at t = 0.125, t = 0.25 and t = 0.5.

On the left in figure 3.13 the normalised force as a function of the reference strain is given for arange of γ from 0 to 5; the peak force is indicated by a circle. The arrow indicates the directionof increased γ. On the right the corresponding normalised moment around the inner surface isgiven as a function of the true reference strain. Here the peak is indicated with a square. Due tothe fact that in this figure the moment is given as a function of the reference strain, one may bemisled by the magnitude of the deformation, but the lowest curve in the figure, γ = 5, is wherethe curvature is increased to h0

2 κref = 0.25, which is considerable. Increasing γ leads to an earlierpeak force and peak moment in terms of the reference strain.

19

0 0.1 0.2 0.3 0.4 0.50

0.4

0.8

1.2

1.6

εref [−]

FA

0σ

y[−

]

γ

0 0.05 0.1 0.15 0.20

1

2

3

4

5

εref [−]

γ

Mh

0

2Iσ

y0

[−]

Figure 3.13: Normalised force (left) and normalised moment (right) as a function of the reference strainfor the simultaneous loading paths.

In the left diagram in figure 3.14 the normalised thickness as a function of the reference strainεref is given upon to the point where 1% relative thickness strain is reached.

0 0.1 0.2 0.30.7

0.8

0.9

1

εref [−]

h h0

[−]

0 0.1 0.2 0.3 0.4 0.50

0.05

0.1

0.15

0.2

0.25

εref [−]

η[−

]

γ

η = 0.01η = 0.11η = 0.22

Figure 3.14: Normalised thickness (left) and relative thickness strain (right) as a function of the referencestrain.

The neck initiation can also be visualised by plotting η versus the strain of the reference surface,as has been done in the right diagram in figure 3.14. From this graph it can be seen that in-creasing γ leads to a retardation of the neck initiation. This trend is opposite to that observedin bend-stretching loading path and it also does not follow the trend of the peak force or peakmoment. This means that for the simultaneous loading the peak force is not a good predictor forthe initiation of a neck.

So the introduction of extra hardening in some parts of the sheet, caused by the bending, providesa stabilizing effect. This trend is also visible when the obtained peak force, peak moment andrelative thickness strain are given in the deformation space for a range of γ, which can be seenin figure 3.15. Increasing γ results in a retardation of neck initiation op to a point, γ > 0.71,where in this diagram necking does not occur at all. In this region a fracture development seemsmore plausibel as a failure criteria. But this mechanism is not implemented in the finite elementsimulations and thus cannot be observed.

20

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

εref [−]

h0 2κ

ref

[−] strain path

max(F )

max(M)

η = 0.01

η = 0.11

η = 0.22

Figure 3.15: Necking indicators given in deformation space.

This figure clearly demonstrates that for the simultaneous deformation path is clearly visible, thepeak force nor peak moment is a good prediction for the neck initiation.

3.3 Discussion

From the different loading paths considered so far we can learn that the application of curvatureto a sheet may increase or decrease its stretchability, depending on whether it is applied in a si-multaneous or sequential fashion. In the latter case, even the order of application is of importance.

In stretch-bending the hardening during stretching takes place uniformly throughout the sheetand the thickness of the sheet decreases. However, during the subsequent bending the thicknesshardly changes anymore and no necking therefore occurs.

In bend-stretching the sheet is hardened during the bending phase. As a consequence the harden-ing rate during stretching is lower than for a straight sheet without prebending. Since the thicknessdecrease during stretching is unaffected by the prior bending, this leads to earlier necking of thesheet. Actually the bend-stretching deformation can be compared with a simple tensile test ofsheet which has been pre-hardened.

Simultaneous bending and stretching results in a higher hardening rate compared with purestretching at the top of the sheet, but a lower hardening at the bottom. The thickness evolu-tion is relatively unaffected by the bending component. The peak force nor peak moment is agood neck initiation predictor. For this reason an analytical will be made in following chapter tofind a better predictor.

21

Chapter 4

Semi-analytical model

In the first part of this chapter (sections 4.1 to 4.4) a semi-analytical model of combined stretch-ing and bending is presented. We develop this model for a twofold purpose. On the one handit delivers expressions for the force and moment for the deformation as characterised by the twoparameters ε and κ. On the other hand a generalised bifurcation condition is derived from them,to be able to predict neck initiation for the simultaneous deformation. Due to the large strainsthe model must be geometrically non-linear. The material behavior is assumed to be rigid-plastic,so the elastic contribution is neglected. The deformation is considered to be uniform along thesheet, i.e. there is no variation of deformation and stress in the tangential direction.

The predictions made by the semi-analytical model are confronted with the numerical resultsfor the simultaneous deformation. Subsequently the sensitivity of the bifurcation with respect tothe parameters and hardening behavior used in the material model is studied. Reaching a certainstate in the deformation space, with corresponding reference stretch and curvature, can be doneusing both sequential and simultaneous deformation. Using a simple example the differences interms of necking between the used deformation paths is shown.

4.1 Kinematics

In figure 4.1 the deformation map between an deformed, current and an undeformed, referencestate is depicted. The kinematic assumptions made are a plane strain condition in z-direction,isochoric deformation, each cross-section remains straight and perpendicular to the sheet faces andthe top face is able to move freely whereas the bottom surface follows the curvature and strainimposed by the support. The deformation of the sheet is characterised by the curvature of itsbottom surface, κ, and the tangential strain of the bottom surface, ε.

In the undeformed reference state a position vector ~X of a material point with respect to theCartesian basis {~ex, ~ey, ~ez}, depicted in figure 4.1, is defined as

~X = x0 ~ex + y0 ~ey + z0 ~ez (4.1)

A cylindrical basis is used in the deformed current state. Its origin is fixed and the sheet movestowards it as its curvature increases. The position vector ~x of a material point is denoted in thecylindrical basis {~er, ~eθ, ~ez} as

~x = r ~er(θ) (4.2)

22

f

~ey

~ex~ez x0

y0

h0

(a) reference state

~eθ(θ)

θ r1κ

~er(θ)

~ez

1κ

+ h

(b) current state

Figure 4.1: The sheet considered in the undeformed reference state and the deformed current state.

where r and θ are functions of the initial coordinates x0 and y0 of the material point. The angle θ

can be written as the current length of the initial segment x0 of the bottom surface , x0eε, divided

by its radius 1κ

in the current state. This leads to

θ = κeεx0 (4.3)

An expression for r may be derived by making use of the conservation of volume of the shadedarea in figure 4.1,

x0y0 = θ

∫ r

1

κ

ρ dρ (4.4)

Substituting equation (4.3) into equation (4.4) and rewriting leads to

r =1

κ

√

1 + 2κy0e−ε (4.5)

Rewriting above expression results in an expression for the current thickness as a function of theinitial thickness

h =1

κ

(

√

1 + 2κh0e−ε − 1)

(4.6)

Substituting (4.3) and (4.5) in (4.2) allows one to express the current position of a material point~x, entirely in terms of its coordinates x0, y0 in the reference configuration.

The velocity of the material point can be determined by differentiating (4.2)

~v = ~x = r ~er + r θ ~eθ (4.7)

23

The velocity vector has two components, namely the rate of change of the radius. Differentiatingthe radius r according to (4.5) leads to

r = − 1

2κ

[(

κr − 1

κr

)

ε +

(

κr +1

κr

)

κ

κ

]

(4.8)

And differentiating the angle θ given by (4.3) to

θ = θ

(

ε +κ

κ

)

(4.9)

Substituting equations (4.8) and (4.9) into equation (4.7) finally leads to the velocity

~v = − 1

2κ

[(

κr − 1

κr

)

ε +

(

κr +1

κr

)

κ

κ

]

~er(θ) + r θ

(

ε +κ

κ

)

~eθ(θ) (4.10)

Taking the gradient in the current configuration and symmetrising the result yields the rate ofdeformation tensor

D =1

2

(

~∇~v + ~∇~vT)

=1

2

[(

1 +1

κ2r2

)

ε +

(

1 − 1

κ2r2

)

κ

κ

]

(~eθ~eθ − ~er~er) (4.11)

From the rate of deformation tensor the equivalent strain rate can finally be determined as

˙ε =

√

2

3D : D =

1√3

[(

1 +1

κ2r2

)

ε +

(

1 − 1

κ2r2

)

κ

κ

]

(4.12)

4.2 Statics

Consider the equilibrium of a small through-thickness segment of the sheet. In figure 4.2 aschematic representation of such a segment in the current state is depicted. In it, we have assumedθ = 0 for convenience. The stresses exerted on it by the remainder of the sheet are characterisedby forces F and F + ∆F and moments M and M + ∆M around the bottom surface of the sheet.Furthermore at the bottom a pressure p is applied by the support to maintain the curvature κ.Expressions for the tangential force F and the bending moment M as a function of the deformationparameters ε and κ can be obtained by integrating the expression for the tangential stress as

F =

∫ 1

κ+h

1

κ

σθθ dr (4.13)

M =

∫ 1

κ+h

1

κ

(

r − 1

κ

)

σθθ dr (4.14)

These expressions are further elaborated in section 4.3.

24

∆θ

p

F + ∆F

FM + ∆M

M

1κ

1κ

+ ∆ 1κ

o

Figure 4.2: Equilibrium of moment and forces working on a cylindrical segment of the sheet.

For a small segment of angle ∆θ (figure 4.2) translational equilibrium in horizontal direction tellsus that

∆F = 0 (4.15)

Dividing by ∆θ and taking the limit of ∆θ going to zero leads to

lim∆θ→0

∆F

∆θ=

∂F

∂θ= 0 (4.16)

i.e. the tangential force F must be constant along the circumference of the sheet.

Rotational equilibrium around o similarly leads to

∆

(

F

κ

)

+ ∆M = 0 (4.17)

which after dividing by ∆θ and taking the limit of ∆θ going to zero yields

lim∆θ→0

∆(Fκ) + ∆M

∆θ=

∂

∂θ

(

F

κ+ M

)

= 0 (4.18)

This equation shows that Fκ

+ M should also be constant.

Vertical equilibrium, finally, leads to a relation between the force F and the pressure p which reads

F =p

κ(4.19)

The static equilibrium equations in tangential direction (4.16) and (4.18), can be written in ma-trix form. If we introduce a column of dimensionless parameters which determine the deformation

25

equations as e˜

= [ε h0

2 κ]T , equilibrium can be written as

∂Φ˜

∂θ= 0

˜(4.20)

where Φ˜

is defined as

Φ˜(e˜) =

[

F2Fκh0

+ 2Mh0

]

(4.21)

4.3 Tangential force and bending moment

A convenient way to obtain expressions for the tangential force and moment in terms of ε andκ is to formulate the internal and external power associated with the deformation process for asegment of sheet as depicted in figure 4.3.

o

θ

p

σθθ

σθθ

Figure 4.3: Segment of sheet with stress distribution on its faces.

Conservation of energy states that the internal power Pi must equal the external power Pe andthus Pi = Pe = P . The external power is given by the traction vector acting on the boundary S

of the segment multiplied by its velocity ~v. Per unit of angle θ this gives

Pe =1

θ

∫

S

~n · σ · ~v dS (4.22)

The vertical edge in figure 4.3 coincides with the symmetry plane and therefore the velocity iszero on it. The top edge is free to move and therefore does not experience any traction. Equation(4.22) may therefore be reduced to

P =1

θ

∫ 1

κ+h

1

κ

σθθ(r)vθ(r, θ) dr +1

θ

∫ θ

0

σrr

(

1

κ

)

vr(r)1

κdθ (4.23)

Substituting the radial and tangential velocity from equation (4.10) into the above equation leadsto

P =

∫ 1

κ+h

1

κ

rσθθ(r) dr

(

ε +κ

κ

)

− p

κ2

κ

κ(4.24)

26

Using the expressions for the force (4.13) and moment (4.14), as well as relation (4.19) between p

and F , finally the external power can be written in terms of the force and moment as

P =

(

F

κ+ M

)

ε +M

κκ (4.25)

This result shows that general expressions for the moment and force are

M = κ∂P

∂κand F = κ

∂P

∂ε− Mκ (4.26)

The internal power per unit of angle can be calculated by integration of the plastic work rate overthe volume V of the segment of sheet as

Pi =1

θ

∫

V

σ ˙ε dV (4.27)

Due to the uniformity in tangential direction and the fact that the out-of-plane width is set tounity, equation (4.27) can be simplified to

P =

∫ 1

κ+h

1

κ

σ ˙εr dr (4.28)

Substituting the equivalent strain rate (4.12) gives

P =1

κ√

3

∫ 1

κ+h

1

κ

σκr dr

(

ε +κ

κ

)

+1

κ√

3

∫ 1

κ+h

1

κ

σ

κrdr

(

ε − κ

κ

)

(4.29)

The above expression is formulated on the current configuration and therefore requires the currentequivalent stress σ as a function of r. The current stress, however, depends on the deformation his-tory of a material point - not a spatial point. It is therefore convenient to transform the integrals in(4.29) to the reference configuration. This can be done by substituting equation (4.5) and results in

P =1

κ√

3

∫ h0

0

σe−ε dy0

(

ε +κ

κ

)

+1

κ√

3

∫ h0

0

σ

eε + 2κy0dy0

(

ε − κ

κ

)

(4.30)

where σ can now be evaluated by integrating the hardening law obtained in section 2.5 for amaterial point y0. By integration of equation (4.12) the equivalent strain can be obtained. Therate of change of the bottom surface strain and curvature, ε and κ, are assumed to be constant.Integrating (4.12) and rewriting to reference state leads to

ε =1√3

(ε + ln(eε + 2κy0)) (4.31)

In the numerical model however the reference surface strain and curvature rate, εref and κref , areassumed to be constant. This leads to some inconsistency, which however turn out to have only a

27

marginal effect on the results.

Evaluating the expressions in (4.26) for the above power relation results in the tangential forceand moment

F =2√3

∫ h0

0

σ

eε + 2κy0dy0 (4.32)

M =1√3

∫ h0

0

σ

κeεdy0 −

1

2κF (4.33)

For the power law hardening considered here, the above expressions cannot be evaluated analyti-cally and thus are evaluated numerically using a trapezoidal method (Adams, 1999).

4.4 Bifurcation condition

In a simple tensile test the bifurcation condition according to Considere is found by searchingfor the point where the axial force F becomes insensitive to variations of strain, i.e. ∂F

∂ε= 0, as

explained in section 2.3. As we have seen in the previous chapter, for a more complex deforma-tion where a curvature is introduced this condition does not necessarily hold anymore. Here wederive a generalised bifurcation condition by introducing perturbations of the strain and curvature.

For the deformation considered here, equilibrium is governed by the generalised force Φ˜

- seeequation (4.20). Bifurcation may occur when a second deformation path e

˜+ ∆e

˜can be found,

next to the primary solution Φ˜(e˜), which does not affect Φ

˜. Similarly as in section 2.3, in the limit

of ∆e˜

going to zero we have

lim∆e˜→0

Φ˜(e˜

+ ∆e˜) − Φ

˜(e˜)

∆e˜

≃ K¯∆e

˜= 0

˜(4.34)

for ∆e˜6= 0

˜, where K

¯is the sensitivity of Φ

˜with respect to e

˜:

K¯

=

[

K11 K12

K21 K22

]

=∂Φ

˜∂e˜

=

[

∂F∂ε

∂2F∂(κh0)

∂∂ε

(

2Fκh0

+ 2Mh0

)

∂2∂(κh0)

(

2Fκh0

+ 2Mh0

)

]

(4.35)

A nontrivial solution ∆e˜

implies that the matrix K¯

must become singular. This means that bifur-cation may occur when the determinant of the matrix K

¯becomes equal to zero.

In order to determine expressions for the elements Kij of K¯

recall the expressions for the force andthe moment expressed in the reference state, (4.32) and (4.33). The variations of the equivalentstress with respect to the strain and curvature can be written as

∂σ

∂ε=

∂σ

∂ε

∂ε

∂ε=

2√3

∂σ

∂ε

[

eε + κy0

eε + 2κy0

]

(4.36)

∂σ

∂κ=

∂σ

∂ε

∂ε

∂κ=

2√3

∂σ

∂ε

[

y0

eε + 2κy0

]

(4.37)

28

With this information we obtain expressions for the elements of K¯

which read

K11 =2√3

∫ h0

0

1√3

∂σ

∂ε

[

2(eε + κy0)

(eε + 2κy0)2

]

− σ

[

eε

(eε + 2κy0)2

]

dy0 (4.38)

K12 =2√3

∫ h0

0

1√3

∂σ

∂ε

[

2y0

h0

2 (eε + 2κy0)2

]

− σ

[

2y0

h0

2 (eε + 2κy0)2

]

dy0 (4.39)

K21 =2√3

∫ h0

0

1√3

∂σ

∂ε

[

2(eε + κy0)2

h0

2 κeε(eε + 2κy0)2

]

− σ

[

eε + 2κy0 + 2y20e

−εκ2

h0

2 κ(eε + 2κy0)2

]

dy0 (4.40)

K22 =2√3

∫ h0

0

1√3

∂σ

∂ε

[

2y0(1 + κy0)h2

0

4 κ(eε + 2κy0)2

]

− σ

[

2y0(2 + y0κe−ε)h2

0

4 κ(eε + 2κy0)2

]

dy0 (4.41)

In the semi-analytical model the bifurcation definite integrals (4.38) to (4.41) have to be evaluatednumerically due to the fact no explicit integration is possible. The integration is done using atrapezoidal method (Adams, 1999). The singularity of matrix K

¯is detected by inspecting the sign

of det(K¯).

4.5 Force and moment curves

In figure 4.4 the force and moment curves are given for γ = 0.25 as obtained a numerical evaluationof equations (4.32) and (4.33). The dashed curves represent the numerical response obtained fromthe finite element simulations. On the left the normalised force as a function of the reference strainis given and on the right the normalised moment; the crosses indicate bifurcation as predicted bythe criterion det(K

¯) = 0.

0 0.1 0.2 0.3 0.40

0.4

0.8

1.2

1.6

εref [−]

FA

0σ

y0

[−]

analyticalnumericalbifurcation

0 0.1 0.2 0.3 0.40

1

2

3

4

5

εref [−]

analyticalnumericalbifurcation

Mh

0

2Iσ

y0

[−]

Figure 4.4: Confrontation of numerical and analytical force (left) and moment (right) curve for simulta-neous loading with γ = 0.25 .

At the end the slopes of the numerical curves deviate from the analytical ones. This is due to thefact the sheet starts necking. This is not captured by the analytical model, which assumes uniformdeformation. In the curves the point of bifurcation, obtained by numerical evaluation of the setof integrals in equations (4.38)-(4.41), are marked by a cross. The analytical model predicts theforce and moment curves quite accurately.

29

In the diagram on the left in figure 4.5 the normalised thickness as a function of the referencestrain is given. Also here the dashed curve represent the numerical response and is confrontedwith the analytical obtained thickness evolution from equation 4.6. The analytical model predictsthe thickness quite accurately. The relative thickness strain extracted from the FE simulationsis given as a function of the reference strain is given in the right diagram. In this figure thebifurcation point clearly lies at the neck initiation, which indicates the bifurcation condition is agood predictor of the onset of necking.

0 0.1 0.2 0.3 0.40.5

0.6

0.7

0.8

0.9

1

εref [−]

h h0

[−]

analyticalnumberical

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

εref [−]

η[−

]

ηbifurcation

Figure 4.5: Normalised thickness (left) and relative thickness strain (right) as a function of the referencestrain for simultaneous loading with γ = 0.25.

4.6 Necking diagram

The analytical bifurcation points for all simultaneous deformation paths are presented in thedeformation space as defined before in figure 4.6. The numerical data presented in this figure forcomparison is that of figure 3.15.

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

εref [−]

h0 2κ

ref

[−]

bifurcationcondition

strain path

η = 0.01

η = 0.11

η = 0.22

Figure 4.6: Necking diagram for the simultaneous loading.

For the tensile test, γ = 0, where the εref -axis is followed, the 1% relative thickness strain andbifurcation point coincide. If a small curvature is introduced this relative thickness strain are aswell as the 10 and 20% relative thickness strains retarded. Increasing the curvature even further, soincreasing γ, the 1% relative thickness strain seem to converge to the bifurcation point again. This

30

trend is not well understood. Overall the bifurcation condition proposed earlier in this chaptershows good agreement with the occurrence of necking as signalled by the relative thickness strainand thus seems to predict the neck initiation. It also picks up the trend that adding a bendingcomponent to he reference strain increases the deformation limit, as well as that no necking at allis observed for γ > 0.71. In this regime an alternative criterion (or multiple criteria) are neededwhich predict the mechanisms governing the deformation limit here.

4.7 Material parameter variation

Because all the simulations so far where performed using the same set of parameters for the mate-rial model it is interesting to study the sensitivity of the bifurcation condition to these parameters.This is done here using the semi-analytical model for the simultaneous deformation. For threedifferent sets of parameters next to the previously used set. An overview is given in table 4.1. Inevery case the material model has been varied by changing one of the parameters considerably.

Table 4.1: Variation of the parameters used in the material model.

σy0 [MPa] K [MPa] n [-]

case 1 284 450 0.4case 2 284 450 0.3case 3 150 450 0.4case 4 284 200 0.4

The corresponding yield stress as a function of the equivalent plastic strain is given in figure 4.7.The previously used material model is represented by case 1. In case 2 the hardening exponentis reduced resulting in a higher initial hardening. Case 3 is changed with respect to the originalmaterial model by significantly lowering the initial yield stress. Finally, in case 4 the hardeningparameter K is lowered, resulting in much lower hardening over the whole domain.

0 0.2 0.4 0.6 0.8 10

200

400

600

800

εp [−]

σy

[MPa]

case 1case 2case 3case 4

Figure 4.7: Yield stress as a function of the equivalent plastic strain for the different sets of parametersused in the material models in the sensitivity analysis.

The points of bifurcation are evaluated for a range of γ from 0 to 5 for every parameter set. Thesepoints form curves in the deformation space as can be seen in figure 4.8. From this figure it can beconcluded that the use of different hardening relations has a considerable effect on the initiationof necking. Parameter sets with high hardening rates, i.e. where the stress is increased at a higherrate, result in a retardation of necking compared with sets using lower rates. Lowering the initial

31

yield stress but keeping the rate of hardening the same with respect to the equivalent strain resultsin a retardation of neck initiation.

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

εref

h0 2κ

ref

case 1

case 2 n ↓case 3 σy0 ↓case 4 K ↓

Figure 4.8: Effect of material model variations on the necking diagram for the simultaneous deformation.

Using the finite element model discussed in chapter 3 the simultaneous deformation of γ = 0,which is a tensile test, is simulated using above sets of parameters to gain more insight in the neckinitiation. On the left in figure 4.9 the strain-force curves are given. The peak force is markedwith a circle. The corresponding normalised thickness as a function of the reference strain is givenon the right.

0 0.1 0.2 0.30

40

80

120

εref [−]

F[M

N]

case 1case 2case 3case 4

0 0.1 0.2 0.30.7

0.8

0.9

1

εref [−]

case 1case 2case 3case 4

h h0

[−]

Figure 4.9: Force (left) and normalised thickness (right) as a function of the strain for γ = 0 usingdifferent sets of parameters in the material model.

As can be seen from the thickness curves, the decrease in thickness is coupled to the referencestrain, εref . The necking initiation is only dependent on the chosen hardening relation since thearea decreases at the same rate. The hardening rate is the lowest in case 4 so this will fail first,followed by case 2 and 1. For low equivalent plastic strains the hardening rate of case 2 is higherthen case 1, but for higher equivalent plastic strain the hardening rate of case 1 is higher. For thisreason case 2 fails earlier than case 1. Case 4 fails last, which can be explained by the low initialyield stress, which results in a relatively high amount of hardening.

For the simple tensile test, assuming rigid-plastic material and a plane strain state, the forceas a function of the strain (setting κ equal to zero in equation (4.32)) can be approximated by

32

F =2√3e−εA0σ(εp) (4.42)

As shown before in section 2.3, the point of bifurcation can be found by determining the strainat which the condition ∂F

∂ε= 0 is valid. Differentiating equation (4.42) with respect to the axial

strain ε and setting the result equal to zero leads to

(n − εp)εn−1p =

σy0

K(4.43)

The bifurcation points can finally be found by solving function (4.43) for the axial strain for agiven set parameters of table 4.1. The points of bifurcation thus obtained coincide with the pointsof peak force obtained with the finite element simulations. The same points are also recoveredfrom the generalised bifurcation criterion for γ = 0 and figure 4.8 shows that the trend observedfor the different sets of material data in pure tension persists as a bending component is added tothe deformation.

4.8 Linear vs. nonlinear hardening

The difference between linear hardening and nonlinear hardening is compared in this subsectionfor the simultaneous deformation. This is done to compare the effect of different hardening rates atthe top and bottom surface. The linear hardening is obtained by setting the hardening exponentin the material model equal to unity. By doing so the hardening rate becomes independent of theequivalent strain. The difference in parameters of the material model is given in table 4.2. Againusing the semi-analytical criterion the points of bifurcation are obtained for a range of γ from 0to 5. These points form curves in the deformation space which are given in figure 4.10.

Table 4.2: Material model parameters used in the comparison between nonlinear and linear hardening.

σy0 [MPa] K [MPa] n [-]

nonlinear 284 300 0.4linear 284 300 1

Although the initial hardening rate is much larger for the nonlinear hardening case necking isinitiated earlier. The linear hardening law gives rise to a stronger retardation of neck initiation,which is the result of the constant hardening rate over the full range of equivalent plastic strain.This effect is increased for higher values of γ, resulting in a larger difference in equivalent plasticstrain rate at the top surface compared with the bottom surface. This does not affect hardeningrate of the linear hardening law but has a large influence on the nonlinear hardening as the ratedecreases for large equivalent plastic strain values.

33

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

εref

h0 2κ

ref

nonlinearlinear

Figure 4.10: Comparison between nonlinear and linear hardening during simultaneous loading in thedeformation space.

4.9 Deformation path dependence; an example

To emphasize the relevance of path dependence in the realisation of a state in the deformationspace without the initiation of necking, three different deformation paths to one single final stateare considered. The point within the deformation space is marked by the dot in figure 4.11 andhas a normalised curvature of h0

2 κref = 0.25 and reference stretch of εref = 0.25. Three differentdeformation paths are used to reach this state, namely by employing both sequential and thesimultaneous deformation. The differences between the proposed deformation paths are depictedin figure 4.11.

εref

h0

2κref

0.25

0.25

Figure 4.11: Proposed routings in the deformation space using different loadings; the dot represents therequired state.

On the left side in figure 4.12 the normalised thickness as a function of the reference strain isgiven. Both the sequential as the simultaneous loadings show the same thickness reduction ratewith respect to εref . The fact that the curve of the simultaneous loading reaches a lower thick-ness is due to the fact that both sequential loadings initiate a neck before the desired state isreached. On the right side the corresponding relative thickness strains are given as a function ofthe imposed reference strain. From this figure it can be seen directly that both sequential loadingsresult in necking before the desired state is reached, whereas the relative thickness strain of thesimultaneous loading remains approximately zero. So only the simultaneous deformation pathreaches the desired state without neck initiation. This is visualised in figure 4.13 where the resultsare shown in the deformation space.

34

0 0.1 0.20

0.2

0.4

0.6

0.8

1

εref [−]

h h0

[−]

simultaneoussequential; bend stretchingsequential; stretch bending

0 0.1 0.20

0.04

0.08

0.12

0.16

simultaneoussequential; bend stretchingsequential; stretch bending

εref [−]

η[−

]

Figure 4.12: Normalised area (left)and relative thickness strain (right) as a function of the referencestrain for the different proposed routings.

0 0.1 0.20

0.1

0.2

εref

h0 2κ

ref

strain path

bifurcation

required state

Figure 4.13: Proposed routings in the deformation space using different loading paths.

The arrows in the figure point in the direction of the followed deformation. In the stretch-bendingdeformation the bifurcation point is depicted by the cross. In the bend-stretching deformation thepeak force is marked by the cross. Under the bend-stretching loading the sheet necks earlier thanin the stretch-bending in terms of the reference strain. This is due to the hardening initiated bythe bending prior to the stretching. The stretch-bending even results in failure before the bendingcan take place. The introduction of a strain gradient in the sheet by the simultaneous bendingduring stretching delays the initiation of necking sufficiently to reach the desired state.

4.10 Discussion

The analytically obtained bifurcation points show good agreement with the neck initiation ob-served in the numerical simulations. The introduction of simultaneous curvature increase duringstretching results in strain gradients trough the thickness of the sheet, which causes a strong delayof necking in terms of applied reference strain.

35

Conclusion and outlook

An analysis of sheet necking under combined stretching and bending has been performed. In orderto gain more insight in the response of sheet-metal during such loading, the deformation is dividedinto bending and stretching contributions. The influence of the deformation path on in particularthe initiation of necking is studied in detail by the use of a numerical model. Within the defor-mation the order of curvature and stretching increase played a crucial role. From the numericalresults we learn that the increase in curvature improves or reduces the stretchability of the sheet.During stretch-bending no necking is observed at all, which is caused by the constant thicknessduring bending. In bend-stretching the introduced prebend promotes neck initiation during thefollowing stretching phase. This is caused by the hardening introduced in the bending phase. Inthis deformation path the peak force gives a good prediction for the initiation of a neck. Stretchingwith simultaneous curvature increase delays the initiation of necking considerably compared tosolely stretching. The deformation is stabilised as a result of the strain gradient introduced bythe continuously increased bending. Here the peak force and moment did not provide a good neckinitiation prediction.

For the simultaneous loading a semi-analytical model has been constructed. This model notonly delivered accurate force and moment relations, but also delivered a generalised necking con-dition derived from the bifurcation condition. Evaluation of this generalised bifurcation condition,for different amounts of curvature increase, showed good agreement with the numerical findings.One of the main goals of this work was to predict the initiation of necking during stretching ofsheet-metal under large curvature bending. This goal is reached by the delivery of this generalisednecking condition which predicts the initiation of necking.

With respect to the long-term objective to develop an enhanced shell element in which the limita-tion of the conventional element is removed, the first necessity is obtained, namely the predictionof neck initiation. However the semi-analytical bifurcation condition must be implemented in theelement. In the proposed bifurcation condition the elements of matrix K

¯, the definite integrals

of (4.38) to (4.41), must be evaluated numerically and the sign of K¯

inspected. The elementsof K

¯depend on the used material model together with the strain and curvature of the bottom

surface and the initial thickness of the sheet. The strain and curvature can be taken from thegeometry of the deformed element. Subsequently when the point of bifurcation is reached duringa deformation, adaptation between the element is needed.

In figure 4.14 a possible solution for the post-necking is shown schematically. Two shell ele-ments are shown with an enhanced shell element in between which forms the coupling betweenthe elements. It can be seen as an extra element with initial a length of zero. The adjacent nodescan be seen as one before the bifurcation is reached. When the sign of K

¯changes, the enhanced

element is able to open and form the coupling between the elements; the coupling between F andM on the one hand and the local displacements, u1 and u2, and local rotations, ϕ1 and ϕ2 on theother.

36

ϕ1, u1 ϕ2, u2

F, M F, M

l∆u, ∆ϕ

F, M

Figure 4.14: Four conventional shell elements with enhanced shell element in between (left) and force-displacement and moment-angle curve (right).

The enhanced shell element substitutes the remaining force and moment curve (dashed line inthe figure) after the point of bifurcation is reached between the thin lines. The conventional shellelements outside the enhanced shell element behave as normal.

37

Bibliography

Adams, R., 1999. Calculus: a complete course, 4th Edition. Ron Doleman.

Broekhuijsen, J., 2003. Ductile failure and energy absorption of y-shape test section. Master’sthesis, Delft University of Technology, Netherlands.

Charpentier, P., 1975. Influence of punch curvature on the stretching limits of sheet steel. Metal-lurgical transactions 6A, 1665–1669.

Ehlers, S., Broekhuijsen, J., Alsos, H., Biehl, F., Tabri, K., 2008. Simulating the collision responseof ship side structures:a failure criteria benchmark study. International Shipbuilding Progress55, 127144.

Fenner, R., 1999. Mechanics of solids. CRC Press, Florida.

Nadai, A., 1950. Theory of flow and fracture of solids. Vol. I. McGraw-Hill.

Parisch, H., 1995. A continuum-based shell theory for non-linear applications. International Jour-nal for Numerical Methods in Engineering 38, 1855–1883.

Rubino, V., Desphande, V., Fleck, N., 2008. The collapse response of sandwich beams with ay-frame core subjected to distributed and local loading. International Journal of MechanicalSciences 50, 233–246.

Shi, M., Gerdeen, J., 1991. Effect of strain gradient and curvature on forming limit diagrams foranisotropic sheets. Journal of Materials Shaping Technology 9 (4), 253–268.

Vredeveldt, A., Wolf, M., Broekhuisen, J., Gret, E., 2004. Safe transport of hazardous cargothrough crashworthy side structures. ICCGS.

Yoshida, M., Yoshida, F., Konishi, H., Fukumoto, K., 2005. Fracture limits of sheet metals understretch bending. International Journal of Mechanical Sciences 47, 1885–1896.

Zhu, H., 2006. Large deformation pure bending of an elastic plastic power-law-hardening wideplate: Analysis and application. International Journal of Mechanical Sciences 49, 500–514.

38

Appendix A

Sensitivity

The finite element model must be studied with respect to the sensitivity to the imperfectionmade and the number of elements. The sensitivity of the plane strain elements with respect toelement density and the imperfection which triggers the localisation will be shown for three typesof loading, namely a tensile test, a bending test and a simultaneous loading test. The finite elementmodel consists of plane strain eight-node quadrilateral elements with reduced integration. Thechoice of using higher order elements is to prevent possible locking and to describe the bendingmore accurately. Conventional elements can produce volumetric locking for nearly incompressiblebehavior. The reduced integration is used to reduce computation costs.

A.1 Mesh density

First the mesh density sensitivity for the stretching deformation will be investigated where astretch of εref = 0.3 is imposed with constant stretch rate εref and the curvature will be keptzero, h0

2 κref = 0. From a rather coarse mesh, 10 x 4 = 40 elements, the sheet will be refined to afiner mesh throughout a number of refinements. With every refinement the number of elements isquadrupled.

0 0.1 0.2 0.30

0.4

0.8

1.2

1.6

εref [−]

FA

0σ

y0

[−]

ne = 40ne = 160ne = 640ne = 2560ne = 10240

0.25 0.30.8

1.2

1.6

εref [−]

FA

0σ

y0

[−]

Figure A.1: Mesh dependence for a tensile test with εref = 0.3 and h0

2κref = 0 comparing different

element densities; on the left side the normalised force is given and on the right an enlargement of the endof the curve.

In figure A.1 the results are shown. Post peak behavior does not depend on the the mesh densitiy.From the force peak on a small mesh dependence is visible. The elements near the symmetry line

39

will enter the softening region first and start necking by a more rapidly decrease of area. Con-tinuously increasing the mesh density results in convergence of the force stretch curves into onesolution. Refining the mesh more than ne = 2560 is unnecessary as one step further only improvesthe solution marginally and is outweighed by the increase in computation cost.

The variable η is introduced to characterise the relative thickness strain between the thicknessoutside the neck, hs, and inside the neck, hn, and is defined as

η = log

(

hs

hn

)

(A.1)

In the following figure the relative thickness strain η is given with respect tot the reference surfacestrain. From this figure can be concluded that for every mesh density necking starts with the sameamount of stretch. Only the thickness of the sheet inside the neck, hn, differs between the meshdensities. Also here the convergence to one solution is clearly visible. So using 2560 elementssuffice for the tensile test.

0 0.1 0.2 0.30

0.5

1

1.5

εref [−]

η[−

]

ne = 40ne = 160ne = 640ne = 2560ne = 10240

Figure A.2: Mesh dependence for tensile test with εref = 0.25 and h0

2κref = 0 comparing different

element densities; relative thickness strain with respect to the reference surface strain is shown.

The following loadcase is the bending case where the reference surface curvature is increased upto h0

2 κref = 0.25 with constant rate κref as where the reference surface stretch is imposed to bezero. In figure A.3 the bending moment with respect to the reference surface is given for increasingmesh densities. The moment is normalised by multiplication with the half thickness of the sheetdivided with the second moment of area multiplied with the initial yield stress.

40

0 0.1 0.20

1

2

3

h0

2κref [−]

Mh

0

2Iσ

y0

[−]

ne = 40ne = 160ne = 640ne = 2560

0.15 0.252.5

3

h0

2κref [−]

Mh

0

2Iσ

y0

[−]

ne = 40ne = 160ne = 640ne = 2560

Figure A.3: Mesh dependence for bending loading case comparing different element densities; normalisedbending moment as a function of the normalised curvature (left) and an enlargement (right).

Also here the conclusion holds that increasing the amount of elements results into convergenceinto one solution. The curves of the lower mesh densities of ne = 40 to ne = 640 show some non-smooth behavior which comes from the smaller amount of integration points through the thicknesscompared with higher densities. This trend seems to disappear when using 2560 elements. Theelement density needed for the stretching suffices for the bending as well. Within the bendingloadcase no necking is observed.

At last the loadcase is tested where the reference surface stretch and curvature is simultane-ously linearly increased to εref = 0.4 and h0

2 κref = 0.1 respectively again with constant rates.

0 0.1 0.2 0.3 0.40

0.4

0.8

1.2

1.6

εref [−]

FA

0σ

y0

[−]

ne = 40ne = 160ne = 640ne = 2560ne = 10240

0 0.02 0.04 0.06 0.08 0.10

1

2

3

4

5

ne = 40ne = 160ne = 640ne = 2560ne = 10240

h0

2κref [−]

Mh

0

2Iσ

y0

[−]

Figure A.4: Mesh dependence for simultaneous test with εref = 0.4 and h0

2κref = 0.1 comparing different

element densities; normalised force as a function of the reference strain (left) and normalised moment asa function of the reference curvature (right).

Above figure shows the results with on the left side the normalised force with respect to the refer-ence surface strain and on the right the normalised moment around the inner surface with respectto the reference surface curvature. The force seems to converge to one solution and in the momentcurves no large differences can be found.

Concluding from figure A.5 again as in the tensile test all the densities used give the same neck

41

0 0.1 0.2 0.3 0.40

0.2

0.4

0.6

0.8

εref [−]

η[−

]ne = 40ne = 160ne = 640ne = 2560ne = 10240

Figure A.5: Mesh dependence for tensile test with εref = 0.4 and h0

2κref = 0.1 comparing different

element densities; relative thickness strain as a function of the reference surface strain.

initiation and the curves converge to one solution so using a mesh density of ne = 2560 sufficesfor all the loadcases thus can be used in the actual simulations.

A.2 Imperfection

In the symmetry plane an imperfection has been made to trigger the localization. This imper-fection is achieved by decreasing the in-plane width of the sheet. In order to study the effect ofsize of this imperfection on total behavior the imperfection is varied in the model using ne = 2560from 0% to 10% width reduction. Results are compared with the behavior using no imperfection.The loadcases used are a tensile test where the reference surface stretch is increased linearly toεref = 0.25, a bending test where the reference surface curvature is increased to h0

2 κref = 0.25

and a simultaneous loading where the curvature and stretch are increased linearly to h0

2 κref = 0.1and εref = 0.4 respectively.

0 0.05 0.1 0.15 0.2 0.250

0.5

1

1.5

2

εref [−]

FA

0σ

y[−

]

δ = 0%δ = 0.1%δ = 1%δ = 10%

0.2 0.251

1.5

2

εref [−]

FA

0σ

y[−

]

δ = 0%δ = 0.1%δ = 1%δ = 10%

Figure A.6: Imperfection dependence comparing different in-plane thicknesses for a tensile test withεref = 0.25 and κref = 0; normalised force as a function of the reference strain (left) and an enlargement(right).

In figure A.6 the results are shown for the tensile test. The most top curve is the curve withoutimperfection. This leads to a equilibrium path with uniform deformation and no necking occurs.

42

Increasing the imperfection increases the softening post to the force peak, i.e. the necking point,where the 10% width reduction leads to very large softening.

0 0.1 0.20

1

2

3

h0

2κref [−]

Mh

0

2Iσ

y0

[−]

δ = 0%δ = 10%

0.2 0.25

3

h0

2κref [−]

Mh

0

2Iσ

y0

[−]

δ = 0%δ = 10%

Figure A.7: Imperfection dependence comparing different in-plane thicknesses for a bending test withεref = 0 and h0

2κref = 0.25; normalised bending moment as a function of the reference curvature (left)

and an enlargement (right).

Because we are also interested in the post necking behavior the imperfection of 1% width reduc-tion can be a safe choice. In the figure A.7 the results are shown for the bending test. Fromthe figures can be concluded that the imperfection has no significant influence on the total de-formation behavior. The 10% width reduction of the first column of elements only marginallylowers the bending moment. So again choosing a width reduction of 1% is a safe choice. At lastthe imperfection sensitivity is tested on a simultaneous loading where the reference curvature ofh0

2 κref = 0.1 and a reference surface strain of εref = 0.5 is chosen.

0 0.1 0.2 0.3 0.40

0.5

1

1.5

εref [−]

FA

0σ

y[−

]

δ = 0%δ = 0.1%δ = 1%δ = 10%

0 0.02 0.04 0.06 0.08 0.10

1

2

3

4

5

εref [−]

δ = 0%δ = 0.1%δ = 1%

Mh

0

2Iσ

y0

[−]

Figure A.8: Imperfection dependence for a simultaneous loading case comparing different in-plane thick-nesses; normalised force as a function of the reference strain(left) and the normalised moment as a functionof the reference strain (right).

From figure A.8 can be concluded that the 10% width reduction results in a much earlier neckinginitiation and large increase in softening. Other smaller reductions of 1% to 0.1% only result inmarginal differences in both behavior and necking initiation. The moment curves do not suffer

43

from the width reduction as can been seen in the right figure below. This can be explained by thefact the largest share of the bending stiffness is obtained from the thickness of the sheet ratherthan the width. This is caused by the second moment of area of a simple beam which is relatedto the thickness by a cubic relation as stated by the simple beam theory (Fenner, 1999).

0 0.1 0.2 0.3 0.40

0.5

1

1.5

εref [−]

η[-]

δ = 0%δ = 0.1%δ = 1%δ = 10%

Figure A.9: Imperfection dependence for a simultaneous loading case comparing different in-plane thick-nesses; relative thickness strain as a function of the reference surface stretch.

Besides the overall behavior the initiation of localisation is observed. This is done by determiningthe relative thickness strain η during the simulation and can be observed in figure A.9 where thisstrain measure is given as a function of the reference strain. Also here can be seen that the 10%thickness reduction results in a necking initiation at a lower strain compared with the 0.1% and1% width reduction. Choosing no imperfection will not lead to neck initiation in the tensile testand therefore concluding that both 0.1% and 1% can be chosen without large sensitivity.

44