Simulations of lateral stress variations in a flexographic print nip

2.1 Paperboard: reference paperboard and model

The paperboard model was built using the same method and reference paperboard as in a previous work by the authors (Rydefalk et al. 2024). The structural thickness and surface roughness of a commercial liquid packaging paperboard were measured to create a structured sheet in the FE model, shown in Figure 1. Since the measured surface was smaller than the sheet in the print press model, the measurement data was mirrored to create a larger surface. The structural data was then mapped over the same grid size as the sheet resolution in the FE model. The top and bottom nodes in the model were then moved in the Z-direction according to the measurement data, and the middle layer nodes were redistributed between the new positions, as illustrated in Figure 2.

Figure 1:

Surface roughness and structural thickness from measurements.

Figure 2:

Node distribution in ZD in the paperboard model.

The material model used was MAT_ORTHOTROPIC_ELASTIC, an orthotropic elastic model already implemented in LS-Dyna. The model allowed us to model the anisotropy of the paperboard without the high computational cost of a more complex model. The input is shown in Table 1.

All the simulations are explicit and performed with solid elements. The hexahedra elements are LS-DYNA’s ELFORM = −2 which are fully integrated S/R solid elements intended for elements with poor aspect ratio. The elements in the paperboard are 0.5 x 0.5 x 0.25 mm in two layers.

2.2 The print press model

An FE model with the dimensions of an IGT F1 laboratory print press was built in LS-PrePost and ran in LS-Dyna, as shown in Figure 3. It consists of a rigid substrate carrier and a rotation print cylinder. The core of the print cylinder can be considered rigid and is covered with a hyperelastic print form. The print cylinder is 50 mm wide and has a radius of 92 mm. Some simplifications have been made to the actual case. The print form in the model is 3 mm thick and consists of a single material. In the actual print press, the print plate used for paperboard is usually 1.14 mm with a Shore A above 70 and would have a soft foam backing of 0.5 mm. To compensate for the effect of the soft backing, but without having to include a second material, the print form in the model is slightly thicker than the combined print plate and backing and has a Shore A stiffness of 50. The material model of the print plate is hyperelastic, MAT_MOONEY-RIVLIN_RUBBER.

Figure 3:

Illustration of the print press and the initial movement of the substrate carrier.

The elements on the print plate have the same xy-resolution as the paperboard model, 0.5 x 0.5 mm. Compensation has not been made for the cylindrical shape of the print plate, and therefore. The simulation starts with aligned nodes between the substrate and the print plate in the centre of the nip. Due to the curvature of the print plate, the alignment between nodes on the substrate and print plate is not maintained during the simulation.

The initial steps of the simulation are shown in Figure 3. The sample carrier is raised 120 µm and is held in position for the duration of the simulation. When the desired impression is achieved, the cylinder starts to rotate and drives the sheet through the nip at a 300 mm/min velocity. The rotation of the cylinder drives the paperboard through the nip, both in the lab press and the FE model. The model has no friction between the paperboard and substrate carrier.

2.3 The simulation cases

Four different parameters have been altered in the present study: the surface roughness, the paperboard ZD stiffness, the impression, and the print form stiffness, as is presented in Table 2. The height maps of the filtered surfaces are presented in Figure 4. The band-passed (BP) surfaces will be referred to by the wavelength not filtered out. All cases have the same bottom surface, also shown in Figure 4, and are run with the same print speed of 300 m/min.

Figure 4:

Height maps [µm] of the band passed surfaces.

3.1 Varying the surface roughness

The surface roughness was changed through band-passing the measurement data, but the bottom surface is the same in all simulations.

The fine-scale surface roughness (0–1 mm) has a larger area within the FWHM stress span, as seen in Figure 9. 80–83 % of the area falls within the FWHM stress span, compared to 68 % for the reference case. The excluded (black) zones on the fine-scaled, filtered cases largely overlap with the excluded regions in the reference case but are much smaller. In the larger wavelength span (1–4 mm), the covered area approaches the same percentage as the reference case, 71 % compared to 68 %. The excluded areas are even more similar to the reference case than to the fine-scale cases.

Figure 9:

Area within FWHM: White. Area either above or below FWHM: black.

The fine-scaled pattern in 0–1 mm wavelengths is evenly distributed over the surface, as shown in Figure 4. The excluded areas do not correspond to any apparent structures on the surface. Instead, they originate from the roughness on the backside of the sheet. As can be noted in Figure 1, the pattern of the local thickness and the surface roughness are similar. Therefore, a large hill on the top is usually mirrored by a larger hill on the back, which is valid for craters.

The “most likely” stress in the band-passed cases is very close to the reference case, as seen in Figure 8D. The FWHM is, however, broader (Figure 8B). The areas with higher or lower stresses are smaller for the smaller wavelength cases of band-passing than the reference. However, in 1–4 mm, there are larger areas above FWMH than in the reference case.

The height map in Figure 10 shows the surface structure within the areas receiving a higher stress than FWHM. The areas coincide with the areas in the reference case, but the height map demonstrates that the surface roughness amplitude is smaller than in the reference case. In the BP 1–4 mm case, the areas are even larger, although they still obviously have a smaller amplitude, similar to those below the FWHM stress in Figure 11. They appear in similar positions to those in the reference case. However, there is a finer pattern in the low-stress areas in BP 0–0.05 mm compared to 0.05–1 mm and 1–4 mm. The high-stress areas had a similar trend, but it is more visible in the low-stress areas. The areas are more scattered in the smaller wavelengths and fewer and larger in 1–4 mm.

Figure 10:

Heightmap [µm] areas above FWMH-stress.

Figure 11:

Heightmap [µm] areas with stress below FWHM.

The results show that the local thickness significantly contributes to the pressure distribution on the surface. Even the smoothest surfaces show patterns originating from the roughness of the bottom surface. This implies that making only one of the surfaces smoother is not enough for an evenly distributed pressure. The wavelength of the bottom side unevenness, compared to the nip length and the size of the top side roughness, will also determine if hills or craters are the most detrimental to achieving even pressure. The effect of local thickness should also explain the similar pressure pulses in Figure 5.

It should be noted that the roughness amplitude of the filtered surfaces has not been adjusted. Therefore, the small wavelengths also have a small roughness amplitude. In another study by the authors, we showed that at a fixed mean compression stress, the stress variation was linear to the surface roughness amplitude but sorted with the wavelengths (Rydefalk et al. 2024). Additionally, we saw that the roughness placement on the top relative to the bottom greatly impacted the distribution. The study concerned compression between the rigid platens. It might be interesting to see the effect of roughness placement when there is the addition of directionality and shear due to the rolling, as well as the compliant nature of the print plate.

3.2 Varying the paperboard stiffness

The MD- and CD-stiffnesses are kept the same, and the only parameter that is varied is the ZD-stiffness. The reference case has a ZD stiffness of 23 MPa, and the variant cases have 20 and 25 MPa, respectively.

The sheet influence of the sheet structure is visible in all three cases, regardless of the compliance of the substrate. Both the FWHM stress range and the most likely stress sorts with the paperboard stiffness for the three cases, as seen in Figure 8B and D. The same goes for the pressure pulses in Figure 5. The area within FWHM is similar in size and falls between 68 and 72 % of the surface, as seen in Figure 9. Both the stiffer and the softer substrate have an increased area within the FWHM stress range, but it should be noted that FWHM is smaller for the softer substrate, i.e., the stress variations are smaller over the surface than for the stiffer case. The area with stress above FWHM is similar for all three cases, as seen in Figures 8A and 10. The redistribution between the three cases mainly takes place in the range of FWHM and the lower tail of the histogram, in Figures 7, 8, and 11.

The narrower FWHM span and larger surface area within the FWHM stress of the softer substrate mean a more even pressure distribution compared to the other two cases. However, the more compliant case has a larger tail on the stress below FWHM, and poor contact in craters could be the outcome. However, substrate compliance should be compared to print plate compliance, and their relative stiffnesses are probably more important than the absolute stiffness of either one.

3.3 Varying the print-form stiffness

The print plate stiffness is increased from Shore A 50 in the reference case to 60 and 70.

The “most likely” stress increases with increasing print plate stiffness, as seen in Figure 8D. Unlike when comparing the substrate stiffnesses the same trend is not seen in Figure 8B, which shows the FWHM. Although it is higher for the highest print plate stiffness than the reference, the Shore A 60 case has the same FWHM. The percentage range of the print plate stiffness cases is about the same size as that of the substrate stiffness cases. Figure 9 show that, the area ranges from 64–68 % for the print plate stiffness cases.

The tail of the distribution above FWHM is increased for the higher stiffness print plates, as seen in Figures 8A and 10. However, it does not sort with increasing stiffness. Instead, the case with Shore A 60 has the largest area with higher pressure. On the other hand, it also has the smallest area with stresses below FWHM, seen in Figure 8C, comparable to the Ez = 20 MPa-case. However, Figure 11 shows that there are more small “low-stress islands” in Ez = 20 MPa than in Shore A 60.

A softer print plate compared to the substrate can conform to larger scale variations (which is why a corrugated post-print flexo uses a thick, soft print plate). However, the conformability is limited by the cylinder’s geometry and the compliant layer’s thickness. By using a print plate that is harder than the substrate, it is instead the substrate that will deform and flatten out large-scale waviness. However, as seen in the present results, it is not entirely straightforward.

3.4 Varying the impression

The literature states that this is the most significant change to the pressure pulse comes from changing the impression. This can also be seen in the mean pulses in this study in Figure 5.

By increasing the impression, the “most likely” stress is increased with increasing impression due to the increased deformation, as seen in Figure 8D. Additionally, the FWHM in Figure 8B decreases with increasing impression. However, the percentage of the area within the FWHM stress range also increases, from 68 % in the reference case to 64 % and 55 %, respectively, for the increasing impressions. This implies that a larger portion of the surface has a stress either above or below the FWHM span. The shift is, however, not consistent with impression. The first increase in impression to 130 µm decreases the percentage of elements below FWHM, as seen in Figure 8C. The percentage is comparable to the filtered cases, but the islands with low pressure in Figure 11 take on different shapes due to the different surfaces. An even larger impression, to 140 µm, has instead the same area below FWHM as the reference case, as seen in Figure 8C and similarly distributed as seen in Figure 11. For both increases in impression, the percentage of elements above the FWHM range increases, as shown in Figure 8A. They increase to a similar level, but the distribution of the high-stress regions shows a marked difference in Figure 10. The fields have different shapes and occur at different ends of the substrate.

In the cases with increased impression, the effect of the non-linearities from the substrate structure and print plate material becomes even more pronounced than in the previously discussed cases. Despite the substrate’s linear elasticity in the Z-direction, the purely compressive response is initially non-linear due to the surface roughness (Hagman and Rydefalk 2024; Rydefalk et al. 2024). The hyperelastic material model used for the print plate also has a non-linear stiffness. Combining these parameters makes the deformation and stress distribution more complex than merely linearly increasing with increasing stiffness.

Changing the impression would have an even larger effect if more layers and more complex material behaviour were involved in the model. E.g. if the progressive stiffness of the paperboard was modelled, or the compressible foam backing was added.

It should also be noted that at reference impression, the sheet is fully in contact; otherwise, it would not move through the nip. A model that includes tension in MD and a different mechanism for moving the sheet could make it possible to consider lower impressions.

3.5 Same pulse, different distribution

Since the most common presentation of the nip pressure is in the form of a mean pulse, it is interesting to note that the several overlapping mean pulses in Figure 5 are resulting from different combinations of parameters.

The most closely similar mean pulses are the reference case and band-passed cases. These have already been discussed in the section “Varying the surface roughness”. Suffice it to say that the structural thickness/bottom side roughness plays a large enough part that the mean pulse becomes the same while the stress distribution on the surface differs significantly.

The stiffnesses in the substrate and the print plate achieve a similar mean pressure pulse when increasing or decreasing one in comparison to the other. The present simulation found it for Ez = 25 MPa and Shore A 60. In a 1D model, it does not matter if the plate or board is stiffer or softer; their combined stiffnesses will be the same. In the present case, we can see that differences are seen despite their similar mean pulse, especially in the size and shape of the fields with stress above the FWHM range in Figure 10. The 3D case shows that the geometry affects the distribution of stress.

Increasing the impression to 130 µm or the Shore A stiffness of the print plate to 70 achieves another similar pulse pair. Differences are noticeable in the maps in Figures 10 and 11, both in size (also seen in Figure 8A and C) and in the distribution pattern on the surface. The areas with stress above the FWHM range are extended with a higher impression. The areas with stress below the FWMH range, while larger for the increased impression, have less mid-scale scatter compared to the higher plate stiffness. The impression and plate stiffness trigger different mechanisms. The impression shifts along the non-linear response between the plate and substrate, while the increased plate stiffness changes the non-linear response. The effect has a similar mean but a different distribution.