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 ® angle ± creaser displacement γ_ plastic multiplier " linear strain tensor ¤ attachment length ¸ period of the fluting ºij Poisson’s ratio of axial strain and lateral strain σ Cauchy stress tensor σ equivalent stress σref reference yield stress ­ 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 r~ 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