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LINDBERG AND KULACHENKO
methods, and the considerably higher number of elements com- pared to earlier studies, allowing for more exact results. This enables to study the effect of creases on the frictional interaction. 4. The use of implicit time integration during the simulations, which has a direct impact on the accuracy of the test. 5. The analysis of the frictional conditions required to achieve the desired shape. With all the aforementioned parts included at once in the model, the possibility to simulate advanced forming operations is enhanced. The objective is to increase the precision of the results coming from a simulation model of tray forming, giving more exact results for stresses, strains and the risk of failure compared to previous models. The verification is based on achieving the correct shape of the tray, something not fully attained in earlier studies of tray forming where the flange of the corners has been larger than what is seen in produc- tion, even for failed trays. The risk of failure is also determined and compared to reality. The paperboard is commercially produced under the name Inverform by Iggesund paperboard, which is a part of the Holmen Group. The paperboard is produced on two board machines, but dif- ferences in converting performance have been observed and to understand this difference the present investigation was performed. The paperboard coming from the first board machine is here called Board A, and the paperboard coming from the second machine is here called Board B. In Figure 1, the failure mode of the studied tray using Board A is shown, where the material fails in the corner under the creases. With the new simulation approach, the tray forming operation is simulated using the different paperboards, and the difference in results is evalu- ated and compared to reality.
a punch, also called male die, to press the blank into the bottom of a die, while deep drawing uses the punch to press the blank into a bot- tomless die. The methods are described in e.g. Hagman et al. 3 Östlund and Söderberg 9 and Lowe et al. 10 Complex products such as egg pack- ages are molded. 11 As a part of the product development process using paperboard, simulations of the forming operations are used. The behaviour of the paperboard can then be studied, and it can be evaluated if and how material properties should be altered to minimize the risk of failure of the paperboard during the forming operation. The present study focuses on the tray forming operation of paperboard. A numerical approach is presented where the simulation of the operation is further developed compared to what has been done previously. Deep drawing and tray forming of paperboard has been simulated in earlier studies, 12 – 14 with different numerical approaches and objec- tives. In Wallmeier et al. 12 the effects of varying the blank holder force, the die temperature, and the thickness of the paperboard were investigated. They showed, amongst other things, that the friction of the blank holder and die have significant effects on the stress in the blank, implying that low-friction dies and blank holders can consider- ably reduce the failure probability. In Awais et al. 13 the effect of the number of creases on the strain levels was numerically investigated for paperboard. The creases were modelled with hinge connector ele- ments, and not explicitly included in the geometry. It was seen that the number of creases aided in lowering the strain levels. It could also be seen that this effect was greater for the first paperboard having a higher Young's modulus compared to the second investigated paper- board, which had a larger strain to failure. The approach with hinge connector elements for the creases is investigated more in detail in Livill et al., 14 where the approach allows for spontaneous wrinkling, that is, the number of creases or the position of them do not need to be known by forehand. The approach is included in deep drawing sim- ulations and shows that the number of creases is about the same as in experiments. The disadvantage of these approaches is the inability to assess the effect of creases on the frictional interaction between the paperboard and the forming unit as the wrinkles were not resolved physically. The inclusion of the creases in a numerical model is impor- tant not only to simulate the correct shape, friction interaction and strain levels, but also to investigate the possibilities to add a lid onto the tray to seal it. For the sake of successful sealing operations, the upper edge of the formed tray should be as smooth as possible.
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| MATERIALS AND METHOD
2.1
The tray
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Paper and paperboard are anisotropic, heterogeneous, and hygroscopic materials. They are built up by a network of cellulose fibers from soft- wood or hardwood, or from a combination of the two. The fibers are randomly distributed over the sheet with, partly, random orientations. The fibers are connected via fibre bonds which form spontaneously when water disappears from the web in the papermaking process. The anisotropic material behaviour must be considered in numerical simula- tions of paperboard. A common way is to model the paperboard as orthotropic, 3,12 – 16 where material properties are specified in the paperboard machine-direction (MD), cross-direction (CD), and in the Z-direction (ZD), that is, through the thickness of the paperboard. In Figure 2 the paperboard blank is shown as it is prepared by the tray manufacturer for the forming operation. The blank is laminated with a polymer that is extruded over the blank since the tray must withstand moisture during usage. The creasing pattern with 30 creases in each corner has been pressed into the paperboard so that it folds
The advances in the current study are as follows:
1. The use of an orthotropic material model with isotropic hardening according to Hill plasticity, fitted to actual tensile tests and com- pared with two sets of data from papers with different failure properties. The fitting accuracy was improved, which allowed a reliable comparison of different materials. 2. Accounting for the difference in tension and compression for yielding and failure. 3. The detailed resolution of the creases which are introduced through geometrical features rather than through the ad-hoc
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