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LINDBERG AND KULACHENKO
statistical distributions of the paper properties on the relevant scales. The latter is outside the scope of this work.
in Figure 18C. The model with no creases renders in a shape very dif- ferent to that observed in the physical samples even for low friction and shows how important the creases are to avoid unwanted shapes. Without the creases, the material is having a hard time to fold leading to a large amount of material left on the upper edge at the end of the forming process.
ACKNOWLEDGEMENTS The authors would like to thank Johan Lindgren, Brita Timmermann and Tommy Ström at Iggesund Paperboard for their inputs and sup- port during this study. The authors are also grateful to Hannes Womhoff at Holmen AB for his input. This work has been carried out within the national platform Treesearch and is funded through the strategic innovation programme BioInnovation, a joint effort by Vinnova, Formas and the Swedish Energy Agency. The computations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at HPC2N
4
CONCLUSIONS
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An implicit non-linear finite element model with full-integrated shell elements is built to simulate the tray forming process of paperboard. Two different boards with different failure propensities are investi- gated. The creases are included in the geometry of the paperboard blank which is a new approach compared to earlier studies. The mate- rial data are extracted from tensile test curves of the investigated paperboards, and the used material model includes different behav- iour in tension and compression. The numerical tensile test curves have a very good fit with the experimental tensile test curves. The results from the tray forming simulation show a very good agreement of the shape between the real tray and the simulated trays, which is a clear improvement compared to earlier studies of the paperboard tray forming process. The results show that including the creases in the geometry is very important to acquire the correct shape observed in reality. The incorrect shape is associated with an increased risk of failure of the tray during the converting operation. Further, the model shows that the friction between the paperboard blank and the die and blank holder has a great effect on the shape of the tray. High friction leads to incorrect shape and an increased risk of failure. The model supports the measures used in the industry to lower the friction during the converting operation. The area with the creases has several locations with large stresses. Although failure can occur over these areas as well, the use of homogenous material prop- erties and ignoring delamination leads to an overestimation of the stresses in these regions. The failure evaluation using Maximum Strain Theory shows small differences between the two paperboards but is deemed too conservative. The failure evaluation using Tsai – Wu the- ory shows that Board A has a very high risk of failure, and that the risk of failure using Board B is lower, something that is in full agreement with what has been seen in production. The results from the stochas- tic failure evaluation based on the numerical model more precisely suggest that the risk of failure in the lower corner using Board A is close to 100%, and for Board B the risk of failure is about 1%. No known problems with Board B are reported from production and the here calculated failure risk is probably conservative. The model may be further used to estimate the most critical material properties for the tray forming such as elongation, strength, and friction. Among these, in this study, we only probed the impact of the friction in the corner. A stochastic approach is used as a part of the post processing to study the impact of strength. However, the stochastic nature of paper makes it difficult to include the elongation and strength directly in the model as these parameters can vary locally and therefore require characterization and modelling which includes
(projects SNIC 2020/5-428 and SNIC 2021/6-51). The authors are grateful for the financial support.
DATA AVAILABILITY STATEMENT The data that support the findings of this study are available from the corresponding author upon reasonable request.
ORCID Gustav Lindberg
https://orcid.org/0000-0001-5033-7611
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