This study reveals the crease deviation behavior through the developed forming simulation. A combination resistance model was expanded and applied to simulate the 180° folding process of a creased paperboard, using the shear-yield detaching resistance and the out-of-plane fluffing resistance which were based on the isotropic elastro-plastic model. When varying the misalignment of the creasing rule against the groove, the eccentricity of the crease bulging of a white-coated paperboard was compared through the experiment and simulation of the 180° folding process. Comparing the experimental deformation and the simulation, it was explained that the deviation of e contributed to making the crease deviation cd. At the folding test, the 180° folding was compared with the experiment and simulation. The rolling pass of the folded zone was considered to intensify the deviation state. The 180° folding simulation revealed that the crease deviation of cd ≈ 2e was assessed as an ideal condition when using the rolling pass and non-rolling pass. In the case of some shallow indentation in the experiment, 2e < cd < 4e was observed. The inside folded corners were quite different between the simulation and experiment, especially for a certain shallow indentation model. In the simulation, the local crushing was not performed under the assumption of any isotropic properties. In the simulation, the deviation of the creased position at the 180° folding was sufficiently predictable, when compared with experimental behavior.
Citation: Weerayut Jina, Shigeru Nagasawa, Tetsuya Yamamoto, Takaomi Nagumo. Analysis of the folding behavior of a paperboard subjected to indentation of a deviated creasing rule using the finite element method[J]. AIMS Materials Science, 2023, 10(2): 313-341. doi: 10.3934/matersci.2023017
[1] | Raimund Bürger, Paola Goatin, Daniel Inzunza, Luis Miguel Villada . A non-local pedestrian flow model accounting for anisotropic interactions and domain boundaries. Mathematical Biosciences and Engineering, 2020, 17(5): 5883-5906. doi: 10.3934/mbe.2020314 |
[2] | Wenshun Sheng, Jiahui Shen, Qiming Huang, Zhixuan Liu, Zihao Ding . Multi-objective pedestrian tracking method based on YOLOv8 and improved DeepSORT. Mathematical Biosciences and Engineering, 2024, 21(2): 1791-1805. doi: 10.3934/mbe.2024077 |
[3] | Enrico Bertino, Régis Duvigneau, Paola Goatin . Uncertainty quantification in a macroscopic traffic flow model calibrated on GPS data. Mathematical Biosciences and Engineering, 2020, 17(2): 1511-1533. doi: 10.3934/mbe.2020078 |
[4] | Wen Li, Xuekun Yang, Guowu Yuan, Dan Xu . ABCNet: A comprehensive highway visibility prediction model based on attention, Bi-LSTM and CNN. Mathematical Biosciences and Engineering, 2024, 21(3): 4397-4420. doi: 10.3934/mbe.2024194 |
[5] | Alessandro Corbetta, Adrian Muntean, Kiamars Vafayi . Parameter estimation of social forces in pedestrian dynamics models via a probabilistic method. Mathematical Biosciences and Engineering, 2015, 12(2): 337-356. doi: 10.3934/mbe.2015.12.337 |
[6] | Veronika Schleper . A hybrid model for traffic flow and crowd dynamics with random individual properties. Mathematical Biosciences and Engineering, 2015, 12(2): 393-413. doi: 10.3934/mbe.2015.12.393 |
[7] | Fengjie Liu, Monan Wang, Yuzheng Ma . Multiscale modeling of skeletal muscle to explore its passive mechanical properties and experiments verification. Mathematical Biosciences and Engineering, 2022, 19(2): 1251-1279. doi: 10.3934/mbe.2022058 |
[8] | Paola Goatin, Matthias Mimault . A mixed system modeling two-directional pedestrian flows. Mathematical Biosciences and Engineering, 2015, 12(2): 375-392. doi: 10.3934/mbe.2015.12.375 |
[9] | H. T. Banks, R. A. Everett, Neha Murad, R. D. White, J. E. Banks, Bodil N. Cass, Jay A. Rosenheim . Optimal design for dynamical modeling of pest populations. Mathematical Biosciences and Engineering, 2018, 15(4): 993-1010. doi: 10.3934/mbe.2018044 |
[10] | Simone Göttlich, Stephan Knapp . Modeling random traffic accidents by conservation laws. Mathematical Biosciences and Engineering, 2020, 17(2): 1677-1701. doi: 10.3934/mbe.2020088 |
This study reveals the crease deviation behavior through the developed forming simulation. A combination resistance model was expanded and applied to simulate the 180° folding process of a creased paperboard, using the shear-yield detaching resistance and the out-of-plane fluffing resistance which were based on the isotropic elastro-plastic model. When varying the misalignment of the creasing rule against the groove, the eccentricity of the crease bulging of a white-coated paperboard was compared through the experiment and simulation of the 180° folding process. Comparing the experimental deformation and the simulation, it was explained that the deviation of e contributed to making the crease deviation cd. At the folding test, the 180° folding was compared with the experiment and simulation. The rolling pass of the folded zone was considered to intensify the deviation state. The 180° folding simulation revealed that the crease deviation of cd ≈ 2e was assessed as an ideal condition when using the rolling pass and non-rolling pass. In the case of some shallow indentation in the experiment, 2e < cd < 4e was observed. The inside folded corners were quite different between the simulation and experiment, especially for a certain shallow indentation model. In the simulation, the local crushing was not performed under the assumption of any isotropic properties. In the simulation, the deviation of the creased position at the 180° folding was sufficiently predictable, when compared with experimental behavior.
Since 2008, more than half of the world population is living in cities [40], which asks for the development of large-scale multimodal transportation systems. At the core of such systems, crowds of pedestrians are to be found in transportation facilities. The massive presence of this population in airports, train stations, metro stations, etc. is a challenge in terms of safety and transportation efficiency [48,56,49]. This paper deals with the experimental analysis and modeling of crowd movements so as, at longer term, to develop operational tools to manage pedestrians in places as various as airports, train or metro stations or streets.
Managing the comfort and safety for pedestrians is difficult first because they are not operated or regulated as simply as vehicles can be through sets of rules and traffic signals [44]. Hence, it is difficult to prevent the formation of dense traffic areas, where compacted crowds are exposed to risks of stampedes [53,16]. For this reason, it is important to design pedestrian traffic management systems similar to those existing for car traffic, and more particularly simulation tools to predict the short-term evolution of the traffic from conditions measure in-situ and in real time.
Today, on the one hand, pedestrian crowd simulators are developed for the purpose of validating the layout and the structure of buildings aimed at hosting a large public. This validation is performed beforehand, at the stage of design [52]. Most are not adapted to an on-line usage. Nevertheless, on the other hand, recent progress in pedestrian tracking technologies, both for detailed trajectories [7,30,46] or global traffic conditions [2,59,51], make possible to initialize a pedestrian traffic simulators with current conditions and to perform short-term traffic predictions. This would open the possibility to anticipate risks of congestion and to react accordingly. Such simulators are lacking today, thus, it is important to focus our attention on simulators which are able to continuously process a flow of input traffic data and which are efficient enough to perform short term predictions on-line. Our work aims at filling this particular gap, by developing online simulation tools to assist the management of pedestrian traffic.
We aim at performing short term predictions in places presenting risks of large attendance and risk of congestion. A specific type of environment retains our attention: corridors. By corridors, we mean elongated areas such as metro corridors, sidewalks or shopping arcade where pedestrians form bidirectional traffic flows. This type of place is particularly interesting to study because corridors generally link large places with high attendance (e.g., a corridor between two platforms). Because the traffic intensity is higher in corridors, they are the place where congestion can more easily initiate. In addition, because corridors are often organized in networks, they offer an opportunity to apply rerouting strategies to locally limit the risk of congestion: pedestrian would be suggested or forced to avoid corridors presenting a risk of future congestion, similarly to car drivers with modern GPS-based assistance systems.
How to perform accurate and fast predictive simulation for bidirectional pedestrian traffic in corridors? Two categories of models are described in the literature: microscopic and macroscopic approaches. The former approach is an ascending one: the motion of individual pedestrians is independently simulated based on multi-agent technique and crowd traffic conditions emerge from combination of interactions between agents. They are fundamentally based on the local models of interactions, which were developed in various disciplines such as physics [19], computer graphics [47] or behavioral and cognitive sciences [43,41,5], and which dictate how agents influence each other's motion. The latter type of approach are macroscopic ones. They model a crowd as a whole, a matter moving like a compressible fluid. Relying on the conservation law, a macroscopic simulation computes the crowd motion by estimating the temporal variations of local density [50]. Algorithmic complexity is one fundamental difference between these two types of simulators. Microscopic simulators consider interactions between all possible pairs of agent and are quadratic by nature. In contrast, macroscopic algorithms are linearly complex with the size of the crowd. This difference makes macroscopic approaches a model of choice. While microscopic models are unadapted to produce real time simulations as soon as the pedestrian number becomes large, macroscopic models can provide fast, simple and yet surprisingly accurate results even at large densities.
In this paper, we propose a macroscopic model to predict bi-directional traffic in corridors. To enhance computational efficiency, we consider only the longitudinal direction of the walkway, ignoring the lateral dimension. We show hereafter that despite this simplification a macroscopic model is able to quantitatively reproduce key features of crowd dynamics. In such category of model, the relevant information encoding crowd dynamics is the fundamental diagram, which defines pedestrian flux as a function of pedestrian densities. More precisely, the proposed method relies on a "Bi-directional Fundamental Diagram" (BFD), which captures situations of a crowd made of people moving in the same and opposite directions. Fundamental diagrams are widely used for one-way traffic (since the pioneering work of Lighthill and Whitham [38]) but two-way pedestrian traffic has been scarcely investigated under the angle of the BFD [1,31,33,3,54,8,15]. Yet, two-way traffic is the most common situation in everyday life and counter-flows played a decisive role in several crowd disasters [17,18].
To characterize the BFD, several experiments have been conducted from which we establish the first result of the present paper: an estimation of the BFD from experimental data. Although Bi-directional flows have already been studied in the literature, it is the first time, up to our knowledge, that it is measured experimentally. The data are acquired in real time tracking experiment using automatic motion capture techniques [42]. Our second result is the validation of the Bi-directional Macroscopic (BM) model which uses the calibrated BFD as its core. We introduce as quantifier the cluster velocity allowing to compare quantitatively experiments and simulations. We demonstrate that the BM model captures essential features of crowd dynamics. We then test our model in a typical forecasting scenario where the crowd is recorded at two distant points. We show that the model accurately predicts the state of the corridor between the recording points and develops traveling waves similar to those observed in the experiments. Our third result is the determination of the optimum of a corridor segregation strategy consisting in separating -or not- antagonist pedestrian fluxes according to the corridor occupancy. The BM model suggests that the segregation is efficient when the two antagonist fluxes are roughly balanced, but is counter-productive in strongly unbalanced cases.
In this section, we describe our experiment on pedestrian traffic. Our objective was to build a Bi-directional Fundamental Diagram (BFD), i.e., to observe the relation between the moving speed of participants under different conditions of density and balance of counter-flows. To this end, we asked participants to walk in a circular corridor delimited by walls (as displayed in Figure 1a). Some participants were instructed to walk in the clockwise direction, some others to walk in the anticlockwise direction. They were assigned a walking direction before each trial. For each trial, participants were initially still in the corridor and their positions randomly distributed. Each replication lasted 60 seconds after starting signal was given.
Participants were not allowed to communicate, and were asked to behave as if they were walking alone in a street to reach a destination. 119 volunteers participated the experiments (we performed two experimental sessions, one with 59 participants and one with 60 participants). They were adults recruited through advertising and with no known pathology which would affect their locomotion. The experiment conformed to the declaration of Helsinki. Participants were naive with respect to the purpose of our experiments.
Experiments took place in 2009 in Rennes, France. The circular corridor internal and external radius was respectively 2 and 4.5 meters. We studied 3 different proportions of fluxes, i.e., the balance of participants walking in a direction versus participants walking in the opposite direction. We studied:
We collected kinematics data. To this end, participants wore white T-shirts and 4 reflexive markers (see Fig. 1b), one on the forehead, one on the left shoulder, and two on the right shoulder. This made the distinction of the left and right shoulders easier. The motion of the markers has been tracked by 8 infra red cameras placed all around the experimental setting. Marker trajectories have been reconstructed using Vicon IQ software (VICON MX-40, Oxford Metrics, UK). The center of mass of the 4 markers projected onto the horizontal plane is computed and recorded as the position of the subjects [42,34].
The collected data give access to the two-dimensional Cartesian coordinates
S˜ρnk=∑iℓnk(i)Δθ( ˜ρnk in m−2 ), | (1) |
where the sum extends over pedestrians
S=Δθ⋅∫r=Routr=Rinrdr=Δθ⋅8.125. | (2) |
By linear interpolation, this procedure gives rise to a continuous piecewise linear reconstructed density
To estimate the flux, we compute a finite-difference approximation of the azimuthal component of the velocity
vni=rni˙θni≈Rmedθn+1i−θniΔt. | (3) |
The densities
S˜fn+,k=∑iℓnk(i)Δθvni( ˜fn+,kin(ms)−1 ), | (4) |
where again the sum extends over pedestrians
To investigate the flow of the pedestrians and make prediction, we consider a bidirectional extension of the traffic model [38,3]. Let
∂tρ++∂xf(ρ+,ρ−)=0, | (5) |
∂tρ−−∂xf(ρ−,ρ+)=0, | (6) |
where
u+=f+ρ+,u−=f−ρ−, | (7) |
λ+=∂f∂ρ+|(ρ+,ρ−),λ−=−∂f∂ρ+|(ρ−,ρ+). | (8) |
Strictly speaking, the quantities
From these data, we estimate the relation between the local densities
Sample set | a | b | c | |
0.944 | ||||
0.972 | ||||
0.982 |
In Fig. 3a, we plot the BFD for the experimental data corresponding to balanced fluxes (
f(ρ+,ρ−)=aρ+(1−bρ+−cρ−), | (9) |
with densities expressed in (pedestrians) m
In order to test the model, we envision a crowd forecasting system where sensors are placed at two distant locations
To compare quantitatively the model with experimental data, we estimate the average and cluster velocities (resp.
We now exploit the BM model to estimate the efficiency of a flow segregation strategy in which the two pedestrian types circulate preferably on one side of the corridor, with half the width of the corridor devoted to each type. In practice, this strategy can be achieved by suitable signaling. On the one hand segregation reduces the influence of counter-moving pedestrians, but on the other hand the available corridor width is smaller and the density of co-moving pedestrians increases.
To measure the efficiency of the strategy, we test the model in the two situations. Let
In this work, we have set the objective to build simulation tools to assist the management of pedestrian traffic. Our attention was focused on the case of bidirectional flows because, on the one hand, it captures many common situations to be found in corridors or sidewalks, and on the other hand, it received relatively poor attention in the literature. One of our work hypothesis is that two main factors influence conditions in bidirectional traffic: density, and proportion of each directional flow.
In the first part of this work, we empirically analyze bidirectional traffic. Our main result is captured by Figure 3: we show an evidence of the influence of the density of each directional flow on the flux. To perform this empirical observation, we chose laboratory conditions. Choosing laboratory conditions allowed us controlling the goals and motivations of walkers, and limiting the effect of the other many factors which influence pedestrian behaviors, such as physiological (e.g., aging) or social ones (e.g., walking together in groups). We clearly isolate the role of the density factors we considered on the resulting traffic conditions. This however limited the amount of observation data due to the required effort to gather them (subjects recruitment, data reconstruction and processing efforts, etc.). Nevertheless, the presented method applies to any new data. As, for instance, the fundamental diagram may depend on the nature of the subjects that are using the walkway [22] (children, adults, aging people, etc.), it would be interesting to update the BFD from dataset with controlled population.
In the second part of this work, we introduce a first order model to estimate the bidirectional fundamental diagram. In spite of the relative simplicity of this model, we demonstrate the ability to adhere to observation data, as illustrated by our results reported in Table 1. We observe that the estimated average velocity
As a result, the friction coefficients
In the last part of this work, we have chosen to evaluate our model based on an analysis of cluster dynamics. The emergence of clusters as a consequence of density fluctuations is a classical example of spatio-temporal patterns in traffic flow phenomena [20,27], also known as Kinematic Waves [38,35]. Comparing the large-scale dynamical features of the clusters in the simulation and the data provides a practical way to assess the validity of the BM model. For a given type of pedestrians (clockwise or anti-clockwise), clusters are defined as regions of space where the density of this type of pedestrian exceeds a certain threshold value (see appendix B). Fig. 6 gives the estimated average pedestrian velocity
We have presented a study on bidirectional flows of pedestrian. We have first presented an experimental measurement of the bidirectional fundamental diagram, with changing conditions of flow proportions. We have then introduced a bidirectional traffic model, directly based on this diagram (a.k.a. "first order models") for efficiency reasons. Finally, we demonstrate a relevant application of our model in the frame of automated traffic management systems, allowing users to estimate the benefit to flow segregation strategy to improve traffic.
The power of our approach is to capture complex phenomenon with a simple framework involving only three parameters. Moreover, one can easily extend the model to encompass more advanced features. As a first research direction, we would like to use real-time estimation of crowd parameters as it has been developed in several studies [39,60,58,26,21,57,13] to update the BFD diagram. The data assimilation strategy outlined above will be able to adapt the BM model quickly to changes in the nature and composition of the crowd.
Another possible extension would be to use models involving a differential relation between the flux and the local densities [4,45] (a.k.a. "second order models"). Those models could generate metastable equilibria and phase transitions, which play an important role in traffic [29,27]. Still, our first exploration of the data does not indicate that metastability plays an important role in pedestrian traffic.
To extend our work in two-dimensions (i.e. take into account the radial component), one has to consider the direction of the flow of pedestrians. Several macroscopic models have been proposed [24,23,6] with the introduction of a minimization principle to select the best route. Since most of bidirectional pedestrian flow have been studied at the microscopic level with Cellular Automata models [10,25,55,14], it will be compelling to compare the two approaches.
Finally, it would be interesting to test the framework in a more complex environment such as a network of corridors. Several strategies could be tested to decongest traffic such as indicating which corridor to use depending on its occupancy. Hence, the efficiency of a network could be measured allowing to improve its design for safer evacuation.
In a one-way traffic, there is only one type of density
u=f(ρ)ρ,λ=f′(ρ). |
In a bi-directional traffic, there are two values for the cluster velocities, denoted by
A(ρ+,ρ−)=(c++c+−−c−+−c−−), |
where
c++=∂ρ+(f(ρ+,ρ−)),c+−=∂ρ−(f(ρ+,ρ−)),c−+=∂ρ+(f(ρ−,ρ+)),c−−=∂ρ−(f(ρ−,ρ+)). |
Cluster velocities
λ±=12(c++−c−−±√Δ),with Δ=(c+++c−−)2−4c+−c−+. |
The eigenvalues
ε=max{|c+−|,|c−+|}max{|c++|,|c−−|}≪1. | (10) |
In general, it is not possible to associate one of these eigenvalues to either one or the other pedestrian species
α+=Eρ1+−c+−ρ1−√E2−c+−c−+,α−=Eρ1−−c−+ρ1+√E2−c+−c−+,with E=12[√Δ−(c+++c−−)]. |
This means that the densities
α+≈ρ1+,α−≈ρ1−. |
In this approximation, the densities
λ+=c++,λ−=−c−−, | (11) |
which gives a situation similar to car traffic.
Cluster analysis provides a lens to analyze emergence of macroscopic structures in pedestrians dynamics. From a macroscopic viewpoint, a cluster at a given time
ρ(X(t),t)=h. | (12) |
To determine a cluster, we first need to construct the cluster boundaries, i.e. the curves
To construct such curves
The
This procedure can be repeated for different levels
From the level curves
To estimate the efficiency of the segregation strategy, we compare the total flux of pedestrians when the two species of pedestrians
dNdt|NS=(f(ρ+,ρ−)+f(ρ−,ρ+))L |
where
If the species
dNdt|S=(f(2ρ+,0)+f(2ρ−,0))L. |
Then, the relative gain of the segregation strategy
G=(dN/dt)S−(dN/dt)NS(dN/dt)NS=(dN/dt)S(dN/dt)NS−1. |
This work has been supported by the French 'Agence Nationale pour la Recherche (ANR)' in the frame of the contracts 'Pedigree' (ANR-08-SYSC-015-01) and 'CBDif-Fr' (ANR-08-BLAN-0333-01). PD acknowledges support by the Engineering and Physical Sciences Research Council (EPSRC) under grant no. EP/M006883/1, by the Royal Society and the Wolfson Foundation through a Royal Society Wolfson Research Merit Award no. WM130048 and by the National Science Foundation (NSF) under grant no. RNMS11-07444 (KI-Net). PD is on leave from CNRS, Institut de Mathématiques de Toulouse, France. SM was supported in part by NSF grants #1107444 (KI-Net) and #1515592.
Data supporting this work are available on the website figshare at the following address:
● Data-set: https://figshare.com/s/a5aa17b81e7167a710bf
● Readme file: https://figshare.com/s/63edf4bd35458d30e31b
[1] |
Nygårds M, Bhattacharya A, Krishnan S (2014) Optimizing shear strength profiles in paperboard for better crease formation. Nord Pulp Pap Res J 29: 510–520. https://doi.org/10.3183/npprj-2014-29-03-p510-520 doi: 10.3183/npprj-2014-29-03-p510-520
![]() |
[2] |
Marin G, Nygårds M, Östlund S (2022) Experimental quantification of differences in damage due to in‐plane tensile test and bending of paperboard. Packag Technol Sci 35: 69–80. https://doi.org/10.1002/pts.2608 doi: 10.1002/pts.2608
![]() |
[3] | Csavajda P, Böröcz P, Mojzes Á, et al. (2017) The effect of creasing lines on the compression strength of adjustable height corrugated boxes. J Appl Packag Res 9: 15–22. |
[4] |
Hämäläinen P, Hallbäck N, Gåård A, et al. (2017) On the determination of transverse shear properties of paper using the short span compression test. Mech Mater 107: 22–30. https://doi.org/10.1016/j.mechmat.2017.01.012 doi: 10.1016/j.mechmat.2017.01.012
![]() |
[5] |
Robertsson K, Borgqvist E, Wallin M, et al. (2018) Efficient and accurate simulation of the packaging forming process. Packag Technol Sci 31: 1–10. https://doi.org/10.1002/pts.2383 doi: 10.1002/pts.2340
![]() |
[6] |
Viguié J, Dumont PJJ (2013) Analytical post-buckling model of corrugated board panels using digital image correlation measurements. Compos Struct 101: 243–254. https://doi.org/10.1016/j.compstruct.2013.01.023 doi: 10.1016/j.compstruct.2013.01.023
![]() |
[7] |
Luong VD, Abbès B, Abbès F, et al. (2019) Experimental characterisation and finite element modelling of paperboard for the design of paperboard packaging. IOP Conf Ser-Mater Sci Eng 540: 1–6. https://doi.org/10.1088/1757-899X/540/1/012014 doi: 10.1088/1757-899X/540/1/012014
![]() |
[8] |
Ristinmaa M, Ottosen NS, Korin C (2012) Analytical prediction of package collapse loads-basic considerations. Nord Pulp Pap Res J 27: 806–813. https://doi.org/10.3183/npprj-2012-27-04-p806-813 doi: 10.3183/npprj-2012-27-04-p806-813
![]() |
[9] |
Tanninen P, Leminen V, Eskelinen H, et al. (2015) Controlling the folding of the blank in paperboard tray press forming. BioResources 10: 5191–5202. https://doi.org/10.15376/biores.10.3.5191-5202 doi: 10.15376/biores.10.3.5191-5202
![]() |
[10] |
Awais M, Sorvari J, Tanninen P, et al. (2017) Finite element analysis of the press forming process. Int J Mech Sci 131–132:767–775. https://doi.org/10.1016/j.ijmecsci.2017.07.053 doi: 10.1016/j.ijmecsci.2017.07.053
![]() |
[11] |
Bonnet N, Viguié J, Beneventi D, et al. (2023) Reinforcing folding board boxes by printing a PLA patterned grid on their panels: A new approach for lightweighting stiff packaging. Packag Technol Sci 36: 211–218. https://doi.org/10.1002/pts.2705 doi: 10.1002/pts.2705
![]() |
[12] |
Leminen V, Tanninen P, Mäkelä P, et al. (2013) Combined effect of paperboard thickness and mould clearance in the press forming process. BioResources 8: 5701–5714. https://doi.org/10.15376/biores.8.4.5701-5714 doi: 10.15376/biores.8.4.5701-5714
![]() |
[13] |
Lindberg G, Kulachenko A (2022) Tray forming operation of paperboard: A case study using implicit finite element analysis. Packag Technol Sci 35: 183–198. https://doi.org/10.1002/pts.2619 doi: 10.1002/pts.2619
![]() |
[14] |
Gu Z, Hui J, Wang J, et al. (2021) Modelling multiple impacts on the out‐of‐plane cushioning properties of honeycomb paperboard. Packag Technol Sci 34: 541–556. https://doi.org/10.1002/pts.2593 doi: 10.1002/pts.2593
![]() |
[15] | Nagasawa S, Fukuzawa Y, Yamaguchi D, et al. (2001) Deformation characteristics on creasing of paperboard under shallow indentation. The 10th International Congress of Fracture (ICF10), Honolulu, USA. |
[16] |
Nagasawa S, Endo R, Fukuzawa Y, et al. (2008) Creasing characteristic of aluminum foil coated paperboard. J Mater Process Technol 201: 401–407. https://doi.org/10.1016/j.jmatprotec.2007.11.253 doi: 10.1016/j.jmatprotec.2007.11.253
![]() |
[17] | Stenberg N, Fellers C, Östlund S (2001) Measuring the stress-strain properties of paperboard in the thickness direction. J Pulp Pap Sci 27: 213–221. |
[18] |
Nagasawa S, Fukuzawa Y, Yamaguchi T, et al. (2003) Effect of crease depth and crease deviation on folding deformation characteristics of coated paperboard. J Mater Process Technol 140: 157–162. https://doi.org/10.1016/S0924-0136(03)00825-2 doi: 10.1016/S0924-0136(03)00825-2
![]() |
[19] | Nagasawa S, Fukushima Y, Sudo A, et al. (2004) Effect of tip shape of creasing rule on creasing deformation characteristics of paperboard. J Japan Soc Technol Plast 45: 178–182 (in Japanese). |
[20] | Nagasawa S, Murayama M, Sudo A, et al. (2004b) Effect of creaser deviation on folding deformation characteristics of creased paperboard. J Japan Soc Technol Plast 45: 239–243 (in Japanese). |
[21] |
Nagasawa S, Nasruddin M, Shiga Y (2011) Bending moment characteristics on repeated folding motion of coated paperboard scored by round-edge knife. J Adv Mech Des Syst Manuf 5: 385–394. https://doi.org/10.1299/jamdsm.5.385 doi: 10.1299/jamdsm.5.385
![]() |
[22] |
Li Y, Edward S, Reese S, et al. (2016) Anisotropic elastic-plastic deformation of paper : In-plane model. Int J Solids Struct 100–101:286–296. https://doi.org/10.1016/j.ijsolstr.2016.08.024 doi: 10.1016/j.ijsolstr.2016.08.024
![]() |
[23] |
Li Y, Edward S, Reese S, et al. (2018) Anisotropic elastic-plastic deformation of paper : Out-of-plane model. Int J Solids Struct 130–131: 172–182. https://doi.org/10.1016/j.ijsolstr.2017.10.003 doi: 10.1016/j.ijsolstr.2017.10.003
![]() |
[24] | Hine DJ (1987) The rigidity/flexability balance in the creasing of paper based boards. Appita 40: 375–378. |
[25] |
Cavlin SI (1988) The unique convertibility of paperboard. Packag Technol Sci 1: 77–92. https://doi.org/10.1002/pts.2770010206 doi: 10.1002/pts.2770010206
![]() |
[26] |
Cavlin SI, Dunder I, Edholm B (1997) Creasability testing by inclined rules—a base for standardized specification of paperboard. Packag Technol Sci 10: 191–207. https://doi.org/10.1002/(SICI)1099-1522(199707)10:4 < 191::AID-PTS402 > 3.0.CO; 2-J doi: 10.1002/(SICI)1099-1522(199707)10:4<191::AID-PTS402>3.0.CO;2-J
![]() |
[27] |
Mrówczyński D, Garbowski T, Knitter-Piątkowska A (2021) Estimation of the compressive strength of corrugated board boxes with shifted creases on the flaps. Materials 14: 1–18. https://doi.org/10.3390/ma14185181 doi: 10.3390/ma14185181
![]() |
[28] |
Garbowski T, Gajewski T (2022) Influence of analog and digital crease lines on mechanical parameters of corrugated board and packaging. Sensors 22: 1–19. https://doi.org/10.3390/s22134800 doi: 10.1109/JSEN.2022.3226932
![]() |
[29] | Sönmez SİNAN, Dölen E, Fleming P (2011) Binder effects on the creaseability of pigment coated paperboard. Asian J Chem 23: 1193–1197. |
[30] |
Nagasawa S, Yamamoto T, Umemoto K, et al. (2020) Estimation of bending characteristics of creased paperboard using 45° tapered groove against unbalanced punch indentation. J Adv Mech Des Syst Manuf 14: 1–11. https://doi.org/10.1299/jamdsm.2020jamdsm0083 doi: 10.1299/jamdsm.2020jamdsm0083
![]() |
[31] | Hine DJ (1959) Testing boxboard creasing. Modern Packaging 8: 122–128. |
[32] |
Huang H, Hagman A, Nygårds M, et al. (2014) Quasi static analysis of creasing and folding for three paperboards. Mech Mater 69: 11–34. https://doi.org/10.1016/j.mechmat.2013.09.016 doi: 10.1016/j.mechmat.2013.09.016
![]() |
[33] |
Leminen V, Niini A, Tanninen P, et al. (2021) Comparison of creasing and scoring in the manufacturing of folding cartons. Procedia Manuf 55: 221–225. https://doi.org/10.1016/j.promfg.2021.10.031 doi: 10.1016/j.promfg.2021.10.031
![]() |
[34] | Carey K (1992) Creasing: Turning Failure into Success. Packaging Productivity 1: 1–20. |
[35] |
Carlsson L, De Ruvo A, Fellers C (1983) Bending properties of creased zones of paperboard related to interlaminar defects. J Mater Sci 18: 1365–1373. https://doi.org/10.1007/BF01111956 doi: 10.1007/BF01111956
![]() |
[36] |
Tanninen P, Matthews S, Leminen V, et al. (2021) Analysis of paperboard creasing properties with a novel device. Procedia Manuf 55: 232–237. https://doi.org/10.1016/j.promfg.2021.10.033 doi: 10.1016/j.promfg.2021.10.033
![]() |
[37] |
Nygårds M, Sundström J (2016) Comparison and analysis of in-plane compression and bending failure in paperboard. Nord Pulp Pap Res J 31: 432–440. https://doi.org/10.3183/npprj-2016-31-03-p432-440 doi: 10.3183/npprj-2016-31-03-p432-440
![]() |
[38] |
Mentrasti L, Cannella F, Pupilli M, et al. (2013) Large bending behavior of creased paperboard. Ⅰ. Experimental investigations. Int J Solids Struct 50: 3089–3096. https://doi.org/10.1016/j.ijsolstr.2013.05.018 doi: 10.1016/j.ijsolstr.2013.05.018
![]() |
[39] |
Gong Y, Zhao L, Zhang J, (2017) Delamination propagation criterion including the effect of fiber bridging for mixed-mode Ⅰ/Ⅱ delamination in CFRP multidirectional laminates. Compos Sci Technol 151: 302–309. https://doi.org/10.1016/j.compscitech.2017.09.002 doi: 10.1016/j.compscitech.2017.09.002
![]() |
[40] |
Beex LA, Peerlings RH (2012) On the influence of delamination on laminated paperboard creasing and folding. Philos T R Soc A 370: 1912–1924. https://doi.org/10.1098/rsta.2011.0408 doi: 10.1098/rsta.2011.0408
![]() |
[41] |
Alam P, Toivakka M, Carlsson R, et al. (2009) Balancing between fold-crack resistance and stiffness. J Compos Mater 43: 1265–1283. https://doi.org/10.1177/0021998308104227 doi: 10.1177/0021998308104227
![]() |
[42] | Carlsson L, Feller C, Westerlind B (1982) Finite element analysis of the creasing and bending of paper. Svensk Papperstidning 85: 121–125. |
[43] | Tryding J, Gustafsson P-J (2000) Characterization of tensile fracture properties of paper. Tappi J 83: 84–89. |
[44] |
Mäkelä P, Östlund S (2012) Cohesive crack modelling of thin sheet material exhibiting anisotropy, plasticity and large-scale damage evolution. Eng Fract Mech 79: 50–60. https://doi.org/10.1016/j.engfracmech.2011.10.001 doi: 10.1016/j.engfracmech.2011.10.001
![]() |
[45] |
Zechner J, Janko M, Kolednik O (2013) Determining the fracture resistance of thin sheet fiber composites—Paper as a model material. Compos Sci Technol 74: 43–51. https://doi.org/10.1016/j.compscitech.2012.10.007 doi: 10.1016/j.compscitech.2012.10.007
![]() |
[46] |
Tryding J, Marin G, Nygårds M, et al. (2017) Experimental and theoretical analysis of in-plane cohesive testing of paperboard. Int J Damage Mech 26: 895–918. https://doi.org/10.1177/1056789516630776 doi: 10.1177/1056789516630776
![]() |
[47] |
Ortiz M, Pandolfi A (1999) Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation analysis. Int J Numer Methods Eng 44: 1267–1282. https://doi.org/10.1002/(SICI)1097-0207(19990330)44:9 < 1267::AID-NME486 > 3.0.CO; 2-7 doi: 10.1002/(SICI)1097-0207(19990330)44:9<1267::AID-NME486>3.0.CO;2-7
![]() |
[48] |
Park H, Kim S-J, Lee J, et al. (2020) Delamination behavior analysis of steel/polymer/steel high-strength laminated sheets in a V-die bending test. Int J Mech Sci 173: 105430. https://doi.org/10.1016/j.ijmecsci.2020.105430 doi: 10.1016/j.ijmecsci.2020.105430
![]() |
[49] |
Carlsson L, Fellers C, De Ruvo A (1980) The mechanism of failure in bending of paperboard. J Mater Sci 15: 2636–2642. https://doi.org/10.1007/BF00550769 doi: 10.1007/BF00550769
![]() |
[50] |
Biel A, Tryding J, Ristinmaa M, et al. (2022) Experimental evaluation of normal and shear delamination in cellulose-based materials using a cohesive zone model. Int J Solids Struct 252: 111755. https://doi.org/10.1016/j.ijsolstr.2022.111755 doi: 10.1016/j.ijsolstr.2022.111755
![]() |
[51] |
Confalonieri F, Perego U (2019) A new framework for the formulation and validation of cohesive mixed-mode delamination models. Int J Solids Struct 164: 168–190. https://doi.org/10.1016/j.ijsolstr.2018.12.032 doi: 10.1016/j.ijsolstr.2018.12.032
![]() |
[52] |
Nygårds M (2022) Relating papermaking process parameters to properties of paperboard with special attention to through-thickness design. MRS Adv 7: 789–798. https://doi.org/10.1557/s43580-022-00282-7 doi: 10.1557/s43580-022-00282-7
![]() |
[53] |
Huang H, Nygårds M (2011) Numerical and experimental investigation of paperboard folding. Nord Pulp Pap Res J 26: 452–467. https://doi.org/10.3183/npprj-2011-26-04-p452-467 doi: 10.3183/npprj-2011-26-04-p452-467
![]() |
[54] |
Nygårds M (2008) Experimental techniques for characterization of elasticplastic material properties in paperboard. Nord Pulp Pap Res J 23: 432–437. https://doi.org/10.3183/npprj-2008-23-04-p432-437 doi: 10.3183/npprj-2008-23-04-p432-437
![]() |
[55] |
Hagman A, Nygårds M (2016) Short compression testing of multi-ply paperboard, influence from shear strength. Nord Pulp Pap Res J 31: 123–134. https://doi.org/10.3183/npprj-2016-31-01-p123-134 doi: 10.3183/npprj-2016-31-01-p123-134
![]() |
[56] |
Huang H, Nygårds M (2010) A simplified material model for finite element analysis of paperboard creasing. Nord Pulp Pap Res J 25: 505–512. https://doi.org/10.3183/NPPRJ-2010-25-04-p505-512 doi: 10.3183/NPPRJ-2010-25-04-p505-512
![]() |
[57] |
Choi DD, Lavrykov SA, Ramarao BV (2012) Delamination in the scoring and folding of paperboard. Tappi J 11: 61–66. https://doi.org/10.32964/TJ11.1.61 doi: 10.32964/TJ11.1.61
![]() |
[58] |
Hagman A, Huang H, Nygårds M (2013) Investigation of shear induced failure during SCT loading of paperboards. Nord Pulp Pap Res J 28: 415–429. https://doi.org/10.3183/npprj-2013-28-03-p415-429 doi: 10.3183/npprj-2013-28-03-p415-429
![]() |
[59] |
Sudo A, Nagasawa S, Fukuzawa Y, et al. (2005) Analysis of exfoliation of laminated layers and creasing deformation of paperboard. Proc Hokuriku-Shinetsu Dist Annu Conf Japan Soc Mech Eng 047-1: 35–36 (in Japanese). https://doi.org/10.1299/jsmehs.2005.42.35 doi: 10.1299/jsmehs.2005.42.35
![]() |
[60] |
Beex LA., Peerlings RH (2009) An experimental and computational study of laminated paperboard creasing and folding. Int J Solids Struct 46: 4192–4207. https://doi.org/10.1016/j.ijsolstr.2009.08.012 doi: 10.1016/j.ijsolstr.2009.08.012
![]() |
[61] |
Nygårds M, Just M, Tryding J (2009) Experimental and numerical studies of creasing of paperboard. Int J Solids Struct 46: 2493–2505. https://doi.org/10.1016/j.ijsolstr.2009.02.014 doi: 10.1016/j.ijsolstr.2009.02.014
![]() |
[62] |
Giampieri A, Perego U, Borsari R (2011) A constitutive model for the mechanical response of the folding of creased paperboard. Int J Solids Struct 48: 2275–2287. https://doi.org/10.1016/j.ijsolstr.2011.04.002 doi: 10.1016/j.ijsolstr.2011.04.002
![]() |
[63] |
Jina W, Nagasawa S, Chaijit S (2017) Estimation of detaching resistance of a peeled in-plane layer of a white-coated paperboard using fluffing resistance and an isotropic elasticity model. J Adv Mech Des Syst Manuf 11: 1–12. https://doi.org/10.1299/jamdsm.2017jamdsm0018 doi: 10.1299/jamdsm.2017jamdsm0018
![]() |
[64] |
Jina W, Nagasawa S (2018) Finite element analysis of the folding process of creased white-coated paperboard using a combined fluffing resistance and shear yield glue model. J Adv Mech Des Syst Manuf 12: 1–12. https://doi.org/10.1299/jamdsm.2018jamdsm0063 doi: 10.1299/jamdsm.2018jamdsm0063
![]() |
[65] | MSC Software (2010a) Marc document volume A: Theory and user information. 567–568. |
[66] | Komiyama Y, Kon W, Nagasawa S, et al. (2013) Effect of structural shape of corrugated medium on flat crush characteristics of corrugated fiberboard. J Chinese Soc Mech Eng 34: 361–369. |
[67] | Wink WA (1961) The effect of relative humidity and temperature on paper properties. Tappi 44: 171A–178A. |
1. | Ahmed Elaiw, Yusuf Al-Turki, Mohamed Alghamdi, A Critical Analysis of Behavioural Crowd Dynamics—From a Modelling Strategy to Kinetic Theory Methods, 2019, 11, 2073-8994, 851, 10.3390/sym11070851 | |
2. | Dorine Duives, Guangxing Wang, Jiwon Kim, Forecasting Pedestrian Movements Using Recurrent Neural Networks: An Application of Crowd Monitoring Data, 2019, 19, 1424-8220, 382, 10.3390/s19020382 | |
3. | Rinaldo M. Colombo, Magali Lecureux-Mercier, Mauro Garavello, 2020, Chapter 5, 978-3-030-50449-6, 83, 10.1007/978-3-030-50450-2_5 | |
4. | Michael R. Lindstrom, Andrea L. Bertozzi, Qualitative features of a nonlinear, nonlocal, agent-based PDE model with applications to homelessness, 2020, 30, 0218-2025, 1863, 10.1142/S0218202520400114 | |
5. | Kaidong Hu, Sejong Yoon, Vladimir Pavlovic, Petros Faloutsos, Mubbasir Kapadia, Predicting Crowd Egress and Environment Relationships to Support Building Design Optimization, 2020, 88, 00978493, 83, 10.1016/j.cag.2020.03.005 | |
6. | 2020, 9781786305022, 313, 10.1002/9781119695004.refs | |
7. | Shiyue Li, Lei Zhang, Qianling Wang, 2021, A revised social force model considering the velocity difference between pedestrians, 978-9-8815-6380-4, 6755, 10.23919/CCC52363.2021.9549244 | |
8. | Sergey V. Kovalchuk, Chao Li, Oleg V. Kubryak, 2024, Towards Explaining Emergent Behavior in Multi-Agent Systems Micro-Parameter Space Structuring with Feature Importance in Heatbug Model, 979-8-3315-0494-6, 824, 10.1109/WI-IAT62293.2024.00134 | |
9. | Cecile Appert-Rolland, Alethea B.T. Barbaro, 2025, 25430009, 10.1016/bs.atpp.2025.04.004 |
Sample set | a | b | c | |
0.944 | ||||
0.972 | ||||
0.982 |