optimal sequential sheet forming - materials technology · optimal sequential sheet forming mirre...

Optimal sequential sheet forming

Mirre Janssen

Master Thesis, MT07.19Material Technology,

Eindhoven University of Technology

Supervisor:

Prof.dr.ir. M.G.D. Geers

Coaches:

Dr.ir. S.H.A. BoersDr.ir. P.J.G. Schreurs

June, 2007

Abstract

Sheet metal is widely used in different industries, for example car manufacturing and(food)packaging. The production of complex geometries is often cheaper and quicker than theassembly of multiple simple parts. Complex manufacturing methods, such as hydroforming andmulti-step deformation processes are necessary to produce these complex shapes. An advantageof multi-step deformation processes is the ability to influence the distribution of deformation.This can lead to mechanically stronger products or products that meet the requirements withless material.

In this research the forming process of two different benchmark geometries, a strip and anaxisymmetrical cup, is optimized using one intermediate deformation step. The geometries areparameterized using a Bezier curve and two design parameters, a and b. The thickness reductionin the center of the geometries is used as the object function to be optimized.

In structure optimization, numerical simulation programs are often used to analyze the situation.The use of a direct optimization routine often leads to programming difficulties between thesimulation program and the optimization package. Another disadvantage is the large number offunction evaluations necessary. A solution is the sequential approximate optimization approachwhere a response surface is constructed.

The two-step deformation process is simulated using the finite element simulation programMSC.Marc. Simulations are done using different intermediate die shapes. The thickness reductionin the center is plotted against the parameters a and b, characterizing the die geometry. Betweenthose points a response surface is constructed. Some intermediate configurations lead to wrinklingof the specimen in the second deformation step. These configurations are excluded from thefeasible domain. The optimal intermediate configuration is found at the minimum of the feasibledomain.

To validate the numerical results, experiments are done. A reconfigurable die is used to performthe forming process and the residual thicknesses are measured. A response surface is createdand experimental and numerical results are compared.

The numerical simulations are in very good agreement with the experiments concerning theprediction of the thickness reduction after the first deformation step, as well as wrinkling duringthe second deformation step. Some deviations between the numerical and experimental resultsare visible after the second deformation step. This can be due to different forming conditions,a.o. the influence of the interpolator on the die shape, the direction of the forming pressure,friction.

1

Samenvatting

Plaatstaal wordt tegenwoordig wereldwijd gebruikt in onder andere de auto-en verpakking in-dustrie. De productie van complexe producten, bestaande uit een deel, is vaak goedkoper ensneller dan het produceren van dezelfde vorm uit meerdere simpele delen. Om complexe pro-ducten te kunnen produceren zijn er complexe deformatie technieken nodig, voorbeelden hiervanzijn hydrovormen en multi-stap productie processen. Een voordeel van multi-stap productie pro-cessen is dat de verdeling van de rek in het eindproduct beınvloed kan worden door keuzes inde tussenstappen. Dit kan leiden tot sterkere producten, of producten die met minder materiaalgebruik aan dezelfde eisen voldoen.

In dit onderzoek is het productieproces van twee verschillende vereenvoudigde producten, eenstrip en een axi-symmetrisch kopje, geoptimaliseerd. De vormen van deze producten zijn met be-hulp van een Bezier curve gekarakteriseerd en twee ontwerp parameters, a en b zijn gedefinieerd.Het productieproces vindt plaats in twee stappen. De vorm van de eerst gebruikte matrijs isvariabel, terwijl de vorm van de tweede matrijs voor alle samples vastligt. De dikte reductie inhet midden van het product is de doelfunctie die geoptimaliseerd gaat worden.

Bij het optimaliseren van producten worden vaak numerieke simulaties gebruikt. Het gebruikvan een optimalisatie routine in combinatie met een simulatie programma leidt geregeld totprogrammeer problemen tussen het simulatie programma en het optimalisatie programma. Eenander nadeel is het grote aantal simulaties dat nodig is om tot een minimum te komen. De”sequential approximate” optimalisatie aanpak biedt een oplossing voor deze problemen. Bijdeze aanpak wordt een oppervlak van de doelfunctie opgespannen als functie van de parametersa en b. De optimale waarden voor a en b kunnen uit dit oppervlak worden gehaald als zijnde delocatie van het minimum.

Het te optimaliseren, twee-staps deformatie proces is gesimuleerd met MSC.Marc. De diktereductie in het middelpunt van de kopjes is uitgezet tegen de ontwerp parameters. Dit resulteertin een oppervlak waaruit het minimum afgelezen kan worden. Tevens is een directe optimalisatieroutine gebruikt om het minimum te vinden. Sommige matrijs vormen leiden tot kreukels inhet materiaal. Deze configuraties voldoen niet aan de gestelde eisen en zijn niet verwerkt in deresultaten.

Experimenten zijn uitgevoerd om de numerieke resultaten te valideren. Hierbij is een verstelbarematrijs gebruikt om de samples te deformeren. Ook de resultaten van de experimenten zijn uit-gezet tegen de ontwerp parameters. Vervolgens zijn deze resultaten vergeleken met de numeriekeresultaten.

De numerieke en experimentele resultaten komen zeer goed overeen betreffende de dikte reductiena de eerste deformatie stap en het kreukelen tijdens de tweede deformatie stap. Na de tweede

2

Samenvatting

deformatie stap, wijken de experimenteel en numeriek bepaalde dikte reducties echter af. Ver-schillende onzekerheden kunnen hiervan de oorzaak zijn, onder andere: de geometrische invloedvan de interpolator, wrijving en de richting van de druk.

3

Contents

1 Introduction 6

2 Theory 10

2.1 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.2 Solution strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Material model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Friction model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Numerical simulations 21

3.1 Geometry - Strip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1.1 Model definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1.2 Numerical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Geometry - Axisymmetric cup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2.2 Numerical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3 Direct Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Experiments 35

4.1 Experimental plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4

Contents

5 Simulations vs Experiments 41

5.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.2 Influence of forming conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.2.1 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.2.2 Interpolator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.2.3 Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6 Conclusions 47

7 Discussion 49

A Experimental results 52

5

Chapter 1

Introduction

Nowadays, sheet metal is widely used in different industries, for example car manufacturingand (food)packaging. Sheet metal can easily be cut or deformed in almost any shape. In figure1.1 examples of complex geometries made from sheet metal are shown. The manufacturing ofcomplex geometries is often cheaper and quicker than the assembly of multiple simple shapesand the appearance of the final product is better. Another tendency is the durable use of rawmaterials, which also saves material costs. Off course the quality of the products is not allowedto suffer under these trends. To meet the above requirements complex manufacturing methods,such as hydroforming and multi-step deformation processes are necessary.

During the deformation of sheet metal large plastic strains may lead to product failure. Themagnitude of local strains can be lowered by distributing the deformation more homogeneously.In multi-step forming, strains in specific parts of the product can be influenced by the choice ofdie shape to improve the homogeneous strain distribution or to locally strengthen a product. Anhomogeneous deformation distribution leads to mechanically stronger products with the sameamount of material or products that meet the requirements with the use of less material.

The object of this research is the optimization of a sequential sheet forming process. Optimumforming is here defined as the adequate selection of a (sequence of) deformation path(s) such

(a) Complex die shape (b) Flank of a car construction

Figure 1.1: Examples of complex geometries

6

Chapter 1 Introduction

Figure 1.2: Parabolic cups made with a two-step (left) and single step (right) deformation process

that plastic strains are reduced and homogeneously distributed during the production process,minimizing the risk of failure. To emphasize the purpose of optimum forming, figure 1.2 showstwo parabolic cups made from sheet metal. The left one is made with an intermediate productionstep in which the die shape was pyramidal. The cup at the right was produced in a singledeformation step. The cup at the right shows fracture in the center while the cup at the left isstill intact [1]. This shows that although the initial and final geometry of the two cups are thesame, the strains in the deformed cups are not. An intermediate deformation step has a greatinfluence on the strains in the final geometry.

It used to be common practice to determine the best design parameters (a.o. die design, blankholder shape and force, forming pressure) experimentally by trial and error. This was a labour-intensive procedure which not necessarily resulted in the best possible option. Since a fewdecades, computational analyses of complex parts can be carried out due to improved com-puter performance and numerical analyses techniques. Already a lot of research has been doneon finding the optimum blank shape often using numerical simulations. Most of this researchis focussed on the backward tracing scheme or inverse method. Chung and Richmond [2][3]proposed a direct design method which is called the ideal forming theory. Others [4][5][6] haveimproved this theory by including anisotropy and boundary conditions as friction. In this re-search the blank shape is set and the optimum forming will be obtained by changing the dieshape in between intermediate forming steps, while the other studies adapt the blank shape tooptimize the final product. Also research has been done on the optimization of die shapes inmetal forging [7][8]. This research uses shape sensitivity analyses and numerical simulations toarrive at an optimum shape.

Assuming homogeneous deformation, it is possible to calculate the path that each material pointhas to travel from its initial position to its position in the final geometry. Unfortunately it isvery difficult to achieve this deformation path in practice. In multi-step forming, it is possible toapproach this ideal deformation path better than in single-step forming. In multi-step forming,deviations from the deformation path can be corrected in a next forming step. Unfortunately,it is not possible to determine the final position of each material point. Complete homogeneous

7


FF F

Figure 1.3: Principle of incremental forming, without (left) and with (right) a supporting die shape

(a) (b)

Figure 1.4: Reconfigurable die

deformation for complex geometries is impossible, but with multi-step forming the deformationdistribution can be optimized.

To realize such an optimum deformation process multiple dies are needed having different shapes.The costs of using multiple conventional dies would in small-batch production abrogate the ben-efits of the optimization. Different techniques can be used to overcome this problem. Incrementalforming [9][10] is a technology where a small tool moves along user-defined paths and incremen-tally develops a shape in the sheet metal. No dies are necessary which makes this technique fastand cheap. The working principle of incremental forming is depicted in figure 1.3. Even betterresults can be achieved if a supporting die is used to prevent undesired deformation of materialat certain places. Another technology is the use of a reconfigurable die. The surface of the dieis formed by a number of pins that can be positioned independent from each other. This tech-nology is already used at large scale in airplane manufacturing and research is done by Boers[1] to make a commercial available version on a smaller scale. A prototype of a reconfigurabledie is developed and manufactured at the Eindhoven University of Technology. Figure 1.4(a)is a picture of the prototype and 1.4(b) shows the pins that define the surface of the die. Therequirements for surface quality and shape accuracy of the products made with this prototypeare satisfied. This prototype will be used to do the experiments in this research.

In conventional sheet metal forming a lower and an upper die are used but with the reconfigurabledie one part is replaced by a rubber block. The pressure on this block forces the metal to deformin the negative or over the positive die. Figure 1.5 shows a positive die (left) and a negative die(right) that can be used to produce the same geometry. So besides the shape and number ofthe intermediate die(s) used, it is also important to know the differences between deformation

8


Figure 1.5: A positive (left) and a negative (right) die for the same final shape

with a positive or a negative die shape. An advantage of a positive die is that the influence offriction is less than during deformation with a negative die. Friction is often one of the causesof inhomogeneous deformation. So a positive die leads to more homogeneous strain distribution.Unfortunately the use of a positive die shape in combination with the reconfigurable die isdifficult because of the sample clamping on the circumference. The sample will buckle duringthe first stage of the deformation, which leads to inappropriate results and wrinkles.

Forming of wrinkles should be avoided, because the shape requirements of the final part are notmet in that case. Whenever a material is in a state of in-plane compression there is a risk forwrinkling. Wrinkles can be detected by a visual inspection of the deformed state or in a FiniteElement Method (FEM) simulation by checking for large out of plane rotations. A lot of researchis done [11][12] to calculate the risk of wrinkling and different wrinkle indicators are defined.

For the optimization of sheet metal forming processes knowledge about sheet metal forming isnot enough, mathematical aspects of optimization techniques should be considered as well. Theformulation of the optimization problem, the choice of object function, types of constraints anddifferent solution methods are discussed by Haftka [13]. The use of finite element programmesto analyse a structure creates opportunities for the optimization. Occasionally a design programincludes an optimization algorithm, but it is more common for an analyst to have a structuralanalyses package and a separate optimization routine. The interfacing between these programmescan lead to two problems. The communication between the two programs can be difficult anddue to the nature of optimization packages a lot of simulations are necessary with the associatedhigh costs. Fortunately, there is an approach that solves both problems: sequential approximateoptimization. A number of points in the design space is used to construct a surface throughthe whole domain, this will be explained further in section 2.1.2. This approach is only usefulwith a small number of design parameters. Kok and Stander have used this approach to find theoptimum die shape for the forming of a automotive wheel rim [14].

In this research the forming process is optimized for two benchmark products, the geometriesare simple because the focus is on the optimization. The first geometry is a strip and thesecond is an axisymmetric cup. Some theory about optimization as well as the used materialand friction model will be discussed in the next chapter. Chapter three is concentrated on thenumerical modelling. The problem is defined, a numerical model is made and the sequentialapproximation approach is used to construct a response surface of the object function withinthe domain of the design parameters. This surface is compared with the results acquired by astandard optimum finder. After that, experiments are done to validate the numerical results.In chapter 5, experimental and numerical results are compared and deviations are explained.Finally, conclusions are drawn.

9

Chapter 2

Theory

2.1 Optimization

Optimization is aimed on achieving the best outcome of a given operation while satisfying certainrestrictions. In this section some theory and terminology about optimization is discussed. Morebackground information can be found in [13].

2.1.1 Problem formulation

The first step towards the optimization of a certain operation is defining the problem. Assump-tions are made to simplify the model.

Design variablesIn order to improve or optimize an operation there must be freedom to change some parameters.Such parameters are called the design variables and are stored in column x= (x1,x2,. . . ,xn). Thedesign variables are the input of the function that is optimized.

Objective functionThe objective or object function is the function that is optimized. Because the object func-tion is a measure of effectiveness of the design, it is very important to choose the appropriatefunction. The object function is denoted by f(x). Optimization with more than one objectiveis called multicriteria optimization and is complicated. There are two ways to avoid multicri-teria optimization. The first is by generating a composite objective function that replaces allthe objectives. The second is to select the most important objective as the only one and to setboundaries on the others so that they are treated as constraints.

ConstraintsConstraints can be imposed on the design variables to prevent surrealistic outcomes or failure.There are inequality constraints, for example stress limits, and equality constraints, for exampledisplacements. In structural optimization, constraints are very important and often affect thefinal solution and force the object function to obtain a higher value than it would take withoutconstraints. The space of the design variables can be divided into a feasible domain and aninfeasible domain. The design points that satisfy all the constraints can be found in the feasible

10

Chapter 2 Theory

domain. In the infeasible domain are the design points that violate at least one of the constraints.It is expected that the constraints have an influence on the optimal design, therefore the optimaldesign points are often situated on the boundary between the feasible and infeasible domain. Theconstraints that limit the optimal design are called active constraints, the others are inactive.

Standard formulationA standard optimization problem with a single objective function is normally formulated as

minimize f(x)

such that gj(x) ≥ 0, j = 1,. . . ,ng,

hk(x) = 0, k = 1,. . . ,ne,

where x is a column with design variables, gj the inequality constraints and hk the equalityconstraints defined above. There are many different types of problems depending on the sortof object function and constraints, linear or non-linear, constraint or unconstraint. The solvingstrategy depends on the problem type.

2.1.2 Solution strategies

There are two fundamentally different solution strategies in structural optimization. A directoptimization technique can be used to find the minimum of an object function or a responsesurface can be created with the sequential approximate optimization approach. Both techniquesare explained in this section.

Direct optimizationDirect numerical search techniques start from an initial guess and proceed in small steps towardsthe minimum object function value. The search is terminated when no progress can be madewithout violating constraints, when progress in improving the object function becomes very slowor tolerances are reached. Although the solution strategy is different for each sort of problem,almost all optimization techniques are based on four steps. First the active constraints aredetermined. Then the search direction is calculated based on the objective function and theactive constraints. The next step determines how far to go in the calculated direction and thefinal step determines whether additional moves are required. For a wide range of optimizationproblems numerical solvers can be used. In Matlab two solvers are available.

A single-variable function on a fixed interval can be optimized with the command fminbnd. Thissolver is based on the golden section search method. It is named after the ideal proportion, 0.382,found in the Fibonacci series. The golden section search method is a very reliable, efficient linesearch method for finding the minimum of a unconstraint function f(α) on a certain interval. Thefunction f(α) should be unimodal, that means that there is only one minimum within the definedinterval a0 ≤ α ≤ b0. The function does not have to be differentiable. The idea behind the goldensection search method is to reduce the interval of uncertainty within which the minimum of thefunction f lies. Consider the function f(α) on the interval (a0,b0). The function is evaluated intwo points within the interval, α1 and α2. The function is unimodel, therefore if f(α1)≥ f(α2) theminimum can not lie within the section (a0,α1) but can be found on (α1,b0). This new interval issmaller than the original interval. The process is repeated until the desired accuracy is reached.The placement of the points α1 and α2 determines the number of function evaluations necessary.

11

Chapter 2 Theory

(a) The initial simplex for the optimizationof parameter a and b

(b) Simplex method

Figure 2.1: The simplex method schematically represented

It is found that the best positions of α1 and α2 are α1 = a0 + 0.382l0 and α2 = b0 - 0.382l0,with l0 the length of the current interval.

The other solver in Matlab is called fminsearch and can be used for the minimization of un-constraint problems with multiple variables. This command uses the sequential simplex method.The function to be minimized needs to be unimodel, but does not have to be differentiable. Thesimplex method is based on an initial design of n+1 trials, where n is the number of variablesto be optimized. This situation is shown in figure 2.1(a) for the optimization of two parameters.The trials are the corners (vertices) of a n+1 geometric figure (the simplex) in a n-dimensionalspace. For n = 2, the simplex is a triangle, for n = 3 a pyramid. The corners of the first simplexare given by the initial guess and points along each of the n coordinate directions. After theinitial trials one new trial, in or near the simplex, is evaluated. The function value of this newpoint is compared to the function values at the vertices of the simplex. If the new function valueis smaller than the highest function value at one of the vertices, this vertice is replaced by thenew trial, giving a new simplex. This step is repeated until the diameter of the simplex is lessthan a set tolerance. In figure 2.1(b), the method is schematically presented.

Sequential approximate optimizationIn structural optimization the object function and constraints are often implemented in astructural analyses package, such as a finite element program. The underlying formulas are toocomplicated to rewrite in a simple optimization problem. Therefore in structural optimizationoften a structural analysis package is used as well as an optimization package. Difficulties thatoccur due to this are programming problems and high computational costs for optimization pro-grams may require to evaluate object functions and constraints hundreds of times. Fortunately,the second solution strategy deals with both problems. This strategy is called the sequentialapproximate optimization approach. In this technique first the structural analyses package isused to analyze the design at a few different design points in the design space. The responseof the object function in those points is then used to construct an polynomial approximationto the object function, the response surface. The minimum of this surface is located at theoptimum design parameters. The polynomial approximation that is obtained through theanalyses of the design at multiple points, is a global optimization. This can be very expensivefor a large number of design variables. It is also common to use local approximations based on

12

Chapter 2 Theory

Figure 2.2: Display of the two different solution methods

derivatives of the object function and its constraints. An advantage is less computational costsbut unfortunately these approximations are only valid in a small area round the calculateddesign point. Following an approximate optimization an exact analyses is performed in theoptimum design point obtained by the approximate approach. The derivatives are calculatedand a new approximation is made. The process repeats until convergence is achieved.

In figure 2.2 the difference between the two different solution methods is depicted. For functionswith multiple minima, the result of direct optimization with function fminbnd or fminsearch isnot always the global solution. The obtained minimum can be a local minimum. The sequentialapproximate optimization technique can be used to obtain shape information of the responsesurface. This information can be used to choose the initial point for direct optimization and torate the solution at its true value.

2.2 Material model

In this research two different materials are used. In the strip forming analyses DC06 steel is usedand the cup forming is done with Tinplate BA quality steel. The behavior of both materials canbe described by an elastoplastic material model [15]. The plastic behavior of the DC06 steel isanisotropic and for this material Hill’s yield criterion is applied. The Tinplate deforms isotropiclyand Von Mises yield criterion is used. The elastoplastic material model with Von Mises yieldwill be discussed here and the Hill yield criterion will be briefly discussed.

The one-dimensional mechanical representation of an elastoplastic model consists of a spring inseries with a parallel arrangement of a spring and friction slider, as can be seen in figure 2.3.The series-spring represents the purely elastic part of the deformation, when stress is below the

13

Chapter 2 Theory

H

σy

εeεp

σE

Figure 2.3: Discrete model for elastopastic material behavior

Figure 2.4: Multiplicative decomposition of total deformation

yield stress. The elastoplastic response becomes manifest when the stress reaches the yield stressσy. After yielding the total strain ε is the sum of the elastic strain εe and the plastic strain εp.The stress is not influenced by the strain rate.

σ < σy → ε = εe ; σ = σy → ε = εe + εp (2.1)

KinematicsTransformation from the undeformed configuration at time t0 (position vector ~x0) to thecurrent configuration at time t (position vector ~x) is described by the deformation tensorF = (~∇0~x)c, where ~∇0 is the gradient operator with respect to the undeformed state. The rightand left Cauchy-Green strain tensors, C = F c · F and B = F · F c, are functions of F as isthe Green-Lagrange strain tensor E = 1/2(C − I ). The deformation rate is described by thevelocity gradient tensor L = (~∇~x)c, where ~∇ is the gradient operator with respect to the currentstate and ~v is the velocity of the material volume. The total deformation F is multiplicativelydecomposed into an elastic and a plastic contribution, as illustrated in figure 2.4. For thevelocity gradient tensor an additive decomposition into the symmetric deformation rate tensorD and the skew-symmetric spin tensor Ω is used. Both D and Ω can be split into an elasticand a plastic part. To make the decomposition unique it is commonly assumed that the plasticrotation rate during the current increment is zero, i.e. Ωp = 0. Superimposed material rotationsare thus fully represented in Fe.

14

Chapter 2 Theory

Elastic behaviorThe stress is related to the elastic strain with an elastic material model. In elastoplasticdeformation problems, it can often be assumed that elastic strains are small, which allowsthe use of a hypo-elastic generalized Hooke’s law, relating the Cauchy stress tensor σ to thelogarithmic strain tensor Λ. The material is assumed to be isotropic in which case the elasticmaterial behavior is characterized by two material constants, the bulk modulus K and the shearmodulus G.

σ = 4C : Λe (2.2)

4C = (K − 23G)II + G( 4I + 4IRT ) (2.3)

Yield criterion and hardeningA yield function F is used to evaluate the stress state and to check whether the deformationis purely elastic (F < 0) or elastoplastic (F = 0). The current stress state, represented by theequivalent stress σ, is compared to a yield stress σy. Its initial value is σy0.

F = σ2 − σ2y(εp) (2.4)

The yield stress changes with plastic deformation and is therefore related to the effective plasticstrain εp. The relation between σy and εp is described by the so-called hardening law:

H(εp) =∂σy

∂εp(2.5)

To decide whether elastic or elastoplastic deformation occurs, the Kuhn-Tucker relations areused:

(F < 0) ∨ (F = 0 ∧ F < 0) → elastic deformation (2.6)

(F = 0) ∧ (F = 0) → elastoplastic deformation (2.7)

Elastoplastic deformationDuring elastoplastic deformation (F = 0) the plastic deformation rate Dp is related to thestress by the flow rule. For a so-called normality or associative flow rule the direction of Dp

is perpendicular to the yield surface in stress space. The length of Dp is characterized by theplastic multiplier λ.

Dp = λ∂F

∂σ= λa (2.8)

The value of the plastic multiplier λ can be determined from the requirement that the stressstate must always reside on the yield surface during elastoplastic deformation, so F = 0. Thisrelation is referred to as the consistency condition.

Constitutive modelThe material model can be summarized as a set of constitutive relations. It must be used todetermine the stress when (an approximation of) the deformation is known. It is also used toderive a relation between the variation of stress and deformation, which is an essential part of

15

Chapter 2 Theory

the later solution procedure. The stress rate is related to the deformation rate, using a suitableobjective time derivative of σ. Use of the objective rate

¯σA is necessary because of the large

rotations, which may occur during elastoplastic deformation. The set of differential equationsmust be integrated over the deformation history to determine the current stress σ(t) when thecurrent deformation F (t) is known. The constitutive relations are summarized as,

elastic behavior:(F < 0) ∨ (F = 0 ∧ F < 0) → D = De (2.9)

¯σA = 4C : D (2.10)

Dp = 0 (2.11)

˙εp = 0 (2.12)

elastoplastic behavior:

(F = 0) ∧ F = 0) → D = De + Dp (2.13)

¯σA = 4C : (D − λa) (2.14)

2σ ˙σ − 2σyH ˙εp = 0 (2.15)

with: a = ∂F∂σ , σy = σy(σyo, εp),

˙εp =√

23Dp : Dp = λ

√23a : a

For the Von Mises yield criterion, which represents the plastic behavior of Tinplate, the yieldsurface is a circular cylinder in principal stress space. When hardening is assumed to be isotropic,the radius of the cylinder will change upon plastic deformation. The normal to the yield surfacecan be expressed in the deviatoric stress σd. The Von Mises equivalent stress is given by:

σV M =√

32σd : σd (2.16)

The plasticity of the DC06 steel is represented with the anisotropic Hill’48 yield criterion [1]:

σH2 = F (s22 − s33)2 + G(s33 − s11)2 + H(s33 − s11)2 + 2Ls23

2 + 2Ms312 + 2Ns12

2 (2.17)

where σH is the equivalent Hill stress

σH =√

M : 4A : M (2.18)

with the rotation neutralized deviatoric stress tensor M based on the Kirchhoff stress tensor Kdefined as

M = QT ·Kd ·Q (2.19)4A is the fourth-order symmetric traceless orthotropic Hill flow anisotropy tensor withcomponents sij.

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Chapter 2 Theory

Figure 2.5: Incremental deformation

Incremental analysesA large plastic strain deformation process is divided into small steps called increments. Fig-ure 2.5 shows the relevant configurations in a large strain plastic deformation process. The totaldeformation gradient tensor F (t) at the current time can be decomposed into the deformationtensor at the beginning of the current increment F (tn) and the incremental deformation tensorFn(t). For time τ in the current increment the deformation gradient tensor is described as

F (τ) = Fn(τ) · F (τn) → Fn(τ) = F (τ) · F−1(τn) (2.20)

Assuming that all variables at the begin-increment time tn are known and satisfy all governingequations, the state at the end-increment time tn+1 has to be determined. The constitutiveequations can be rewritten in an incremental setting and by integrating these equations over thecurrent increment the end-increment state can be determined. In the incremental setting theDienes stress tensor σD is used. This tensor is invariant for the incremental rigid body rotations.The material time derivative of σD is related to the objective Dienes rate

¯σD of the Cauchy

stress tensor.

σD = RTn · σ ·Rn → σD = RT

n ·¯σD ·Rn (2.21)

¯σD = σ − (Rn ·RT

n) · σ − σ · (Rn ·RTn)T (2.22)

It is assumed that there is no rigid body rotation during the increment and that all rigidincremental rotation is concentrated at the end increment time τ = tn+1.

Rn(tn ≤ τ < tn+1) = I ; Rn(tn+1) = F(tn+1) ·U−1(tn+1) (2.23)

When it is also assumed that the incremental principal strain directions are constant during theincrement, the tensors Un and U−1

n are commuting and we have

D = Un ·U−1n = U−1

n ·Un =3∑

i=1

(λni

λni)~nni(tn)~nni(tn) = Λn (2.24)

were ~nni are the incremental principal strain directions and λni the associated principle strains.The constitutive equations that have to be integrated over the current increment are:

σ = 4C : (Λn − λa) (2.25)

17

Chapter 2 Theory

3σdD : 4C : Λn − λ(3σd

D : 4C : a + 2σyH

√23a : a) = 0 (2.26)

Integration procedureThe stress integration procedure is always started with the calculation of an elastic stress pre-dictor. It is assumed that the increment is fully elastic and that the begin-increment elasticitytensor can be used to calculate the rotation neutralized Cauchy stress tensor.

σDe = σ(tn) + 4C(tn) : Λn → σe = Rn · σDe ·RTn (2.27)

Subsequently the elastic Cauchy stress tensor is calculated and used to evaluate the yield cri-terion with two possible outcomes. The increment can be fully elastic or the yield criterion isviolated which implies that elastoplastic deformation has taken place. When the increment isfully elastic, the end-increment Cauchy stress equals the calculated elastic Cauchy stress. As noplastic deformation has occurred during the increment, the effective plastic strain and the yieldstress have not changed.

σ(tn+1) = σe ; εp(tn+1) = εp(tn) ; σy(tn+1) = σy(tn) (2.28)

If the elastic stress predictor violates the yield criterion, the increment is elastoplastic. The stressand the elastic multiplier must be determined such that at the end of the increment, the stressresides on the yield surface. In this research the stress is integrated according to an explicitintegration algorithm. In the explicit integration procedure the begin-increment equations aretaken as a starting point to determine σ(t). The total increment ∆t = tn+1 − tn is subdividedinto a part ∆te where the deformation is purely elastic and the remaining part ∆tp where thedeformation is elastoplastic. The elastic trial stress has been shown to violate the yield criterion.The fraction β of the incremental logarithmic strain which gives F = 0, can be determined. Thestress which brings us to the yield surface is σβ. The fraction (1− β)Λn of the total logarithmicstrain represents elastoplastic deformation. In the by MSC.Marc used mean-normal method, thestress σm is calculated as the average of σβ and the elastic trial stress σe. It is used as startingvalue for an iterative integration procedure. This procedure is repeated until convergence. Theprocedure is schematically represented in figure 2.6

σm =12(σβ + σe) (2.29)

2.3 Friction model

Friction is a complex physical phenomenon that is difficult to simulate. Friction between the dieand the sample is dependent on surface roughness, lubrication, temperature, relative velocitybetween the sample and die and multiple other phenomena. MSC.Marc provides two simplifiedmodels, the Coulomb friction model and a shear friction model. The Coulomb model is themost popular for different applications with the exception of forging simulations where theshear model is widely used. The Coulomb model is discussed here.

18

Chapter 2 Theory

Figure 2.6: Mean-normal method

ft

Vr

Stick

Slip

(a) stick-slip

ft or σt

Vr

(b) arctangent

Figure 2.7: Coulomb friction model

Stick-slip Coulomb friction modelCoulomb friction depends on the friction coefficient between the two touching surfaces andthe normal force on these surfaces. This model is also known as the stick-slip model and ischaracterized by:

‖ft‖ < µfn (stick) and ft = µfn · t (slip), (2.30)

where f t is the tangential friction force, f n is the normal force, µ the friction coefficient andt is the tangential vector in the direction of the relative velocity. For a given normal force,the friction force has a step function behavior upon the relative sliding velocity v r. This stepfunction is depicted in 2.7(a).

Unfortunately, the discontinuity in the step function causes numerical instabilities. Several vari-ations on the stick-slip model are made that are continuous and improve the convergence of themodel. Examples are the arctangent model, the modified step function and the bilinear model.The arctangent model is used in the numerical simulations of this research.

Arctangent Coulomb friction modelThe arctangent model, which can be seen in figure 2.7(b), is based on a continuously differentiable

19

Chapter 2 Theory

relation between the relative sliding velocity v r and the friction force:

ft = µfn2π

arctan(γ) · t, (2.31)

where γ is the relation between the relative velocity threshold value and the relative slidingvelocity. The relative velocity threshold value can be seen as the value of the relative velocitybelow which sticking occurs. A too large γ results in a reduced effective friction, while a toosmall value can result in poor convergence. Normally a value between 0.1 and 0.01 is chosen. Insection 3.1.2 simulations are made to analyse the effect of friction on the results.

20

Chapter 3

Numerical simulations

In this research, the operation that is optimized is the forming of sheet metal into a final geome-try. The forming of two different geometries, a strip and an axisymmetrical cup, is optimized. Thechosen geometries are relatively easy, this to focus on the optimization strategy. An additionaladvantage is the shorter computation time of the numerical models.

3.1 Geometry - Strip

The first forming geometry to be optimized is a strip. The design parameters, object functionand constraints are formulated. Then a numerical model is made, simplifications and boundaryconditions are discussed. Finally, results are shown.

3.1.1 Model definition

Design variablesThe undeformed sample is a metal strip with dimensions 20× 100× 0.7 mm3. As can be seen infigure 3.1. The dimensions are chosen such that the product can be produced with the reconfig-urable die constructed by Boers [1]. The reconfigurable die has a working area of 40×50×25 mm3.The metal that is used is the commonly used DC06 steel. The geometry of the final productis symmetric about the y=0 plane en determined by four design points. These points are thecontrol points of a Bezier curve that is the basis of the final product shape. This Bezier curveis depicted in figure 3.2. The depth of the geometry is set to be 15 mm and the width of the

y

x

z

Figure 3.1: Specimen geometry

21

Chapter 3 Numerical simulations

P1

P3

P2

P4

YY

ZZ

00 12−1200

15

Figure 3.2: Bezier curve with control points

y

x

z

Figure 3.3: Final geometry

curvature is 24 mm. Due to this, control points 1 and 4 have a fixed position. By changing they-position of control points 2 and 3 the shape of the Bezier curve can be modified. To limitall the possible curve shapes, points 2 and 3 are fixed in the z-direction and two parametersare defined. Parameter a is defined as the distance in the y-direction between points 2 and 3,a = yp3 − yp2. Parameter b describes the y coordinate of the bending point of the Bezier curveand is defined by b = yp2+yp3

2 . By using the symmetry and extracting the Bezier curve a 3Dgeometry is formed. The final geometry is determined by (a, b) = (0, 0) and shown in figure 3.3.The parameters a and b are restricted to a certain domain. This means that a can vary between-24 and 24 and b between -24 and 12.

The most extreme die shapes are found on the borders of this domain and are used as firstgeometry of a sequential forming series, which can be seen in figure 3.4. In the simulated formingprocess, nine intermediate die shapes are used. The sequential die shapes are determined by aninitial set (a, b)i where the values for a and b proportionally change. Figure 3.5 shows theevolution of the die shape for two different choices of (a, b)i. In figure 3.5, the die geometry ischanged in six sequential steps, ending in the proposed final die geometry (a, b) = (0, 0). Thedesign variable is x = (a, b)i.

Object functionThe function to be minimized is the difference between the thickest and thinnest point ofthe sheet metal product after deformation. When this function is minimized, the deformationthroughout the whole product is most homogeneous. Because the problem is solved with a nu-merical model in a finite element program the geometry of the product is divided into a limitednumber of elements. The object function can be defined as

f(x) = max[tp1 , tp2 , . . . , tpn ]−min[tp1 , tp2 , . . . , tpn ], (3.1)

with tpn the thickness at a point within element n depending on x. In practice, this definitioncorresponds with the thickness reduction in the center of the geometry.

22


Figure 3.4: Final geometry

Figure 3.5: Evolution of the die for initial shape 2 ((a,b)i=(0,-24)) and 5 ((a,b)i=(12,6)) from figure 3.4with 6 intermediate steps

23


0 0.1 0.2 0.3 0.4 0.5100

200

300

400Material Hardening

Effective plastic strain [−]Y

ield

str

ess

[MP

a]

Figure 3.6: Material hardening relation

ConstraintsThe domain of the design variables is limited due to geometric properties. These simple upperand lower limit constraints on the design variables are called (inequality) side constraints andare defined by,

g1 = 24− |a|, (3.2)

g2 = 12− b, (3.3)

g3 = 24 + b, (3.4)

withgj(x) ≥ 0, j = 1, 2, 3. (3.5)

Other constraints are more difficult to define. During the deformation process some intermediatedie shapes cause the sheet metal to form wrinkles in the final steps of the deformation process.One of the requirements is that the product meets the set final geometry. If wrinkling occurs,this requirement is not met. Wrinkling can be detected by visual inspectation of the specimenor by out of plane rotations in the simulations.

3.1.2 Numerical model

The numerical modelling is done with the finite element program MSC.Marc. The specimenis modelled with shell elements. The theory behind the elastoplastic material model is alreadydiscussed in chapter two. The yield criterion that is used is the anisotropic Hill yield criterion.Although a more complex material model, as for example the Teodosiu model, gives more ac-curate results, the disadvantages abrogate the advantages for simple geometries [1]. Numericalsimulations that use the Teodosiu model are even for simple geometries very time demanding.The initial yield stress is 120 MPa and the parameters for the Hill yield criterion are F=0.269,G=0.347, H=0.653, L=1.5, M=1.5, N=1.56. These parameters are experimentally determinedin [1]. For the material hardening the relation between effective plastic strain and effective stressis implemented, this relation can be seen in figure 3.6.

The reconfigurable die with rubber interpolator is modelled as a very stiff deformable body. Dueto the very high Young’s modulus and yield stress of the die material, there is no deformationin the die during the deformation of the metal strip. Deformation of the die is done by applyingdisplacements to the nodes of the die. A cycle of the sequential deformation process consists

24


Pres

sure

y−di

spla

cem

ent

Die

def

orm

atio

n

00 2211 33 44 55

Figure 3.7: A cycle of the sequential forming process

(a) (b) (c)

Figure 3.8: Deformation process at the beginning (a), halfway (b) and at the end (c), a=24 b=0

of applying the pressure and keeping the pressure constant during the deformation. After thedeformation, the pressure is removed and the strip is taken out of the die by moving it in thepositive y-direction. The die shape is adapted and the deformed strip is placed back in theblankholder. This cycle is repeated for each intermediate die shape and schematically presentedin figure 3.7. Figure 3.8 shows the evolution of the deformation process.

To analyse the influence of the friction model two forming configurations ((a, b)i = (0, 0) and(a, b)i = (24, 0)) are investigated with several different values for γ. The smaller the value of γ,the larger the friction between the interpolator and the sample. Picture 3.9 shows the results forthe (a, b)i = (0, 0) configuration. At the left the thickness of the sample is plotted as a functionof the distance from the center of the strip to the edge. It can be seen that for decreasing γthe thickness reduction in the center increases. At the right, the lateral displacement of thematerial on the sides of the sample is plotted against the distance from the center. The lateraldisplacement increases as well for decreasing γ. Combining these two results it can be concludedthat a decreasing γ results in less material in the middle region after deformation. Anotherforming configuration that is analyzed for the effect of friction is (a, b)i = (24, 0). Figure 3.10shows the thickness reduction and lateral displacement in these simulations. Friction has lessinfluence during the deformation of these geometries. The influence of the friction is geometrydepended. Further simulations are done, using the Coulomb arctangent friction model with a

25


0 10 20 30 40 500.55

0.6

0.65

Distance from center [mm]

Thi

ckne

ss [m

m]

γ = 1γ = 0.1γ = 0.01

(a) Thickness reduction

0 10 20 30 40 50−2

−1.5

−1

−0.5

0


Late

ral d

ispl

acem

ent [

mm

]

γ = 1γ = 0.1γ = 0.01

(b) Lateral displacement

Figure 3.9: Influence of friction for geometry a=0, b=0

0 10 20 30 40 500.55

0.6

0.65


Thi

ckne

ss [m

m]

γ = 1γ = 0.1γ = 0.01

(a) Thickness reduction

0 10 20 30 40 50−2

−1.5

−1

−0.5

0


Late

ral d

ispl

acem

ent [

mm

]

γ = 1γ = 0.1γ = 0.01

(b) Lateral displacement

Figure 3.10: Influence of friction for geometry a=24 b=0

gamma of 0.1.

3.1.3 Numerical results

For the eight most extreme and the (a, b)i = (0, 0) configurations, sequential deformation processsimulations are made. Figure 3.11 shows the forming sequence with an initial value (a, b)i =(−24, 0). During this sequence a pressure of 30 MPa is used to press the steel into the die. Itis observed that a wrinkle is formed approximately half way the forming process at (a, b) =(−12, 0). The in-plane compression during the second half of the forming process causes thematerial to wrinkle. A forming pressure of 30 MPa is not enough to prevent this wrinkling.Other initial configurations that showed wrinkling are (a, b)i = (0, 12) and (a, b)i = (−12, 6).In order to avoid this problem, the forming pressure is increased to 100 MPa at the moment awrinkle starts to form. During two of the three simulations, increasing the pressure effectivelysolves the problem of wrinkling. However, such an high forming pressure is often undesirable.

For two forming sequences (a, b)i = (24, 0) and (a, b)i = (−12, 6), the thickness distributions

26


(a) (b) (c)

Figure 3.11: Deformation process at the beginning (a), halfway (b) and at the end (c), a=-24 b=0

0 10 20 30 40 500.5

0.55

0.6

0.65

0.7


Thi

ckne

ss [m

m]

a=24 b=0, 30MPaa=24 b=0, 100MPa

0 10 20 30 40 500.5

0.55

0.6

0.65

0.7


Thi

ckne

ss [m

m]

a=−12 b=6, 30MPaa=24 b=0, 100MPa

Figure 3.12: Thickness reduction

(figure 3.12) and the contractions of the specimens width (figure 3.13) are presented. It isobserved that for the (a, b)i = (24, 0), the residual thickness is minimal in the center and at thetransition region (distance=25 mm) of the discrete die surface and the blank holder. Althoughthe (a, b)i = (−12, 6) sequential forming process with a forming pressure of 30 MPa causeswrinkling, the (a, b)i = (−12, 6) sequential forming process with a forming pressure of 100 MPadoes not show wrinkling and results in an almost constant residual thickness. The simulationswith an higher forming pressure generally result in a smaller local residual thickness comparedto simulations with a lower forming pressure. Figure 3.13 represents the lateral displacement ofone edge of the specimen. It can be seen that the lateral displacement for the simulation with

0 10 20 30 40 50−2

−1.5

−1

−0.5

0


Late

ral d

ispl

acem

ent [

mm

]

a=24 b=0, 30MPaa=24 b=0, 100MPa

0 10 20 30 40 50−2

−1.5

−1

−0.5

0


Late

ral d

ispl

acem

ent [

mm

]

a=−12 b=6, 30MPaa=24 b=0, 100MPa

Figure 3.13: Lateral displacement

27


(a) 30MPA (b) 100MPa

Figure 3.14: The object function in the specified domain for two different pressures

(a, b)i = (24, 0) predicts a more monotonic decrease of the specimen width towards the center,compared to the simulations with (a, b)i = (−12, 6) where the transition region of the die surfaceand the blank holder is clearly noticeable.

The object function is defined as the difference between the maximum and minimum thicknessof the final product. In figure 3.14 the results are presented by discretely sampling the objectfunction for the simulations. In 3.14(a) a forming pressure of 30 MPa is used and in 3.14(b) theforming pressure is increased to 100 MPa to avoid wrinkling. It can be seen that the simulation(a, b)i = (−12, 6) gives almost identical results to (a, b)0 = (0, 0). The rest of the sequentialdeformation processes leads to a larger thickness reduction than (a, b)i = (0, 0). Accordingto these results it is expected that the minimum value for the object function lies between(a, b)i = (−12, 6) and (a, b)i = (0, 0).

3.2 Geometry - Axisymmetric cup

The second forming process that is analyzed is the forming of an axisymmetric cup.

3.2.1 Problem formulation

Design variablesThe blank is a circular piece of sheet metal with a diameter of 100 mm. The material is Tin-plate BA quality steel. This steel is approximately isotropic and it deforms according to theelastoplastic material model with the Von Mises yield criterion as discussed in section 2.2. Thethickness of the sheet metal is 0.28 mm. This is much thinner than the previous used DC06steel. Using DC06 steel with a thickness of 0.7 mm would require too high forming pressures.Again the Bezier curve is used to define the geometry. For the axisymmetric cup the depth ofthe die is set to be 8 mm and the width of the curvature is 18 mm. The shape of the curvecan be altered by varying a and b. The domain for the parameters a and b is, −18 < a < 18and −9 < b < 9. The cup is formed by rotating the Bezier curve 360o around the z-axis, sothe forming will take place in a circle with a diameter of 36 mm. The reconfigurable die has aworking area of 40× 50× 20 mm3.

During the previously analyzed strip forming process, nine intermediate die shapes were used.The first die shape was determined by design parameters and the rest was set by proportionally

28


100 mm

36 mm

8 mm

Figure 3.15: Axisymmetric geometry

changing a and b. It is found that the first intermediate die shape is determining the result.Therefore, only one intermediate step is used to form the cup (figure 3.15), which final geometryis set on (a, b) = (0, 0). The intermediate step can have any shape defined by (a, b)i within thespecified domain. By altering (a, b)i the best intermediate forming step can be found, so thedesign variable is x=(a, b)i.

Object functionThe object function is the difference between the thickest and thinnest point of the final product:

f(x) = max[tp1 , tp2 , . . . , tpn ]−min[tp1 , tp2 , . . . , tpn ], (3.6)

with tpn the thickness of a point within element n depending on x.

ConstraintsThe domain of the design parameters a and b is limited by geometrical properties, these con-straints are defined by

g1 = 18− |a|, (3.7)

g2 = 9− |b|, (3.8)

withgj(x) ≥ 0, j = 1, 2. (3.9)

Another constraint is that the final geometry must meet the shape requirements. Intermedi-ate shapes that cause wrinkling in the second deformation step are excluded from the feasibledomain.

3.2.2 Numerical model

The numerical modelling is done with the finite element program MSC.Marc. The blank ismodelled with shell elements. The material behavior is captured by an elastoplastic materialmodel (section 2.2) with the Von Mises yield criterion. The material has an initial yield stressof 287 MPA and the hardening of the material is experimentally determined with an uniaxialtensile test. The relation between the effective strain and the yield stress is implemented inMSC.Marc. The hardening of the material is almost lineair as can be seen in figure 3.16. Theinterpolator between the die surface and the specimen has an influence on the forming geometry.Depending on the contact forces (figure 3.17) during deformation, the rubber will or will notcompletely move around the edges of the pins and in between the cavities of the pin surface. Onthe edges, where the contact force is the largest, the influence of the interpolator on the die shapewill be minimal. In the middle, where contact forces are smaller, the interpolator will be thicker.

29


0.05 0.1 0.15 0.2 0.25 0.3 0.35300

350

400

450

500

550

Effective plastic strain []Y

ield

str

ess

[MP

a]

Figure 3.16: Material hardening

specimen

interpolator

die

pressure

F =0c

F >0cF >0c

Figure 3.17: Contact forces on the interpolator

The influence of the interpolator on the geometry of the specimen is modelled by correctingthe depth of the Bezier curve by 1 mm. This situation is presented in figure 3.18. Insteadof modelling the reconfigurable die as a deformable body and changing the shape during thedeformation, the reconfigurable die with the rubber interpolator is modelled as a rigid surface.Two of these surfaces are present in each simulation representing the intermediate die shapeand the final die shape. Figure 3.19 shows an overview of the specimen and the two dies. In acontact table, the contact between the different contact bodies is determined. For the frictionbetween the interpolator and the blank, the Coulomb arctangent friction model is used with afriction coefficient of 0.2 and γ = 0.1.

3.2.3 Numerical results

The response surface method is used to analyze the influence of the design parameters on theobject function. In 181 points in the design space, the forming process is simulated and the

Figure 3.18: Geometry correction

30


die 1

die 2

specimen

Figure 3.19: Overview of the specimen and the two dies

−20 −15 −10 −5 0 5 10 15 20−10

−5

0

5

10

a

b

Figure 3.20: Distribution of design parameters

corresponding value of the object function is calculated. The positions of the 181 simulatedpoints can be seen in figure 3.20. Figure 3.21 shows the evolution of the forming process withintermediate configuration (a, b)i = (0,−9). The thickness reduction of the material in thecenter of the formed cup is plotted as a function of the design parameters (a, b)i and a surfaceis constructed between the calculated points. The thickness reduction after the intermediatedeformation step can be seen in figure 3.22 and figure 3.23 shows the response surface after thecomplete deformation process.

Unfortunately, some of the design points show wrinkling during the second deformation step.Two different wrinkling modes can be distinguished. The first mode occurs in the circumferentialdirection and can be seen at the edges of the cup. This wrinkle is due to too much extension in

(a) Undeformed specimen (b) Deformation process 1 (c) Deformation process 2

Figure 3.21: Overview of the deformation process for intermediate configuration (a, b)i = (0,−9)

31


−200

20−10 −5 0 5 10

0.02

0.04

0.06

0.08

0.1

a

b

Thi

ckne

ss r

educ

tion

[mm

]

b −200

20

−10−505100.02

0.04

0.06

0.08

0.1

a

Thi

ckne

ss r

educ

tion

[mm

]

b

Figure 3.22: Response surface after the intermediate deformation step for two different view angles

−20

0

20−10 −5 0 5 10

0.02

0.04

0.06

0.08

0.1

a

b

Thi

ckne

ss r

educ

tion

[mm

]

b −200

20

−10−505100.02

0.04

0.06

0.08

0.1

a

Thi

ckne

ss r

educ

tion

[mm

]

b

Figure 3.23: Final response surface for two different view angles

the circumferential direction during the intermediate forming step. The second mode can be seenat the transition region between the discrete die surface and the blankholder in radial direction.The cause of this wrinkle is a too large radial width of the intermediate shape. Figure 3.24shows the different wrinkling modes. Although the intensity of the wrinkles differs for eachdesign point, any notice of wrinkling makes the value of the object function in the relevantdesign point invalid. The design space can be divided into a feasible (without wrinkles) and aninfeasible domain (with wrinkles), see figure 3.25.

If only the feasible domain is taken into account it can be seen that parameter a has less influenceon the object function than parameter b.The minimum of the object function is situated in theregion b > 0 and restricted by the border of the infeasible domain. In this region all design points

Mode 1Mode 2

Figure 3.24: Simulation of the configuration (a, b)i = (0, 9) that shows severe wrinkling

32


−20

0

20−10 −5 0 5 10

0.02

0.04

0.06

0.08

0.1

ab

Thi

ckne

ss r

educ

tion

[mm

]

feasibleinfeasible

b

Figure 3.25: Response surface divided into the feasible domain and the infeasible domain

MSC.Marc Matlabx=(a,b)i f( )x x’=(a,b)i

Figure 3.26: Structure direct optimization

give a better result than the direct deformation of the blank into the (a, b) = (0, 0) configuration.The intermediate step that results in the least thickness reduction in the center of the cup afterdeformation is (a, b)i = (4, 5). The thickness after this deformation sequence is 0,1950 mm, this isa reduction of 30,4% referred to the original thickness of 0,28 mm. The thickness of the materialif only one deformation step is used is 0,1647 mm, a thickness reduction of 41,2%. This showsthat a considerable improvement can be realized by using an intermediate deformation step.

3.3 Direct Optimization

In the direct optimization procedure an optimization tool is used in combination with a nu-merical simulation program. The two major difficulties in direct optimization are programmingdifficulties and the large number of function evaluations required for the optimization process. Inthis case, the programming difficulty is solved by defining a function f(x) in Matlab that invokesthe numerical simulation in MSC.Marc. The input of the function is the value for x=(a, b)i,the simulation is done and the output is the associated value of the object function. The inMatlab build-in optimization tool fminsearch is used to minimize the function. This structureis schematically shown in figure 3.26. The number of function evaluations necessary to find(a, b)optimal depends strongly on the convergence tolerance and the initial set of parameters, butit is common to have more than 50 function evaluations. A specific disadvantage of the usedoptimization tool is that it does not take constraints into account. This means that the found

33


Table 3.1: Results direct optimization(a, b)initial f(x) Function evaluations (a, b)optimal f(x)

(0,-5) 0.0987 mm 89 (0,7.75) 0.0602 mm(10,0) 0.0924 mm 49 (10.4,0) 0.0922 mm

(a, b)optimal can be positioned outside the specified domain. Unfortunately, no optimization toolfor constraint multi-variable optimization problems is available in Matlab. Another problem isthat configurations that cause wrinkling are not excluded.

For two different initial sets of parameters the optimization procedure is started. The results arelisted in table 3.1. The direct optimization that is started in point (a, b)i = (0,−5) results in anoptimal configuration (a, b)i = (0, 7.75). This corresponds well with the response surface foundin section 3.2.3 , where a minimum is found around (a, b)i = (0, 7). The second procedure startsin the point (a, b)i = (10, 0) and finds a minimum at (a, b)i = (10.4, 0). The found minimumis a local minimum and not the optimal solution. One of the requirements of the used simplexmethod is that the object function has only one minimum. The response surface shows besidesthe global minimum a few local minima. Depending on the starting position, the optimizationroutine will find the optimal solution.

34

Chapter 4

Experiments

To verify the numerical simulations made in the previous chapter, experiments are done. Theexperimental plan and the results of the experiments will be discussed here.

4.1 Experimental plan

For a number of points in the design space, the sequential deformation process will be executed.In the numerical approach 181 points are used to construct a response surface. In the experimen-tal approach the deformation is done for 25 design points, representing 25 different intermediatedie shapes. The position of these design points in the design space is presented in figure 4.1. Thefinal configuration is (a, b) = (0, 0). The deformation process corresponding to a single designpoint is done three times to investigate the reproducibility of the forming process. The materialthat is used is Tinplate BA quality steel with a stated thickness of 0.28 mm. The specimens arecircular shaped with a diameter of 100 mm. After the production of the 75 specimens, the thick-ness of each specimen is measured. This is done with an accuracy of 1 µm using a micrometergauge. The mean measured thickness is 0.275 mm ± 0.0017 mm, this thickness will be used asreference. The reconfigurable die designed by Boers [1] is used to perform the deformations. Tosmoothen the discrete nature of the die surface an interpolator between the die surface and thespecimen is used. The interpolator is a 1 mm thick piece of rubber with the same geometry as the

−20 −10 0 10 20−10

−5

0

5

10

a

b

Figure 4.1: Distribution of the 25 design parameters

35

Chapter 4 Experiments

Initial state Intermediate state Final state

Figure 4.2: An undeformed sample (left), the 25 deformed intermediate shapes (middle) and the finalgeometry (right). The arrows show three different sequential deformation processes.

undeformed specimen. The interpolator is replaced after three deformations. The reconfigurabledie uses a rubber pad setup, where one of the dies in conventional die forming is replaced by arubber pad. The rubber pad is positioned in an open ended cylinder. At one side, a hydraulicram is pushing against the rubber pad. This pressure is transmitted through the pad and actson the specimen. It is assumed that in this technique, the pressure always acts perpendicular tothe specimen surface. During the experiment a pressure of 30 MPA is applied to the specimen.After the first deformation step, the specimen is taken out of its holder and the die surface isadjusted. The specimen is placed back and deformed in the final shape. Picture 4.2 is taken afterthe intermediate deformation. At the left an undeformed specimen is visible, in the middle the25 different intermediate shapes are presented and at the right the final geometry can be seen.In the next deformation step the specimens, which now all have their own intermediate shapes,will be deformed into the same final shape.

4.2 Experimental results

First, all the specimen are deformed into the intermediate shape. After the deformation allspecimen are optically examined for irregularities. Configurations with a sharp-edged transitionbetween the discreet die and the blankholder showed dimples on the surface of the transitionregion. These dimples are caused by the pins of the discrete die surface. The interpolator couldnot prevent the sample surface from dimpling because the pins penetrated the rubber inter-polator. It is found that for configurations where the angle α (figure 4.3) is larger than 45o,a rubber interpolator of 1 mm thickness is not sufficient. One configuration, (a, b)i = (0, 9),fractured during the deformation and is therefore excluded from the results. In figure 4.4, thedimples are visible and the fracture is shown. The remaining two specimen to be deformed inthis configuration, are deformed with the use of two interpolators. The use of two interpolatorsincreases the geometrical deviation from the intensional shape, but decreases the dimples andprevents fracture in the (a, b)i = (0, 9) configuration.

36


Figure 4.3: Die shape for configuration (a, b)i = (12, 3)

DimplesFracture

Figure 4.4: Picture of the intermediate step with (a, b)i = (0, 9)

37


−200

20−10 −5 0 5 10

0.02

0.04

0.06

0.08

0.1

a

Thi

ckne

ss r

educ

tion

[mm

]

b

Figure 4.5: Response surface after the intermediate deformation step

After the first deformation step, the thickness in the center of each cup is measured with themicro gauge. The results are listed in tabel A.1. Although three experiments per configuration istoo less to make statistical statements, the reproducibility is sufficient. The standard deviationof the mean residual thickness of the 25 configurations is between 0,0006 and 0,0058 mm. Largedeviations from the mean residual thickness can be explained by wear of the interpolator. Infigure 4.5 the mean center thickness reduction of the cups in their intermediate state is plottedas a function of the design variables (a, b)i. It can be seen that the thickness reduction of theconfiguration (a, b)i = (−12, 3) is the largest and that configuration (a, b)i = (0,−9) results inthe thickest center point. The thickness reduction of the configuration (a, b)i = (0, 0) is 0,068 mm.Although this configuration already has its final shape, it is placed back in the die and deformedfor the second time, like all the other specimens.

In the second deformation step all the specimens are deformed in the same final geometry,(a, b) = (0, 0). Large in plane stresses caused some specimens to wrinkle (figure 4.6). Twotypes of wrinkles are visible: the wrinkling in the circumferential direction (mode 1) and thewrinkling in the radial direction (mode 2). All the residual thicknesses are measured and listedin table A.2. Figure 4.7 shows the mean thickness reduction for each configuration. Again theexperiments are reproducible. In figure 4.8 the feasible domain and infeasible domain are shown.It is observed that the response surface is relatively flat, despite some trends are visible. All theconfigurations where b > 0 are wrinkled and thinner in the center than the reference configuration(a, b)i = (0, 0). The cup with intermediate configuration (a, b)i = (0, 9) shows less thicknessreduction than the other configurations with b > 0, this is probably due to the double interpolatorthat is used during deformation. The best cups are produced with an intermediate configurationwithin the domain (a < 0) and (b < 0). Configuration (a, b)i = (−6,−6) results in a thicknessreduction of 25 %. When the cup is directly formed the center thickness is reduced with 29 %.

The residual thickness of the (a, b)i = (0, 0) configuration is 0,197 mm. After the first deformationstep the thickness of this configuration was 0,207 mm. From this can be concluded that althoughthe shape of the die is not changed, deformation in the specimen has taken place during thesecond process. This unexpected result can be due to influences of the interpolator. Figure 4.9shows the situation in the first deformation step and during the second deformation step. In thesecond deformation step, the rubber of the interpolator is differently distributed over the surface

38


Mode 1

Mode 2

Figure 4.6: Wrinkles in the specimen after the final deformation, (a, b)i = (0, 9)

−200

20−10 −5 0 5 10

0.02

0.04

0.06

0.08

0.1

ab

Thi

ckne

ss r

educ

tion

[mm

]

Figure 4.7: Final response surface

−200

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0.02

0.04

0.06

0.08

0.1

a

Thi

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ss r

educ

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[mm

]

feasibleinfeasible

b

Figure 4.8: Final response surface, divided in a feasible and infeasible domain

39


specimen

interpolator

die

pressure

(a) First step

specimen

interpolator

die

Pressure

(b) Second step

Figure 4.9: Situation after clamping of the specimen

of the pins than in the first step. This results in a slightly different forming shape and a smalldeformation of the specimen during the second step.

40

Chapter 5

Simulations vs Experiments

In this chapter the results of the numerical simulations and the experiments are compared.

5.1 Results

First deformation stepThe numerical and experimental results obtained after the intermediate forming step can beseen in figure 5.1. The response surfaces in the graphs are almost identical and it is concludedthat the numerical model approaches the actual situation very well during the first deformationstep.

Second deformation stepDuring the second deformation step, wrinkling occurred for some configurations in both theexperiments and simulations. In figure 5.2, a picture of a specimen is compared to the numericalresult. The intermediate configuration for this deformation process is (a, b)i = (0, 9). Bothwrinkling modes are clearly visible and even the number of wrinkles in the simulation correspondswith the number of wrinkles in the produced cup.

The thickness reductions in the center of the cups are compared in figure 5.3. The most strikingdifference is found in the feasible domain, where the thickness reduction in the simulations is

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0.02

0.04

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a

b

Thi

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(a) Simulation

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a

Thi

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[mm

]

b

(b) Experiment

Figure 5.1: Numerical and experimental response surface after the intermediate deformation step

41

Chapter 5 Simulations vs Experiments

Mode 1Mode 2

(a) Simulation

Mode 1

Mode 2

(b) Experiment

Figure 5.2: Wrinkling in cup with (a, b)i = (0, 9)

−20

0

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0.02

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0.1

a

b

Thi

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]

b

(a) Simulation

−200

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0.1

ab

Thi

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[mm

]

(b) Experiment

Figure 5.3: Numerical and experimental response surface after the second deformation step

larger than in practice. In the region b < −2, the response surface of the simulations and theexperiments can be considered flat. In that region the simulations show a thickness reduction of0,95 mm, which is 34% of the initial thickness. In practice, the center of the cups reduced only0,8 mm, which is 27%. Multiple factors can cause the difference between the experimental andnumerical results. In section 5.2, some possible causes for this deviation are further elaborated.

Another difference can be seen when the infeasible domains are compared (figure 5.4). Interme-diate configurations with b = 3 did not wrinkle in the simulations, while after the experimentssmall shape irregularities are detected in the specimens. This is disappointing because theseconfigurations resulted in the largest residual thickness in the simulations, but the experimentsgive no satisfactory results.

In both response surfaces, the object function value in point (a, b)i = (−6, 0) is lower than thefunction value in point (a, b)i = (0, 0). In the experiment this intermediate configuration resultsin a final thickness reduction of 0.073 mm (26%), the configuration (a, b)i = (0, 0) leads to areduction of 0.079 mm (29%). The numerical results also show a small improvement, respectivelya thickness reduction of 0,92 mm (33%) and 0,94 mm (34%).

42


(a) Simulation (b) Experiment

Figure 5.4: Feasible and infeasible domain of the response surfaces

0 0.5 1 1.5 20.182

0.184

0.186

0.188

0.19

0.192


Thi

ckne

ss [m

m]

γ =10γ = 0.1γ = 0.0001

(a) (a, b)i = (0, 0)

0 0.5 1 1.5 20.182

0.184

0.186

0.188

0.19

0.192

Thi

ckne

ss [m

m]


γ = 10γ = 0.1γ =0.0001

(b) (a, b)i = (24, 0)

Figure 5.5: Influence of friction

5.2 Influence of forming conditions

Some factors that cause errors in the numerical model are discussed here. These points mayrequire more research to know their exact influence.

5.2.1 Friction

Friction in deformation processes is difficult to model. The influence of the relative slidingvelocity of two bodies in contact is difficult to estimate. And as one can see in section 3.1.2, theinfluence of friction on the thickness reduction is dependent on the shape of the intermediatedie. The influence of friction is analyzed for several configurations using numerical simulations.Figure 5.5(a) and 5.5(b) show for different values of γ, the thickness of the cup as a function ofthe distance from the center. With decreasing γ, the influence of friction increases. But for bothconfigurations there is only a very small variation in the center thickness noticeable.

To verify these results, two forming experiments are done where six specimens are deformedby using the intermediate configuration (a, b)i = (0, 0). Three of the specimens are coated

43


Table 5.1: Influence of lubricationsample nr t0[mm] t1[mm] t2[mm]

1 0.274 0.209 0.200without lubrication 2 0.275 0.210 0.196

3 0.275 0.209 0.2054 0.276 0.214 0.201

with lubrication 5 0.274 0.209 0.2006 0.275 0.211 0.200

(a) (b)

Figure 5.6: Geometrical influence of the interpolator

with a lubricant which is normally used in deep drawing processes. It is fair to assume thatno friction is present during the forming process of these specimens. The other three samplesare deformed without lubrication. Although the samples already have the final shape afterthe intermediate step, the second deformation step is done as well. In table 5.1 the resultsare presented. In agreement with the simulations, lubrication does not have a considerableinfluence. It is therefore concluded that friction is not the main cause of the difference betweenthe numerical and experimental results.

5.2.2 Interpolator

Due to the interpolator between the die surface and the specimen, the actual die shape differsfrom the set geometry. The numerical results are obtained by assuming the geometry as depictedin figure 3.18. In this assumption the thickness of the interpolator remains 1 mm in the centerof the cup, as in figure 5.6(a), while its thickness is ignored on the edges. However, it is morelikely that the interpolator moves around the edges and in between the cavities of the pinsurface (figure 5.6(b)). Simulations with and without an interpolator, figure 5.7, show that thegeometrical effect of an interpolator has a large influence on the thickness reductions but not onthe present trend in the response surface. In section 4.2 it is found that an interpolator of 1 mmthickness fails if the angle between the die surface and the blankholder is larger than 45o. Moreresearch has to be done to find the appropriate method to correct for the geometrical deviationdue to the interpolator.

44


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Thi

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]

with interpolatorwithout interpolator

ba

Figure 5.7: Response surface for simulations with and without the interpolator

Die

Specimen

Pressure

(a) follower force

Die

Specimen

Pressure

(b) no follower force

Figure 5.8: Pressure direction

5.2.3 Pressure

The reconfigurable die uses a rubber pad setup, where one of the dies in conventional die form-ing is replaced by a rubber pad. It is assumed that in this technique, the pressure always actsperpendicular to the specimen surface (follower force) as in hydroforming. This situation is pre-sented in figure 5.8(a). However, it is possible that this assumption is not correct. Figure 5.8(b)shows the situation where the pressure remains in the applied direction. Simulations represent-ing situation 5.8(a) and 5.8(b) are done for two different configurations, (a, b)i = (0, 0) and(a, b)i = (24, 0). The results are shown in figure 5.9. Without follower force, the thickness afterdeformation is more homogeneous and in the center the thickness reduction is less. A residualthickness of 0.216 mm is calculated for configuration (a, b)i = (0, 0), this is a reduction of 23%.In the simulations with the follower force the residual thickness is 0.186 mm, a reduction of34%. The experiments showed a reduction around 29%. The direction of the applied pressureis probably one of the main error points in the current used model. The assumption that thepressure remains perpendicular to the specimen surface during deformation may not be correct.Research is necessary to find the direction of the applied pressure in the reconfigurable die setup.

45


0 10 20 30 40 500.15

0.2

0.25

0.3

0.35


Thi

ckne

ss [m

m]

follower forceno follower force

(a) (a, b)i = (0, 0)

0 10 20 30 40 500.15

0.2

0.25

0.3

0.35T

hick

ness

[mm

]


follower forceno follower force

(b) (a, b)i = (24, 0)

Figure 5.9: Influence of applied pressure direction

46

Chapter 6

Conclusions

In the search for an optimization procedure, the forming process is optimized for a strip and anaxisymmetrical cup. The geometries are parameterized, using a Bezier curve, with two param-eters a and b. The object function that is optimized is the thickness reduction in the center ofthe geometry.A numerical model is made to simulate the sequential forming process of a strip. From thesesimulations some conclusions are drawn:

• The best configuration lies between (a, b)i = (0, 0) and (a, b)i = (−12, 6).

• Some intermediate configurations result in wrinkles in the final geometry. Increasing theforming pressure prevents wrinkling for some configurations.

• Friction has a great influence on the forming process. This influence is shape depended.

The forming process of an axisymmetric cup is simulated for different intermediate configura-tions. The sequential approximate optimization method is used to construct a response surfaceof the object function and a direct optimization routine is used to find the optimum solution.The simulations showed that:

• The residual thickness is the largest for configuration (a, b)i = (4, 5). This configurationresults in a thickness reduction of 30,4 %, compared to 41,5 % reduction in direct forming.

• Parameter b determines the separation between the feasible and infeasible domain. Con-figurations with large positive values of b are prone to wrinkle.

• The result of the direct optimization procedure is depending on the initial guess due tothe presence of multiple minima.

With the use of a reconfigurable die, experiments are done to verify the simulations of the cupforming process. The experiments lead to the following conclusions:

• In practice, the best intermediate configurations are found in the region a < 0 and b < 0. Athickness reduction of 25% is achieved using intermediate configuration (a, b)i = (−6,−6),compared to 29% in direct forming.

47

Chapter 6 Conclusion

• All the configurations with b > 0 result in wrinkling.

• For sharp transitions between the die surface and the blankholder (α > 45o), an interpo-lator of 1 mm thickness is not enough to correct the discrete nature of the reconfigurabledie.

Comparing numerical and experimental results, it can be concluded that:

• Simulations can be a tool to predict wrinkling, the feasible and infeasible domains corre-spond well.

• Friction has little influence during the forming of this geometry.

• The numerical and experimental response surfaces, constructed after the first deformationstep, are in very good agreement.

• The numerical and experimental response surfaces, constructed after the final deformationstep, deviate. The numerical results predict too much thickness reduction. This is probablydue to errors in the modelled pressure direction.

48

Chapter 7

Discussion

In this chapter, some assumptions are discussed.

The chosen optimization procedures are feasible only for a limited number of design parameters.For more complex products, the parametrization will result in more parameters, which will leadto very long computation times. The use of more intermediate production steps leads not onlyto more design parameters, but also to coupled design parameters. The sequential deformationsteps can not be optimized independently because the different intermediate deformation stepsdirectly influence each other.

The exact influence of some forming conditions is unknown. It is assumed that the rubber padin rubber pad forming acts like water in hydroforming. Water and rubber are both incom-pressible, but rubber has a friction coefficient and a stiffness. Perhaps this difference causes theforming pressure to deviate from its perpendicular direction to the specimen. This could be anexplanation for the difference between the numerical and experimental results.

The interpolator has an non negligible influence on the geometry of the die surface. However,the exact influence of the interpolator on the geometry is unknown. It is assumed that itsinfluence is minimal on the circumference at the transition region between the die surface andthe blankholder and maximal at the center of the axisymmetrical shape. This assumption maybe right in the first deformation step, but due to the shape of the blank at the beginning ofthe second deformation step, the situation is different. Research is necessary to find a relationbetween the shape of the die surface and the actual forming shape.

Another point that needs more attention is the clamping of the blank in the second deformationstep. Depending on the shape of the blank the clamping leads to difficulties. Because the blanksin this research are relatively flexibel and have a diameter of 100 mm, while only the center witha diameter of 36 mm is deformed, the clamping was not a problem.

49

Bibliography

[1] S.H.A.Boers, Optimum forging strategies with a 3D reconfigurable die, UniversiteitsdrukkerijTU eindhoven, 2006.

[2] K.Chung, O.Richmond. Ideal forming-I. Homogeneous deformation with minimum plasticwork, Int. J. Mech. Sci. 34-7:575-591, 1992.

[3] K.Chung, O. Richmond. Ideal forming-II. Sheet forming with optimum deformation, Int. J.Mech. Sci. 34-8:617-633, 1992.

[4] V.Pegada, Y.Chun, S. Santhanam. An algorithm for determining the optimal blank shapefor the deep drawing of aluminium cups, Journal of materials processing technology, 125-126:743-750, 2002.

[5] J.Y.Kim, N.Kim, M.S.Huh. Optimum blank design of an automobile sub-frame, Journal ofmaterials processing technology, 101:31-43, 2000.

[6] C.H.Lee, H.Huh. Blank design and strain estimates for shet metal forming processes by afinite element inverse approach with initial guess of linear deformation, Journal of materialsprocessing technology, 82:145-155, 1998.

[7] G.Zhao, E.Wright, R.V.Grandhi, Forging preform design with shape complexity control insimulating backward deformation, Int. J. Mach. Tools manufact., 35-9:1225-1239, 1995.

[8] G.Zhao, E.Wright, R.V.Grandhi, Preform die shape design in metal forming using an opti-mization method, Int. J. for numerical methods in engineering, 40:1213-1230

[9] M.S.Shim, J.J.Park. The formability of aluminium sheet in incremental forming,Journal ofmaterials processing technology,113:654-658, 2001.

[10] J.J.Park, Y.H.Kim. Fundamental studies on the incremental sheet metal forming tech-nique,Journal of materials processing technology, 140:447-453, 2003.

[11] P.Nordlund, B.Haggblad. Prediction of wrinkle tendencies in explicit sheet metal-formingsimulations, Int. J. for numerical methods in engineering, 40:4079-4095, 1997.

[12] X.Wang, J.Cao. On the prediction of side-wall wrinkling in sheet metal forming processes,Int. J. of mechanical sciences, 42:2369-2394, 2000.

[13] R.T.Haftka, Z.Gurdal. Elements of structural optimization. Kluwer Academic Publishers,ISBN 0-7923-1505-7, third edition, 1992.

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[14] S.Kok, N.Stander, Optimization of a sheet metal forming process using succesive multipointapproximations, Structural optimization, 18:277-295, 1999.

[15] S.H.A.Boers, P.J.G.Screurs, M.G.D.Geers, Operator-split damage-plasticity applied togroove forming in food can lids, Int. J. of soldids and Structures, 42:4154-4178, 2005

51

Appendix A

Experimental results

52

Appendix A Experimental results

Table A.1: Values of the design parameters and the center point thickness after the intermediate stepThickness sample 1 Thickness sample 2 Thickness sample 3

[mm] [mm] [mm]18 0 0.216 0.226 0.22412 3 0.218 0.219 0.21612 0 0.216 0.222 0.21912 -3 0.226 0.225 0.2306 6 0.215 0.215 0.2176 3 0.213 0.211 0.2076 0 0.216 0.215 0.2176 -3 0.226 0.224 0.2266 -6 0.236 0.236 0.2330 9 * 0.223** 0.226**0 6 0.214 0.211 0.2110 3 0.203 0.202 0.2000 0 0.204 0.210 0.2050 -3 0.225 0.222 0.2180 -6 0.235 0.236 0.2350 -9 0.245 0.248 0.247-6 6 0.207 0.204 0.202-6 3 0.196 0.195 0.195-6 0 0.205 0.203 0.202-6 -3 0.221 0.220 0.221-6 -6 0.236 0.234 0.238-12 3 0.191 0.183*** 0.190-12 0 0.204 0.204 0.194***-12 -3 0.223 0.224 0.223-18 0 0.209 0.208 0.211

* sample fractured during deformation** deformed with two interpolators*** deformed with damaged interpolator

53

Appendix A Experimental results

Table A.2: Values of the design parameters and the center point thickness after the final deformation stepThickness sample 1 Thickness sample 2 Thickness sample 3

[mm] [mm] [mm]18 0 0.195 0.195 0.19412 3 0.197* 0.197* 0.195*12 0 0.195 0.194 0.19112 -3 0.193 0.191 0.1926 6 0.200* 0.200* 0.200*6 3 0.196* 0.194* 0.195*6 0 0.199 0.196 0.1956 -3 0.200 0.198 0.1986 -6 0.200 0.198 0.1970 9 ** 0.210* 0.208*0 6 0.202* 0.200* 0.194*0 3 0.198* 0.195* 0.194*0 0 0.197 0.198 0.1960 -3 0.206 0.208 0.197***0 -6 0.202 0.202 0.1960 -9 0.202 0.203 0.198-6 6 0.202* 0.198* 0.193*-6 3 0.193* 0.192* 0.190*-6 0 0.203 0.201 0.201-6 -3 0.200 0.198 0.197-6 -6 0.208 0.205 0.204-12 3 0.185 0.180* 0.188-12 0 0.203 0.201 0.194***-12 -3 0.204 0.204 0.196***-18 0 0.205 0.201 0.203

* wrinkles formed during deformation** sample fractured during previous deformation step*** deformed with damaged interpolator

54

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