Crushing of Single-Walled Corrugated Board During Converting: Experimental and Numerical Study

energies

Article Crushing of Single-Walled Corrugated Board during Converting: Experimental and Numerical Study

Tomasz Garbowski 1 , Tomasz Gajewski 2,* , Damian Mrówczy ´nski 3 and Radosław J˛edrzejczak 4

1 Department of Biosystems Engineering, Poznan University of Life Sciences, Wojska Polskiego 50, 60-627 Poznań,Poland; [email protected] 2 Institute of Structural Analysis, Poznan University of Technology, Piotrowo 5, 60-965 Poznań,Poland 3 R&D Department, FEMat Sp. z o.o., Romana Maya 1, 61-371 Poznań,Poland; [email protected] 4 Werner Kenkel Sp. z o.o., Mórkowska 3, 64-117 Krzycko Wielkie, Poland; [email protected] * Correspondence: [email protected]

Abstract: Corrugated cardboard is an ecological material, mainly because, in addition to virgin cellulose fibers also the fibers recovered during recycling process are used in its production. However, the use of recycled fibers causes slight deterioration of the mechanical properties of the corrugated board. In addition, converting processes such as printing, die-cutting, lamination, etc. cause micro- damage in the corrugated cardboard layers. In this work, the focus is precisely on the crushing of corrugated cardboard. A series of laboratory experiments were conducted, in which the different types of single-walled corrugated cardboards were pressed in a fully controlled manner to check the impact of the crush on the basic material parameters. The amount of crushing (with a precision of 10 micrometers) was controlled by a precise FEMat device, for crushing the corrugated board in the range from 10 to 70% of its original thickness. In this study, the influence of crushing on Citation: Garbowski, T.; Gajewski, T.; bending, twisting and shear stiffness as well as a residual thickness and edge crush resistance of Mrówczyński,D.; J˛edrzejczak,R. corrugated board was investigated. Then, a procedure based on a numerical homogenization, taking Crushing of Single-Walled Corrugated Board during Converting: into account a partial delamination in the corrugated layers to determine the degraded material Experimental and Numerical Study. stiffness was proposed. Finally, using the empirical-numerical method, a simplified calculation model Energies 2021, 14, 3203. https:// of corrugated cardboard was derived, which satisfactorily reflects the experimental results. doi.org/10.3390/en14113203 Keywords: corrugated cardboard; converting; numerical homogenization; strain energy equivalence; Academic Editor: Peter Foot finite element method; shell structures; transverse shear

Received: 6 May 2021 Accepted: 27 May 2021 Published: 30 May 2021 1. Introduction Paper and cardboard are made of cellulose fibers that mainly come from trees. Some Publisher’s Note: MDPI stays neutral of the fibers circulate repeatedly in the production-recycling loop. The material is, therefore, with regard to jurisdictional claims in environmentally friendly, but the quality of the produced material from recycled fibers published maps and institutional affil- iteratively declines. This requires a deeper understanding if one wants to optimize the iations. product and at the same time keep the material eco-friendly. It becomes even more important if the final product is a corrugated cardboard, which consists of two to seven alternating flat (liners) and corrugated (fluting) layers of paperboard. The particular orientation of the fibers resulting from the cardboard production process Copyright: © 2021 by the authors. causes the material to have different mechanical properties along the mutually perpendicu- Licensee MDPI, Basel, Switzerland. lar directions. Such materials are called orthotropic materials, as opposed to isotropic ones, This article is an open access article which exhibit the same physical properties independent of the direction. The material ori- distributed under the terms and entation along the fibers that follow the direction of the web during production is called the conditions of the Creative Commons machine direction (MD), the direction perpendicular to it is called the cross direction (CD). Attribution (CC BY) license (https:// As the material is much stiffer and stronger in MD (along fibers), the corrugated board creativecommons.org/licenses/by/ production method compensates for its poorer behavior in CD by using the corrugated 4.0/).

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layers. The main task of which is to increase the load capacity and stiffness in the CD, but most importantly, to keep the flat layers at an appropriate distance from each other, which allows obtaining significantly increased bending stiffness in both directions. The optimal corrugated board should be characterized by maximum strength and stiffness, and at the same time be light and cheap to produce. In assessing the quality of corrugated board, the compressive or tensile and flexural or torsional stiffnesses/strengths are important; the former due to the load-bearing capacity, while the latter due to the resistance to loss of stability. The maximum stiffness or strength can be obtained by selecting the appropriate materials for the individual layers and/or by selecting the appropriate geometry (wave height and period) for the corrugated layers. There are hundreds of papers on the market with different grammages and mechanical properties, produced in a different proportion of virgin to recycled fibers. This gives an enormous number of combinations of possible layer arrangements and layer geometry of the corrugated cardboard, which does not stop producers from constantly trying to find the best solution. Unfortunately, their efforts may be wasted if the carefully designed quality of the corrugated cardboard does not achieve the assumed strength parameters after production, i.e., converting processes. Before corrugated board becomes a typical transportation box, or a color, branded shelf-ready box (SRP) or a display-ready box it goes through a number of converting processes, e.g., printing, lamination, die-cutting, etc. All these processes cause crushing of the corrugated layers, which in turn leads to a reduction of strength and stiffness of the material and therefore affects the performance of structures made of corrugated boards. The deterioration of the mechanical properties can be observed in all crucial laboratory test results, e.g., four-point bending test, torsional test, shear test, edge crush test, etc. Even though, the degraded stiffnesses and strengths could easily be measured by cutting specimens from the converted cardboard, this is rarely done in practice. Typically, this effect is accounted for by adding safety factors to the equations that estimate the load capacity of the package. Since the middle of the last century, scientists and engineers have tried to find robust and simple tools for estimating the strength of corrugated board boxes. Among them the analytical tools [1–10] are the simplest, but unfortunately less precise and limited only to the typical box structures. Numerical models of corrugated board packaging [11–16], although much more precise, require specific knowledge and a full set of material parameters to correctly simulate the behavior of the box. In the finite element-based tools, a procedure called homogenization [17–25] is very often used, which allows for significant time savings in the analysis and at the same time guarantees the correct behavior of simplified models. This is especially important when the computational models are complex (they consist of many layers of cardboard) or the analysis concerns geometries with complex shapes, such as cardboard furniture [26]. In our previous works, we presented analytical-numerical methods [27–29] for assessing the strength of corrugated board boxes, which allow to obtain quick and accurate results for the packaging with different geometries (ventilation/hand holes [28] and perforations [29]). In order to properly estimate the strength of a corrugated box, special attention should be paid to transverse shear properties [30–32]. This is because the decrease in this material parameter significantly affects the load-bearing capacity of the package. In general, any deteriorated stiffness (i.e., flexural, torsional or transversal) and the strength of the corrugated board can be readily implemented in any estimation method. The only requirement is to determine the relationship between the amount of damage (stiffness and strength degradation) and the amount of physical crushing of the cardboard. The article presents the results of the research on the crushing of various single-walled corrugated boards. In the first step, the relationship between intentionally induced crushing with very high accuracy to different types of corrugated boards and the measured loss of stiffness in various laboratory tests was checked. Since the crushing of the sample takes place in the thickness direction, the greatest decrease in bending and twisting stiffness (due to the reduction of the corrugated board thickness and, therefore, a significant reduction of Energies 2021, 14, x FOR PEER REVIEW 3 of 17

crushing with very high accuracy to different types of corrugated boards and the meas‐ ured loss of stiffness in various laboratory tests was checked. Since the crushing of the Energies 2021, 14, 3203 sample takes place in the thickness direction, the greatest decrease in bending and twist3‐ of 16 ing stiffness (due to the reduction of the corrugated board thickness and, therefore, a sig‐ nificant reduction of the section moment of inertia) was expected. Less load capacity re‐ duction was expected in the edge crush test. However, due to the high resilience of the corrugatedthe section board, moment in some of inertia) cases, especially was expected. for a Less small load amount capacity of reductioncrushing, wasthe sample expected in recoversthe edge its original crush test. thickness However, (because due toof thethe highrelaxation), resilience which of the allow corrugated the observation board, in of some interestingcases,especially findings. All for aissues small will amount be discussed of crushing, in the the Results sample section. recovers its original thickness (because of the relaxation), which allow the observation of interesting findings. All issues In the second step, a numerical model based on the homogenization that can easily will be discussed in the Results section. be used to predict all the stiffness parameters of the corrugated board with different de‐ In the second step, a numerical model based on the homogenization that can easily be grees of crushing was proposed. The numerical model, based on the finite element method used to predict all the stiffness parameters of the corrugated board with different degrees (FEM), was used to build a global stiffness matrix of the full detailed 3D finite element of crushing was proposed. The numerical model, based on the finite element method (FE) model of the corrugated board. The model (based on the numerical homogenization (FEM), was used to build a global stiffness matrix of the full detailed 3D finite element procedure presented in [25]) takes into account the crushing of the corrugated board. This (FE) model of the corrugated board. The model (based on the numerical homogenization extension of the homogenization procedure allows to determine the degraded corrugated procedure presented in [25]) takes into account the crushing of the corrugated board. This board stiffness matrix, which ultimately enables a robust simulation of the real laboratory extension of the homogenization procedure allows to determine the degraded corrugated tests using the simplified analytical formulas. The results obtained from the simulation of board stiffness matrix, which ultimately enables a robust simulation of the real laboratory four‐point bending and torsion (twist) tests [31,32] are in good agreement with the results tests using the simplified analytical formulas. The results obtained from the simulation of obtainedfour-point from the bending laboratory and torsion tests. (twist) tests [31,32] are in good agreement with the results obtained from the laboratory tests. 2. Materials and Methods 2.1. Mechanical2. Materials Tests and of Methods Corrugated Cardboard 2.1.The Mechanical strength and Tests stiffness of Corrugated of a corrugated Cardboard board can be determined by performing various typesThe strengthof tests. andThe stiffnessmost commonly of a corrugated used include: board (a) can BNT—stiffness be determined in by the performing four‐ pointvarious bending types test; of(b) tests. ECT—edge The most crush commonly test, (c) SST—shear used include: stiffness (a) BNT—stiffness testing and (d) in TST— the four- torsionalpoint stiffness bending test. test; (b) ECT—edge crush test, (c) SST—shear stiffness testing and (d) TST— torsionalThe measurement stiffness test. of the bending stiffness is a laboratory test which is based on the four‐pointThe bending measurement method of(BNT), the bending see Figure stiffness 1. This is test a laboratory is usually testcarried which out is on based a sam on‐ the ple withfour-point a dimension bending of method 50 × 250 (BNT), mm. seeIt is Figure important1. This that test in is usuallythis test carriedthere is out a constant on a sample momentwith and a dimension zero shear of force 50 × 250in the mm. sample It is important between the that internal in this testsupports. there is However, a constant there moment is stilland a shear zero shear forceforce between in the the sample outer betweenand inner the supports—this internal supports. allows However, to take thereinto con is still‐ a siderationshear forcethe shear between stiffness theouter aspect. and It inneris worth supports—this noting that a allows bending totake stiffness into considerationmeasure‐ mentthe is very shear sensitive stiffness to aspect. sample It isdamage worth (crushing noting that or a creasing), bending stiffnessso the results measurement for samples is very crushedsensitive by more to sample than 50% damage are usually (crushing unreliable. or creasing), so the results for samples crushed by more than 50% are usually unreliable.

(a) (b) (c)

FigureFigure 1. The 1. fourThe‐point four-point bending bending test: (a)test: cardboard (a) cardboard testing device; testing (b device;) loaded (b sample;) loaded (c sample;) de‐ (c) de- formedformed sample. sample.

The Thesample sample compressive compressive strength strength in the in Edge the Edge Crush Crush Test Test(ECT) (ECT) is obtained is obtained for rela for‐ rela- tivelytively stocky stocky samples samples (conventionally (conventionally thicker thicker than than 1 mm) 1 mm) with with dimensions dimensions of 25 of × 25 100× mm,100 mm, see Figuresee Figure 2. In2 .the In thecase case of slender of slender specimens, specimens, the themain main failure failure mechanism mechanism is the is theloss loss of of stability,stability, and andnot the not crushing the crushing of a ofsample. a sample. The TheECT ECTof the of corrugated the corrugated cardboard cardboard is one is oneof of the mostthe most known known and andimportant important (from (from the thepracti practicalcal point point of ofview) view) parameter, parameter, often often used used in an analytical [1,10] or an analytical-numerical [27–29] determination of the load capacity of a corrugated board packaging. The shear stiffness (SST) of a corrugated board is measured on a sample 80 × 80 mm loaded with a pair of forces at opposite corners, see Figure3. Measuring the displacements and reaction forces at the other two corners allows to determine the cardboard shear stiffness. Only the linear part of the load-displacement curve is used in an iden- tiﬁcation of the sample shear stiffness. The SST parameter is very sensitive to a sample Energies 2021, 14, x FOR PEER REVIEW 4 of 17 Energies 2021, 14, x FOR PEER REVIEW 4 of 17

in an analytical [1,10] or an analytical‐numerical [27–29] determination of the load capac‐ in an analytical [1,10] or an analytical‐numerical [27–29] determination of the load capac‐ ity of a corrugated board packaging. ity of a corrugated board packaging.

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in an analytical [1,10] or an analytical‐numerical [27–29] determination of the load capac‐ crushing. The results obtained in the SST laboratory tests are valid also for highly crushed ity of a corrugated board packaging. or broken(a) samples. (b) (c) (a) (b) (c) Figure 2. The edge crush test: (a) corrugated board testing device; (b) loaded sample; (c) deformed Figure 2. The edge crush test: (a) corrugated board testing device; (b) loaded sample; (c) deformed sample. sample. The shear stiffness (SST) of a corrugated board is measured on a sample 80 × 80 mm The shear stiffness (SST) of a corrugated board is measured on a sample 80 × 80 mm loaded with a pair of forces at opposite corners, see Figure 3. Measuring the displacements loaded with a pair of forces at opposite corners, see Figure 3. Measuring the displacements and reaction forces at the other two corners allows to determine the cardboard shear stiff‐ and reaction forces at the other two corners allows to determine the cardboard shear stiff‐ ness. Only the linear part of the load‐displacement curve is used in an identification of the ness. Only the linear part of the load‐displacement curve is used in an identification of the sample shear stiffness. The SST parameter is very sensitive to a sample crushing. The re‐ sample shear stiffness. The SST parameter is very sensitive to a sample crushing. The re‐ sults obtained in the(a) SST laboratory tests are valid(b) also for highly crushed or broken(c) sam‐ sults obtained in the SST laboratory tests are valid also for highly crushed or broken sam‐ ples. Figureples.Figure 2. The 2. edgeThe crush edge crushtest: (a test:) corrugated (a) corrugated board testing board device; testing (b device;) loaded (b sample;) loaded (c) sample; deformed (c) de- sample.formed sample.

The shear stiffness (SST) of a corrugated board is measured on a sample 80 × 80 mm loaded with a pair of forces at opposite corners, see Figure 3. Measuring the displacements and reaction forces at the other two corners allows to determine the cardboard shear stiff‐ ness. Only the linear part of the load‐displacement curve is used in an identification of the sample shear stiffness. The SST parameter is very sensitive to a sample crushing. The re‐

sults obtained in the SST laboratory tests are valid also for highly crushed or broken sam ‐ (a) (b) (c) ples. (a) (b) (c) Figure 3. The shear stiffness testing: (a) cardboard testing device; (b) loaded sample; (c) deformed Figure 3. The shear stiffness testing: (a) cardboard testing device; (b) loaded sample; (c) deformed sample. Figure 3. The shear stiffness testing: (a) cardboard testing device; (b) loaded sample; (c) de- sample. formed sample. In the torsional stiffness (TST) measurement, a twisting of a 25 × 150 mm sample by In the torsional stiffness (TST) measurement, a twisting of a 25 × 150 mm sample by a few degreesIn in the both torsional directions stiffness is conducted, (TST) measurement, see Figure 4. aThe twisting results of obtained a 25 × 150are valid mm sample a few degrees in both directions is conducted, see Figure 4. The results obtained are valid even forby highly a few crushed degrees or in broken both directions samples. isTherefore, conducted, the seeTST Figure parameter4. The has results a high obtained sen‐ are even for highly crushed or broken samples. Therefore, the TST parameter has a high sen‐ sitivity validto crushing even for of highlythe corrugated crushed orboard broken sample. samples. Only Therefore, the linear thepart TST of a parameterdiagram (i.e., has a high sitivity to crushing of the corrugated board sample. Only the linear part of a diagram (i.e., the anglesensitivity of rotation to crushingvs bending of moment) the corrugated is used board to determine sample. Onlythe sample the linear torsional part of stiff a diagram‐ the angle of rotation vs bending moment) is used to determine the sample torsional stiff‐ ness. The(i.e., reliable the angle (measurementsa) of rotation are vs bendingassured by: moment)( b(1)) a stable is used method to determine of holding the( thec) sample sample, torsional ness.stiffness. The reliable The measurements reliable measurements are assured are by: assured (1) a stable by: (1) method a stable of methodholding the of holding sample, the (2) aFigure static 3.method The shear of measuringstiffness testing: the angle(a) cardboard of rotation testing and device; torque (b )and loaded (3) asample; relatively (c) deformed large (2) asample, static method (2) a static of measuring method of measuringthe angle of the rotation angle of and rotation torque and and torque (3) a relatively and (3) a relatively large widthsample. of the sample, thanks to which the sample behaves as a homogenized material. widthlarge of widththe sample, of the thanks sample, to thanks which to the which sample the behaves sample behavesas a homogenized as a homogenized material. material. In the torsional stiffness (TST) measurement, a twisting of a 25 × 150 mm sample by a few degrees in both directions is conducted, see Figure 4. The results obtained are valid even for highly crushed or broken samples. Therefore, the TST parameter has a high sen‐ sitivity to crushing of the corrugated board sample. Only the linear part of a diagram (i.e., the angle of rotation vs bending moment) is used to determine the sample torsional stiff‐ ness. The reliable measurements are assured by: (1) a stable method of holding the sample, (2) a static method of measuring the angle of rotation and torque and (3) a relatively large

width of the sample, thanks to which the sample behaves as a homogenized material. (a) (b) (c) (a) (b) (c) Figure 4. The torsional stiffness test: (a) cardboard testing device; (b) loaded sample; (c) deformed FigureFigure 4. The 4. torsionalThe torsional stiffness stiffness test: (a) test: cardboard (a) cardboard testing device; testing (b device;) loaded ( bsample;) loaded (c) sample; deformed (c ) de- sample. sample.formed sample.

The crushing device (CRS) is used to assess the impact of converting processes such as laminating, stamping, creasing or printing on the quality and load-bearing capacity of the corrugated board, see Figure5. In this research, a fully controlled manner of

crushing cardboard in the range from 10% to 70% was precisely obtained by using the CRS laboratory device (fematsystems.pl/services/crs [33]), which assured the crushing accuracy of(a±) 10 µm. (b) (c) Figure 4. InThe Figures torsional1– 5stiffness, the devices test: (a) forcardboard different testing testing device; methods (b) loaded of corrugated sample; (c) deformed cardboards sample.are presented. Furthermore, the loading scheme of the sample (Figure5b), as well as the

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The crushing device (CRS) is used to assess the impact of converting processes such Energies 2021, 14, 3203 as laminating, stamping, creasing or printing on the quality and load‐bearing capacity of5 of 16 the corrugated board, see Figure 5. In this research, a fully controlled manner of crushing cardboard in the range from 10% to 70% was precisely obtained by using the CRS labora‐ tory device (fematsystems.pl/services/crs [33]), which assured the crushing accuracy of deformed shape of the sample (Figure5c) with exemplary displacement ﬁelds (obtained 10 μm via ﬁnite. element method modelling and analytical approximation) are illustrated.

(a) (b) (c)

FigureFigure 5. The 5. cardboardThe cardboard crushing: crushing: (a) cardboard (a) cardboard testing testing device; device; (b) loaded (b) loaded sample sample scheme; scheme; (c) de‐ (c) deformed sample scheme. formed sample scheme.

2.2.In Figures Estimation 1–5, Error the devices by Coefficient for different of Determination testing methods of corrugated cardboards are presented.To Furthermore, explore the relationship the loading between scheme a of crushing the sample and a(Figure decrease 5b), of as the well corrugated as the de board‐ formedstiffness shape values, of the sample the coefficient (Figure of5c) determination with exemplary was displacement computed forfields each (obtained board quality,via finitedefined element by method the formula: modelling and analytical approximation) are illustrated. n 2 2 ∑i=1(xi − yî) 2.2. Estimation Error by Coefficient of DeterminationR = 1 − , (1) (n − 1)·var(x) To explore the relationship between a crushing and a decrease of the corrugated where: x —the expected ratio of the measured value of the crushed sample to the initial board stiffnessi values, the coefficient of determination was computed for each board qual‐ value (CRS = 0% → x = 1.0, CRS = 10% → x = 0.9, etc.), yˆ —the values computed on ity, defined by the formula:i i i the basis of the formula (2) describing the linear regression, var(x) — the variance of the expected ratio of the measured value∑( of𝑥 the −𝑦 crushed) sample to the initial value: 𝑅 =1− , (1) (𝑛−1) ∙var(𝑥) yî = a(xi − x) + y, (2) where: 𝑥—the expected ratio of the measured value of the crushed sample to the initial valuewhere: (CRSx =—the 0% mean→ 𝑥 = value 1.0, CRS of the = expected 10% → 𝑥 ratio =0.9, of theetc.), measured 𝑦 —the value values of the computed crushed sampleon the basisto the of initial the formula value, y —the(2) describing mean value the oflinear the measuredregression, quantities var(𝑥) – (SST-MD,the variance TST-CD of the etc.), expecteda—the ratio slope of ofthe the measured linear regression: value of the crushed sample to the initial value:

( n ) 𝑦 =𝑎 𝑥 ∑−𝑥̅ (+𝑦x ,− x)(y − y) (2) a = i=1 i i . (3) 𝑥̅ n 2 where: —the mean value of the expected∑ ratioi=1(x ofi − thex) measured value of the crushed sample to the initial value, 𝑦—the mean value of the measured quantities (SST‐MD, TST‐ CD etc.), The𝑎—the higher slope the of value the linear of the regression: coefficient of determination, R2, the better the fit of the regression line and the estimation error is considered to be smaller. ∑(𝑥 −𝑥̅)(𝑦 −𝑦) 𝑎= . (3) 2.3. Numerical Approach for Modelling∑( Crushing𝑥 −𝑥̅) The Inhigher this paper,the value apart of fromthe coefficient laboratory of tests determination, on crushed 𝑅 corrugated, the better cardboards, the fit of the also a regressionsimple line numerical and the approach estimation to er considerror is considered the crushed to propertiesbe smaller. of the corrugated board is proposed. The derived method does not require the modelling of the plasticization of the 2.3. fluting,Numerical because Approach its analytical for Modelling equivalent Crushing (presented later) in FE model can be used. The aim ofIn this this part paper, of the apart study from is tolaboratory validate thetest approachs on crushed by using corrugated torsion cardboards, test modelling also [31 a, 32]. The numerical study consists of several steps, illustrated by scheme in Figure6: simple numerical approach to consider the crushed properties of the corrugated board is proposed.• Building The derived initial method geometry does of thenot intactrequire corrugated the modelling cardboard of the (stageplasticization a). of the fluting,• becausePerforming its analytical FE analysis equivalent of corrugated (presented cardboard later) in crushing FE model with can plasticity be used. included The aim of this(stage part b)—substituted of the study is withto validate an analytical, the approach simplified by using crushed torsion flute test shape modelling approxima- [31,32]. Thetion. numerical study consists of several steps, illustrated by scheme in Figure 6: • Using the crushed geometry to build the classical material stiffness matrix of corru- • Building initial geometry of the intact corrugated cardboard (stage a). gated cardboard (stage c,d). • Homogenizing the crushed corrugated cardboard by composite properties, based on Garbowski and Gajewski method [25] (stage d,e). • Computing torsion/bending response of crushed corrugated cardboard sample using composite properties (stage f). Energies 2021, 14, x FOR PEER REVIEW 6 of 17

• Performing FE analysis of corrugated cardboard crushing with plasticity included (stage b)—substituted with an analytical, simplified crushed flute shape approxima‐ tion. • Using the crushed geometry to build the classical material stiffness matrix of corru‐ gated cardboard (stage c,d). • Homogenizing the crushed corrugated cardboard by composite properties, based on Energies 2021, 14, 3203 Garbowski and Gajewski method [25] (stage d,e). 6 of 16 • Computing torsion/bending response of crushed corrugated cardboard sample using composite properties (stage f).

FigureFigure 6.6. TheThe schemescheme representingrepresenting the steps of numericalnumerical study conducted in in this this study: study: ( (aa)) unde unde-‐ formedformed RVE, RVE, ( b(b)) loaded loaded and and deformed deformed RVE, RVE,(c) ( crushedc) crushed geometry geometry extracted, extracted, (d) material (d) material stiffness stiffness matrix ofmatrix RVE, of (e) RVE, the representative (e) the representative shell stiffness shell matrix stiffness and matrix (f) tests and outcomes (f) tests from outcomes analytical from estimation. analytical estimation. Initial geometry of the corrugated cardboard used in the numerical study represents an intactInitial (i.e., unconverted geometry of or the uncrushed) corrugated geometry cardboard of theused cardboard in the numerical and it was study assumed represents from thean intact literature (i.e., [unconverted19,25]. The fraction or uncrushed) of a single geometry wall corrugated of the cardboard cardboard and wasit was simulated, assumed namely,from the the literature in-plane [19,25]. section The of 8fraction× 8 mm of. Thea single fluting wall period corrugated was also cardboard8 mm; the was fluting simu‐ wavelated, “starts”namely, fromthe in the‐plane liner. section The thicknessof 8×8 mm of the. The liners fluting and period fluting was are also0.29 8mm mm;and the 0.30fluting mm, wave respectively. “starts” from The axial the liner. spacing The between thickness the of liners the liners is 3.51 and mm. fluting are 0.29 mm and The0.30 materialmm, respectively. parameters The of axial intact spacing corrugated between cardboard the liners were is 3.51 also mm taken. from the literatureThe material [19,25]. Theparameters classical of orthotropic intact corrug constitutiveated cardboard law was were assumed also taken for from each the layer lit‐ witherature a perfect [19,25]. plasticity The classical (no orthotropic hardening). consti The orthotropictutive law was material assumed data for (E each1, E2 layer, v12, withG12, G and G , i.e., Young moduli in both directions, Poisson’s ratio and 3 shear moduli, a 13perfect plasticity23 (no hardening). The orthotropic material data (𝐸, 𝐸, 𝑣, 𝐺, 𝐺 respectively) and yield stress, σ , for liners and fluting are presented in Table1. and 𝐺, i.e., Young moduli in both0 directions, Poisson’s ratio and 3 shear moduli, respec‐ tively) and yield stress, 𝜎, for liners and fluting are presented in Table 1. Table 1. Material properties of liners and fluting of intact corrugated cardboard. Table 1. Material properties of liners and fluting of intact corrugated cardboard. E E ν G G G σ Layers 1 2 12 12 13 23 0 𝑬 𝑬 𝝂 𝑮 𝑮 𝑮 𝝈 Layers (MPa)𝟏 (MPa)𝟐 (-)𝟏𝟐 (MPa)𝟏𝟐 (MPa)𝟏𝟑 (MPa)𝟐𝟑 (MPa)𝟎 (MPa) (MPa) (‐) (MPa) (MPa) (MPa) (MPa) liners 3326 1694 0.34 859 429.5 429.5 2.5 flutingliners 26143326 15321694 0.320.34 724859 362429.5 362429.5 2.52.5 fluting 2614 1532 0.32 724 362 362 2.5 To acquire the crushed geometry of the corrugated cardboard the static FE analysis To acquire the crushed geometry of the corrugated cardboard the static FE analysis was performed. In the numerical study, five cases were considered, see Figure7, in which was performed. In the numerical study, five cases were considered, see Figure 7, in which the induced crushing of the cardboard were 10%, 20%, 30%, 40% and 50%. For instance, the induced crushing of the cardboard were 10%, 20%, 30%, 40% and 50%. For in‐ 10% of crushing means here that the corrugated cardboard was enforced by kinematic 10% constraintstance, to decrease of crushing its thicknessmeans here to that be 9 0%the ofcorrugated the intact cardboard geometry was (see enforced Figure6a,b). by kin In ‐ 0% theematic numerical constraint model to decrease (to the upper its thickness and lower to be liner 9 surfaces) of the intact the kinematic geometry constraints(see Figure were applied assuming that 50% of the crushed deformation is elastic and the other 50% comes from the plastic and/or damage deformations. Therefore, in the numerical analysis, to obtain the geometry from plastic deformation only (i.e., after unloading), the actual kinematic constraints were 5%, 10%, 15%, 20% and 25%. The output geometries of the FE analysis using those constraints were later considered to be the ones coming from the crushing of 10%, 20%, 30%, 40% and 50%. For the FE analysis the Abaqus Unified FEA from Dassault Systems was used, in which 4-node general-purpose shell finite elements were utilized (S4 according to [34]). Single model had about 3280 shell elements with linear shape functions and about 3160 nodes. The fluting was represented by 64 segments, since this value is important to retrieve Energies 2021, 14, x FOR PEER REVIEW 7 of 17

6a,b). In the numerical model (to the upper and lower liner surfaces) the kinematic con‐ straints were applied assuming that 50% of the crushed deformation is elastic and the other 50% comes from the plastic and/or damage deformations. Therefore, in the numer‐ ical analysis, to obtain the geometry from plastic deformation only (i.e., after unloading), the actual kinematic constraints were 5%, 10%, 15%, 20% and 25%. The output geome‐ tries of the FE analysis using those constraints were later considered to be the ones coming from the crushing of 10%, 20%, 30%, 40% and 50%. For the FE analysis the Abaqus Unified FEA from Dassault Systems was used, in which 4‐node general‐purpose shell finite elements were utilized (S4 according to [34]). Energies 2021, 14, 3203 7 of 16 Single model had about 3280 shell elements with linear shape functions and about 3160 nodes. The fluting was represented by 64 segments, since this value is important to re‐ trieve the correct transvers shear stiffnesses of a representative volume element (RVE) as shownthe correct by our transvers recent shearwork stiffnesses[25]. Here, ofthe a representativenumber of segments volume was element doubled (RVE) due as to shown mod‐ ellingby our contact recent between work [25 the]. Here, top liner the numbervs. flutin ofg and segments fluting was vs. doubledbottom liner. due toIn modellingtangential direction,contact between the frictionless the top liner contact vs. ﬂuting was assumed; and ﬂuting and vs. in bottom normal liner. direction In tangential the Herz direction, type contactthe frictionless was assumed. contact Boundary was assumed; conditions and in al normallowed directionto deform the the Herz RVE type in out contact of plane was direction,assumed. blocking Boundary from conditions movement allowed the external to deform (side) thenodes. RVE in out of plane direction, blocking from movement the external (side) nodes.

(b) (c) (a)

(d) (e) (f)

Figure 7. TheThe crushed crushed geometries geometries of of the the corrugated corrugated cardboard cardboard ob obtainedtained from from the the static static finite ﬁnite element element analysis: analysis: (a) 0%, (a) 0%,(b) (10%b) 10%,, (c) (20%c) 20%,, (d) ( d30%) 30%,, (e ()e )40% 40% and ( f) 50%.50%.

BasedBased on on FEM FEM computations computations performed performed to to ob obtaintain different different crushing crushing levels, levels, the the alter alter-‐ nativenative approach maymay bebe usedused toto determine determine the the crushing crushing shapes shapes of of fluting fluting to to reconstruct reconstruct its itscrushed crushed geometry geometry (this (this is validis valid for for different different flute flute amplitudes amplitudes and and periods). periods). The The analytical analyt‐ icalformula formula is proposed is proposed here, here, which which accounts accounts for the for vertical the vertical coordinates coordinates of a half-wave of a half fluting:‐wave fluting: 1 1 f (x) = ts 1 1 − (4) 𝑓(𝑥) =𝑡𝑠 1 + e−2wx−/L 2 (4) 1+𝑒/ 2 inin which which t𝑡 is is the the amplitude amplitude of of intact intact fluting, fluting,L 𝐿is isthe the period period length length of of intact intact fluting, fluting,x is 𝑥 the is thehorizontal horizontal coordinate, coordinate,w is the𝑤 is parameter the parameter related related to inclination to inclination and curviness and curviness of the flutingof the flutingvertical vertical wall, while wall,s whileis the parameter𝑠 is the parameter scaling the scaling crushing the crushing thickness; thickness;w and s should 𝑤 and be 𝑠 shouldused to be fit used the fluting to fit the shape fluting to particular shape to levelparticular of crushing. level of The crushing. parameters The parameters of w and s for of 𝑤cases and used 𝑠 for in cases thisstudy used arein this summarized study are insummari Table2.zed The in fluting Table shapes 2. The offluting half-waves shapes for of different crushing levels obtained from the formula proposed are presented in Figure8a . It should be noted that the fluting length in the analytical approach was preserved by reproducing the geometry from FE analyses. The example of comparison between the fluting shape computed with the FE model and the analytical formula for 40% of crushing is presented in Figure8b; a perfect agreement can be observed. In the next stage of the study, the output geometries (without any residual stresses) were imported to Abaqus software to build the initial material stiffness matrix of the structure, see Figure6c–d. Before this, for each case, namely, crushing of 10%, 20%, 30%, 40% and 50%, the geometries were inspected in order to determine which regions of the fluting were actually plasticized. For those finite element models (along CD), all elastic properties (apart Poisson’s ratio) were deteriorated by scaling factor. Two regions of fluting were distinguished for each case; thus, two scaling factors were considered, see Figure9 . The first region is the contacting area of liners and fluting (region A) and the second Energies 2021, 14, x FOR PEER REVIEW 8 of 17

half‐waves for different crushing levels obtained from the formula proposed are pre‐ sented in Figure 8a. It should be noted that the fluting length in the analytical approach was preserved by reproducing the geometry from FE analyses. The example of compari‐ son between the fluting shape computed with the FE model and the analytical formula for 40% of crushing is presented in Figure 8b; a perfect agreement can be observed.

Table 2. Geometrical parameters of 𝑤 and 𝑠 for fluting to determine analytically the crushed geometries of corrugated cardboard used in the numerical part of this study.

Crushing (%) 𝒘 (‐) 𝒔 (‐) 0 (intact) 3.5 1.06 10 4.0 0.98 20 4.8 0.92 Energies 2021, 14, 3203 8 of 16 30 6.2 0.86 40 9.0 0.80 50 15.5 0.75 region is in span, which clearly would evolve into the plastic joint (region B) for larger crushingIn the loads. next stage The elements, of the study, in which the output the material geometries was identiﬁed(without any to beresidual plasticized, stresses) i.e., wereregions imported A and to B, Abaqus were obtained software from to build FE computations,the initial material see stiffness Figure9 .matrix Since of 50% the ofstruc the‐ ture,deformation see Figure was 6c–d. assumed Before to bethis, elastic, for each new geometriescase, namely, for crushing further computations of 10%, 20% with, 30% 50%, 40%less crushingand 50% were, the geometries generated by weref function, inspected see in Equationorder to determine (4), but in which A and regions B regions of the the flutingmaterial were properties actually were plasticized. deteriorated. For those finite element models (along CD), all elastic properties (apart Poisson’s ratio) were deteriorated by scaling factor. Two regions of flut‐ ingTable were 2. Geometrical distinguished parameters for each of case;w and thus,s for tw ﬂutingo scaling to determine factors analyticallywere considered, the crushed see Figure geome- 9.tries The of first corrugated region cardboard is the contacting used in the area numerical of liners part and of this fluting study. (region A) and the second region is Crushingin span, (%)which clearly would evolvew (-) into the plastic joint (regions (-) B) for larger crushing loads. The elements, in which the material was identified to be plasticized, i.e., 0 (intact) 3.5 1.06 regions A and B, were obtained from FE computations, see Figure 9. Since 50% of the 10 4.0 0.98 deformation was20 assumed to be elastic, new 4.8 geometries for further computations 0.92 with 50% less crushing30 were generated by 𝑓 function, 6.2 see Equation (4), but in 0.86A and B regions the material properties40 were deteriorated. 9.0 0.80 50 15.5 0.75

Energies 2021, 14, x FOR PEER REVIEW 9 of 17

The local material deterioration was defined independently for two regions: A and B. Acquired material stiffness matrix of RVE with embedded orthotropic and locally de‐ teriorated properties (due to decreasing of elastic properties for two regions by scaling factor) were subjected to Garbowski and Gajewski homogenization method [25], see Fig‐ ure 6d,e. The method based on the stiffness matrix of the RVE enables computing 𝐀 ma‐ trix, which is the overall stiffness matrix for the laminate shell element. The great ad‐ vantage of the method is that it captures also the effective transvers shear stiffnesses in

CD and MD of the input RVE structure, i.e., 𝐴 and 𝐴, respectively. (a) To summarize, the overall algorithm presented in (thisb) subsection enables to compute Figure 8. Half‐waves of flutingthe representative due to crushing transvers obtained shear stiffnesse from thes for analytical corrugated formula cardboard (4) samples and pa with‐ dif‐ Figure 8. Half-waves offerent ﬂuting intensity due toof its crushing crushing obtainedincluded. Results from the section analytical will prove formula that the (4) torsion and test rameters presented in Table 2: (a) shapes of the flute corresponding to different levels of crushing parameters presented in Tablemay be2:( useda) shapesto effectively of the vali ﬂutedate corresponding this algorithm to to determine different the levels local of deterioration crushing of and (b) the comparison for 40% of crushing: FEM (magenta circles) vs analytical formula (solid and (b) the comparison fora corrugated 40% of crushing: board sample FEM with (magenta particular circles) intensity vs analytical of the crushing. formula (solid line). line).

FigureFigure 9. 9.A A and and B regions regions for forall geometries all geometries of corrugated of corrugated cardboard considered, cardboard i.e., considered, with crushing i.e., of (a with) 10% crushing, (b) 20%, of (a(c)) 10%,30%, ( (db) )40% 20%, and (c )(e 30%,) 50%. ( d) 40% and (e) 50%.

The local material3. deterioration Results was deﬁned independently for two regions: A and B. Acquired material3.1. stiffness Experimental matrix Study of RVE with embedded orthotropic and locally In the experimental part of our study, four corrugated boards of different grammage were selected and tested, namely, two B flutes: 285 g/m, (B‐285), 410 g/m (B‐410) and two C flutes: 340 g/m (C‐340), 440 g/m (C‐440). Those corrugated boards were subjected to a series of measurements and laboratory tests to check: (a) sample thickness before and after crushing—THK and THK2; (b) sample resistance to edge crushing–ECT; (c) bending stiffness in machine direction (BNT–MD) and in cross‐direction (BNT–CD); (d) shear stiff‐ ness in machine direction (SST–MD) and in cross‐direction (SST–CD); and (e) torsion stiff‐ ness in machine direction (TST–MD) and in cross‐direction (TST–CD). In most cases, each type of corrugated board was subjected to three to four series of tests at the same crushing level. Each corrugated board was crushed in the range of 10% to 70% of its original thick‐ ness with 10% increments. Due to the elastic relaxation of the corrugated cardboard, the measurement of the crushed sample thickness was performed a few minutes after crush‐ ing, which guaranteed initial stabilization of the relaxation of the material. In Figures 10–13, the results of the measured parameters for the four analysed corru‐ gated boards are presented. Based on the data, the regression lines for each test, according to Equation (2) and (3), were determined. The BNT, SST and TST values were presented in a normalized manner, the values demonstrated are the ratio of the value obtained for

Energies 2021, 14, 3203 9 of 16

deteriorated properties (due to decreasing of elastic properties for two regions by scaling factor) were subjected to Garbowski and Gajewski homogenization method [25], see Figure6d,e . The method based on the stiffness matrix of the RVE enables computing Ak matrix, which is the overall stiffness matrix for the laminate shell element. The great advantage of the method is that it captures also the effective transvers shear stiffnesses in CD and MD of the input RVE structure, i.e., A44 and A55, respectively. To summarize, the overall algorithm presented in this subsection enables to compute the representative transvers shear stiffnesses for corrugated cardboard samples with different intensity of its crushing included. Results section will prove that the torsion test may be used to effectively validate this algorithm to determine the local deterioration of a corrugated board sample with particular intensity of the crushing.

3. Results 3.1. Experimental Study In the experimental part of our study, four corrugated boards of different grammage were selected and tested, namely, two B ﬂutes: 285 g/m2, (B-285), 410 g/m2 (B-410) and two C ﬂutes: 340 g/m2 (C-340), 440 g/m2 (C-440). Those corrugated boards were subjected to a series of measurements and laboratory tests to check: (a) sample thickness before and after crushing—THK and THK2; (b) sample resistance to edge crushing–ECT; (c) bending stiffness in machine direction (BNT–MD) and in cross-direction (BNT–CD); (d) shear stiffness in machine direction (SST–MD) and in cross-direction (SST–CD); and (e) torsion stiffness in machine direction (TST–MD) and in cross-direction (TST–CD). In most cases, each type of corrugated board was subjected to three to four series of tests at the same crushing level. Each corrugated board was crushed in the range of 10% to 70% of its original thickness with 10% increments. Due to the elastic relaxation of the corrugated cardboard, the measurement of the crushed sample thickness was performed a few minutes after crushing, which guaranteed initial stabilization of the relaxation of the material. Energies 2021, 14, x FOR PEER REVIEW 10 of 17 In Figures 10–13, the results of the measured parameters for the four analysed corrugated boards are presented. Based on the data, the regression lines for each test, according to Equation (2) and (3), were determined. The BNT, SST and TST values were presented in a normalizedcrushed sample manner, to the the value values obtained demonstrated from an areintact the corrugated ratio of the cardboard value obtained (i.e., for for CRS crushed = sample0%). to the value obtained from an intact corrugated cardboard (i.e., for CRS = 0%).

(a) (b) (c)

(d) (e)

FigureFigure 10. The10. The decreases decreases in in measured measured values values forfor B-285B‐285 corrugated corrugated cardboard cardboard in: in:(a) (BNT;a) BNT; (b) SST; (b) SST; (c) TST; (c) TST;(d) ECT (d) and ECT and (e) THK2. (e) THK2.

(a) (b) (c)

(d) (e) Figure 11. The decreases in measured values for B‐410 corrugated cardboard in: (a) BNT; (b) SST; (c) TST; (d) ECT and (e) THK2.

Energies 2021, 14, x FOR PEER REVIEW 10 of 17

crushed sample to the value obtained from an intact corrugated cardboard (i.e., for CRS = 0%).

(a) (b) (c)

(d) (e) Energies 2021, 14, 3203 10 of 16 Figure 10. The decreases in measured values for B‐285 corrugated cardboard in: (a) BNT; (b) SST; (c) TST; (d) ECT and (e) THK2.

(a) (b) (c)

Energies 2021, 14, x FOR PEER REVIEW 11 of 17 (d) (e)

FigureFigure 11.11. The decreases in measured values values for for B B-410‐410 corrugated corrugated cardboard cardboard in: in: (a ()a BNT;) BNT; (b ()b SST;) SST; (c) ( cTST;) TST; (d) ( dECT) ECT and and (e) ( THK2.e) THK2.

(a) (b) (c)

(d) (e)

FigureFigure 12.12. The decreases in in measured measured values values for for C C-340‐340 corrugated corrugated cardboard cardboard in: in: (a ()a BNT;) BNT; (b ()b SST;) SST; (c) ( cTST;) TST; (d ()d ECT) ECT and (ande) THK2. (e) THK2. The relationship between crushing and decrease of parameter values was investigated by computing the coefﬁcient of determination for each quantity, according to Equation (1). In Table3, the values obtained are presented. Determining the crushing level from the stiffness tests is more precise when the average values from the MD and CD are used to compute the linear regression. This fact may be observed in Figure 14, in which the normalized parameter values were averaged from two directions and the crushing lines are shown. In Table4, the corresponding coefﬁcients of determination for the averaged values from two directions are presented.

(a) (b) (c)

(d) (e) Figure 13. The decreases in measured values for C‐440 corrugated cardboard in: (a) BNT; (b) SST; (c) TST; (d) ECT and (e) THK2.

The relationship between crushing and decrease of parameter values was investi‐ gated by computing the coefficient of determination for each quantity, according to Equa‐ tion (1). In Table 3, the values obtained are presented.

Table 3. The coefficient of determination 𝑅 between crushing and normalized values.

Cardboard THK2 ECT BNT‐MD BNT‐CD SST‐MD SST‐CD TST‐MD TST‐CD Index B‐285 0.154 0.481 0.928 0.705 0.995 0.964 0.914 0.932

Energies 2021, 14, x FOR PEER REVIEW 11 of 17

(a) (b) (c)

(d) (e) Energies 2021, 14, 3203 11 of 16 Figure 12. The decreases in measured values for C‐340 corrugated cardboard in: (a) BNT; (b) SST; (c) TST; (d) ECT and (e) THK2.

(a) (b) (c)

(d) (e)

FigureFigure 13.13. The decreases in in measured measured values values for for C C-440‐440 corrugated corrugated cardboard cardboard in: in: (a) ( aBNT;) BNT; (b ()b SST;) SST; (c) ( cTST;) TST; (d) ( dECT) ECT and and (e) (THK2.e) THK2.

Table 3. The coefﬁcientThe relationship of determination betweenR2 between crushing crushing and de andcrease normalized of parameter values. values was investi‐ gated by computing the coefficient of determination for each quantity, according to Equa‐ Cardboard Index THK2tion (1). ECT In Table BNT-MD3, the valuesBNT-CD obtained are SST-MD presented. SST-CD TST-MD TST-CD B-285 0.154 0.481 0.928 0.705 0.995 0.964 0.914 0.932 Energies 2021B-410, 14, x FOR PEERTable 0.001REVIEW 3. The coefficient 0.596 of determination 0.986 𝑅 0.687 between crushing 1.000 and normalized 0.992 values. 0.535 0.98812 of 17 C-340 0.111 0.016 0.517 0.861 0.719 1.000 0.542 0.993 CardboardC-440 0.000 0.452 0.967 0.841 0.855 0.999 0.290 0.968 THK2 ECT BNT‐MD BNT‐CD SST‐MD SST‐CD TST‐MD TST‐CD Index B‐285 0.154 0.481 0.928 0.705 0.995 0.964 0.914 0.932 B‐410 0.001 0.596 0.986 0.687 1.000 0.992 0.535 0.988 C‐340 0.111 0.016 0.517 0.861 0.719 1.000 0.542 0.993 C‐440 0.000 0.452 0.967 0.841 0.855 0.999 0.290 0.968

Determining the crushing level from the stiffness tests is more precise when the av‐ erage values from the MD and CD are used to compute the linear regression. This fact may be observed in Figure 14, in which the normalized parameter values were averaged from two directions and the crushing lines are shown. In Table 4, the corresponding coef‐ (a) (b) ficients of determination for the averaged values from two directions are presented.

FigureFigure 14. 14. TheThe decreases decreases in in measured measured values values (average (average from from two two directions) directions) for for corrugated corrugated board board withwith an an index: index: ( (aa)) B B-285;‐285; ( (bb)) B B-410;‐410; ( (cc)) C C-340‐340 and and ( (dd)) C C-440.‐440.

Table 4. The coefficient of determination 𝑅 between crushing and average normalized values.

Cardboard Index BNT SST TST B‐285 0.823 0.983 0.996 B‐410 0.885 0.998 0.846 C‐340 0.815 0.970 0.876 C‐440 0.972 0.957 0.742

We found that there is a relationship between the decreased ECT and THK2 value. A reference line, which capture the relationship between the decrease of these normalized parameters and crushing of the corrugated cardboard was established. The equation de‐ scribing the reference line has a following form: 𝑦 = 1 − 0.53𝑥 (5) where: 𝑦—the normalized parameter value and 𝑥—the crushing value. In Figure 15, the normalized ECT and THK2 values for the analysed corrugated card‐ boards fitting to the reference lines represented by the Equation (5) are presented.

(a) (b)

Energies 2021, 14, x FOR PEER REVIEW 12 of 17

(a) (b)

(c) (d) Energies 2021, 14, 3203 12 of 16 Figure 14. The decreases in measured values (average from two directions) for corrugated board with an index: (a) B‐285; (b) B‐410; (c) C‐340 and (d) C‐440.

TableTable 4. 4. TheThe coefficient coefﬁcient of of determination determination 𝑅R2 between between crushingcrushing andand averageaverage normalizednormalized values.values.

CardboardCardboard Index Index BNT BNT SST SST TST TST BB-285‐285 0.823 0.823 0.983 0.983 0.996 0.996 BB-410‐410 0.885 0.885 0.998 0.998 0.846 0.846 CC-340‐340 0.815 0.815 0.970 0.970 0.876 0.876 CC-440‐440 0.972 0.972 0.957 0.957 0.742 0.742

WeWe found found that that there there is is a a relationship relationship betw betweeneen the the decreased decreased ECT ECT and and THK2 THK2 value. value. A A referencereference line, line, which which capture capture the the relationship relationship between between the the decrease decrease of of these these normalized normalized parametersparameters and and crushing crushing of of the the corrugated corrugated cardboard cardboard was was established. established. The Theequation equation de‐ scribingdescribing the thereference reference line line has has a following a following form: form:

𝑦 =y 1= − 10.53𝑥− 0.53 x (5) (5) where: 𝑦—the normalized parameter value and 𝑥—the crushing value. where: y—the normalized parameter value and x—the crushing value. In Figure 15, the normalized ECT and THK2 values for the analysed corrugated card‐ In Figure 15, the normalized ECT and THK2 values for the analysed corrugated boards fitting to the reference lines represented by the Equation (5) are presented. cardboards ﬁtting to the reference lines represented by the Equation (5) are presented.

Energies 2021, 14, x FOR PEER REVIEW 13 of 17

(a) (b)

Figure 15. The decreases in normalized ECT and and THK2 THK2 va valueslues for for corrugated corrugated board board with with an an index: index: (a) B B-285;‐285; (b) B-410;B‐410; (c) C-340C‐340 and (d) C-440.C‐440.

For the the reference reference line line and and normalized normalized ECT ECT and and THK2 THK2 values, values, the thecoefficients coefﬁcients of de of‐ terminationdetermination 𝑅R were2 were computed computed (Table (Table 5),5 ),according according to to Equation Equation (1) (1) for for the the two two sets sets of dataof data simultaneously, simultaneously, where: where: 𝑥x—referencei—reference line line value value calculated calculated from from the the Equation Equation (5), 𝑦yˆi—normalized ECT and THK2 values and varvar(𝑥)(x)—variance—variance ofof thethe referencereference lineline value.value.

2 Table 5. TheThe coefficient coefﬁcient of ofdetermination determination 𝑅 R betweenbetween reference reference line line andand normalized normalized ECT ECTand and THK2 values.

CardboardCardboard IndexIndex 𝑹R2𝟐 BB-285‐285 0.985 BB-410‐410 0.982 CC-340‐340 0.965 C-440 0.995 C‐440 0.995

3.2. Numerical Appraoch for Modelling Crushing

The shell stiffnesses (𝐀 matrix) were computed for five crushing levels of corru‐ gated cardboard considered, i.e., with different amount of crushing: 10%, 20%, 30%, 40% and 50%. The local deterioration factor for region A was assumed 0.5 and for region B was assumed 0.9 (due to severe delamination in these regions as shown by [35,36]). Se‐ lected values of 𝐀 matrices for different level of crushing were presented in Table 6. Having computed 𝐀 stiffnesses which represents the overall material properties of RVE, the values were used to determine the behavior of the corrugated cardboard samples from different tests. Torsion and bending stiffness tests in both directions were considered here, see its analytical formulas derived in [31,32]. The dimension of the torsion sample was 80 × 80 mm, in bending the sample had 100 mm length between the internal sup‐ ports. The decreases of stiffness of the corrugated sample in the cases of torsion, bending in CD and bending in MD for assumed scaling factors (see non deteriorated values in Table 1) are presented in Table 7. It may be observed that the values are close to the in‐ duced crushing values (first column), especially for torsion test (second column), what proves that the method was validated. This method may be used in other applications for modelling crushed corrugated cardboard samples for deriving its material and mechani‐ cal properties.

Table 6. The stiffnesses of the representative shell element computed for different crushing of corrugated cardboard sam‐ ple for local deterioration scaling factors, separately for region A and region B.

Stiffness 10% Crushing 20% Crushing 30% Crushing 40% Crushing 50% Crushing

𝐴,(kPa⋅m) 2101.4 2092.8 2088 2085.3 2078.7 𝐴,(kPa⋅m) 1591.5 1568.1 1548.5 1531.6 1477.4 𝐴,(kPa⋅m) 371.3 367.8 365.5 363.8 361.5 𝐴,(kPa⋅m) 603.0 586.3 573.4 562.3 543.9

Energies 2021, 14, 3203 13 of 16

3.2. Numerical Appraoch for Modelling Crushing

The shell stiffnesses (Ak matrix) were computed for ﬁve crushing levels of corrugated cardboard considered, i.e., with different amount of crushing: 10%, 20%, 30%, 40% and 50%. The local deterioration factor for region A was assumed 0.5 and for region B was assumed 0.9 (due to severe delamination in these regions as shown by [35,36]). Selected values of Ak matrices for different level of crushing were presented in Table6.

Table 6. The stiffnesses of the representative shell element computed for different crushing of corrugated cardboard sample for local deterioration scaling factors, separately for region A and region B.

Stiffness 10% Crushing 20% Crushing 30% Crushing 40% Crushing 50% Crushing

A11, (kPa · m) 2101.4 2092.8 2088 2085.3 2078.7 A22, (kPa · m) 1591.5 1568.1 1548.5 1531.6 1477.4 A12, (kPa · m) 371.3 367.8 365.5 363.8 361.5 A33, (kPa · m) 603.0 586.3 573.4 562.3 543.9 3 D11, Pa · m 5.79 5.19 4.64 4.11 3.61 3 D22, Pa · m 3.6 3.23 2.89 2.57 2.21 3 D12, Pa · m 1.01 0.90 0.81 0.71 0.63 3 D33, Pa · m 1.50 1.34 1.19 1.06 0.92 A44,(Pa · m) 45.18 31.12 22.01 15.41 10.14 A55, (Pa · m) 82.13 72.55 65.58 60.11 51.45

Having computed Ak stiffnesses which represents the overall material properties of RVE, the values were used to determine the behavior of the corrugated cardboard samples from different tests. Torsion and bending stiffness tests in both directions were considered here, see its analytical formulas derived in [31,32]. The dimension of the torsion sample was 80 × 80 mm, in bending the sample had 100 mm length between the internal supports. The decreases of stiffness of the corrugated sample in the cases of torsion, bending in CD and bending in MD for assumed scaling factors (see non deteriorated values in Table1 ) are presented in Table7. It may be observed that the values are close to the induced crushing values (ﬁrst column), especially for torsion test (second column), what proves that the method was validated. This method may be used in other applications for modelling crushed corrugated cardboard samples for deriving its material and mechanical properties.

Table 7. The decreases in stiffness of torsion, bending in CD and bending in MD for scaling factors separately identiﬁed for region A and region B – obtained for different input geometry due to induced crushing (see Figure9).

Induced Torsion Stiffness CD Bending Stiffness MD Bending Stiffness Crushing (%) Decrease (%) Decrease (%) Decrease (%) 10 12 10 12 20 21 19 21 30 30 28 29 40 38 36 37 50 46 44 46

4. Discussion In the first part of our work, we investigated the relationship between the intentional level of flat corrugated cardboard crushing and the drops in the measured parameters of various laboratory tests. The results in Tables4 and5 show the coefficients of determination for linear trends (y = x) across all tests. Values closer to 1 indicate the better correlation between the experiment result and the regression lines. It has been observed that the decrease in both bending and torsional stiffness is more or less correlated with the amount of crush. Energies 2021, 14, 3203 14 of 16

The results obtained show that the SST parameters best catch the amount of intentional crushing, the TST parameters are slightly worse and the BNT parameters are the least accurate. Mean coefficients of determination for all four corrugated board types were: SST— 0.977; TST—0.874, BNT—0.865. The correlation of crush amount with the results of the SST tests at the level of 97.7% means that if there is a need to check a posteriori the amount of the unknown damage caused by the converting machines, it is sufficient to measure the SST parameter before and after the converting process and its ratio will indicate the crushing level. This is because the amount of decrease of the SST parameter precisely reflects the degree of corrugated board crushing. The decrease in the parameter measured in SST does not depend on the residual thickness of the crushed corrugated cardboard, which often returns to its original value due to relaxation. Therefore, organoleptic inspection (e.g., measuring thickness) often does not reveal the problem hidden inside the delaminated cardboard fibers. Therefore, the new insight from this part of study is the following: one may use shearing test (SST) for determining the crushing level of the single wall cardboard, other tests such as bending or torsion are less indicative. This conclusion opens new possibilities to design similar test/machines in other types of materials in which crushing is observed and unwanted, for instance in civil engineering or automotive industry. On the other hand, the drop in ECT parameter and the residual thickness, THK2 correlate well with each other. The measured thickness after crushing decreases almost by the same value as the measured ECT value (again after crushing). The amount of deterioration of measured ECT and the permanent thickness reduction is approximately 50% of the actual crush amount of the corrugated board. Thus, thickness/ECT reduction may be used to roughly estimate the amount of crushing in a single wall corrugated boards but is not recommended for cases in which high accuracy is expected. In the numerical part of the study, the results show that the numerical procedure presented can reproduce the deterioration of the samples stiffnesses due to crushing by decreasing the elastic parameters of the RVE, see Table6. The regions for which the deterioration must be included were identified as the contacting area of liners and fluting (region A) and in the span area (region B), see Figure9, also the deterioration factors were acquired, i.e., 0.5 for region A and 0.9 for region B. This property seems to be significantly important if one would like to compute the deteriorated properties of the cardboard, basing only on its crushed shape and knowing intact properties of the corrugated cardboard (Ak matrix). As shown, by introducing the analytical formulation, Equation (4), with adequate parameters, see Table2, to determine the crushed shape one does not have to perform costly FE analysis [37,38], nor have to use advanced (plastic) material constitutive models, see Figure6. The analytical formulation may be therefore adopted in optimization or inverse problem frameworks, in which computational cost should be limited. It should be underlined that the numerical study presented enables obtaining the transversal shear stiffness properties of the sample by using the torsion test for different levels of crushing, see Table6. This is extremely important since the crushing of the cardboard is often suspected to cause the biggest amount of unintended decrease of the load capacity of corrugated cardboard boxes. However, up to this point, there are no systematic and implemented in the industry methods that would be able to determine how the crushing influences the deterioration of the transversal shear stiffnesses. Thus, if this phenomenon may be modelled easily, by the procedure proposed (without using FE computations, nor advanced material models, but adopting an analytical/algebraic rapid and versatile approach), we are one step closer to deliver the scientific-based methodology for dealing with the crushing issue for the corrugated board packaging industry.

5. Conclusions The article presents extended laboratory tests of single-walled corrugated cardboard, consisting in checking the impact of crushing on its mechanical properties. The intentional crushing of the cross-section from 10 to 70% of the original height was fully controlled and initiated with high precision. During the tests, a number of parameters of the corrugated Energies 2021, 14, 3203 15 of 16

board were measured, i.e., (a) residual thickness after elastic relaxation, (b) decrease in bending stiffness, (c) decrease in torsional stiffness, (d) decrease in shear stiffness and (e) decrease in edge crush strength. The reduction in stiffness was checked in two mutually perpendicular directions (i.e., in machine direction and cross direction), while the strength reduction was checked in cross direction only. A correlation was found between intentional and controlled crushing and the corresponding reduced stiffness. It was observed that the best match between the amount of crush and the reduction in stiffness for all specimens tested occurred in the shear stiffness test (SST). The paper also presents numerical and analytical tools for quick and reliable calculations of degraded stiffness of crushed corrugated cardboard samples. For selected crushing levels of the corrugated cardboards the stiffnesses of the representative volume elements were computed. In further research, the impact of the crushing of corrugated board for double-walled structures on the packaging performance will be studied.

Author Contributions: Conceptualization, T.G. (Tomasz Garbowski) and T.G. (Tomasz Gajewski); methodology, T.G. (Tomasz Garbowski); software, T.G. (Tomasz Gajewski) and D.M.; validation, T.G. (Tomasz Garbowski), T.G. (Tomasz Gajewski) and D.M.; formal analysis, T.G. (Tomasz Garbowski), T.G. (Tomasz Gajewski) and D.M.; investigation, T.G. (Tomasz Garbowski), T.G. (Tomasz Gajewski), D.M. and R.J.; writing—original draft preparation, T.G. (Tomasz Garbowski), T.G. (Tomasz Gajewski) and D.M.; writing—review and editing, T.G. (Tomasz Garbowski), T.G. (Tomasz Gajewski) and R.J.; visualization, T.G. (Tomasz Gajewski) and D.M.; supervision, T.G. (Tomasz Garbowski); project administration, T.G. (Tomasz Garbowski); funding acquisition, R.J. All authors have read and agreed to the published version of the manuscript. Funding: The APC was funded by the National Center for Research and Development, Poland, grant at Werner Kenkel Sp z o. o., grant number POIR.01.01.01–00-1306/15–00. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available on request from the corresponding author. Acknowledgments: The authors thank AQUILA VPK Wrzesnia for providing samples of corrugated cardboard for the study. The authors also thank to Femat Sp. z o. o. for providing the laboratory equipment and commercial software. Conﬂicts of Interest: The authors declare no conﬂict of interest. The funders had no role in the design of the study; in the collection, analyses or interpretation of data; in the writing of the manuscript or in the decision to publish the results.

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