Finite element analysis of e-commerce cushioning in corrugated board

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DEGREE PROJECT IN MECHANICAL ENGINEERING, SECOND CYCLE, 30 CREDITS

STOCKHOLM, SWEDEN 2018

Finite element analysis of e-commerce corrugated board cushioning

DOMINYKAS GUDAVI Č IUS

KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES

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Abstract

Corrugated board and other paper products are widely used in product packaging due to good mechanical properties, recyclability, price etc. Due to the growing e–commerce industry the dynamical behaviour of corrugated board is of particular interest.

Paper exhibits an orthotropic behaviour and is a non–linear material.

In this work, corrugated board is analyzed from mechanical point of view, when subjected to a short time span impact load. Edge crush test (ECT), flat crust test (FCT), bending stiffness and static deflection tests which are part of standards used in the industry are performed. Both experimental and numerical approaches are used in the analysis and the results are compared and discussed.

The elastic part of a corrugated board material model predicts the physical behaviour well and provides reliable results. Bending stiffness model results have approximately 1 % difference from physical experiments and the static deflection test together with FCT has a 6 % difference. Dynamical drop test gives 13 %, 11 % and 27 % relative error for different drop heights. Elastic–plastic behaviour requires further investigations, especially in ECT test which has around 45 % discrep- ancy from physical experiment. The possible difference in the dynamical model might arise from the boundary conditions which were not fully controlled during the physical experiment and the difference in the ECT can be possibly explained by layer thickness approximations together with glue line width assumptions between fluting and the liners in the numerical models.

Results suggest that the finite element approach is a reliable way to model corrugated board but it poses challenges especially in complex loading conditions. The elastic behaviour of the corrugated board is well predicted by assuming an orthotropical material model. The dynamic behaviour when subjected to an impulse force was well predicted for low drop heights. but requires more investigation for high impulse impacts.

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Sammanfattning

Wellpapp och andra pappersprodukter används ofta i olika typer av förpackningar på grund av materialets goda mekaniska egenskaper, återvinningsbarhet, pris per prestanda, m.m. I den snabbt växande e- handelsbranschen är wellpapps egenskaper under dynamiska laster av särskilt intresse.

Papper uppvisar ett ortotropt beteende och är ett olinjärt material. I detta arbete analyseras wellpapp ur mekanisk synpunkt när det ut- sätts för belastningar med korta tidsförlopp. För att verifiera den mo- dell för analys med FEM som tas fram utförs och analyseras även kant- krossningsprov (ECT), plantryckningsprov (FCT), böjstyvhetsmätning- ar och statiska utböjningsprov, vilka delvis ingår i de standarder som används i branschen. I analysen används både experimentella och nu- meriska analysmetoder och resultaten från desssa jämförs och disku- teras.

I analysen används en elastisk-plastisk materialmodell för liner och fluting och analysen visar att den elastiska delen av materialmodellen kan användas för att väl beskriva wellpapps styvhetsegenskaper ger tillförlitliga resultat. Modellens böjstyvhet skiljer sig från de fysiska experimenten bara med ungefär 1 % och för det statiska utböjnmings- provest och för FCT är skillnaden ca. 6 %.

För dynamiska dropptest är det relativa felet 13 %, 11 % och 27 % när en stålkula släpps mot wellpappanelen från olika dropphöjder. Ma- terialens plastiska egenskaper kräver ytterligare undersökningar, sär- skilt för ECT som har en avvikelse på omkring 45 % från det fysiska experimentet. En möjlig förklaring till skillnaden mellan numerisk analys och experiment för den dynamiska modellen kan vara bristan- de kontroll på randvillkoren i de fysiska experimenten och skillnaden vid analysen av ECT kan möjligen förklaras av skillnader mellan well- pappanelens och FE-modellens tjocklek samt antagandet att limskiktet mellan fluting och liner i FE-modellen är begränsat till enbart en rad med noder.

Sammanfattningsvis kan konstateras att resultaten tyder på att FEM är ett tillförlitligt sätt att modellera wellpapp under dynamiska laster, men det finns utmaningar, speciellt vid komplexa lastförhållan- den. Wellpapps mekaniska egenskaper förutsägs väl genom att anta ett ortotropt material. Det dynamiska beteendet när wellpapp utsätts för impulskrafter var väl predikterat för låga dropphöjder, men en ut-

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ökad analys krävs för att beakta högimpulseffekter.

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3.1 General results . . . 32 3.2 Thickness measurement results . . . 33 3.3 Tensile testing results . . . 34 3.4 Static deflection test numerical and physical results . . . 35 3.5 Dynamic deflection test numerical and physical results . 35 3.6 Bottle impact test numerical and physical results . . . 37 3.7 Different fluting shape results . . . 39

4 Discussion 41

5 Conclusions 43

6 Future work 44

7 Acknowledgement 45

Bibliography 46

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List of Abbreviations

In this master thesis, the following abbreviations are used ν_ij Poisson’s ratio in ij–plane

E_i Young’s modulus in i–direction G_ij Material shear modulus in ij–plane CAD Computer Aided Design

CD Cross machine direction ECT Edge Crush Test

FCT Flat Crush Test

FEM Finite element method MD Machine direction

ZD Through–thickness direction

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Chapter 1 Introduction

Huge increase in consumption during the past century has led to needs for effective ways of transporting goods, let it be in the form of liquids or solids. Due to good mechanical integrity with low weight, cheap price and recyclability corrugated board proved to be a reliable way of storing and protecting goods. What is more, recent increase in e–

commerce business affects the use of corrugated boards. However, a lot of challenges exist in the value chain of corrugated board which goes from initial manufacturing to the moment when the corrugated board reaches the final costumer. One of the biggest challenges comes when the corrugated containers together with content are distributed on conveyors. Not seldom, the corrugated box might fall down on the conveyor or impact other containers. During those moments, the load is applied for fractions of a second. Not only does the corrugated board has to protect the content but also to maintain outside neatness and keep an aesthetic look. Therefore, in this master thesis, corrugated board is analyzed from a mechanical point of view during impulse loading.

1.1 Background

Since the first patent of corrugated paper in 1856, the material proved to be useful and manufacturing techniques quickly developed. The manufacturing process starts by disintegrating the wood fibres from wood chips by using either mechanical or chemical process [1]. The process is called pulping and it removes bonds in the wood fibres and so a substance called pulp is created [1]. After additional clean-

1

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2 CHAPTER 1. INTRODUCTION

ing, pulp is pumped into the paper machine where paper sheets gain their structure and shape. Newly made substance goes through heated rollers which remove the moisture from the pulp and shape the direction of wood fibres [1]. After roller forming and drying, kraft paper is made which is stiffer in the machine direction (MD) than the cross–machine direction (CD). The trough–thickness direction (ZD) is the least stiff. Segment of the roller forming process in illustrated in Figure 1.1.

Figure 1.1: Rolling process which influences presence of MD and CD in paper.

The next big step is combining kraft papers into corrugated board.

This involves gluing two outer layers, called liners, together with a wavy middle layer called fluting [1]. Before this can be done, the paper for fluting has to go through corrugating rolls where it is pre–heated and mechanically processed so it obtains wave like shape. Only then the fluting can be combined with liners using an adhesive. The combined liners and fluting, which gives the corrugated board is presented in Figure 1.2. The red circles indicate the points where the fluting is glued to the liners.

The success of corrugated board is dependent on humidity, impact forces, static forces, corrugated board design and other factors which influence the mechanical integrity. In order to improve and under- stand the mechanical properties of corrugated board, numerous inves-

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CHAPTER 1. INTRODUCTION 3

Figure 1.2: Final product after combining fluting with liners.

tigations are being reported in the literature.

1.1.1 Corrugated board storage environmental factors

Transportation of quickly spoiling goods such as dairy products, veg- etables, fruits usually requires them to be stored in cold environments together with the container. Change in temperature alters moisture content in the corrugated board and as a result of that, mechanical properties can change. The increase of moisture content in the paper material is usually an undesirable phenomena as the mechanical properties degrades [1].

It is reported [2] that increasing humidity decreases the load corrugated board can take in the flatwise direction. What is more, increase in temperature can change the dry solid content which will influence the corrugated board performance. Based on empirical data, a mathematical model is provided [3]. Study tries to analyse the influence of dry solid content on the mechanical paper performance [3]. It was found that the paper drying history has a significant influence on dry samples.

Also, the yield stress and yield strain decrease with decreasing dry solid content in the paper material. It is reported that that the increase of relative humidity from 30 % to 90 % decreases edge compression strength by around 19 %, burst strength by around 3 % and other me-

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chanical parameters significantly [4].

Kellicutt found a negative exponential relation between compression strength and moisture content of corrugated boxes [5]. A hygro – elasto – plastic model is developed which takes fibre orientation, anisotropy and dry solid content as an input and outputs plastic strain and shrink- age [6]. The advantage is that it can be solved analytically in one dimension but the full potential is achieved by using numerical methods [6].

1.1.2 Corrugated board design factors

Different corrugated board applications require different box designs.

The most common changes are adding holes in order to make the handling more comfortable and efficient or just to provide air flow to the product stored in the box. However, altering the structure might lead to poor structural integrity.

Carrying slots made for boxes are studied and conclusion state that location of the hand holes and radius of curvature are very important parameters that should be taken into account when designing a corrugated box [7].The ventilation holes placed at a bad location might decrease the box compression strength up to 20 % [7]. Another study also concludes that the slot position on the corrugated board is an important design factor. Slots which are placed at the center of the corrugated box faces show better compressive strength results than any other position. Circular hand slots are the most effective shape [8].

A similar conducted study investigates the hand hole shape size and location influence on box compression test. A linear relation is found between hole area and compression strength [9]. However, the obtained results disagree with other studies where the most optimal shape was concluded to rectangular or parallelogram [9].

1.1.3 Corrugated board modelling using the finite el- ement method

In applications, some of the structural engineering problems that are faced have complex geometry, varying load, non–linear material behaviour and not straight–forward boundary conditions. Such problems involve solving differential equations that would be really hard or even impossible to solve analytically. For that reasons, numerical

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methods were developed which provide approximate solutions of suf- ficient accuracy. The most popular numerical method in solid mechan- ics is finite element method (FEM). The main idea of this method is to create finite element mesh which discretises the problem. This allows the problem to be solved in multiple computational steps by using computers. Development of numerical tools allows modeling of corrugated board and boxes using the FEM.

Corrugated board modelling reported on the literature uses elastic — plastic material models. Two models are proposed [10], [11]. One of them uses perfect bonding between fluting and liner, the other one uses multi point constraints. Plasticity is modelled with hyperbolic tangent functions. The model was solved by using four point bending and the results were compared with empirical experiment. They con- clude that results obtained by using FEM underestimate glue bonding as added stiffness by the glue is neglected [10]. Other research group performs numerical 3D buckling modelling where classical laminate theory is applied [11]. Elastic parameters are used in the model, therefore, the model is only valid until the buckling point and post — buckling is not taken into account. Constructed model agrees well with experimental results [11].

Another research group made a 3D model which takes material and geometrical non–linearities into account in order to simulate the edge crust test [12]. The material is assumed to act bilinearly. Hill‘s plasticity model is used for the hardening and Tsai — Wu criterion for the failure. Ultimate failure agrees well with empirical results [12].

A different study uses 3D and 2D finite element models to perform a 3 point bending testing of corrugated board. The result suggest that there is no big difference between 2D and 3D models and that could save a lot of computational time [13].

An approach to construct a finite element model which allows to ex- tract stiffness parameters for corrugated board is presented in [14]. A homogenization procedure is described which can be used to solve buckling problems.

One study states that most of the research which is based on laminate and sandwich theories have problems regarding the behaviour of corrugated board under transverse shear and torsional moments [15].

They have proposed a new shell torsion model with different torsional stiffness in MD and CD. The model turns out to be computationally effective.

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1.1.4 External force factors

Corrugated boards and boxes, just like any other structures are damaged due to external forces acting on them. The scientific literature makes a separation between long lasting forces and the damage they generate (creep etc.) and short lasting forces (drop test etc.) The combination of long and short term forces is seldom investigated.

Coffin proposes an equation to calculate creep modulus which can be implemented into the finite element formulation [16]. Similar study investigates creep which is influenced by cyclic humidity environment [17]. The experiment is performed in a environmental chamber where five specimens are placed in a tensile test machine. Three cases are tested: constant load with constant humidity, constant load with varying humidity and varying load with constant humidity. The research concludes that total creep of material is larger for cyclic load that ac- companies humidity cycling than constant load [17]. There is no new phenomena happening in the material but creep. Also, a mathematical model is provided which predicts creep strain. However, it has a lim- itation that it can not be extrapolated between creep strains in tension and compression [17].

Empirical drop tests of the corrugated board suspended in foam are performed and acceleration is checked [18]. The experimental set up includes placing the corrugated box with accelerometers on mechanical fork. The fork is quickly removed and the corrugated box starts to fall until it hits the rigid surface and acceleration is recorded. Re- sults are compared with finite element model which used 2D shells and homogenized laminate theory. Researchers have concluded that the contribution of the cardboard box to the protection cannot be neglected and that acceleration is sensitive to the foam surface flatness [18].

A Study according to ASTM is performed in order to research dynamic cushioning and vibration transmissibility of corrugated paper board pads with different number of layers [19]. Experiment is conducted according to ASTM D 1596. Test specimen is impacted with predeter- mined drop block from different drop heights. The drop shock acceleration and time curve is recorded. Also, experiment according to ASTM D 4168 is performed. It includes placing two test specimens on top of each other in the vibration machine and starting vibrations with low sine sweep. The results suggest that the minimum peak acceleration of

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three layer pad is lowest when comparing with single or double layer pads [19].

Series of empirical experiments are performed where multi–layered corrugated board is pre–compressed to different strains and sets of dynamical loads are applied [20]. The experiment includes pre–compressing pads and applying 20 dynamic load sets. The pads then are placed under special conditions for 48 hours. After that, 15 mode dynamical load sets are applied. Results show that corrugated paperboard has the best peak acceleration characteristics when the pad has 95 % plastic strain of the original configuration. Conclusion states that selection of optimal static stress should be made in conjecture with expected number of impacts. The corrugated layers then can perform exceptionally well [20].

The observation is made that increasing drop height influences box protection significantly. Researchers have inserted inclusion of paperboard cushion which significantly extend protection and ability to cope with extreme drop heights which have low statistical probability to happen [21].

An analytical model which uses beam theory with two sets of boundary conditions for out of plane compression is suggested [22]. One set accounted for perfect fluting, the other one for the damaged fluting. A finite element model is also proposed that helps to identify if there is a problem in the analytical model. Empirical results suggest that the fluting is damaged during the manufacturing process [22].

A quite different approach is taken in [23] where dynamic tests on corrugated boxes by using vibration machine is performed. The pressure mapping system provided a lot of relevant information and turns out to be an useful experimental method.

1.2 Problem formulation

Handling of corrugated boards during the transportation phase is an important topic as product protection is one of the main tasks of corrugated boxes. Quite often the force acts for a short time span and the packaging needs to be well designed to withstand such forces. There- fore, the task of this thesis was to make a model of corrugated board which could predict the mechanical behaviour under impact loads.

This model, then can be used to improve the design and manufactur-

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ing process of the corrugated board.

1.3 Solution method

In order to solve the problem successfully a two–step procedure is adopted. The first step required to obtain all relevant mechanical prop- erty data from empirical experiments and the second step involves using the obtained data to construct a numerical simulation model. The model then has to be validated with different empirical experiments.

To start with, materials had to be tested and all required mechanical properties recorded. The corrugated board type in this thesis consisted of two papers called paper A and paper B. The thickness of the papers together with the thickness of the whole corrugated board was measured by using the ISO 534 standard. Material tensile parameters were obtained by performing tensile experiments with different paper specimens according to ISO 124-3 standard. Data from the tensile experiments were used in the curve fitting algorithm in MATLAB to obtain the yield stress, and all required coefficients. After that, averages of the coefficients were calculated and they were used to calibrate the FE model.

After the papers were analyzed, tests were performed on the corrugated boards. The first test was the ECT according to ISO 3037. This test determines how much force on the edge of corrugated board is required to crush it. In real life applications, knowing the value of ECT can determine how many boxes can be stacked on top of each other before collapsing. Another experiment which helps to determine elastic properties of corrugated board is the bending stiffness test which was performed according to ISO 5628. Bending stiffness test determines the mechanical properties of corrugated board under bending load.

Higher bending stiffness is desired as it increases mechanical integrity of the corrugated box. Finally, two types of drop tests were performed.

The first involved dropping a steel ball in the center of the corrugated board and then measuring the deflection. The second test involved impact of a bottle to the wall of corrugated box. The first test was performed to obtain dynamical response and observe the damping of the corrugated board used so the FE model could be calibrated. The second, bottle, test simulated a real life corrugated box handling problem.

The impact of the bottle involves high impact velocities, therefore plas-

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tic deformation was visible. The data was compared with simulations using the FE model.

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Chapter 2 Methods

A corrugated board specimen with two different types of papers (A and B) as used in this thesis. Both paper types had different geometrical and mechanical properties. The sketch of corrugated board with different types of paper materials is presented in Figure 2.1.

Figure 2.1: Corrugated board structure used in this thesis.

2.1 Experiments

All of the physical experiments except the drop test and the static load test were performed at RISE Bioeconomy in Stockholm, Sweden. The tests included thickness measurements, tensile tests in MD and CD, ECT and bending stiffness tests. Samples were pre-conditioned according to ISO 187, which defines standard laboratory climate with temperature of 23^◦Cand relative humidity of 50 %.

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CHAPTER 2. METHODS 11

2.1.1 Thickness measurements

Thickness measurements were performed according to ISO 534 standard with 10 measurements per paper grade and board. The measuring equipment is shown in Figure 2.2.

Figure 2.2: Thickness measurement procedure. a) paper specimen measurement b) corrugated board measurment.

2.1.2 Tensile testing

Paper as a material to a good approximation can be treated as orthotropic which means that tensile tests in MD and CD have to be performed. Tensile tests were conducted according to ISO 1924-3. The testing equipment together with the resulting tensile stress–strain curve for paper material is presented in Figure 2.3.

Figure 2.3: Tensile test equipment and common results. a) tensile test equipment, b) typical stress–strain curve.

The stress strain curve can be approximated into elastic part and plastic part. The elastic part can be defined by a linear function which

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12 CHAPTER 2. METHODS

shows that there is no permanent damage in the paper. Plastic part occurs when the yield stress is reached and paper starts going to plastic region which is no longer described by a linear function. The Failure of the specimen occurs when the paper can no longer withstand the loading and rutpures.

2.1.3 Edge Crust Test

To get a better insight into corrugated board and how it behaves under compressive loading, an ECT was performed according to ISO 3037.

The test equipment and the test sketch is shown in Figure 2.4.

Figure 2.4: ECT equipment with schematic drawing. a) ECT equipment, b) Schematic drawing of ECT where red arrows indicate movement of the crushing surface.

2.1.4 Flat Crush Test

Another common test to perform is the FCT. The test allows to know the load a corrugated can handle when the force is applied perpendicular to the liner. The FCT was performed according to ISO 3035. The schematic set up is presented in Figure 2.5.

Figure 2.5: FCT sketch.

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The red arrows indicate the direction of movement of the rigid plate which crushes the board. The corrugated board is placed on a rigid surface.

2.1.5 Bending stiffness test

In order to obtain the corrugated board behaviour in bending, bending stiffness test was performed according to ISO 5628. The test equipment together is presented in Figure 2.6. Schematic drawing of the experiment is presented in Figure 2.7. The red arrows indicate points where the force was applied. Blue dashed line shows deformation of the original corrugated board.

Figure 2.6: Four–point bending stiffness test equipment.

Figure 2.7: Bending stiffness test.

2.1.6 Deflection test

Deflection test of full corrugated board was conducted by placing it on cylindrical supports which allow rotation at the boundaries. Then, to remove the initial curvature of the plate in order to measure the deflection more easily, the board was pre–stressed by a mass. After that, another mass of 191 g was placed on top in order to measure the deflection. Test set up is presented in Figure 2.8.

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Figure 2.8: Deflection test equipment.

2.1.7 Drop test

The purpose with the drop test was to observe the corrugated board behaviour under dynamical shock loading and get the response in the form of the deflection. The projectile dropped on the board is spher- ical steel ball while the plate was placed on cylindrical beams which allowed only rotation. Also, a small amount of adhesive tape was applied in order to prevent the board from sliding and jumping off the supports during the impact. Drop height was changed three times in order to observe deflection dependency on the drop height. The corrugated board with the steel ball and it’s movement direction is in Figure 2.9. The problem sketch with simply supported boundary conditions are presented in Figure 2.10.

The red square on the corrugated board indicates the location where the deflection was measured using a laser Doppler system. Arrow on the corrugated board indicates MD. Ruler is placed for a reference.

2.1.8 Bottle impact test

The drop test provides useful information about the model. To study the phenomena which happens during transportation phase, a new experiment is performed. The experiment consists of placing a standard 750 ml wine bottle filled with water in a corrugated box and then slide the box of a specific ramp with a rigid wall at the end. Dur- ing the crash, the bottle impacts one of the walls of the corrugated box

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Figure 2.9: Drop test set up.

Figure 2.10: Sketch of the drop test with simply supported boundary conditions.

which has dimensions of 9.5 cm in MD and 35 cm in CD. An accelerometer is placed on the bottle which outputs acceleration as function of time. Having the acceleration graph allows assessing of the corrugated board behaviour during impact. A schematic illustration of the experimental set up is presented in Figure 2.11.

Table 2.1 provides dimensions for the bottle impact test set up. Note that the angle was changed after every impact and a new acceleration curve was recorded.

Table 2.1: Bottle impact test length and inclination angle values.

Dimension Value

Length L, m 2.11

Inclination angle Θ,^◦ 10, 15, 20

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Figure 2.11: Bottle impact test set up.

Accelerometer was placed on the middle of the bottle by using a epoxy adhesive. After the accelerometer was glued firmly to the bottle, the upper part of the corrugated box was removed as it does not have firm connection to the impact surface and does not contribute significantly to the final result. A slit was made on the back of the corrugated box for the cable of the accelerometer. The bottle and box set up before experiment started is presented in Figure 2.12.

Figure 2.12: Accelerometer attachment to the bottle.

After set up, the box was allowed to slide down the slope and the acceleration was recorded.

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2.2 Numerical modelling

After getting results from the empirical experiments a finite element (FE) model, which allows to predict mechanical behaviour of corrugated board, was constructed. The model should be able to verify previous physical experiments by producing similar results as well as predict the performance for design purposes.

2.2.1 Geometry generation

The FE model geometry was generated with the help of MATLAB and CAD software. As the geometry of the corrugated board is already known, a script is created which outputs the coordinates for the CAD software. The liners and fluting were made in CAD software and then exported to FEM software where they were meshed.

Depending on the required results, either implicit or explicit FE formulations were used. The difference between implicit and explicit is that implicit method of solving differential equations is always stable but sometimes hard to implement and involves inversion of the stiffness matrix which can be very time consuming [24]. Explicit, however, is easier to implement and does not require inversion of the stiffness matrix but it is conditionally stable as the time step has to be picked manually [24]. The explicit method works well when the time du- ration of the simulation is short and a lot of dynamics is involved while implicit works well when steady–state solutions are required.

A flowchart showing the steps taken in the simulation modelling is presented in Figure 2.13.

Model generation starts with having corrugated board dimensions, including width and height of the corrugated board, fluting height and fluting periodicity. The data are used in a MATLAB script which outputs text file compatible for the AutoCAD software. AutoCAD then outputs geometry files for the both liners and fluting. Depending on what type of analysis that is required, the geometry files are imported either to Abaqus or LS–Dyna where it is meshed and material, loads and boundary conditions are applied, etc.

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Figure 2.13: Flowchart of the FE model generation.

2.2.2 Geometry and simplifications

The shape of the fluting was assumed to be a perfect sine wave. How- ever, because of the element discretization the sine wave shape consists of many small straight lines. Also, the thickness assignment for the fluting in the FE software was defined from the center of the liners and fluting which makes the thickness of the whole corrugated board model slightly lower than in the real board. The thickness approximation is presented in Figure 2.14. This slightly influences the results as the bending stiffness around MD and CD becomes slightly lower than in practice.

2.2.3 Material model and simplifications

Paper as a material consists of a fibre network which makes the paper to behave differently in MD and CD The constitutive modelling of paper consists of elastic and plastic parts. In this work, directions 1, 2, 3 refer to MD, CD and ZD, respectively.

Elastic part

The fibre orientation distribution makes the paper to a good approximation orthotropic. For that reason, a total of 9 independent coeffi-

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Figure 2.14: Fluting thickness approximation.

cients are needed in the compliance matrix. The are E1, E2, E3, G12, G₁₃, G23, ν12, ν13 and ν23. The compliance matrix for the orthotropical material involving elastic parameters is

C =







1

Ex −^ν_E²¹

2 −^ν_E³¹

3 0 0 0

−^ν_E¹²

1

E2 −^ν_E³²

3 0 0 0

−^ν_E¹³

1 −^ν_E²³

2

1

E3 0 0 0

0 0 0 _G¹

23 0 0

0 0 0 0 _G¹

31 0

0 0 0 0 0 _G¹

12







. (2.1)

Since the compliance matrix C is symmetric, important relations are ν_ij

E_i = ν_ji

E_j. (2.2)

However, when talking specifically about paper, some empirical ob- servations are made which add further simplifications. It is noticed that (2.3) is valid to a good approximation and helps to easily obtain Poisson’s ratio without the need of fairly complex experiments [25].

√ν₁₂ν₂₁ ≈ 0.297. (2.3)

Another useful equation which allows to avoid tedious testing is presented in (2.4). It can be used together with (2.3) to obtain useful results [26].

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G₁₂≈

√E1E2

2(1 +√

ν₁₂ν₂₁) = 0.65p

E₁E₂. (2.4) The out of plane properties are hard to measure and certain approximations can be used. It is noted that material stiffness modulus in MD can be around 250 times higher than in ZD [12], [27]. Also, the out of plane shear moduli can be approximated as half of the stiffness modulus in the out of plane direction. These assumptions are expressed in (2.5).

E₃ ≈ E₁

250, G₁₃≈ G₃₂≈ E₃

2 . (2.5)

What is more, paper tends to delaminate without influencing strains in other directions. This means that the out of plane Poisson’s ratio can be approximated. This assumption in confirmed by experiments [28], [29] and is expressed as

ν₁₃= ν₂₃ = ν₃₁= ν₃₂= 0. (2.6) Equation (2.6) then make in–plane and out–of–plane deformations un- coupled. Taking into account Equations (2.3), (2.4), (2.5) and (2.6) allows for full definition of an elastic orthotropic material by only using E₁and E2 obtained from experiments.

Plastic part

A plasticity model for paper materials is implemented in the LS–Dyna FE solver and is based on [30], [31]. The model has multiple yield surfaces and takes into account different material yielding points in tension and compression. Due to the lack of yield data in compression, the compressive yield stress was taken to be 60 % of the yield stress in tension [32]. The nature of material hardening is assumed to be in the shape of a hyperbolic tangent as proposed by [30], [31]. The mathematical expressions are

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σ_sÎ = σÎ_s0+ AÎtanh(BÎε^p_{ef f}) + CÎε^p_{ef f} (2.7) σ_sÎI = σ_s0ÎI+ AÎItanh(BÎIε^p_{ef f}) + CÎIε^p_{ef f} (2.8) σ_sÎII = σÎII_s0 + AÎIItanh(BÎIIε^p_{ef f}) + CÎIIε^p_{ef f} (2.9) σ_sÎV = σ_s0ÎV + AÎVtanh(BÎVε^p_{ef f}) + CÎVε^p_{ef f} (2.10) σ_s^V = σ_s0^V + A^Vtanh(B^Vε^p_{ef f}) + C^Vε^p_{ef f} (2.11) σ_s^{V I} = σ_sÎII(ε^p_{ef f}), (2.12) where σ_s0ⁱ is the initial yield stress the parameters Aⁱ, Bⁱ, Cⁱare hardening constants and i refers to the yield plane. I,II are tension and compression in MD, respectively, III, VI are positive and negative shear, respectively and, finally, IV, V are tension and compression in CD, respectively.

Due to the lack of research done in the compressive post–yield behaviour of paper, perfect plasticity was assumed with failure at the same strain as in tension. A simplified schematic stress–strain curve is presented in Figure 2.15.

Figure 2.15: Visualization of uniaxial stress – strain curve for paper where ε is strain, σ is stress, σ_y⁺ is the yield stress in tension and σ_y⁻is the yield stress in compression.

For the corrugated board used in this thesis, Table 3.4 provides hardening coefficients and Table 3.5 provides yield stresses in tension and compression in both MD and CD. The coefficients were obtained by plotting the data obtained from the experiment and then fitting one

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of the curves from Equations (2.7–2.12) through the data points. It was done by using the cftool function in MATLAB.

Note that in compression perfect plasticity was assumed, thus Equa- tion (2.13) was implemented in the model.

AÎI = A^V = BÎI = B^V = CÎI = C^V = 0 (2.13) Also, all other material parameters which are required for this LS–

Dyna model including shear were obtained from [30] for a paperboard material as experiments were not performed to explicitly determine those parameters.

2.2.4 FE model for ECT

Implicit model

The FE model of ECT was constructed by using a linear elastic material model in combination with the Tsai–Wu failure criterion in Abaqus without using plasticity. Plasticity was not implemented as ABAQUS does not have build in constitutive equation for paper materials. In- stead, it was treated as linear orthotropic just to observe the buckling and failure. All movements at one edge which goes in the MD direction of the corrugated board were prevented. The nodes of the other MD direction edge were controlled by the master node to which displacement is prescribed simulating the empirical experiment. The Tsai–Wu criterion [33] which accounts for different material strengths in tension and compression is expressed as

F_ijσ_iσ_j + F_iσ_i = 1, (2.14) where i, j = 1, 2 . . . 6.

The Tsai–Wu criterion failure stresses in tension were obtained from the experiments and the shear strength with cross–product term coefficients were taken from [12]. A buckling analysis was performed in order to acquire the buckling load. When the load was obtained, the buckling shape with small imperfection was exported to a Rik’s analysis. The load was increased until the Tsai–Wu criterion reached the value of 1 indicating failure. The load at that point was taken as the failure load. The first buckling mode of the corrugated board is presented in Figure 2.16.

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Figure 2.16: First buckling mode of the corrugated board during simulation of the ECT test.

Note, that the displacement in the legend is not physical as the buckling analysis solves an eigenvalue problem and does not produce physical displacements but only the deformed shape.

Sometimes it is hard to experimentally determine the failure point as there is a range between damage initiation and ultimate failure as demonstrated in [12]. To get a better understanding of the problem, an explicit model using LS–Dyna was created in order to compare it with the implicit model.

Explicit model

The same geometry as used in the implicit model was used in LS–Dyna for the buckling analysis. The reason for doing the test in LS–Dyna was because this software supports an elastic–plastic paper material model. The same boundary conditions were applied and the only difference from the implicit model was that the punch was modelled as a rigid body impacting the shell edge of the corrugated board. The peak reaction force on the punch was recorded which was taken as the ECT value. The post buckling shape of the corrugated board is presented in Figure 2.17.

2.2.5 FE model for FCT

The numerical FCT model was constructed by using an explicit FE formulation that included plasticity. The punch was modelled as a rigid body which impacts the corrugated board made out of shell ele- ments. The adhesive line between liners and fluting was assumed to be perfect and no delamination behaviour was modelled. The boundary conditions used allowed the lower liner to move in the plane but

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Figure 2.17: Post buckling shape of corrugated board during simulation of an ECT.

prevented any other motion including any rotations of the nodes. The FE model of the FCT is presented in Figure 2.18.

Figure 2.18: FE model for FCT.

By recording the reaction force exerted on the rigid punch, the FCT value can be evaluated. However, no failure criterion was implemented, so the FCT value was calculated by taking the maximum pressure the board could carry before starting to collapse. The FCT value was calculated and is presented in Table 3.1. The shape of the fluting just before the collapse is presented in Figure 2.19.

Figure 2.19: Shape of the fluting during simulation of the FCT just prior collapse.

2.2.6 FE model for bending stiffness test

For the bending stiffness an Abaqus model was constructed. Since the ISO 5628 test tries to avoid shear, pure bending at the edges was ap-

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plied on the FE model by connecting all edge nodes to a master node and applying bending moment on a master node. The boundary conditions are presented in Figure 2.20. The red arrows indicate movement which was allowed for the bending stiffness test.

Figure 2.20: FE simulation of the bending stiffness test and allowed movement.

Since the curvature of the corrugated board is proportional to the bending stiffness, this parameter can be extracted. A deformed corrugated board after pure bending is presented in Figure 2.21. Note, that the shape is not completely cylindrical. Due to Poisson’s ratio, deformation in perpendicular direction is invoked. Therefore, the curvature was measured at the very center line of the corrugated board.

Figure 2.21: Deformed shape after bending moment was applied in the simulation of the bending stiffness test.

2.2.7 FE model for the static deflection test

Finally, a FE analysis of the static test was performed in order to verify the elastic orthotropic material behaviour. A static load was applied to the FE model which simulated the real experiment. Boundary conditions were applied at the edges parallel to MD which allowed rotation around MD axis but prevented any other motion.

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2.2.8 FE model for the dynamic deflection test

The final test was to observe the dynamic behaviour of the corrugated board. For this reason an explicit FE formulation was used with the same boundary conditions as in the static test. By observing the frequency of oscillations which did not match the FE model, frequency range from 30 Hz to 60 Hz was picked in the model and was damped by 0.5 of the critical damping. The FE set–up is presented in Figure 2.22.

Figure 2.22: Impact model for corrugated board.

The model simulated a steel ball impacting the corrugated board from three different heights (6.3 cm, 12.3 cm and 16.3 cm) and the initial velocity of the ball when hitting the corrugated board was applied in the model.

2.2.9 Bottle impact test

To make the bottle impact test more predictable and understandable a FE model simulating the bottle impact test was created. The model simulates a standard 750 ml wine bottle impacting the corrugated board.

The bottle was taken as a rigid body with a certain initial velocity. Only the box part which takes the direct impact was modeled due to computational limitations. The connections between walls were assumed to have a simply supported boundary condition. The wall was modeled by assuming it as rigid and then it was placed near the corrugated board so that after the impact only motion in positive ZD was possible. An illustration showing boundary conditions and initial velocity

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of the simulation is presented in Figure 2.23.

Figure 2.23: Initial conditions for the FE bottle impact test.

Estimation of bottle impact velocity

Before the actual FE model was created, some simple analyses were made in order to predict the magnitude of acceleration that the bottle experiences during the impact. The fluting height was 1.7 mm and it was assumed to be the distance through which the impacting body should come to a complete rest. The glass bottle was much stiffer than the corrugated board and was assumed to be a rigid body with mass m and initial velocity v0. The initial velocity was calculated using equation (2.15) which comes from conservation of energy and the geometry represented in Figure 2.11. The mass body with initial velocity is presented in Figure 2.24.

v₀ =p

2glsinθ = 2.6812 ms⁻¹, (2.15) where g is 9.81 ms⁻², θ is 10^◦and l is 2.11 m.

It is assumed that the body decelerates with a constant retardation and the velocity as function of time can be expressed as a linear function as illustrated in Figure 2.25.

Since the acceleration is assumed to be constant, the velocity of the rigid body during the breaking period is expressed by Equation (2.16).

v(t) = v₀− at. (2.16)

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Figure 2.24: Distance between corrugated board liners and impacting body.

Figure 2.25: Velocity during the impact as a function of time.

Knowing that the velocity has to be equal to zero, at the final time tf

gives

t_f = v₀

a. (2.17)

Integrating the velocity function (2.16) gives the distance covered.

Z tf

0

v(t)dt = Z tf

0

(v₀− at)dt = t_f

v₀−at_f 2

. (2.18)

Assuming that the entire distance was used while breaking, Equation (2.18) has to be equal to the distance between liners h. Therefore, sub- stituting Equations (2.17) in (2.18) and equating everything to h allows for calculation of the acceleration a. The final equation for acceleration a is

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a = v₀²

2h. (2.19)

After inserting the initial values, the acceleration was calculated to be around 2100 ms⁻² which is around 210 times more than the acceleration of free fall! However, this is only an assumption and is only valid if the velocity changes linearly. Also, all of the fluting has to be fully compressed during the impact.

FE model of bottle impact test

The problem with FE modeling can directly be seen as the initial velocity during the experiment was too high. The through–thickness distance was too small to bring the bottle to a rest. Due to large deformations, the numerical model failed to converge as at some point all three layers of the corrugated board were completely flattened out. This is a reasonable behaviour as the total mass of the corrugated board was around 15 g and was impacted by 1.3 kg mass bottle at around 10 kmh⁻¹.

Just to make sure if the model works properly, and to observe the response, the initial velocity was reduced to 2.3 kmh⁻¹. Nodal values were extracted from FE model with reduced velocity (Figure 2.23). The node was chosen to be the one which was closest to the actual place of the accelerometer in the physical experiment. Figure 2.26 shows part of the mesh, without the bottle, that was used for this simulation.

Figure 2.26: FE mesh for bottle impact simulation.

The extracted nodal displacement velocity and acceleration is presented in Figure 3.6. Due to equipment limitations in the physical test, numerical simulations was constructed in order to investigate how the

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bottle orientation with respect to the fluting direction influences the impact. Figure 2.27 shows the initial set up for simulations that were used. The geometry and boundary conditions for both simulations are identical to the previous bottle impact test.

Figure 2.27: Comparison between different fluting alignments during a bottle impact test. a) MD test during which bottle was aligned along MD direction, b) CD test during which bottle was aligned along CD direction.

After applying an initial velocity of 1.9 ms⁻¹to the bottle, the node where the accelerometer would be placed, located on the top surface of the bottle was selected.

2.3 Investigation of different fluting shapes

After performing both numerical and physical experiments with corrugated board, it was decided to make a comparison between the classical sine wave fluting and a sine wave which periodically changes amplitude and period. The study was performed only numerically with no physical experiments. The shape of the modified fluting is presented in Figure 2.28.

Figure 2.28: Shape of the new flute profile.

Every second sine wave in the fluting has an amplitude and a period reduced by 25 % compared with the original sine wave used in all previous experiments in this work.

Two explicit FE models were created with identical set ups. The board was rectangular with a length of 93 mm in MD and CD. The

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boundary conditions allow the lower liner to translate in the plane, but prevent any movement in the through–thickness direction. The rigid impact body was taken to be made out of a steel material and have a shape of a cylinder with a diameter of 10 mm and length of 50 mm. No friction was assumed between the upper liner and the impacting body.

The FE model is presented in Figure 2.29.

Figure 2.29: FE set up for the experiment with the new fluting.

The model was analyzed by taking three different initial velocities for the rigid body, 1.0 ms⁻¹, 1.5 ms⁻¹ and 1.7 ms⁻¹.

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Chapter 3 Results

3.1 General results

Table 3.1 provides a relative difference between the results from the FE models and physical tests. For the impact test, the maximum experimental deflection was compared with the maximum deflection predicted by the FEM.

The difference between the implicit numerical and experimental ECT results can be explained by the fact that plasticity was not implemented in the model. However, even with plasticity included in the explicit model, lower values were still observed as presented in Table 3.1. The error in both the implicit and explicit methods can, possibly, be explained by the fact that the corrugated board layers are thicker in real life than in the FE models due to the thickness approximations in the models as discussed in Section 2.2.2. Also, in real corrugated board the adhesive lines are thicker and should perhaps be modelled with more than one line of nodes. A schematic drawing showing the adhesive surfaces in red is shown in Figure 3.1.

The numerical bending stiffness results are presented in Table 3.1. Re- sults are in good agreement with experiments, which shows that the elastic parameters were well–calibrated.

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CHAPTER 3. RESULTS 33

Figure 3.1: Adhesive surfaces joining liners and fluting.

Table 3.1: Result summary of FE and physical experiments.

Exp. res. FE res.

ECT (implicit) 4.82 kN/m 44% lower ECT (explicit) 4.82 kN/m 45% lower

FCT 404.5 kPa 6% lower

Bending stiffness in MD 1.745 Nm 0.8% lower Bending stiffness in CD 0.828 Nm 1.0% higher Static deflection 4.9 mm 5.7% lower Impact test (h = 6.3 cm) 3.1 mm 13.2% higher Impact test (h = 12.3 cm) 4.4 mm 11.4% lower Impact test (h = 16.3 cm) 5.6 mm 26.8% lower

3.2 Thickness measurement results

The measured thickness for corrugated board is presented in Table 3.2. The wavelength L and flute height H were already known to be 4.87 mmand 1.7 mm respectively from previous measurements.

Table 3.2: Summary of thickness measurement for paper materials A, B and corrugated board.

Paper A Paper B Corrugated board Average thickness (µm) 150.00 166.10 2040.00

Standard deviation 0.00 3.67 10.00

Coefficient of variation (%) 0.0 2.2 0.5

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34 CHAPTER 3. RESULTS

3.3 Tensile testing results

Table 3.3 shows the average tensile stiffnesses obtained from 10 tensile experiments of each paper grade. The hardening coefficients were obtained by performing data fitting in MATLAB and are presented in Table 3.4. Material yield stresses in tension were identified by plotting experimental data in MATLAB and observing the yield stress from a graph. Obtained yield stresses are presented in Table 3.5.

Table 3.3: Tensile stiffness of paper materials A and B.

Paper material name Paper A Paper B Elastic modulus in MD, E1 (MPa) 642 506

Standard deviation 16.7 19.7

Coefficient of variation (%) 14.9 1.73 Elastic modulus in CD, E2 (MPa) 238 221

Standard deviation 12.87 10.64

Coefficient of variation (%) 3.6 2.3

Table 3.4: Hardening coefficients of papers A and B.

Paper A Paper B A^I 1.574 · 10⁷ 1.080 · 10⁷ A^IV 8.462 · 10⁶ 6.596 · 10⁶

B^I 217.6 243.7

B^IV 153.4 192.0

C^I 4.488 · 10⁹ 1.185 · 10⁹ C^IV 1.746 · 10⁸ 1.905 · 10⁸

Table 3.5: Yield stresses in tension and compression for papers A and B in MD and CD.

Paper material name Paper A Paper B

Yield stress in MD tension (MPa) 22.2 16.8 Yield stress in MD compression (MPa) 13.3 10.1 Yield stress in CD tension (MPa) 8.9 8.2 Yield stress in CD compression (MPa) 5.36 4.9

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3.4 Static deflection test numerical and phys- ical results

The numerical deflection test results in comparison with average of 10 physical measurements are presented in Table 3.1. During physical experiment, the whole corrugated board was initially curved which adds stiffness. Also, the measuring tool exerted additional force during the measurement. This could have added additional deflection which FE model does not account for. The obtained deflection is presented in Figure 3.2.

Figure 3.2: Deformed shape of corrugated board after application of a pure static load.

3.5 Dynamic deflection test numerical and physical results

The graphs comparing empirical results and FE solutions are presented in Figure 3.3. Table 3.6 shows the drop height of the steel ball and maximum deflection together with time taken to reach the maximum deflection. Higher drop height influenced the boundary conditions more and this can be seen from Figures 3.3b and 3.3c.

It can be seen that the solution for a drop height of 6.3 cm presented in Figure 3.3a almost perfectly matches the first maximum deflection peak determined experimentally both with respect to deflection and time. However, when the drop height increases the dissimilarity between the FE and experiment results start to increase. This can be explained by the initial upward curvature observed on the corrugated

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Table 3.6: Drop height and maximum deflection with corresponding time.

Drop height (mm) Maximum deflection (mm) Time (ms)

63 3.5 19.5

123 4.4 24.8

163 5.6 21.0

(a) Drop height 6.3 cm. (b) Drop height 12.3 cm.

(c) Drop height 16.3 cm.

Figure 3.3: Comparison between experimental result and FE solution for the impact tests.

board which made it stiffer. Also, the boundary conditions assumed perfect contact between support and corrugated board in the FE model.

However, reality showed that the corrugated board separated at the edges after the impact so the duct tape did not fully prevent board separation when the drop height was increased. Figure 3.4 shows how the edges separate after the impact.

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Figure 3.4: Separation of edges after ball impact the board.

3.6 Bottle impact test numerical and physi- cal results

Figure 3.5 shows acceleration peaks which occur during the wall impact with different initial angles Θ.

Figure 3.5: Bottle impact test acceleration peaks.

From Figure 3.5 it can be be seen that the acceleration peaks between 15^◦and 20^◦is not very large. In reality is it not true as the accelerometer equipment is not able to cope with these accelerations as they are over the measurable limit. Due to this, the conclusions that can be drawn from the impact with the highest incline angle is limited.

Other limitations of the experiment include that it was hard to make the corrugated box hit the rigid wall perfectly flat during the impact. This problem existed in two planes as the box would wobble on the rollers and the wall had a small tilt as the angle between surface with rollers and wall was obtuse and not perfectly 90^◦. Since the

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accelerometer could only measure acceleration in one direction, this factor has effected acceleration curves since an impact which is not perfectly flat causes accelerations in other directions which were not measured.

Numerical results were processed by extracting displacement, velocity and acceleration from a single node in the bottle impact test. The results are presented in Figure 3.6. The extracted nodal velocities ac- counting for different fluting pipe orientations are presented in Figure 3.7. The results obtained suggest that there is a difference between fluting orientation during the bottle impact test. The results in the MD direction showed that the acceleration peaks when the time was approximately around 0.55 ms – 0.60 ms. However, the same fluting demonstrates lower acceleration at the end of the crash. These initial results suggest that the fluting orientation can be used to engineer corrugated board with best properties for a specific impact case.

Figure 3.6: FE simulation nodal displacement, velocity and acceleration.

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Figure 3.7: The velocity of the impacting bottle hitting a corrugated board with the fluting pipes orientated in MD and CD, respectively.

3.7 Different fluting shape results

Results illustrating differences in displacement compared to classical uniform fluting are presented in Figure 3.8. The blue vertical lines indicate the time when the smaller fluting got in the contact with the upper liner in the proposed modified fluting.

The results suggest that different fluting geometries respond to the impact differently. The modified fluting takes more time to completely reduce the velocity of the impacting body. This indicate, that the average acceleration was lower for the newly proposed fluting when compared with the classical fluting. However, the peak acceleration is higher in the new proposed fluting for the 1.5 ms¹ and 1.7 ms¹ initial velocity cases. This indicates that the rigid body had larger retardation when the lowered fluting waves got in contact with the upper liner.

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Figure 3.8: Comparison between classical fluting and the new fluting for a) velocity of the rigid body after impact, b) acceleration of the rigid body after impact.

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Chapter 4 Discussion

In this work, physical experiments and numerical modelling of corrugated board was performed. Experiments included ECT, FCT, bending stiffness test, deflection tests, drop test and bottle impact test. For all the physical experiments, a corresponding FE models were created and the results were compared. Close agreement between experimental and FE results can be seen from the bending stiffness tests both in MD and CD. Small relative errors indicate that the model was well–

calibrated in the elastic region. This statement is supported by the static deflection test which gives a relative error of 5.7 % between experiment and FE model. The error is in a reasonable range as the board was not pre–conditions according to the ISO requirements. Another source of the error is within the deflection measuring device as it adds additional force during the measurement which was not included in the FE model. The comparison between FCT test and FE model suggest that the out of plane parameters used for the corrugated board are close to real values.

Larger errors can be seen from the impact test which gave 13.2 % and 11.4 % relative errors from drop heights 6.3 cm and 12.3 cm, respectively. However, the largest error can be seen from the results for a drop height of 16.3 cm which was around 26.8 %. Increasing drop height provides more energy to the board and the boundary conditions in the physical experiment changes. The corrugated board starts to separate from the cylindrical supports as the adhesive tape did not ensure perfect contact between the corrugated board and the supports when the drop height was increased. Also, looking from the result plots, it can be seen that not all of the response frequencies were

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42 CHAPTER 4. DISCUSSION

captured in the simulations. Less damping in the higher frequency band might have improved the results but this needs more investigation. Least agreement between physical test and the FE model was noted for the ECT test which gives around 44 % and 45 % relative errors in the implicit and explicit formulations, respectively. The mechanical behavior of corrugated board during the ECT test is complex. In the physical experiment, corrugated board is thicker in the ZD which means that the critical buckling load should be slightly higher. Also, the glue line between liners and fluting can be thicker and does not necessarily mean that the contact between a single node line and a surface simulates the behaviour correctly. Another aspect to which the results are sensitive are the boundary conditions. Modified boundary conditions might change the results by the factor of two or even more.

Similar errors might indicate that the plasticity algorithm was not fully engaged during the explicit analysis.

Numerical results that can not be compared with the empirical experiments for the crash test results and results obtained with a modified fluting geometry. The crash test results can not be compared as the measuring device was unable to capture the accelerations properly due to physical limitations. Physical experiments of the modified flute were not performed. However, a difference can be observed between bottle orientations in the MD and CD directions. This difference might help to orientate the product inside the corrugated box so it would ex- perience as small acceleration peaks as possible. Also, a modified fluting geometry might be implemented in real corrugated boards which would further change current packaging practice.

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Chapter 5 Conclusions

Comparison between results obtained from FE models and empirical experiments suggest that the FEM is a reliable way to solve static or dynamic problems for paper structures. However, the complexity of paper materials makes it a challenging task. From the work performed in this thesis, the following conclusions can be drawn:

• The elastic behaviour of corrugated board is well predicted by assuming an orthotropical material with different elastic stiff- nessess in MD and CD.

• The dynamic behaviour of corrugated board when subjected to an impulse force was well predicted for low drop heights.

• Higher drop heights gives rise to rise an uncertainties in the experimental set–up and FE model, but the difference can be explained with sound physical arguments.

• A modified fluting geometry with different sine wave period and amplitude influences the results and reduces average acceleration when compared with a classical fluting shape.

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Chapter 6 Future work

To improove understanding of paper and corrugated boards, the following work might be considered for the future:

• Include implementation of plasticity in the implicit FE models.

This would allow for better understanding of the influence of plasticity in the ECT test and would possibly allow for more reliable results.

• Include viscoelastic effects. Paper can be modelled as a viscoelastic material which deforms differently under high or low load–

rates. This might lead to better agreement between experimental and numerical results in some of the tests performed in this study, which would improve the understanding of paper as an engineering material.

• More investigation are also suggested for the influence of different fluting shapes.

• Implementation of pre–stress in liner and fluting as it would ap- pear from the manufacturing process.

• Implementation of moisture and temperature dependencies as they influence the constitutive behaviour.

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