Rebrasti karton se sve više traži ne samo za proizvodnju različitih pakovanja već i za izradu promidžbenih materijala, umjetničkih djela, pribora pri prijevozu skladišne robe i mnogo drugih stvari. S porastom potražnje za rebrastim kartonom javlja se i potreba detaljnijeg poznavanja njegovih mehaničkih svojstava kako bi se poboljšala kvaliteta i pouzdanost finalnih proizvoda. Korišten je model zasnovan na metodi konačnih elemenata te su analizirana mehanička svojstva slojeva rebrastog kartona i njihov utjecaj na njegova ukupna mehanička svojstva.The corrugated paperboard has become a widely used material not only for producing different packages but also for making advertising materials, pieces of art, accessories used for transportation of a number of stock items and many others. With the growing demand for corrugated paperboard comes the need for more detailed knowledge about its mechanical characteristics which would serve for improving the quality and reliability of final products. A model based on the finite element method is used to study how the mechanical characteristics of the layers of the corrugated paperboard affect its complex mechanical characteristics

Delyan Gospodinov

Stefan Stefanov

Vilhelm Hadjiiski

Croatian

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1IntroductionUvodCorrugated paperboard is a multi layered structurewhich is widely used in the packaging industry to producevarious boxes. In the past few years the demand for thismaterial has grown by hundreds of times worldwide. Apartfrom packages, designers around the globe are starting tomake different products out of it such as pieces of art,advertising products, furniture, shelters, accessories forstoring and transportation of goods and many more [1, 4, 5,8, 9]. With this widening use of the corrugated paperboardcomes the need for better knowledge of its mechanicalcharacteristics.Corrugated paperboard normally consists of threelayers – two external layers which are flat and are called"liners" and one internal corrugated layer which is called"flute" (Fig. 1).357ISSN 1330-3651UDC/UDK 676.274.017:519.6USE OF THE FINITE ELEMENT METHOD IN STUDYING THE INFLUENCE OF DIFFERENTLAYERS ON MECHANICAL CHARACTERISTICS OF CORRUGATED PAPERBOARDDelyan Gospodinov, Stefan Stefanov, Vilhelm HadjiiskiThe corrugated paperboard has become a widely used material not only for producing different packages but also for making advertising materials, pieces of art,accessories used for transportation of a number of stock items and many others. With the growing demand for corrugated paperboard comes the need for moredetailed knowledge about its mechanical characteristics which would serve for improving the quality and reliability of final products. A model based on thefinite element method is used to study how the mechanical characteristics of the layers of the corrugated paperboard affect its complex mechanicalcharacteristics.Keywords: analysis, corrugated paperboard, FEM, layered materials, mechanical characteristicsOriginal scientific paperRebrasti karton se sve više traži ne samo za proizvodnju različitih pakovanja već i za izradu promidžbenih materijala, umjetničkih djela, pribora pri prijevozuskladišne robe i mnogo drugih stvari. S porastom potražnje za rebrastim kartonom javlja se i potreba detaljnijeg poznavanja njegovih mehaničkih svojstavakako bi se poboljšala kvaliteta i pouzdanost finalnih proizvoda. Korišten je model zasnovan na metodi konačnih elemenata te su analizirana mehanička svojstvaslojeva rebrastog kartona i njihov utjecaj na njegova ukupna mehanička svojstva.Ključne riječi: analiza  materijali slojeva a svojstva, , mehaničk , MKE  rebrasti karton,Izvorni znanstveni članakKorsvojstva rebrastog kartonaištenje metode konačnih elemenata u proučavanju utjecaja različitihslojeva na mehaničkaKor svojstva rebrastog kartonaištenje metode konačnih elemenata u proučavanju utjecaja različitih slojeva na mehaničkaTechnical Gazette 18, (2011),3 357-361properties of the paper and paperboard [2, 6, 7] Thisundoubtedly brings changes in the properties of thecorrugated paperboard as well. Temperature itself does notbring serious changes but relative humidity does.Paper and paperboard are hygroscopic materials. If theyare transferred from conditions of lower relative humidity toan environment with higher moisture, within a certainperiod of time they will reach balance with these newconditions absorbing more water [2]. Schematic graphicalrepresentation of this process is shown in Fig. 2.D  Gospodinov. et al.Figure 1Slika 1.Structure of the corrugated paperboardStruktura rebrastog kartonaSince the layers are typically made of paperboard, thecomplex mechanical characteristics of the corrugatedpaperboard will be in dependence of the characteristics ofthe material used for the liners and the flute. During their lifecycle most of the corrugated paperboard products aresubjected to the influence of different environmentalconditions like variable temperatures and relative humidity.It is known that the moisture causes severe changes in theConducted researches show that the effect of water issmaller in the so called machine direction. [2] This is thedirection of the orientation of the cellulosic fibers. Thedirection perpendicular to it is called cross direction. Thestronger effect of the water in this direction is explained bythe fact that the mechanical properties are determinedmainly by the hydrogen-based bonds between the cellulosicFigure 2Slika 2.Water absorption by paperPapir i upijanje vode358Use of the finite element method in studying the influence of different layers on mechanical characteristics of corrugated paperboardTehni ki vjesnikč ,18, 3(2011)  357-361fibers which the water comprises. Since the characteristicsin machine direction depend on the mechanical propertiesof the fibers themselves, the effect of the water in thisdirection is smaller.D  Gospodinov. et al.The dimensions of the used model are as follows: 10mm. of width and 64 mm. of length. This is visuallyrepresented in Fig. 5.The model is divided (meshed) into 2399 finiteelements while the total number of nodes is 7636. The typeof used element is "shell" – quadratic. It has 8 nodes and 6degrees of freedom is given to each node. Schematicrepresentation of the element is shown in Fig. 6.The size of the elements on the external layers is 2 mm.each while the ones used for the internal layer are with nonconstant size. This is because of its rather complex form.The meshed 3D model is shown in Fig. 7.Figure 3Slika 3."Stress-strain" curves of paper at different levels of moistureKrivulje "naprezanje-deformacija" kod papira prirazličitim nivoima vlageBasically, moisture causes lowering of the strengthparameters of the paper such as modulus of elasticity, tensilestrength, compressive strength, minimal stress whichinitiates plastic deformation, etc. Until about 70 % RH(relative humidity) these parameters undergo smallerchange but it becomes bigger once the relative humidityrises above this level [2]. Fig. 3 schematically shows howthe "stress – strain" curves of the paper differ from oneanother in conditions of lower and higher relative humidity.In order to study how the changes of the mechanicalcharacteristics of the individual layers of the corrugatedpaperboard affect its complex mechanical characteristics, amodel based on the finite element method (FEM) is used.The used 3D FEM based model is made of a piece ofcorrugated paperboard with "C" flute type. This is the mostwidely used type of flute. The pitch of the flute is 8 mm. andthe total thickness is 4,1 mm. Visually these parameters arerepresented in Fig. 4.2FEM based model and conducted studiesModel zasnovan na MKE i provedena ispitivanjaFigure 4Slika 4.–Geometrical parameters of “C” flute type used forthe modeling – dimensions are in millimetersGeometrijski parametri rebara tipa “C”korištenih zamodeliranje dimenzije su u milimetrimaFigure 5Slika 5.Geometrical parameters of the used model – dimensions arein millimetersGeometrijski parametri korištenog modela – dimenzije suu milimetrimaFigure 6Slika 6.Schematic representation of the used shell elementShematski prikaz upotrebljenog elementa "" ""kvadratićaFigure 7Slika 7.Meshed 3D model with finite "shell" elements" "Umreženi 3D model s konačnim elementima kvadratićimaThe thickness of the liners is 0,26 mm while the one ofthe flute is 0,21 mm. These are common thicknesses forpaperboard used to make these layers.The model is subjected to tensile loading. This is due tothe fact that normally the mechanical property such asmodulus of elasticity, which is a crucial parameter forcommon FEM simulations used by designers, is determinedby conducting tensile tests. On the other hand theexperimental "stress – strain" curves which are available forpaperboards and corrugated paperboard are also taken byrunning such an experiment. [3]The model is loaded using the force which rises from 0N to 160 N by steps of 8 N each. Visually the applied loadand the boundary conditions for the model are shown in Fig.8.,359Technical Gazette ,18, 3(2011)  357-361D  Gospodinov. et al.This gives us the opportunity to conclude that theproposed model is reliable enough and can be used to carryout more detailed studies of the corrugated paperboardcomplex properties.In order to research how the layers individually affectthe complex characteristics of the paperboard we change theproperties of their assigned materials by predefining their"stress – strain" curves. At first we change only the curvesfor the material used for the liners. Each curve is defined by20 points (excluding the zero point). Each point has twocoordinates – the coordinate for the strain and thecoordinate for the stress. We multiply the coordinate ofeach point by 0,997 and by 0,82 we multiply theircoordinates. In this way we generate 20 new points(excluding the zero point) which stand for a new curve. Byfollowing this procedure we produce 5 additional curves.They are all shown in Fig. 11.All this is done because of twomain reasons. The first is that we imitate the changing ofmechanical behavior of the paperboard when it absorbswater from the environment. The second is that there are noenough experimental curves available, so we can seek areliable and accurate enough dependency.X YXYThe aim of the first conducted study is to examine thereliability of the proposed model. One material is assignedto both external layers and another is assigned to the internalfluted layer. "Stress – strain" curves for these materials areshown in Fig. 9.After the calculation procedures have been carried outwe examine the stress distribution in the model as well as thestrain in different areas of the corrugated paperboard panel.This allows us to see which of its parts undergo elasticdeformation and where plastic deformation occurs. [10]We use the obtained results to create a theoreticalcomplex "stress – strain" curve of the corrugatedpaperboard panel. We compare this curve to theexperimentally determined one. [3] What we see issatisfactorily good coincidence of both curves. Visually thiscomparison is shown in Fig. 10.Kor svojstva rebrastog kartonaištenje metode konačnih elemenata u proučavanju utjecaja različitih slojeva na mehaničkaFigure 8Slika 8.Applied load (2) and defined boundaries (1)Primijenjeno ) i definirane granice (1)opterećenje (2Figure 9Slika 9."Stress – strain" curves used for different layersKrivulje "naprezanje-deformacija"slojeve[3][3]korištene za različiteFigure 10Slika 10.Comparison between the experimental and FEM complexcurves for the corrugated paperboardeksperimentalnih i nihkrivulja MKE rebrastog kartonaUsporedba između složeFigure 11Slika 11.Used "stress – strain" curves for the linersKorištene krivulje "naprezanje-deformacija" za  vanjske slojeveCalculation procedures are carried out for each of these6 materials and the results are examined. Basically wereceive 6 different complex "stress – strain" curves for thecorrugated paperboard. They are shown in Fig. 12.Figure 12Slika 12.Resulting "stress – strain" complex curvesDobivene ne krivulje "naprezanje – deformacija"složeFirst we study how the change of the modulus ofelasticity of the material of the liners affects the equivalentmodulus of elasticity of the corrugated paperboard panel.360 Tehni ki vjesnikč ,18, 3(2011)  357-361Graphically this is shown in Fig. 13 depicted with dashedline.liners is unchanged this time. These 6 curves, which areshown in Fig. 15, are defined using the same procedure as inthe previous case − by multiplying their coordinates by0,997 and their coordinates by 0,82.XYUse of the finite element method in studying the influence of different layers on mechanical characteristics of corrugated paperboard D  Gospodinov. et al.Figure 13Slika 13.Effect of "liners" on the modulus of elasticityDjelovanje "vanjskih slojeva" na modul elastičnostiWe can see from the chart that this is almost a straightline, so we find the linear equation of regression which iswritten as follows:.11625,312808,0)(  XXY (1)The coefficient of regression for this equation is0,99949.The change of the maximal load is also examined. Thevalue for this load is considered to be the magnitude of theforce corresponding to the stress value for the last pointfrom the "stress − strain" curve. We determine this load forthe material of the liners itself from the 6 charts which areused (Fig. 11) and for the curves obtained for the corrugatedpaperboard model (Fig. 12). The dependency is graphicallyrepresented in Fig. 14. The solid line on this chart depicts thelinear equation of regression.R2Figure 14Slika 14. PChanging of the maximal load of the liners andthe corrugated paperboard panelromjena maksimal inog opterećenja vanjskih slojevaploče od rebrastog kartonaThe equation of regression for this case is written asfollows:.82213,406114,2)(  XXY (2)The coefficient of regression for this equation is0,99657.The study continues with the examination of theinfluence of the internal corrugated layer. Just as in the casewith the liners, 6 different "stress − strain" curves are givenfor the material of the fluted layer while the material for theR2Figure 15Slika 15."Stress – strain" curves used for the fluteKrivulje "naprezanje-deformacija" upotrebljene za rebroFigure 16Slika 16."Stress – strain" complex curves resultingfrom the second studyKompleksne krivulje "naprezanje – deformacija" dobiveneiz druge analize" "After the process of calculation is over we examine theresults. We analyze the 6 complex stress − strain curveswhich we have obtained for the corrugated paperboardpanel. What we get is almost coincident curves. The averagedifference between each of the points is 0,22 %. Thisindicates the insignificant effect of the flute on the complexcharacteristics of the corrugated paperboard. The curves inquestion are shown in Fig. 16.Following the previously used procedure, we examinethe change of the equivalent modulus of elasticity for theFigure 17Slika 17Effect of the internal layer on the modulus of elasticity. Djelovanje unutarnjeg sloja na modul elastičnosti361Technical Gazette ,18, 3(2011)  357-361Kor svojstva rebrastog kartonaištenje metode konačnih elemenata u proučavanju utjecaja različitih slojeva na mehaničkaD  Gospodinov. et al.corrugated paperboard. This change is shown on the chart inFig. 17.We determine the linear equation of regression which isdepicted with solid line. It is written as follows:.92304,55400146,0)(  XXY (3)After analyzing the results we notice that the change ofthe maximal load is so insignificant that it can be ignored.The coefficient of regression for equation (3) isdetermined to be 0,98485.1. The FEM based 3D model of the corrugatedpaperboard has been proposed. The results obtained fromthis model have been compared to experimentally obtaineddata [3] and they show good coincidence. This gives us thechance to use the FEM for a more detailed study of thecomplex properties of the corrugated paperboard and howits parameters, components and geometrical parametersaffect its resultant mechanical characteristics.2. A study has been carried out to investigate how thechange of the properties of the materials which are used tomake individual layers affects the complex mechanicalbehavior of the corrugated paperboard. Such changesnormally occur in conditions of changing environmentalfactors such as temperature and relative humidity.3. The obtained results open the opportunity for thedesigners of packaging products to predict how the changeof the properties of corrugated paperboard would affect thestrength of the final products.4. The investigation shows that the liners affect muchstronger the complex characteristics of the corrugatedpaperboard than the flute, whose influence is almostinsignificant. This is clearly seen by analyzing equations (1)and (3). The difference of the slope coefficients (0,12808 forequation (1) and 0,00146 for equation (3)) is approximately88 times.R23Conclusions4ReferencesZaključciLiteratura[1] Arzoumanian, M. Marketing compression-based linerboard:What's really changed? // Official Board Markets, 77,16(2001), pp. 14-16.[2] Allaoui, S. Aboura, Z. Benzeggagh, M. L. Effects of theenvironmental conditions on the mechanical behaviour of thecorrugated cardboard. // Composites Science andTechnology, 69, (2009) 104–110.[3] Allaoui, S. Aboura, Z. Benzeggagh, M. L. Phenomenagoverning uni-axial tensile behaviour of paperboard andcorrugated cardboard. // Composite Structures, 87, (2009)80–92.[4] El Damatty, A. A.; Mikhail, A.; Awad, A. A. Finite elementmodeling and analysis of a cardboard shelter. // Thin-WalledStructures, 38, 2(2000), pp. 145-165.[5] Corrugated board packagingmore and more "valuable" (Opakowania z tektury falistejcoraz "cenniejsze"). // Przeglad Papierniczy, 63, 8(2007), pp.450-452.[6] Gazeta Ardito, E. D. F.; J. D. A.; Urbanik, T. Performance ofthe corrugated fibreboard in different conditions of relativemoisture. // O Papel (Brazil), 61, 7(2000), pp. 69-75.; ;, pp.; ;, ppFornalski, Z.; KoŁodziejski, R.[7] Haslach Jr., H. W. Moisture and rate-dependent mechanicalproperties of paper: a review. // Mechanics Time-DependentMaterials, 4, 3(2000), pp. 169-210.[8] Masood, S. H.; Haider Rizvi, S. An investigation of palletdesign using alternative materials for cold room applications.// International Journal of Advanced ManufacturingTechnology, 29, 1-2(2006), pp. 1-8[9] Isler, W. Latest developments in the paper industry //International Paperworld IPW (10-11), pp. 32-34, 2010[10] Gospodinov, D. Hadjiiski, V.; Stefanov, S. Modeling ofcorrugated paperboard by using the finite element method //University of Ruse "Angel Kanchev" Proceedings, Volume49, Book 9.2, (2010), pp.114-118...;.Authors' addressesAdrese autoraStefan Stefanov, Assoc. Prof.University of Food Technologies26 Mariza Boul., Plovdiv, BulgariaContact +359 32 603 814stvstefanov@yahoo.com

Use of the finite element method in studying the influence of different layers on mechanical characteristics of corrugated paperboard

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