Creasing behaviour of corrugated board An experimental and numerical approach L.G.J. Gooren Report MT06.06

Master’s thesis

Coach: Dr.Ir. R.H.J. Peerlings

Supervisor: Prof.Dr.Ir. M.G.D. Geers

Technische Universiteit Eindhoven Department Mechanical Engineering Materials Technology

Eindhoven, February 2006

Abstract

The present study focusses on the creasing behaviour of corrugated board. To fold a board in a proper way, crease lines are applied to define the folding line and to reduce the necessary moment for folding. The purpose of this study is to understand and predict cracking of corrugated board during the creasing process by means of experiments with microscopic techniques combined with numerical simulations.

A test set-up is designed which allows one to perform creasing on small samples of paperboard in the field of view of an Optical Microscope or a Scanning Electron Microscope (SEM). The basis for the setup is formed by a Micro-Tensile Stage which fits in the vacuum chamber of the SEM. Special tools have been manufactured for this stage, which mimic flatbed-creasing as performed in an industrial environment. In this report we consider only the case in which the creaser is oriented in the cross direction, i.e. perpendicular to the machine direction. This orientation is the most critical in practice.

In the experiments no cracking is observed. However, it is observed that the inside liner is damaged, depending on the position of the creaser with respect to the flute.

Numerical simulations are used to provide insight in the stress and strain distribution. An or- thotropic (hypo-)elasticity model, as well as an anisotropic yield criterion due to Hill48, is em- ployed. Tensile tests are performed in order to determine the material properties. Due to measure- ment difficulties the out-of-plane properties are not determined experimentally, instead empiric relations are used.

The deformations predicted by the finite element (FE) model show a good agreement with the experiments. However, the load-displacement response predicted by the model deviates from that in experiments. The lack of compressibility of the Hill48 yield criterion and the out-of-plane shear stress and shear yield stress have a significant influence on the creasing reaction force. The reaction force is mainly governed by the fluting stiffness.

It is demonstrated that creasing exactly between two peaks of the fluting is the most critical case. The stress in machine direction is again influenced by the value of the out-of-plane properties, therefore a realistic stress distribution can not be given. The washboard effect, i.e. the waviness of the liner, appear to reduces the stress in the inside liner.

i ii Samenvatting

Dit rapport beschrijft het rilgedrag van golfkarton. Een ril wordt gemaakt in golfkarton om een vouwlijn aan te brengen en om het benodigde vouwmoment te verlagen. Het doel van dit onderzoek is het begrijpen van het rilgedrag en vervolgens het voorspellen van het scheuren van golfkarton. Hiervoor zijn microscopische technieken in combinatie met numerieke simulaties gebruikt.

Een experimentele opstelling is bedacht waarmee kleine proefstukjes gerild kunnen worden in het zichtveld van een optische of een elektronen microscoop (SEM). Een op de industriële gebaseerde rilsimulator is ontworpen waarmee deze rilproeven op kleine schaal uitgevoerd kunnen worden. Als basis is een Micro-Tensile Stage genomen welke in de SEM geplaatst kan worden. De tool kan op een Micro-Tensile Stage gemonteerd worden. In dit rapport zal alleen gefocust worden op rillen in de dwars richting van het papier, met andere woorden haaks op de machine richting. In de praktijk is dit de meest kritische ril.

Daadwerkelijk scheuren is niet geconstateerd tijdens de experimenten. Desalniettemin, laat het rillen duidelijk meer vrijgekomen vezels op de rillijn zien, wat de groei van schade impliceert.

Numerieke simulaties zijn gebruikt om inzicht te krijgen in de spanning- en rekdistributie. Een or- thotroop (hypo-)elastisch met een anisotroop vloeicriterium volgens Hill48 is gebruikt. Trekproeven zijn uitgevoerd om de materiaal eigenschappen te bepalen. Omdat de materiaal eigenschappen in dikte richting moeilijk te bepalen zijn, zijn hiervoor empirische relaties gebruikt.

De vervormingen van het eindige elementen (EE) model tonen een goede overeenkomst met de experimenten. Echter, de kracht-verplaatsing respons wijkt aanzienlijk af van de experimenten. Het ontbreken van compressibiliteit van het Hill48 vloeicriterium en de invloed van de afschuif- modulus en de afschuifvloeigrens beïnvloeden het verloop van de reactie kracht. De reactie kracht wordt voornamelijk bepaald door de stijfheid van de golf.

Het is aangetoond dat rillen tussen twee golftoppen het meest kritisch is. De spanning in machine richting wordt mede bepaald door de materiaal eigenschappen in dikte richting. Hierdoor kan geen realistisch beeld van spanning worden gegeven. Het zogenaamde wasbordeffect, de golving van de liner, blijkt een positieve invloed te hebben op de spanning in de liner.

iii iv Nomenclature

Roman symbols

Symbol Description

4C elasticity tensor D deformation rate tensor E Young’s modulus F deformation gradient tensor F, G, H, L, M, N Hill48 yield surface constants f yield function G shear modulus L velocity gradient tensor R ratio of yield in orthotropic direction and σref T corrugated board thickness tf flute thickness tl liner thickness ~x position vector ~v velocity vector

Greek symbols

Symbol Description

α angle δ creaser displacement γ˙ plastic multiplier ε linear strain tensor Λ attachment length λ period of the fluting νij Poisson’s ratio of axial strain and lateral strain σ Cauchy stress tensor σ equivalent stress σref reference yield stress Ω spin tensor

v General subscripts and superscripts

Symbol Description

(.)e elastic part (.)ii component in i-direction (.)ij component in ij-direction (.)p plastic part (.)ref reference state

Operators

Symbol Description

[.] components of a tensor in matrix form x˙ material time derivative of x a · b single tensor contraction A : B double tensor contraction ∇~ x gradient of x

Abbreviations

Symbol Description

CD cross direction DIC digital image correlation MD machine direction ZD thickness direction

vi Contents

Abstract i

Samenvatting iii

Nomenclature v

Contents vii

1 Introduction 1 1.1 Corrugated board ...... 1 1.2 Creasing of corrugated board ...... 2 1.3 Aims and scope ...... 3

2 Mechanical behaviour of paper 5 2.1 Paper as a fibrous material ...... 5 2.2 Bonds in paper ...... 6 2.3 Stress-strain response ...... 7 2.3.1 In-plane behaviour ...... 7 2.3.2 Out-of-plane behaviour ...... 8

3 Constitutive model 11 3.1 Kinematics ...... 11 3.2 The constitutive model ...... 12 3.2.1 Elastic deformation ...... 12 3.2.2 Elasto-plastic deformation ...... 12 3.3 Hill48 yield criterion ...... 13

4 Experimental identification of the orthotropic material constants 17 4.1 Tensile test set-up and preparation ...... 17 4.2 In-plane behaviour ...... 18 4.2.1 Elastic parameters ...... 19 4.2.2 Plastic parameters ...... 20 4.3 Out-of-plane behaviour ...... 21 4.3.1 Elastic parameters ...... 21 4.3.2 Plastic parameters ...... 21 4.4 Summary of material constants ...... 22 4.5 Numerical in-plane tensile test ...... 22

vii CONTENTS

5 Creasing experiments 25 5.1 Creasing in real-life and experimental creasing ...... 25 5.2 Creasing set-up ...... 26 5.3 Experimental creasing ...... 26 5.3.1 Position I ...... 27 5.3.2 Position II ...... 27 5.3.3 Position III ...... 28 5.4 Creasing with liner plate ...... 29

6 Confrontation of simulations vs experiments 35 6.1 Model definition ...... 35 6.2 Simulations with reference parameters ...... 36 6.2.1 Position I ...... 37 6.2.2 Position II ...... 38 6.2.3 Position III ...... 39 6.3 Comparison of inner liner stress ...... 39 6.4 Influence of material and geometric properties ...... 40 6.4.1 Material properties ...... 40 6.4.2 Geometric properties ...... 42 6.5 Influence of boundary conditions ...... 43 6.6 Summary of observations ...... 44

7 Conclusion and recommendations 47

Bibliography 49

Appendices 51

A Hill48 detailed calculations 53

B Calculation of in-plane shear 55

C Crease simulations overview 57

Dankwoord 59

viii Chapter 1

Introduction

1.1 Corrugated board

We all know corrugated board as used in boxes, but its origin is surprisingly different. In the 19th century, hand-cranked corrugated roller presses were used to generate corrugated paper. Corrugated paper replaced the plain paper which was used to keep the shape of the tall, stiff hats worn by gentlemen. The new cylinder was stronger than plain paper. Later, corrugated paper was first used to wrap bottles and slowly the first boxes were introduced. These boxes were much lighter and less expensive than the original ones made out of solid board.

Nowadays, paper and paperboard are commonly used materials in nearly every industry. World- wide about 300 million metric tons of paper and paperboard are produced each year. Probably the most important structural application are corrugated containers. Corrugated packaging is a versatile, economic, light, robust, recyclable, practical and yet appealing form of packaging. It is typically lightweight and inexpensive, with high stiffness-to-weight and strength-to-weight ratios. In figure 1.1 a piece of corrugated board is presented.

Figure 1.1: An example of a cardboard box at different scales

Corrugated board panels are sandwich structures consisting of two flat sheets, called liners, which are separated by a wave shaped core, called the fluting. These three layers of paper are assembled according the procedure presented in figure 1.2. Its specific structure gives the board a much better resistance against bending than that of each distinct layer.

1 CHAPTER 1. INTRODUCTION

Figure 1.2: The manufacturing of a single wall corrugated board [1].

1.2 Creasing of corrugated board

Creasing of corrugated board is an important technique used in the production of e.g. boxes. To fold a board in a proper way, crease lines are applied to define the folding line and to reduce the necessary moment to create a fold.

When board is creased by a creaser, permanent deformation is induced. Locally it decreases the stiffness and the board is easily bent along the crease line. The local deformation causes an increase of stress and eventually may result in cracking of the board. Figure 1.3 shows a part of an unfolded box which contains a crack in the inside liner.

Cracking of the outside liner also occurs. This is not caused by the creasing process. It occurs after the folding process. Board sheets are folded in a way to minimize storage space, so 180 degrees folds are commonly applied. When board sheets are piled up, the 180 degrees folds may cause cracking of the outside liner.

Both failures are not desired, because they strongly decrease the stiffness of the structure [1]. Humidity and temperature vary during the seasons in the papermaking process. In practice it is noticed that this influences the mechanical behaviour of paper and causes more cracking during the winter season.

Figure 1.3: A piece of corrugated board with creases and cuts. Cracking has occurred in the crease in the inside liner. The problem mainly occurs normal to the fluting. MD and CD represent respectively the machine direction and the cross direction.

2 CHAPTER 1. INTRODUCTION

1.3 Aims and scope

This project is carried out as a part of the project ’Lightweight Paper and Board’ which aims at reducing the necessary grammage for paper and board for packaging applications. This will be done by focussing on the parameters which influence the creasing and folding of the board and the influence of the microstructure of the paper.

The present study focusses on the creasing behaviour of corrugated board. The purpose of this study is to understand and predict cracking of corrugated board during the creasing process by means of experiments with microscopic techniques combined with numerical simulations. With the use of microscopic techniques the creasing process can be visualized, thus providing insight in the process at various scales. The numerical simulations are used to model the experiments. Varying the parameters of the numerical model reveals the relevant parameters of its creasing process.

This report starts with an extensive introduction into the mechanical behaviour of paper (chapter 2). First paper is presented as a fibrous material. The structure and bonding of paper is explained. Subsequently, stress-strain curves are shown to provide insight in the mechanical behaviour.

Chapter 3 starts with the necessary kinematics and describes the constitutive model used in the simulations. The model combines orthotropic elasticity and orthotropic plasticity. Tensile test are performed in order to obtain the material constants. Chapter 4 describes the experimental techniques used to obtain the constants.

Chapter 5 describes the crease experiments which are carried out to provide insight in the process. In order to enable direct observations at various scales a creasing tool is designed, which allows one to perform creasing operations on small samples of corrugated board in the field of view of a Scanning Electron Microscope (SEM).

Finite element simulations are conducted to model the experiments in chapter 6. The simulations are compared with the experimental creasing tests in order to investigate the ability of the model to capture the experimentally observed response.

In chapter 7, finally, conclusions are drawn based on a confrontation of experimental and compu- tational results.

3 CHAPTER 1. INTRODUCTION

4 Chapter 2

Mechanical behaviour of paper

This chapter describes the mechanical behaviour of paper. This behaviour is of importance for the constitutive modelling of the paper layers in the corrugated board. First, the (molecular) structure and bonding of paper is presented. Subsequently, its stress-strain response is discussed.

2.1 Paper as a fibrous material

Paper mainly consists of wood fibers; however nonfiberous materials are added to provide addi- tional properties. Figure 2.1 shows a Scanning Electron Micrograph of fibers in a cracked paper sheet. The precise composition depends on the grade of paper being manufactured. Wood pulps are classified according to their method of manufacture and the wood species used. These two factors determine the chemical composition, structure, and condition of the pulp fibers which, in turn, influence the properties of paper produced from the fibers.

Wood pulp is by far the most important source of fibers. Two main groups can be specified: softwoods (or conifers) and hardwoods (or deciduous trees). They differ in their structure, such as fiber length. Softwoods and hardwoods have fiber lengths of respectively 3-5 mm and 1-2 mm and are respectively 20 - 80 µm and less than 20 µm in diameter [2]. Fibers in softwood provide strength and conduct fluids. Hardwood species gain strength from fibers, but have other elements to conduct fluids (called vessel elements). Hardwood and softwood fibers have similar shapes.

Figure 2.2 shows the structure of a wood fiber. The fiber wall consists of four layers. Layer P (primary wall), layer S1 (secondary 1 layer), S2 (usually the thickest layer) and S3. The hole in the middle is called lumen. The region surrounding the fiber is the middle lamella. In the S2 layer fibrils spiral around the fiber axis. Fibrils are small bundles of primarily cellulose molecules.

Wood fibers consist mainly of three classes materials: cellulose, hemicellulose and lignin.

Cellulose

Wood fibers are mainly built of cellulose (C6H10O5)n. Cellulose is a polymer which is made of a linear repetition (about 5000 to 10000 times) of the monomer glucose (β-glucose). The attraction between cellulose molecules in different fiber surfaces is the principal source of fiber-to- fiber bonding in paper.

5 CHAPTER 2. MECHANICAL BEHAVIOUR OF PAPER

Figure 2.1: A Scanning Electron Microscope Figure 2.2: Fine structure of wood fibers [13] image of cracked paper

Hemicellulose

Hemicellulose molecules have much smaller chains than cellulose molecules. They promote the development of fiber-to-fiber bonding through their influence on the ability of fibers to take up water during processing and their direct participation in the bonding.

Lignin

Another important chemical in wood fibers is lignin. Lignin is found at the outside of cells walls and between cells and acts as a binding agent. Lignin has a complex three dimensional structure and is insoluble in water. All chemical pulping processes focus on removing the lignin. Lignin gives a brownish color to papers and hinders the formation of inter-fiber bonds and therefore decreases the paper strength.

2.2 Bonds in paper

Bonding in paper arises from the tendency of cellulose molecules to bond to each other when dried. Two types of bonds can be distinguished: bonding within a fiber and bonding between fibers.

Bonding within a fiber is provided by chemical bonds (like hydrogen bonds), intermolecular van der Waals bonds and entanglements of cellulose molecules. If two fibers are so close to each other that chemical bonding, intermolecular bonding and entanglement may occur, this is called inter- fiber bonding. Mechanical interlocking and surface tension forces of fibrils results in attraction forces between the fibers.

A hydrogen bond is a type of intermolecular force that exists between two partial electric charges of opposite polarity [19]. Although stronger than most other intermolecular forces, the hydrogen bond is much weaker than both the ionic bond and the covalent bond. Hydrogen bonds between fibers are identical to the hydrogen bonds within the fibers. Each glucose unit can form three hydrogen bonds (red in figure 2.3). Two bonds can be formed between two cellulose molecule chains and one bond forms within the chain. When three bonds are formed, the cellulose molecule is constrained.

6 CHAPTER 2. MECHANICAL BEHAVIOUR OF PAPER

CH2OH CHOH2 H C O H C O OOCCOH H CCOH H O C C H C C H H OH H OH

Figure 2.3: Cellulose molecule Figure 2.4: Hydrogen bonding in dry and wet state

Figure 2.5: Definitions of the directions in a paper or paperboard. MD is the machine direction, CD the cross machine direction and ZD the thickness direction. The notations 1, 2 and 3 are also frequently used.

In a wet state, more hydrogen bonds between cellulose fibrils can form than in a dry state, see figure 2.4. Therefore wet paper is more flexible. This explains why paper is so sensitive to moisture, humidity and temperature.

2.3 Stress-strain response

Because of the continuous papermaking process, wood fibers are oriented more in machine direction than in other directions. Three symmetry axes can be distinguished in a paper sheet. Figure 2.5 defines the three symmetry directions (machine direction = MD, cross direction = CD, thickness direction = ZD) in ordinary paper.

2.3.1 In-plane behaviour

Paper is an anisotropic material and exhibits different stress-strain behaviour in each direction. Figure 2.6 shows typical in-plane stress-strain curves in MD and CD. The MD-direction has an approximately 2-4 times higher stiffness and failure stress than the CD-direction [24], see chapter 4 for detailed information. In CD-direction paper has a larger maximum strain than in MD.

Paper is a hygroscopic material, changes of relative humidity and temperature affect properties such as stiffness. The moisture content depends on the current ambient conditions, but also on earlier conditions. Figure 2.6 also shows the trend observed for an increase of relative humidity and temperature. It results in lower tensile strength and a higher failure strain. It is important to control the ambient conditions to determine the material properties.

7 CHAPTER 2. MECHANICAL BEHAVIOUR OF PAPER

40 45

35 40 MD

MD σ 35 f 30

30 25

25 20 CD

increase of RH and T force [N] stress [MPa] 20 15 E 15 1 CD σ f 10 10

5 5 E 2 0 0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0 100 200 300 400 500 strain [−] displacement [µm]

Figure 2.6: Typical in-plane stress-strain Figure 2.7: Initial non-linearity of the load- curves of paper. Ei indicates stiffness mod- displacement curves ulus in i-direction, σf is the failure strength. The arrows and dashed curves show the trend observed for an increase in humidity or temperature.

Near the origin, the curves exhibit non-linearity, figure 2.7. During the initial phase of the tensile test not all the fibers are under tension. Gradually all the fibers are loaded and the load increases according to a more or less constant slope. It is advised to pre-tension a tensile specimen to remove this effect [10].

Linearity at the beginning of the stress-strain curves can be noticed. After a certain point the stress-strain curve gradually deviates from the initial slope. After the yielding point non-linear plasticity is invoked, see figure 2.8. The figure shows a stress-strain curve in MD with sequential loading and unloading of the tensile specimen. The figure shows that paper exhibits no clear yield point. Instead, the stress-strain curve gradually deviates from the initial slope. The tensile curves can be decomposed in an elastic and plastic part (ε = εe + εp). The plastic strain and the yield stress, σy, increase with ongoing tensile loading.

Yielding in paper is not symmetric in tension and compression. In compression, paper is less stiff than in tension [24] and exhibits about 65% and 25% of the tensile yield stress in, respectively, MD and CD.

Paper material is strain rate dependent. It is stiff and brittle at high strain rates and weak and ductile at low strain rates. However, [7] shows that in case of creasing the influence of strain rate can be neglected.

2.3.2 Out-of-plane behaviour

The in-plane tensile properties of paper are relatively easy to determine by tensile tests. Due to the small thickness of paper sheets, the out-of-plane properties are harder to obtain.

Reference [21] shows that the Young’s modulus in ZD-direction is about 200 times lower than the MD-direction. [22] observed that the amount of lateral in-plane strain generated during the through-thickness tensile loading is negligible. Poisson’s ratios ν31 and ν32 therefore are close to zero.

8 CHAPTER 2. MECHANICAL BEHAVIOUR OF PAPER

50

45 σ f 40 ε ε p e 35

30 σ 25 y1

Stress [MPa] 20

15 σ y0 10

5

0 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 Strain [−]

Figure 2.8: Stress-strain curve illustrating elasto- plastic behaviour in MD by loading and unloading

Reference [3] used empiric approximations to determine the out-of-plane parameters as

E E E 1 = 200 ; 1 = 55 ; 2 = 35 (2.1) E3 G13 G23 The out-of-plane behaviour in tension and compression is not symmetric. The tensile yield stress in thickness direction is about 0.5-2 MPa, while the compression yield stress is about 3-7 MPa.

9 CHAPTER 2. MECHANICAL BEHAVIOUR OF PAPER

10 Chapter 3

Constitutive model

This chapter describes the kinematics and constitutive model which will be using in numerical simulations. The model consists of an elastic part and a plastic part. To describe plastic yield, the Hill48 orthotropic yield criterion is used. The behaviour of paper is assumed to be uniform and homogenous in elastic and plastic deformation.

3.1 Kinematics

Initially, a deformable body is in the unloaded and stable reference state. The deformation gradient tensor F of the current state with respect to the initial configuration is defined by [5]:

T F = (∇~ 0~x) (3.1) where ~x is the position vector in the current configuration and ∇~ 0 the gradient with respect to the initial configuration. F is multiplicatively decomposed into an elastic part F e and a plastic part F p according to F = F e · F p.

The deformation rate is described by the velocity gradient tensor L, which can be determined by

L = (∇~ ~v)T = F˙ · F −1 (3.2) where ~v = ~x˙ is the velocity field. The deformation rate tensor D and the spin tensor Ω are, respectively, the symmetric and the skew-symmetric part of L and they are also decomposed into elastic and plastic parts according to:

1 D = (L + LT ); D = D + D (3.3) 2 e p

1 Ω = (L − LT ); Ω = Ω + Ω (3.4) 2 e p

This decomposition can be made unique by setting Ωp = O. The assumption implies that rotations superimposed on the original deformation process are fully attributed to the elastic deformation gradient tensor F e, while the plastic deformation gradient tensor F p is not affected.

11 CHAPTER 3. CONSTITUTIVE MODEL

3.2 The constitutive model

The constitutive model consists of two parts, an elastic and a plastic part. In the initial state, only elastic deformation can occur. While the stress rises a yield criterion, f, indicates the transition to plasticity. Kuhn-Tucker relations are used to decide whether elastic or elasto-plastic deformation occurs. {(f < 0) ∨ (f = 0 ∧ f˙ < 0)} → elastic deformation {(f = 0) ∧ (f˙ = 0)} → elasto-plastic deformation (3.5)

3.2.1 Elastic deformation

When the deformation is purely elastic, Dp = 0, so De = D. The objective Jaumann rate of the Cauchy stress tensor is related to the elastic deformation rate De by

◦ 4 σ= C : De (3.6) 4C is the fourth order stiffness tensor.

As argued in chapter 2, paper is made of oriented wood fibers and elastic stiffness and strength are therefore anisotropic. If we assume the fiber distribution to be symmetric, the stiffness prop- erties are orthotropic. The orthotropic linear elastic stiffness tensor is defined (in compact matrix notation) as

 −1 −1 −1 −1 E1 −ν21E2 −ν31E3 0 0 0  −1 −1 −1   −ν12E1 E2 −ν32E3 0 0 0   −1 −1 −1   −ν13E1 −ν23E2 E3 0 0 0  C =  −1  (3.7)  0 0 0 G12 0 0   −1  0 0 0 0 G23 0 −1 0 0 0 0 0 G31 Due to the symmetry of C, the material parameters must obey the three Maxwell relations ν ν ν ν ν ν 12 = 21 ; 23 = 32 ; 31 = 13 (3.8) E1 E2 E2 E3 E3 E1

νij is equal to minus the ratio of the transverse strain in the j -direction and the axial strain in the i-direction when the material is uniaxially stressed in i-direction. Additionally, positive definiteness of C requires [26]   E1,E2,E3,G12,G23,G31 > 0  ¡ ¢ 1  E1 2  |ν12| <  E2 ¡ ¢ 1 E1 2 |ν13| < (3.9) E3  ¡ ¢ 1  E2 2  |ν23| <  E3 1 − ν12ν21 − ν23ν32 − ν31ν13 − 2ν21ν32ν12 > 0

The orthotropic elasticity tensor now contains nine independent parameters, E1, E2, E3, ν12, ν13, ν23, G12, G13, G23, which have to be measured.

3.2.2 Elasto-plastic deformation

In elasto-plastic material models a yield function defines whether the response is elastic or elasto- plastic. As long as the stress state has not reached the threshold f = 0 defined by the yield

12 CHAPTER 3. CONSTITUTIVE MODEL

function f (σ, εp) the behaviour is elastic. f is defined as

2 2 f (σ, εp) = σ − σy(εp) (3.10)

2 with σ a quadratic form of the stress tensor and σy the yield stress defined as

σy = σy(σy0, εp) (3.11)

The hardening modulus H is defined as

∂σy = H(εp) (3.12) ∂εp

The yield stress increases due to hardening as the plastic deformation continues and is therefore related to the effective plastic strain εp: Z r t 2 εp = ε˙ pdτ; ε˙ p = Dp : Dp (3.13) τ=0 3

Plastic flow is assumed to obey normality, i.e. we use the associative flow rule:

∂f D =γ ˙ (3.14) p ∂σ

Dp is characterized by the plastic multiplier, γ˙ . In order to obtain the plastic multiplier the consistency condition for plasticity (f˙ = 0) is used.

3.3 Hill48 yield criterion

Several material models to describe paper have been developed in the past years, [3][21][24]. A popular criterion which describes orthotropic plasticity is the Hill48 quadratic yield criterion.

Hill [11] proposed an orthotropic yield function as an extension of the Von Mises yield function. This yield criterion is commonly applied in sheet metal applications and is insensitive to hydrostatic pressure, or mean stress. The plastic strain rate is normal to the yield surface and therefore no plastic volume change is invoked. Hill48 does not distinguish between tension and compression, so tensile and compressive yield occur at ±σiiy.

[21] and [24] propose a model with orthotropic elastic behaviour and an orthotropic yield surface constructed from sub-surfaces with orthotropic hardening. An extended Hill48 criterion which allows volume change during plastic deformation is proposed. Each proposed model assumes uniform and homogenous elastic and plastic deformation.

Hill48 is provided as an option in MSC.MARC/MENTAT 2005 and is used as a first approximation for describing orthotropic yield. The general yield criterion can be written as [11][15]

2 2 f = F(σ22 − σ33) + G(σ33 − σ11) + 2 2 2 2 H(σ11 − σ22) + 2Lσ23 + 2Mσ31 + 2Nσ12 − 1 (3.15) where the subscript ii indicates the stress and ij indicates the shear stress in the principal directions of orthotropy. F, G, H, L, M and N characterize the anisotropy of the material. Note that by taking F, G and H equal to 1 and L, M and N equal to 3, the Hill function degenerates to the

13 CHAPTER 3. CONSTITUTIVE MODEL

Von Mises yield function. The constants F, G, H, L, M and N are defined as 1 1 1 1 2F = 2 + 2 − 2 ; 2L = 2 σ22y σ33y σ11y σ23y 1 1 1 1 2G = 2 + 2 − 2 ; 2M = 2 (3.16) σ11y σ33y σ22y σ31y 1 1 1 1 2H = 2 + 2 − 2 ; 2N = 2 σ11y σ22y σ33y σ12y with σijy the yield stress values in ij-direction. Rearranging (3.16) gives 1 1 1 2 = G + H ; 2 = F + H ; 2 = F + G (3.17) σ11y σ22y σ33y It is clear that the constants F-N must satisfy

(F + G), (G + H), (F + H) > 0, L, M, N > 0 (3.18)

[12] also shows that the constant have to obey

FG + GH + HF > 0 (3.19) in order for the yield surface to be convex. Note that this condition leaves open the possibility that one of F, G and H may be negative.

Instead of the constants of (3.16) MSC.MARC/MENTAT 2005 uses ratios, Rij (see appendix A for more details), of actual yield stress to isotropic yield stress, which are defined in the orthotropic directions according to

σ11y √ σ12y R11 = ; R12 = 3 σref σref σ22y √ σ13y R22 = ; R13 = 3 (3.20) σref σref σ33y √ σ23y R33 = ; R23 = 3 σref σref where σref is the initial yield stress derived from the stress-strain curve in a preferential direction. Here we will use σref = σ11y, which implies that R11 = 1.

Hill48 limits the degree of orthotropy in three directions. For given σ11y and σ22y, σ33y depends on σ22y and cannot be chosen arbitrarily. By substituting (3.16) and (3.21) in (3.19), rewriting the result in terms of Rii and setting R11 = 1 it can be shown that R33 has to satisfy (see also appendix B) R22 R22 < R33 < (3.21) (1 + R22) (1 − R22)

Figure 3.1 shows a range of R22 with the limitations imposed on R33 by these inequalities.

14 CHAPTER 3. CONSTITUTIVE MODEL

1

0.9

0.8

0.7

0.6

0.533 R

0.4

0.3

0.2

0.1

0 0 0.2 0.4 0.6 0.8 1 R 22

Figure 3.1: Limitations on R33, when R11 = 1 and R22 is known. For a high degree of or- thotropy, i.e. small R11, we have R22 ≈ R33.

15 CHAPTER 3. CONSTITUTIVE MODEL

16 Chapter 4

Experimental identification of the orthotropic material constants

The previous chapters described the material response of paper and proposed a constitutive model to characterize it. This model contains several material constants which are to be determined in this chapter. First the test set-up and preparation are described. The material constants identification is divided into elastic and plastic properties. The plastic hardening is also derived from the tensile tests. The in-plane parameters are used in the material model and compared with the experiments.

4.1 Tensile test set-up and preparation

A single-wall corrugated board with B flute is considered. This type of board consists of three different papers. Although the three papers are not entirely identical, it was found that there is little difference in their tensile response. Therefore only the inside liner is used for parameter identification and the results are also used for the flute and outside liner. The specimens were kindly provided by Kappa Containerboard.

The tests are carried out using a Micro Tensile Stage with a calibrated load cell of 100 N. The specimens are carefully mounted along the tensile direction. A specimen with tangentially blending fillets between the uniform test section and the ends is used to determine relevant mechanical properties of the material. Figure 4.1 presents the specimen geometry, which has a gauge length of 12 mm and width of 5 mm.

Figure 4.1: Tensile test specimen. Specimen thickness is 0.186 mm and it is cut out of a sheet of paper

17 CHAPTER 4. EXPERIMENTAL IDENTIFICATION OF THE ORTHOTROPIC MATERIAL CONSTANTS

The strain in the gauge section is determined by dividing the clamp displacement by an effective gauge length of 18 mm. According to international standards for paper testing, tests have to be performed at a constant 23oC, a constant relative humidity of 50% and a constant elongation rate of 20 mm/min ± 5 mm/min.

Since the creasing tests which we aim to model are done in a SEM, the tensile tests are also done in the SEM. One should notice that the climate in the sample chamber of the SEM will be different from the international standards. The condition in the SEM is approximately 30oC and RH 0%. The paper will respond to the vacuum by evaporating all moisture.

Figure 4.2 displays the difference between a tensile test in vacuum and tests conducted in an environment according to the standards. Without moisture the paper is stiffer and more brittle. The tensile tests and crease experiments performed in the sample chamber of a SEM therefore represent the worst case scenario in terms of fracture strain.

Figure 4.3 shows the increase of stress in time while the SEM creates a high vacuum. The tensile specimen is fixed in the clamps of the tensile stage. The high vacuum causes the moisture to evaporate and the specimen starts to develop stress. This results in a tensile force. Before the tensile test starts the developed stress is released and set to zero. The initial non-linearity, as seen in chapter 2, is removed. This procedure is applied in all tensile tests.

A sequence of images are taken of the paper surface during the test. Digital Image Correlation (DIC) is used to determine Poisson’s ratios, which is discussed further below. The image processing software of the SEM is too slow to keep up with the elongation rate of 20 mm/min prescribed by the testing standards. An elongation rate of 5 µm/s is sufficient to get a series of images, but this elongation rate is significantly lower than the standards.

50 4.5 MD SEM test 45 4 MD standard test 40 3.5 35 3 30 2.5 25 2 Stress [MPa] 20 Stress [MPa] CD SEM test 1.5 15 CD standard test 10 1

5 0.5

0 0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0 500 1000 1500 Strain [−] Time [s]

Figure 4.2: Difference between tensile tests Figure 4.3: Evaporation of moisture causes a in SEM mode and according to standards at increase of stress in time in a clamped spec- 5 µm/s imen

4.2 In-plane behaviour

Figure 4.4 represent three different tensile tests to determine the in-plane properties of the paper sheet. Tests in MD and CD are used to determine Young’s modulus and Poisson’s ratio. The specimen with orientation of the fibres at 45o from the MD direction is used to determine the in-plane shear modulus.

18 CHAPTER 4. EXPERIMENTAL IDENTIFICATION OF THE ORTHOTROPIC MATERIAL CONSTANTS

Figure 4.4: Experimental method to define the in-plane properties. The horizontal lines indicate the fiber orientation. δ is the applied displacement and F is the measured reaction force.

The uniaxial stress-strain curves for the MD, CD and orientation of the fibres at 45o from the MD direction are plotted in figure 4.5. The stress is determined as the load divided by the initial area of the cross section. Strain is determined as the elongation divided by the effective gauge length. These measures are thus engineering stress and strain and we limit ourselves to a geometrically linear analysis.

The curves in figure 4.5 clearly show the expected anisotropic behaviour of paper. There is a factor 2-3 difference between Young’s modulus in MD and CD direction. The failure stress σf for MD is 3-4 times higher than that for CD. The yield stress and hardening in MD is higher than that in CD. The Young’s modulus, flow stress and hardening for 45o are between the two other directions.

−3 50 x 10 0 45 σ f −1 40 −2 35 MD −3 ν 21 30 −4 25 σ −5 f 45o MD−CD Stress [MPa] 20 −6 σ Lateral strain [−] 15 CD f −7 ν 12 10 −8

5 −9

0 −10 0 0.005 0.01 0.015 0.02 0.025 0 0.005 0.01 0.015 0.02 0.025 Strain [−] Axial strain [−]

Figure 4.5: In-plane tensile stress-strain Figure 4.6: Lateral strain vs. axial strain for curves. tensile loading in the MD and CD.

4.2.1 Elastic parameters

The Young’s moduli E1 and E2 are directly determined by measuring the initial slope of the stress-strain curves, which are in MD and CD respectively 3950 MPa and 1650 MPa.

The Digital Image Correlation (DIC) method is used to determine the Poisson’s ratios. DIC is based on the correlation of gray values of digital images of the undeformed and the deformed

19 CHAPTER 4. EXPERIMENTAL IDENTIFICATION OF THE ORTHOTROPIC MATERIAL CONSTANTS specimen. Using the change of gray values DIC can compute local strain.

Once the specimens are in the SEM and the tensile test starts, a sequence of images are taken of the paper surface. These images are used to compute local strain fields. DIC is a time-consuming procedure. The computed strain is averaged in each direction over the total image and plotted in figure 4.6. This figure shows lateral strain vs. axial strain data for tensile loading in the MD and CD; the slopes of the curves fitted through the data are ν12 and ν21 respectively. The Maxwell relation (3.8) holds for the values of ν12 and ν21. ν12 and ν21 are determined at respectively 0.65 and 0.27 (solid lines).

With the above four known parameters, the shear modulus is deduced from the 45o curve. Using elasticity theory and the assumption of plane stress, G12 is determined at 760 MPa, see appendix B.

4.2.2 Plastic parameters

A clear yield point is not visible in figure 4.5 because the deviation from the linear path grows gradually as the strain increases. One way to determine the yield point is to identify the stress at which the smooth curve deviates from a straight line beyond a certain percentage. Another way is by loading and unloading a specimen and inspecting whether there is any remaining deformation at zero stress, see figures 4.7 and 4.8. Both methods have been used and the yield stresses are approximated at σ11y0 = 13 MPa and σ22y0 = 4 MPa. A small deviation of the yield stress can be compensated in the hardening relation without affecting the total stress-strain response.

50 20

45 18

40 16

35 14

30 12

25 10 Stress [MPa] Stress [MPa] 20 8

σ 15 y 6 σ y 10 4

5 2

0 0 0 0.005 0.01 0.015 0.02 0 0.005 0.01 0.015 0.02 Strain [−] Strain [−]

Figure 4.7: Stress-strain curve illustrating Figure 4.8: Stress-strain curve illustrating elastoplastic behaviour in MD under loading elastoplastic behaviour in CD under loading and unloading and unloading

o The yield stress of 45 tensile test, σ45y, can be used to determine σ12y. Using σ45y in equation A.1 for f = 0 results σ12y, see appendix B. σ12y is determined at approximately 5.6 MPa.

Figure 4.9 shows a stress-strain curve for the tensile test in MD direction. The total strain can be decomposed as εe + εp. The yield stress as a function of the plastic strain, figure 4.10, is obtained by the elastic strain σ εp = ε − (4.1) E1 Paper exhibits non-linear hardening behaviour. An exponential relation between the yield stress

20 CHAPTER 4. EXPERIMENTAL IDENTIFICATION OF THE ORTHOTROPIC MATERIAL CONSTANTS

45 45

40 40

35 35

30 30

25 25

20 20 stress [MPa] stress [MPa]

15 15 σ 11y0 10 10

5 5 E 1 0 0 0 0.005 0.01 0.015 0.02 0 0.005 0.01 0.015 0.02 strain [−] plastic strain [−]

Figure 4.9: Elastic strain and total strain Figure 4.10: Yield stress vs plastic strain (εp = ε − εe) and the effective plastic strain describes this post-yield behaviour (in MD) very well:

m σy = σy0(1 + Aε¯p) (4.2) with A = 3900 and m = 0.37. To simulate the most critical case, equation (4.2) is fitted to the stress-strain curve with the lowest failure strength in figure 4.5. This equation is used in MSC.MARC/MENTAT 2005.

4.3 Out-of-plane behaviour

As described in chapter 2 it is difficult to measure the mechanical behaviour in thickness direction of the paper. In our creasing experiments the load in ZD will be mainly compressive. Reference [21] provides insight in the out-of-plane behaviour of paper, in tension and compression for similar paper.

4.3.1 Elastic parameters

Recalling the empiric relations (2.1)[3] gives us approximated values for E3 = 20 MPa, G13 = 71 MPa and G23 = 47 MPa.

DIC is used to determine the Poisson’s ratios ν13 and ν23. However, the results are not reliable. The paper sheet thickness is too small to apply DIC. Focussing at smaller scales with a SEM provides more surface to apply DIC, but the strain field becomes too localized to get global results.

[22] observed that the amount of lateral in-plane strain generated during through-thickness tensile loading is negligible. Poisson’s ratios ν31 and ν32 are close to zero.

4.3.2 Plastic parameters

The out-of-plane shear yield stress σ23y and σ31y are not determined. In the literature typical shear yield stress of 0.3-1.1 MPa can be found [21]. However, as a first approximation R31 = R23 = 1 is applied, for which we have σ31y = 7.5 MPa.

21 CHAPTER 4. EXPERIMENTAL IDENTIFICATION OF THE ORTHOTROPIC MATERIAL CONSTANTS

Table 4.1: The reference parameters Parameter Value S.I. unit Parameter Value Unit E1 3950 MPa σ11y 13 MPa E2 1650 MPa σ22y 4 MPa E3 20 MPa σ33y 4 MPa ν12 0.65 - σ12y 5.6 MPa ν31 0.0035 - σ23y 7.5 MPa ν32 0.0055 - σ31y 7.5 MPa G12 760 MPa A 3900 - G23 47 MPa m 0.37 - G31 71 MPa

Section 3.3 introduced the Hill48 yield criterion for describing yield in orthotropic materials. Equation (3.21) displays the stress coefficients which are used by MSC.MARC/MENTAT 2005. They form the ratio between yield stress in one direction and the reference yield stress. The yield stress in MD direction is chosen as reference yield stress.

Since σ11y and σ22y have been determined, R11 and R22 are known. As discussed in chapter 3, R33 must obey equation (3.21). This restriction implies

3.06 < σ33y < 5.78 MPa (4.3)

In the literature typical values σ33y of 1-2 MPa in tension and 5-7 MPa in compression are found. Because of the uncertainty, σ33y is chosen to be equal to σ22y as a first estimate.

4.4 Summary of material constants

All material parameters are now determined and are summarized in table 4.1. It is emphasised one more that the out-of-plane parameters are hard to determine. The influence these parameters in the creasing simulations will be investigated in chapter 6. The reference material parameters are recapitulated in table 4.1.

4.5 Numerical in-plane tensile test

The uniaxial tension data in MD, CD and for 45o has been used to fit the material properties, as described before. In order to validate these fits, comparisons of the experiments and the simulated stress-strain curves for uniaxial tensile tests in MD, CD, 45o are shown in figure 4.11. Figure 4.12 displays experimental and simulated lateral strain vs. axial strain curves for MD and CD tension. A single four-node isoparametric quadrilateral element is used. The in-plane tensile tests are simulated assuming plane stress (σ33 = σ23 = σ13 = 0) and homogeneous deformation. This assumption invokes only the in-plane constants.

Note that the computational lateral strain vs axial strain curve for the CD (upper dashed curve in figure 4.12) deviates somewhat from the experimental data. This deviation starts at the onset of plastic flow and is caused by the fact that the constants F − H of the Hill48 model, together with incompressibility, fix this ratios and it cannot be defined independently.

The results demonstrate that the proposed model can describe the material behaviour of paper

22 CHAPTER 4. EXPERIMENTAL IDENTIFICATION OF THE ORTHOTROPIC MATERIAL CONSTANTS

−3 x 10 50 0

45 −1 MD 40 −2

35 −3 ν 21 30 −4 45o MD−CD 25 −5

Stress [MPa] 20 −6 Lateral strain [−] CD 15 −7 ν 12 10 −8

5 −9

0 −10 0 0.005 0.01 0.015 0.02 0.025 0 0.005 0.01 0.015 0.02 0.025 Strain [−] Axial strain [−]

Figure 4.11: Tensile test in several loading Figure 4.12: Comparison of experimental cases. The solid lines represent the experi- and simulated lateral strain vs. axial strain mental curves and the dashed lines are from for MD and CD tension the numerical model over the full range of stress-strain in in-plane tension. The compression curve will be exactly the opposite of the tension curve, because incompressibility is invoked in the Hill48 criterion. In reality this is not true.

23 CHAPTER 4. EXPERIMENTAL IDENTIFICATION OF THE ORTHOTROPIC MATERIAL CONSTANTS

24 Chapter 5

Creasing experiments

After the material properties have been determined, crease experiments are performed. This chapter describes the crease tests which are carried out to verify the simulations presented further on in this report. A creasing set-up is designed, which allows one to perform creasing operations on small samples of paperboard in the field of view of a Scanning Electron Microscope (SEM) in order to enable direct observations at various scales.

5.1 Creasing in real-life and experimental creasing

A certain force is needed to crease a board panel. The reaction force depends on the orientation of the creaser with respect to the corrugated board. Here we consider only the case in which the creaser is oriented in the CD and a more or less two-dimensional deformation pattern is thus obtained. This orientation is usually the most critical in practice.

During the manufacturing process of board there is no factor that regulates the position of the crease line with respect to the fluting. The reaction force is different at each location and one location may be more damaging in terms of cracking than another. Several experiments are therefore carried out at different positions. Three positions of the creaser with respect to the period of the fluting are considered:

Position I Exactly between two peaks of the fluting Position II At the peak of the fluting Position III Exactly between the peak and the valley

Figure 5.1: Creasing of corrugated board at three positions, with α as angle and δ as crease depth.

In [8] an extensive study on failure and crease depth of the liner with a similar fluting is performed.

25 CHAPTER 5. CREASING EXPERIMENTS

By increasing the crease depth δ, the angle α as indicated in figure 5.1 becomes larger and the stress in the inside liner increases.

If the fluting is infinitely stiff a crease at position III would be the most critical, because it leads to the largest α and therefore the largest strain in the liner. In practice the flute has a finite stiffness and may collapse. [7] already suggested that the flute around the crease zone must collapse to prevent failure of the liner. Then creasing at position I is the most critical crease, according [8]. In that case cracking of the liner occurs at a depth of approximately 0.57 mm.

5.2 Creasing set-up

The creaser tool and the corrugated board which is to be creased are presented in figure 5.2. The tool is mounted on a Micro Tensile Stage which can be placed in the vacuum chamber of a SEM. The anvil (1) and creaser (2) are each mounted on one clamp of a micro tensile tester, which can be moved towards each other. Moving the clamps to each other forces the board (4) (dimensions hxl = 20x40 mm) against the anvil, resulting in local deformation of the board. A crease is thus made.

In real-life creasing the board panel has a certain width which prevents folding of the board. In the experiments the specimens are too small to resist bending and down holders are added to the creaser tool (3). Pieces of corrugated board are placed between the pins and the board specimen to prevent folding. Another test is performed with use of a liner plate (5). The liner plate is a portal, i.e. a plate with a cut-away in the middle through which the creaser freely moves. Both ends of the inside liner are fixated with double sided tape on the liner plate. The liner plate models the stiffening influence of the board panel from which the sample has been cut.

The creaser is based on a commercially used tip with a radius of 0.5 mm. The crease tests are performed by a Micro-Tensile Stage with a calibrated load cell (6) of 100 N. The extensiometer (9) measures the displacement of both traverses (7). According to [7] the influence of creasing velocity is negligible. The load is applied through a uniform displacement of 5 µm/s. At the start of each experiment the specimen is carefully placed between the crease tip and the anvil. The creaser is displaced until the board is fully creased.

All the tests are performed in the field of view of a (E)SEM to visualize creasing at various scales. If the electron microscope is used in environmental mode, ESEM or low vacuum, we can use the chamber as a sort of a climate chamber and reach the recommended environment. However the field of view is limited to approximately 500 µm, which is insufficient compared with the height of the corrugated board. In SEM mode the field of view is sufficient and therefore high vacuum mode is used.

5.3 Experimental creasing

Figure 5.3 shows the load-displacement curves for crease tests at three different positions. The load is defined as the reaction force divided by the specimen height, 20 mm. The curves represent crease experiments with use of the down holders. Each experiment is repeated three times. Without the down holders the board specimen folds, which results in lower a reaction force. This situation is not considered any further.

One can notice different maximum displacements in the diagrams of figure 5.3. This is partially caused by the fact that locally the board thickness varies due to the washboard effect of the liner,

26 CHAPTER 5. CREASING EXPERIMENTS

Figure 5.2: Creasing set-up with corrugated board specimen mounted in an Scanning Electron Microscope see figure 5.4. This effect is generated when joining the fluting and the liners during the production process. It appears on both sides, but is more severe on the inside liner. This explains that position I creases have less creaser displacement than position II creases. Folding of the fluting hinders further displacement during creasing at position III.

5.3.1 Position I

Figure 5.5 shows four stages of the creasing process at position I, which correspond with stages A-D in the top diagram of figure 5.3. Initially, the fluting has little resistance against deformation while δ increases. At about 800 µm the fluting buckles and unwinds itself along the lower liner, figure 5.5, image B. Beyond this point several mechanisms have been observed, which influence the reaction force. In image C and D of figure 5.5 the flute remains straight under approximately 90 degrees. In another case the flute collapsed in an S shape. In that case the reaction force would be higher. Image C also shows released fibers. The high tension stress at the outside of the liner causes failure of the bonding of the fibers and fiber ends are released from the fiber network. The decreasing number of bonds in the liner results in a rapid decrease of stiffness and the liner will crack. In this experiment no further crack propagation was noticed. This is discussed further below.

5.3.2 Position II

Figure 5.6 displays SEM images of a crease at the top of the fluting. The corresponding diagram of figure 5.3 shows three distinguished peaks. One may notice similarity with a classical Flat Crush Test, where a board piece is compressed between two parallel surfaces. The response of the board is not symmetrical. During the creasing process it slants to one direction. In an exceptional case the deformation is symmetrical. The first 200 µm the load increases at an approximately linear slope. Then the first peak is reached. The flute starts to buckle and this fixates the flute into the lower liner. Then the load slowly decreases until the next peak is reached at 900-1000 µm. The

27 CHAPTER 5. CREASING EXPERIMENTS

3000 position I D C 2000 B

1000 load [N/m] A 0 0 500 1000 1500 2000 2500

3000 D position II 2000 C B 1000 A load [N/m]

0 0 500 1000 1500 2000 2500

3000 position III C D 2000 B

1000 A load [N/m]

0 0 500 1000 1500 2000 2500 displacement [µm]

Figure 5.3: N/m vs displacement curves of crease tests at three dif- ferent positions

Figure 5.4: An optical microscope image at the side of corrugated board. Notice the washboard effect of the liner, which causes sagging and crushing of the fluting.

fluting is buckled and fixated between the upper and lower liner like a strut. The second buckle of image C unwinds itself along the upper liner and displaces itself. Delamination of the flute is also encountered (not shown). The reaction force decreases again and then starts to increase until the other half of the flute buckles. The increasing tensile stress in the inside liner exceeds the buckle limit of the flute top alongside the center top and forces the flute to participate in the deformation. The load decreases a bit, but shortly after that the top of the flute contacts the lower liner and the experiment ends.

No cracking or released fibers are encountered in position II creasing. α is not large enough to reach the critical stresses.

5.3.3 Position III

The third diagram of figure 5.3 displays the response for creasing between the peak and the lowest point. Figure 5.7 shows SEM images of the crease experiment. Initially, the stiffness is a bit higher than at position II, but it deviates shortly after the start of the experiment. The flute

28 CHAPTER 5. CREASING EXPERIMENTS

Figure 5.5: Four different SEM images during a creasing experiment at position I buckles and unwinds itself along the lower liner. The reaction force remains constant until 1000 µm. The third image shows the start of delamination. The now nearly vertical flute part comes under pressure. Finally, it buckles and more delamination is caused. Delamination in position III is encountered more often than in the other positions. In position III creasing released fibers are also encountered, see image C.

5.4 Creasing with liner plate

Despite the presence of down holders the previous experiments exhibit a little folding of the specimen, which is believed to be largely absent in practice. As a consequence, no cracking was observed. Creasing with a liner plate prevents this effect. The main difference between creasing with and without liner plate is the free displacement of the inner liner. While creasing, the inside liner wraps itself around the creaser and it deforms the fluting. In creasing with a liner plate the displacement of the inner liner is restricted. If the flute is stiff enough to resist buckling, the inside liner will crack. The overall load-displacement curves are higher than the curves without the liner plate, see figure 5.8. The curves representing position I and III deviates shortly after the start. Especially position I creasing is sensitive to folding. The liner plate prevents this folding, so the liner is subjected more to stress.

The consequence of the higher stress is depicted in figures 5.9 and 5.10. They show a microscopic

29 CHAPTER 5. CREASING EXPERIMENTS

Figure 5.6: Four different SEM images during a creasing experiment at at position II side view image and a surface view of the inner liner after position I creasing. Due to the higher tensile stress, it can be seen that more fibers are released, i.e. the inside liner is more damaged than in the experiment without liner plate. The through thickness material failure can be observed with the naked eye as a narrow bright band, about 1 mm wide.

Initially, there are no differences in the curves of position II creasing with and without liner plate. After the first buckling peak the inside liner plays a more important role. Tensile stress in the liner cause a deviation of the curve at 500 µm.

The problem of cracking of the liner did not occur during the experiments, only fiber ends are released from the fiber network. Creasing with a liner plate subjects the inner liner to a higher stress. However, it is not sufficient to cause cracking. An artificial phenomenon which is observed in the experiments with a liner plate is buckling of the outside liner.

Figure 5.11 shows a position I crease with liner plate. The board specimen is creased completely. The arrows indicate buckling of the outside liner caused by slip between the tape and the inside liner which results in a motion of the inside liner towards the creaser. The creaser forces the flute to shear, which consequently creates a moment in the attachment zone. Bending of the outside liner is also promoted by the washboard effect.

30 CHAPTER 5. CREASING EXPERIMENTS

Figure 5.7: Four different SEM images during a creasing experiment at bat position III

31 CHAPTER 5. CREASING EXPERIMENTS

3000

2000

1000 load [N/m]

0 0 500 1000 1500 2000 2500

3000

2000

1000 load [N/m]

0 0 500 1000 1500 2000 2500

3000

2000

1000 load [N/m]

0 0 500 1000 1500 2000 2500 displacement [µm]

Figure 5.8: Load vs displacement curves of crease tests at three dif- ferent positions. (—) represents the reference crease, (- -) represents the crease with liner plate and (-·-) represents the crease without any restriction.

Figure 5.9: Electron Micrograph of the in- Figure 5.10: A top view of the surface of the ner liner at the crease tip. The crease released fibers of figure 5.9. The damaged depth is about 2000 µm. Released fibers are and the untainted surface is visible. clearly visible. Note that the image is upside down compared with the other SEM images of creasing experiments

32 CHAPTER 5. CREASING EXPERIMENTS

Figure 5.11: A complete position I crease. The arrows indicate buckling of the outside liner.

33 CHAPTER 5. CREASING EXPERIMENTS

34 Chapter 6

Confrontation of simulations vs experiments

Finite element simulations are performed and compared with the experimental creasing tests in order to investigate the ability of the model to capture the experimentally observed response.

6.1 Model definition

A close look at a piece of corrugated board shows that corrugated board has a non-symmetrical structure and that the exact shape is not well defined. In Finite Element (FE) modelling the fluting is often approximated by a sine or a combination of arcs, [4] and [23]. In this study we adopt the sinusoidal shape. The so called washboard effect, i.e. non-flatness of the inner liner, is not included in the the models used initially. The influence of this effect is described further below.

The detailed FE model of a single wall corrugated board of type B is presented in figure 6.1. The left side is without the washboard effect, the right side includes this effect. The washboard effect is modelled as three arcs which are connected and is characterized by s, the amount of sagging of the liner. The flute period λ is 6.3 mm, the total height T is 3.0 mm, tl is 0.186 mm and tf is 0.155 mm. The length on which the fluting and the liner are attached is represented by Λ = r, where r is the creaser tip radius; these connections are assumed to be rigid. The mechanical behaviour of the adhesive is not known and is neglected in this study. It is assumed that the drying procedure causes no residual stress.

The corrugated board is modelled as a deformable body. The crease tip is modelled as a rigid circular body with a radius of r=0.5 mm and undergoes a negative linear displacement, δ. The anvil is also modelled as a rigid body and is constrained in each direction. The influence of friction is neglected. The down holders are placed 6.5 mm from the center line. The total model is presented in figure 6.2.

The model assumes a plane strain state in CD direction, so ε22 = ε23 = ε12 = 0. Due to this assumption the constants G12, G23, σ23y and σ12y do not influence the results. The fiber orientation is aligned with the elements as indicated in figure 6.1.

The simulation results presented in this chapter have all been obtained with the finite element

35 CHAPTER 6. CONFRONTATION OF SIMULATIONS VS EXPERIMENTS

Figure 6.1: Single wall corrugated board of type B. At the left side no washboard effect is included. The right side includes the washboard effect.

Figure 6.2: Loading and boundary conditions imposed on the finite element model discretisation shown in figure 6.1. This discretisation has six linear quadrilateral elements across the thickness. The total mesh consists of 11300 nodes and 9840. Refining the mesh did not result in significant changes in the response, so that this mesh can be considered to be sufficiently fine.

6.2 Simulations with reference parameters

We will first present the results obtained with the reference material parameters as defined in chapter 5; the relevant reference parameters are recapitulated in table 6.1.

The simulated deformation of the board for the three creasing positions used in the experiments are compared with images taken with an optical microscope, asumming that the influence of different ambient conditions (between optical and SEM) can be neglected. The model without the washboard effect has been used for these simulations.

For each position, one representative experimental load-displacement curve has been plotted along with the corresponding simulation in figure 6.3. The important points in the figures are noted with a roman character.

36 CHAPTER 6. CONFRONTATION OF SIMULATIONS VS EXPERIMENTS

3000 position I 2000 B A 1000 Load [N/m]

0 0 500 1000 1500 2000 2500

3000 position II B 2000 A

1000 Load [N/m]

0 0 500 1000 1500 2000 2500

3000 position III 2000 B A

1000 Load [N/m]

0 0 500 1000 1500 2000 2500 Displacement [µm]

Figure 6.3: Experiments vs simulations at the three positions. (—) represent the experiments and (- -) represent the simulations

Table 6.1: Material parameters used in the reference simulations Parameter Value S.I. unit Parameter Value Unit E1 3950 MPa G31 71 MPa E2 1650 MPa σ11y 13 MPa E3 20 MPa σ22y 4 MPa ν12 0.65 - σ33y 4 MPa ν31 0.0035 - σ31y 7.5 MPa ν32 0.0055 -

Figures 6.4-6.6 show for each position the simulated and experimentally observed deformation at the creaser displacements which have been marked in figure 6.3. The deformations predicted by the FE model show a good agreement with the experiments. The stress in MD direction is also depicted in the deformation figures and will be discussed further below. Appendix C shows a overview of creasing at the three positions. The results are discussed in more detail for each of the creaser positions below.

6.2.1 Position I

The top diagram in figure 6.3 shows the load-displacement curve at position I. The initial stiffness from the simulation slightly deviates from the experiment. The experiment without down holders also show a lower initial stiffness. At (A) the flute collapses under the increasing stress, which is visualized in figure 6.4.

In the experiments fixation of the flute between the inside and outside liner is observed. While the creaser displacement proceeds the buckled flute unwinds itself along the inside and outside liner. Also delamination of the flute caused by the compressive load is observed. In the simulations, the

37 CHAPTER 6. CONFRONTATION OF SIMULATIONS VS EXPERIMENTS

Figure 6.4: Experimental vs simulation deformation at position I. σ11 is pre- sented in the numerical part of the image. compressive load in the flute also initiates buckling. Once a buckle is initiated, however the flute thickness locally decreases and a plastic hinge is formed which can not unwind itself. As a result, the load continues to increase in the simulation, where it already decreases in the experiment. Delamination as encountered in the experiments weakens the flute compression stiffness. Since this is not included in the FE model an increase of creaser load is enforced.

Beyond point (B) the inside liner and fluting rotate around the plastic hinge, which prevents fixation and thus an increase of load. Fixation in position II is more important.

6.2.2 Position II

One can see a qualitative similarity between the experimental curve and the simulation curve of position II. In the simulation curve we can clearly distinguish the two buckling peaks of the experimental curve which are associated with the two hinges that are formed in the flute (figure 6.5. However, the buckling load is a factor two larger. The initial stiffness agree quite well, however shortly after the start the simulated curve deviates for the experimental curve. The first buckling peak is reached after a displacement of 400 µm in the simulation whereas it occurs already at approximately 150 µm in the experiment.

Next, the load increases and another buckle initiates which results in a decrease of load. Beyond (A) the creaser reaction force increases. Fixation of the flute is shown in figure 6.5 image B, see also the displacement of the buckle in the experiment.

38 CHAPTER 6. CONFRONTATION OF SIMULATIONS VS EXPERIMENTS

Figure 6.5: Experimental vs simulation deformation at position II. σ11 is pre- sented in the numerical part of the image. The figure suggests symmetric defor- mation, but this is due to the fact that only the most relevant part is visualized.

6.2.3 Position III

The initial stiffness in the simulation curve is approximately the same as the experimental curve. A first buckle appears at 700 µm, see image A of figure 6.6. However, unlike the buckles observed in position I and II this buckling does not result in a force peak.

6.3 Comparison of inner liner stress

Position I creasing subjects the liner to the highest tensile stress. The highest tensile stress in MD direction can be found at the crease line. With respect to the creaser tip, the outside of the liner is under tension and the inside of the liner is under pressure. Stress of each position is plotted in figure 6.7. The simulations predict a stress of about two times the failure strength. Some material and geometric properties are maybe overestimated. In position II creasing the model assumes a rigid connection between the liners and fluting. In real corrugated board the flute and the liners are joined with starch adhesive. In the model this means that the attachment zone is about twice the liner thickness. The highest stress in the liner is plotted. The fact that the stress is substantially higher for positions I and III than for position II corresponds well with our experimental observation that no damage occurs in position II.

39 CHAPTER 6. CONFRONTATION OF SIMULATIONS VS EXPERIMENTS

Figure 6.6: Experimental vs simulation deformation at position III. σ11 is presented in the numerical part of the image.

6.4 Influence of material and geometric properties

The differences between the experiments and the simulations are in some cases quite large. Some parameters may have been overestimated. This section describes the influence and sensitivity of the known and unknown material parameters. This will help us to determine the governing set of parameters of the crease process.

For this investigation the crease simulations are performed at position II. Cracking of the liner is mainly governed by the flute stiffness. The flute stiffness is best tested at this position. To reduce cpu time the analysis is assumed to be symmetric with respect to the creaser displacement path.

6.4.1 Material properties

In our model orthotropic elasto-plasticity is assumed. Figure 6.8 shows the difference between isotropic elasto-plasticity and orthotropic elasto-plasticity. The used parameters are presented in table 6.2. In case of isotropic elastic behaviour the initial structural stiffness is higher. Only one buckling peak can be noticed in the resulting curve. The rapid increase of load beyond 1200 µm observed in the reference computation does not occur in case of isotropic elastic behaviour. No noticeable thickness reduction of the flute at the buckle zone is observed. No fixation of the flute between the liners occurs and while the creaser moves downward the fluting unwinds itself along the liners. There is little difference between ISO-ISO and ISO-ORTHO curves, but a large difference with the reference curve, so the elastic parameters are important for the overall response, even in the plastic regime. Large plastic strains mainly occur in the fluting in the buckle zones.

40 CHAPTER 6. CONFRONTATION OF SIMULATIONS VS EXPERIMENTS

120 4000

100 position I 3500 iso−iso

3000 80 position III 2500 iso−ortho 60

[MPa] 2000 11

σ 40 Load [N/m] 1500 position II reference 20 1000

0 500 exp

−20 0 0 500 1000 1500 2000 2500 0 500 1000 1500 2000 Displacement [µm] Displacement [µm]

Figure 6.7: The highest stress (σ11) of each Figure 6.8: Load-displacement curve of position at the crease line at the bottom of isotropic and orthotropic elasto-plasticity; the inside liner see table 6.2 for material data used.

Table 6.2: Parameters used for different material models model abbreviation parameters

isotropic elasto-plasticity iso-iso E = E1, ν = 0.33, Rii = 1, Rij = 1 isotropic elasticity, orthotropic plasticity iso-ortho R11 = 1, R22 = R33 = 0.30, Rij = 1 orthotropic elasto-plasticity reference reference parameters

In the reference set there is a factor of 200 between E1 and E3 and a factor of 20 between the isotropic G31 (about 1400 MPa) and orthotropic G31. Simulations with an unrealistic E3 = E1 (the other parameters equal the reference parameters) exhibit a qualitatively similar shape as the ISO-ISO case, yet at a somewhat lower reaction force. A simulation with an unrealistic G31 = 1400 MPa shows a qualitatively similar curve as the reference curve. The shape of the load-displacement curve is mainly governed by the buckling behaviour of the flute, which in turn depends predominantly on the out-of-plane elastic modulus E3.

Shortly after the start of the simulation curve the initial stiffness deviates from the experiments. This can be corrected by taking for G31 = 34 which can be found in literature [22]. This value is about a factor of two lower than the shear modulus of the reference parameters. In figure 6.9 these curve are presented. A more accurate stiffness is obtained with the lower shear modulus. Variations in shear stress mainly affect the flute when it is deformed.

As described in chapter 4 the out-of-plane Poission’s ratios ν are close to zero (ν31 = 0.0035 and ν32 = 0.0055). Let us assume ν in thickness direction is zero. The result is also presented in figure 6.9. Little difference can be noticed in the curve. Thus the out-of-plane ν has no significant influence on the response.

Due to the incompressibility of the Hill48 yield criterion the yield stress is symmetric in tension and compression. Since the fluting is mainly in compression, the yield stress is overestimated. [3] suggested to multiply the reference yield stress with 0.65 to approximate the compressive yield stress. Figure 6.10 presents the curve with the adjusted reference yield stress, σref = 8.5 MPa, for the flute only. Again a better initial stiffness and lower overall load is obtained. The first buckling peak is reached earlier.

As described before, in the reference simulation a first approximation of R31 = 1 is used, so

41 CHAPTER 6. CONFRONTATION OF SIMULATIONS VS EXPERIMENTS

4000 4000

3500 3500

3000 3000 ν =ν =0 32 31 2500 2500 reference σ =8.5 ref 2000 2000 Load [N/m] Load [N/m] 1500 1500 G =35 31 1000 reference 1000

500 500 exp exp

0 0 0 500 1000 1500 2000 0 500 1000 1500 2000 Displacement [µm] Displacement [µm]

Figure 6.9: Variation of G31 and ν31 = Figure 6.10: Variation of reference yield ν32 = 0 stress

σ31y = 7.5 MPa. In chapter 4 a typical shear yield stress of 1 MPa is stated. The obtained curve is presented in figure 6.11. Shortly after the start the curve deviates from the experimental curve. Only the first buckling peak is observed.

4000 4000

3500 3500

3000 3000

reference 2500 reference 2500 washboard effect

2000 2000 Load [N/m] Load [N/m] 1500 1500

Λ=0 1000 1000

σ =1 500 31y 500 exp exp

0 0 0 500 1000 1500 2000 0 500 1000 1500 2000 Displacement [µm] Displacement [µm]

Figure 6.11: Influence of shear yield stress Figure 6.12: Simulations with different ge- ometry compared with the experiments at po- sition II

6.4.2 Geometric properties

First the washboard effect is added to the model. The washboard effect decreases the first peak in the load-displacement response, see figure 6.12. The initial stiffness is not influenced by the washboard effect. Initially, there is little resistance from the inside liner to the crushing of the flute. In fact, the washboard liner is longer than a straight liner. When the liner is fully stretched the liner starts to play a role, i.e. in the second peak. The length difference of the liner results in a postponed as well as a higher second peak.

The attachment length Λ is hard to measure, it differs along the board and the mechanical be- haviour of the adhesive is unknown. If it is set to Λ = 0 no noticeable first peak is obtained (figure 6.12) and the initial stiffness is closer to the experimental value. The effect on the remainder of the load-displacement curve is similar to that of a longer inner liner: the second peak is again

42 CHAPTER 6. CONFRONTATION OF SIMULATIONS VS EXPERIMENTS higher and postponed.

The fact that increasing Λ results in a higher first buckling peak can be understood as follows. For Λ = 0 the flute unwinds itself along the inner liner immediately. Subsequently, the flute starts to buckle and unwinds itself along the inside liner. The consequence of Λ > 0 is that in the crease zone the liner is effectively about twice as thick and the effective period of the flute becomes smaller. This projects the flute more normal to the liners, which results is a higher buckle load.

6.5 Influence of boundary conditions

As described before the boundary conditions in the experiments are not sufficiently restricting to cause cracking of the liner, even for complete penetration of the creaser. A new boundary condition is added to model the influence of the surrounding board, cf. the liner plate in the experiments. The board ends are fully constrained at x = L. Figure 6.13 presents the resulting load-displacement curves. The solid curves represent the experiments, the dash-dot curves represent the reference simulations and the dashed curves represent simulations where the board ends in MD at x = L are constrained. Each of the latter curves strongly deviates from the corresponding reference curve.

5000 position I 4000 3000 2000 Load [N/m] 1000 0 0 500 1000 1500 2000 2500

5000 position II 4000 3000 2000 Load [N/m] 1000 0 0 500 1000 1500 2000 2500

5000 position III 4000 3000 2000 Load [N/m] 1000 0 0 500 1000 1500 2000 2500 Displacement [µm]

Figure 6.13: Experiments vs simulations at the three independent positions. (—) represent the experiments, (-·-) represent the simula- tions according the reference state and (- -) represent the simulations with constrained board ends.

Figure 6.14 shows a comparison of the deformations observed in the constrained situation and with the down holders. The right side shows one buckle, while the left side shows three buckles. The new boundary condition prevents folding of the board and subjects the liner to a higher stress. The higher liner tension causes the flute to collapse more than in the reference case.

Figure 6.15 shows a comparison of the liner stress in the reference simulation and the fully con- strained simulation. The top curves represent the highest σ11 at the crease line and the bottom 1 curves represent σ11 at x = 4 λ in the inner liner. There is little difference between the top curves.

43 CHAPTER 6. CONFRONTATION OF SIMULATIONS VS EXPERIMENTS

Figure 6.14: Deformation for constrained edge (left) and reference situation using down holders (right) at position I

The stress here is mainly governed by bending of the inside liner. The sudden change in the dashed curve at 2000 µm is caused by the initiation of the second and third buckle in the flute. The bottom curves clearly show that the inside liner is subjected to a much higher stress in case of the new boundary conditions. This corresponds well with the observation in the experiments that the more constrained configuration (using the liner plate) results in more damage.

120

at crease line (x=0) 100

80

60 [MPa] 11 σ

40 at x=1/4 λ

20

0 0 500 1000 1500 2000 2500 Displacement [µm]

Figure 6.15: Comparison of the highest stress at the crease line for position I creasing. (—) rep- resents the reference simulation and (- -) repre- sents real-life creasing. The top curves represent the highest σ11 at the crease line and the bottom 1 curves represent σ11 at x = 4 λ in the inner liner.

6.6 Summary of observations

The previous investigation of the material and geometric properties and boundary conditions can be recapitulated as follows

• The deformations predicted by the finite element (FE) model show a qualitative good agree- ment with the experiments. However, a quantitative agreement cannot be given. The load response deviates about max. a factor of two. • Plasticity mainly occurs in the buckle zones. The buckles act as plastic hinges and can- not move. The low stiffness in thickness direction causes the buckling peaks. The lack of compressibility of the Hill48 yield criterion overestimates the stiffness.

44 CHAPTER 6. CONFRONTATION OF SIMULATIONS VS EXPERIMENTS

• Out-of-plane Poisson’s ratios have no significant influence on the load-displacement curve.

• The out-of-plane shear elastic and plastic parameters influence the load response. • The attachment length influences the first buckling peak. • The washboard effect has a positive effect on preventing cracking of the liner.

• Real-life creasing subjects the inside liner to a higher stress.

45 CHAPTER 6. CONFRONTATION OF SIMULATIONS VS EXPERIMENTS

46 Chapter 7

Conclusion and recommendations

The creasing behaviour of corrugated board has been studied by a combined experimental-numerical approach. The crease simulations show a good qualitative agreement with the observed exper- imental deformation, buckling and load response. However, the quantitative agreement is less satisfactory. Difference in load of max. 100 % is encountered. This is caused by the lack of compressibility of the Hill yield criterion, the out-of-plane elastic and plastic components and de- lamination of the flute, which have a significant influence. The flute stiffness plays an important role in the deviation. A constitutive model which allows to model compressibility and delamina- tion should be applied and a method to determine the out-of-plane shear modulus and shear yield stress has to be developed.

The present finite element model assumes a rigid connection between the flute and the liner. The length of this connection has a significant influence on the load response. Peel tests should be carried out in order to get insight in the strength of the joining of the liner and the flute and whether the assumption of the connection being rigid is sufficient.

Cracking of the inside liner is not observed during the experiments, only some damage of the liner is encountered. If new boundary conditions which take the influence of the surrounding board into account are applied, the inner liner is subjected to a higher tensile stress. Experiments and simulations confirm this. A new tool should be made which constrains the board ends. One way to achieve this is to inject a hardening substance between the liners at the board ends, which makes it possible to clamp the ends.

An observation which may be of interest for the industrial practice is that the cracked board sheets (unfolded boxes including creases and cuts) provided by Kappa Containerboard exhibit cracking predominantly near cuts. The cut ends are points where stress concentrates and which may cause cracking. Cutter/creaser designs which avoid these concentrations, e.g. by punching a corner relief (a circular hole at the transition from a cut to a crease), may give rise to less cracking. A corner relief is often applied in sheet metal applications. In terms of further research, the influence of a cut should be investigated, which implies that a two dimensional model is not sufficient and a three dimensional model must be developed.

47 CHAPTER 7. CONCLUSION AND RECOMMENDATIONS

48 Bibliography

[1] Allansson, A., (2001) Stability and collapse of corrugated board; Numerical and Experimental Analysis, Structural Mechanics LTH, Sweden

[2] Bergen van, W.J.C., (1990) Het papierboek, Stam Technische Boeken, Culemborg (in Dutch)

[3] Beldie, L., (2001) Mechanics of paperboard packages, performance at short term static loading, Div. of Structural Mechanics, Lund university

[4] Biancolini, M. E., (2005) Evaluation of equivalent stiffness properties of corrugated board, Composite Structures 69 (3)

[5] Brekelmans, W.A.M., (2002) Mechanical Characterisation of Materials: Constitutive Mod- elling, dictaat 4797, TU/e Eindhoven

[6] Castro, J., Ostoja-Starzewski, M., (2003) Elasto-plasticity of paper, International Journal of Plasticity, Volume 19, Issue 12, December 2003, Pages 2083-2098

[7] Dekker, J.C., Rilbaarheid, Rapport 2A, PRONTA, periode 1 oktober 1995 tot 1 januari 1996, 15 januari 1996, ATO-DLO, Wageningen (in Dutch)

[8] Dekker, J.C., Rilbaarheid, Rapport 2B, PRONTA, periode 1 januari 1996 tot 1 juli 1996, 30 juli 1996 ATO-DLO, Wageningen (in Dutch)

[9] Dekker, J.C., Rilbaarheid, Rapport 2C, PRONTA, periode 1 juli 1996 tot 1 januari 1997, 7 april 1997 ATO-DLO, Wageningen (in Dutch)

[10] D828-97(2002) Standard Test Method for Tensile Properties of Paper and Paperboard Using Constant-Rate-of-Elongation Apparatus, ASTM International

[11] Hill, R. (1950) The mathematical theory of plasticity, Oxford University Press, London

[12] Hill, R., Constitutive Modelling of Orthotropic Plasticity in Sheet Metals, J. Mech. Phys. Solids, 38 (1990), pp. 405-417.

[13] http://www4.ncsu.edu/ hubbe/FIBR.htm

[14] Mäkelä, P., Östlund, S., (2003) Orthotropic elasticŰplastic material model for paper materials, International Journal of Solids and Structures, Volume 40, Issue 21, October 2003, Pages 5599-5620

[15] Meinders T., (2000) Developments in numerical simulations of the real life deep drawing process, Ph.D. Thesis , University of Twente, Enschede

[16] Nagasawa, S. (2003) Effect of crease depth and crease deviation on folding deformation char- acteristics of coated paperboard, Journal of Materials Processing Technology, Volume 140, Issues 1-3, 22 September 2003, Pages 157-162

49 BIBLIOGRAPHY

[17] Netz, E., (1998) Washboarding and print quality of corrugated board, Packaging Technology and Science Volume 11, Issue 4, 1998. Pages 145-167 [18] Niskanen, K. (1998) Paper and paperboard converting, Fapet Oy, Helsinki

[19] Niskanen, K. (1998) Paper physics, Fapet Oy, Helsinki [20] Scott, William E. (1995) Properties of paper, an introduction, TAPPI PRESS, Atlanta [21] Stenberg, N., (2003) A model for the throug-thickness elastic-plastic behaviour of paper, In- ternational Journal of Solids and Structures 40, 7483-7498. [22] Stenberg, N., Fellers C., Östlund, S. (2003) Plasticity in the thickness direction of paperboard under combined shear and normal loading, Journal of Engineering Materials and Technology 123(4) 184-190.

[23] Urbanik, T.J., (2001) Effect of Corrugated Flute Shape on Fibreboard Edgewise Crush Strength and Bending Stiffness, J. Pulp and Pap. Sci. 27(10): 330-335.

[24] Xia, Q.S., Boyce, M.C., Parks, D.M., (2002) A constitutive model for the anisotropic elasticŰ- plastic deformation of paper and paperboard, International Journal of Solids and Structures 39, 4053Ű4071.

[25] Xia, Q.S., (2002) Mechanics of inelastic deformation and delamination in paperboard, MIT- THESES:2002-381 [26] Xue, Z., Hutchinson, J.W., 2004. Constitutive model for quasi-static deformation of metallic sandwich cores, International Journal for Numerical Methods in Engineering. 61, 2205-2238.

50 Appendices

51

Appendix A

Hill48 detailed calculations

Hill48 in MSC.MARC/MENTAT 2005

The Hill48 yield function as provided in MSC.MARC/MENTAT 2005 reads 1£ f = a (σ − σ )2 + a (σ − σ )2 + 2 1 22 33 2 33 11 2 2 2 2 ¤ 2 a3(σ11 − σ22) + 6a4σ23 + 6a5σ31 + 6a6σ12 − σref (A.1)

The constants a1, a2, a3, a4, a5 and a6 are 1 1 1 1 a1 = 2 + 2 − 2 ; a4 = 2 R22 R33 R11 R23 1 1 1 1 a2 = 2 + 2 − 2 ; a5 = 2 (A.2) R11 R33 R22 R31 1 1 1 1 a3 = 2 + 2 − 2 ; a6 = 2 R11 R22 R33 R12

Limitation of orthotropy

Hill48 impedes a high degree of orthotropy in three directions. The constants must obey FG + GH + HF > 0 (A.3) Substituting 1 1 1 2F = 2 + 2 − 2 σ22y σ33y σ11y 1 1 1 2G = 2 + 2 − 2 (A.4) σ11y σ33y σ22y 1 1 1 2H = 2 + 2 − 2 σ11y σ22y σ33y in equation (A.3) and rewriting it in terms of Rii with R11 = 1, results in µ ¶ µ ¶ µ ¶ 1 2 1 1 1 2 − 2 + 2 1 + 2 2 − 1 − 2 > 0 (A.5) R33 R22 R33 R22

53 APPENDIX A. HILL48 DETAILED CALCULATIONS

This inequality can be rearranged as · µ ¶¸ 1 1 2 4 2 − 1 + 2 < 2 (A.6) R33 R22 R22 Extraction of the root gives · µ ¶¸ 2 1 1 2 − < 2 − 1 + 2 < (A.7) R22 R33 R22 R22 or after, again rearranging with respect to R33

2 2 (R22 − 1) 1 (R22 + 1) 2 < 2 < 2 (A.8) R22 R33 R22

Since Rii has to be positive we have

R22 R22 < R33 < (A.9) (1 + R22) (1 − R22)

Which implies that upon increasing the degree of orthotropy R22 ≈ R33.

54 Appendix B

Calculation of in-plane shear

Shear modulus

With the use of elasticity theory, the assumption of plane stress and knowing E1, E2 and ν12 it o is possible to extract G12 from the tensile test at 45 . The plane stress matrix representation of Hooke’s law is ε = Sσ (B.1) e e with      −1 −1  σ11 ε11 E1 −ν21E2 0      −1 −1  σ = σ22 ε = ε22 S = −ν12E1 E2 0 (B.2) e e −1 σ12 2ε12 0 0 G12 The stress tensor σ is 1 σ = σ(~e + ~e )(~e + ~e ) 2 1 2 1 2 1 = σ(~e ~e + ~e ~e + ~e ~e + ~e ~e ) (B.3) 2 1 1 1 2 2 1 2 2 where σ denotes³ the axial stress´ in loading direction. The normal strain ε of the strain tensor ε = ε11~e1~e1 + ε12 ~e1~e2 + ~e2~e1 + ε22~e2~e2 along the normal vector ~n is determined as

1 ε = ~n · ε · ~n with ~n = √ (~e1 + ~e2) (B.4) 2 Using (B.1, the normal strain can be expressed in terms of the stress components as 1³ ´ ³ ³ ´ ´ ³ ´ ε = ~e + ~e · ε ~e ~e + ε ~e ~e + ~e ~e + ε ~e ~e · ~e + ~e 2 1 2 11 1 1 12 1 2 2 1 22 2 2 1 2 1³ ´ = ε11 + ε22 + 2ε12 (B.5) 2µ ¶ 1 1 ν 1 ν 1 1 = − 21 + − 12 + σ 2 E E E E G 2 µ 1 2 2 1 ¶ 12 1 1 − ν 1 − ν 1 = 12 + 21 + σ 4 E1 E2 G12 The initial slope of the tensile test under 45o with respect to the machine direction in figure 4.5 should thus equal the inverse of the compliance given by (B.5). Since E1, E2, ν12 and ν21 are known, G12 can be determined from this relation.

55 APPENDIX B. CALCULATION OF IN-PLANE SHEAR

In-plane shear yield stress

o The constant N can be determined from the tensile test of a specimen at 45 if only σ33y is known. For a plane stress state the yield criterion (3.15) reduces to

2 2 2 (G + H)σ11 − 2Hσ11σ22 + (F + H)σ22 + 2Nσ12 = 1 (B.6)

1 The previous section showed that σ11 = σ22 = σ12 = 2 σ. At yield we have σ45y and equation (B.6) thus further reduces to 1 1 (F + G)σ2 + Nσ2 = 1 (B.7) 4 45y 2 45y

−1/2 (F + G) can be derived from a biaxial tensile test, σ11 = σ22 = σ. Note that (F + G) is also the yield stress in thickness direction, see equation (3.17). Finally, N can be derived from

−2 2N = 4σ45y − (F + G) (B.8)

56 Appendix C

Crease simulations overview

Figure C.1: Creasing at different positions

57 APPENDIX C. CREASE SIMULATIONS OVERVIEW

58 Dankwoord

Bij deze wil ik graag iedereen bedanken voor de hulp tijdens en bij mijn afstuderen. Mijn ouders die de eerste (en laatste) docenten zijn waar ik ooit onderricht van heb genoten en hun steun op een diversiteit aan vlakken tijdens mijn leven en tijdens deze en vorige opleidingen.

Op het professioneel technische vlak wil ik graag mijn begeleider, Ron Peerlings, bedanken voor zijn begeleiding, hulp, correcties en tijd. Ook wil ik Marc van Maris bedanken voor zijn praktische input tijdens de experimenten.

Verder wil ik, ondanks de lange productietijd, de gemeenschappelijke technische dienst (GTD) bedanken voor het vervaardigen van de benodigde gereedschappen.

Ook mijn mede VKO-ers wil ik bedanken voor de gezellige momenten die ik de afgelopen tijd met hen heb mogen beleven.

De TU/e heeft mij mijn basiskennis aanzienlijk verbreed voor de rest van mijn (werkzame) leven, doch was het niet de Britse natuurkundige Sir Isaac Newton die zei:

"Wat wij weten is een druppel, wat wij niet weten een oceaan."

Na de studies in mijn leven kan ik mijzelf niet geheel onttrekken aan het idee dat ik de wijsheid nog niet in pacht heb. Het gevoel dat mijn educatie nog lang niet ten einde is overheerst sterk. Desalniettemin is een spreekwoordelijke mijlpaal bereikt.

59