Rate-dependent tensile properties of paperboard and its plies

1 Introduction

Cellulose based papers are thin lightweight materials whose properties can be changed based on the mechanical and chemical treatment of the fibers and types of fiber used. Multiple layers of such plies with controllable and desirable properties can be produced as laminated multi-layered structures, such that the paperboard performance can be engineered, e.g. to maximize the bending stiffness and the out-of-plane strength or to control the converting performance see e.g. Coffin and Nygårds (2018) or Nygårds (2022).

A paperboard is a multi-layer sandwich construction with stiffer outer plies and a more compliant and bulkier middle ply to create a structure equivalent to an I-beam. This construction allows for controlled inter-laminate failures and delamination to avoid surface cracking when creasing and folding at high speeds (Borgqvist et al. 2015; Nazarinezhad et al. 2022; Nygårds et al. 2009). The outer plies that are stiff and strong are normally made of kraft fibers, where refining is used to consolidate the plies and increase the strength. The outer plies are often also clay coated for better printing surface and improved visual appeal. The middle ply is normally made bulkier using chemi-thermo-mechanical pulp (CTMP), which are stiffer fibers. The middle ply pulp that is used during production is a mixture of CTMP and kraft fibers and broke. Moreover, the orientation of the top, middle and bottom plies are different since orientation is used to engineer the functionality in different applications.

It is a well-established fact that properties of paper are dependent on the rate at which load is applied to the material. The rule of thumb is that the in-plane strength and stiffness increase by approximately 10 %, when the strain rate increases by a factor of 10 per decade increase in strain rate (Andersson and Sjöberg 1953; Bajpai 2018; Haslach 2000; Panek et al. 2020; Robertsson et al. 2021). Converting of paperboard into packages requires creasing of the paperboard followed by folding to make three dimensional packages. During these operations paperboard is loaded in tension at very different rates; the crease response is e.g. controlled by in-plane properties, since it contributes to the loading and the spring back of the crease which gives the paperboard its final geometry.

In the current environment, there is a marked lack of data on the behavior of the paperboard and both the influence and behavior of the individual plies which make up the paperboard at rates at which converting is performed. Usually during material characterization of these materials, tensile tests, are usually performed at crosshead velocities of 100 mm/min, but during the converting process there are regions of the materials which are strained at rate much higher than the rates the materials are tested at. Converting of paperboard is performed at high rates, creasing is normally done at speeds in the range 300–600 m/min, but in literature, there is limited data on how different paper types behave at these rates. Lab creasing rates up to 300 s⁻¹ has been reported by Nazarinezhad et al. (2022). At this rate paperboard is not in quasi-static conditions, as in Figure 1. There is limited knowledge of how paper material behaves at such high loading rates. Yet, there is a strive to constantly increase converting and filling speed of packages. This can lead to problems if rate dependent material properties are unknown.

Figure 1:

Strain rates used in material characterization (Field et al. 2004; Siviour and Jordan 2016).

Viscosity and rate effects in paper materials plays a vital role in understanding the flow and deformation characteristics of paper as a network and its constituent fibers and fiber bonds under loading. Investigating the rate effects and viscosity of paper provides insights into its suitability for applications where dynamic loading or time-dependent behavior is encountered, such as during creasing, folding and similar converting processes.

The creasing process has been analyzed and finite element simulation have been used to analyze the process, see reviews by Coffin and Nygårds (2018) and Simon (2021). In these studies, creasing was done in the lab, at low rates. Paperboards that were used in these studies had also been characterized using standardized methods at low rates. The creasing process was then predicted with good accuracy (Beex and Peerlings 2009, 2012; Borgqvist et al. 2015; Huang and Nygårds 2010; Huang et al. 2014; Li et al. 2014; Nygårds et al. 2009). Lately, studies of creasing at higher rates have been reported (Borgqvist et al. 2015; Nazarinezhad et al. 2022; Robertsson et al. 2021). These results show that the crease response change, at high rates the response is stiffer. This effect was also predicted by when stiffer and stronger material properties were used, as reported by Nazarinezhad et al. (2022). When higher loading rates are used other mechanisms can be activated. To be able to investigate such changes of mechanisms it is of interest to have rate dependent data that can be implemented into material models used in finite element simulations of converting processes. Of particular interest would be to see if individual plies in paperboard have another behavior than the whole paperboard.

This mismatch in the strain rate in the material between industrial processing and material characterization has implications on various parameters such as crease performance and cracking. It also affects the process equipment used for converting, such as the reached peak load, maintenance, and wear of the equipment.

In this work we aim to characterize the strength, stiffness, and strain at break of the paperboard and its constituent free-laid plies, at varying strain rates under tensile loading. This is being done to make more data about the plies available to better optimize the in-plane properties of the paperboard.

2 Materials and methods

2.2 Methods

Rate-controlled tensile testing was used in both in the principal in-plane material directions viz. MD and CD to characterize the rate dependency of the paperboards and its plies. Two tests were used to test rate dependent properties, long and short tensile tests. In both tests strips with width 15 mm was tested, in accordance with the standard (ISO 1924-3), illustrated in Figure 2. The long tensile tests had testing length, L = 50 mm. This was shorter than the standard, which prescribe 100 mm long samples. However, due to machine constraints with the current grips only 50 mm could only be tested. However, 50 mm would still be judged to be representative to a long standard specimen. One would however assume that the strain at break can differ for between 50 and 100 mm long samples (Hagman and Nygårds 2012). The short tensile tests had length L = 5 mm and was suggested as a convenient test to test local stress and strain behavior that would be relevant to analyze crack sensitivity (Tryding et al. 2017). Both lengths were tested to be able detect difference because of the specimen size. The long specimen will have a uniform stress field, while the short specimen will have a biaxial stress state.

Figure 2:

Schematic of specimens tested; shaded portion indicates clamping area. It is reiterated that the 100 mm long sample was not tested and is presented for visualization only.

By varying the strain rate during testing, the ultimate strength, Young’s modulus, and strain at break was evaluated under different loading conditions. The results obtained from rate-controlled tensile tests will be used to establish rate dependent stress-strain relationships and contribute to the understand of the rate-dependent behavior of paper materials.

The tensile test was performed using an Instron ElectroPuls E1000 electro-mechanical testing machine, with a 2 kN load cell, as shown in Figure 3, available at the laboratory at Material and Structural Mechanics at the KTH-Royal Institute of Technology in Stockholm. To control the testing machine Instron Bluehill software was used to set the rate dependent properties.

Figure 3:

Testing machine and the clamps used to do tensile testing.

Since the machine was electro-mechanical very high crosshead velocities, v _f , could be achieved in comparison to the screw based or hydraulic operated tensile testing machines that is more common. The crosshead velocity was altered from 1 to 100,000 mm/min, with a factor 10 between each tested rate. Hence, a total of six target rates were defined in the control software and the sampling rate of the Data Acquisition (DAQ) system was adjusted to acquire enough data points to obtain reliable and repeatable results.

The ultimate strength, σ _f was calculated as using the measured maximum force F,

σ f = F w t ,

where w = 15 mm was specimen width and t is the specimen thickness, presented in Table 1. The strain at break, ε _b , was calculated using the elongation, u _f , at failure and the specimen length,

ε b = u f L .

The Young’s modulus was evaluated for each sample by making linear fit to the stress-strain curve in the interval 5–25 MPa. The crosshead velocity, v _f , is used to calculate the strain rate, ε ˙ , which is calculated as the slope prior to failure using the strain ε and time, t,

ε ˙ = v f L = ε b − ε b − 1 t b − t b − 1 .

Specimens were tested until failure, which was defined as either,

1. An unstable fracture, where the sample failing catastrophically and drop the load, or,

2. In case of a stable fracture, when the load dropped to 1.5 N after exceeding 5 N.

Two failure criterions were defined to accommodate both stable and unstable fractures characteristic of short and long specimen failures respectively. The ultimate strength was defined as the maximum strength prior to the failure.

3 Results

Mentioned previously we aim to test the specimens at six different rates from 1 to 100,000 mm/min in steps of 10 times increments. During testing it was revealed that the maximum crosshead velocity of the frame was 62,500 mm/min, so the highest rate was unachievable. This, although an apparent obstacle, will only have the effect that the maximum velocity is somewhat lower than anticipated. However, the crosshead velocity was yet higher compared to available screw-driven or hydraulic machines.

Since testing was done close to the velocity limits of the machine, the specimens failed before the crosshead could reach the target velocity. The crosshead velocity versus the strain was evaluated to determine if the machine was still accelerating when the specimen failed. In Figure 4, the average curves of the crosshead velocities for all tests done at the specified rate can be seen, and in Table 2 the range of achieved velocities are tabulated. It should be mentioned that the curves are quite rough since they are average curves. It was observed that for the two highest velocities the machine was still accelerating when the sample failed. Obviously for high rates the target velocity could not be reached. Therefore, the velocity at failure was evaluated for each specimen.

Figure 4:

Velocity attained, averaged over all tested MD paperboard samples.

Figure 5:

Stress strain curves at different rates for Paperboard A from testing of long specimens in MD and CD.

The stress-strain curves were evaluated for all the individual plies and samples, and in Figures 5 and 6 some example curves for Paperboard A are presented. It should be noted that the curves are only plotted to the ultimate strength, which was the maximum load experienced by the samples. It should be noted that the long and short specimens measured similar strength, while the strain at break becomes higher in the short specimens. Consequently, the initial slope of the short tensile test curves does not represent the Young’s modulus, and it is therefore not evaluated. The curves for the long samples fail with unstable fracture behavior with the load dropping to zero which satisfies the criteria for a failed sample. For the short samples, which exhibit stable fracture behavior, the load does not immediately drop to zero but slowly reduces.

Figure 6:

Stress strain curves at different rates for Paperboard A from testing of short specimens in MD and CD.

3.1 Rate dependency for strain at break

From the tensile tests the strain at break from the long samples of whole paperboard was evaluated, as shown in Figure 7. The strain at break in MD had no obvious/significant rate dependent effects. When paper materials are tested in MD, the fibers dominate the behavior, and at failure a localization occurs where the fibers are pulled out of the network. Hence, even when altering the loading rate, no major change of mechanisms of failure occurs. In Figure 7 the strain at break in MD is 2–3 %, which correspond to a deformation of 1–1.5 mm. This is essentially half the fiber length; hence the samples fail when the fibers have been torn out of the network.

Figure 7:

Strain at break evaluated from long samples for the two paperboards as function of strain rate. For (a) whole paperboard, (b) top plies, (c) middle plies and (d) bottom plies.

In CD on the other hand, it was observed that the strain at break decreased when the strain rate exceeded 10⁻¹ s⁻¹. It is believed that this rate effect was due to localization of the failure. Normally CD straining initially occurs on more locations along the sample (Hagman and Nygårds 2012), but localization of the deformation occurs close to failure at higher rates. This would lower the strain at break, as seen in Figure 7. In addition, it was concluded that the crack path became straighter for both MD and CD samples with increasing loading rate, as in Figure 8.

Figure 8:

Examples of failed samples that show that the fracture surface becomes straighter with increasing loading rate. Target crosshead velocity and strain rate at failure for (a) MD samples v _f  = 1 mm/min ( ε ˙ = 0.0004  s⁻¹), (b) MD samples v _f  = 15,420 mm/min ( ε ˙ = 8.1  s⁻¹). (c) CD samples v _f  = 1 mm/min ( ε ˙ = 0.0005  s⁻¹), (d) CD samples v _f  = 16,625 mm/min ( ε ˙ = 8.9  s⁻¹).

In the short specimens, no rate effect was observed for the strain at break either for MD or CD samples.

3.2 Rate dependency for strength and elastic modulus

The evaluated ultimate strength and Young’s modulus for the paperboards, using the long specimens can be found in Figure 9. Both the ultimate strength and Young’s modulus increases with increasing strain rate. In fact it can be seen that the increase is independent of loading direction. When the values are normalized with the corresponding strength and modulus measured at the standard rate 100 mm/min the curves coinsides, and linear fit can be made for both strength and modulus. Hence,

(1) σ b = σ standard b ( 0.091 × log ε ˙ + 1.1 )

(2) E = E standard ( 0.071 × log ε ˙ + 1.1 )

Figure 9:

Failure strength and Young’s modulus evaluated from long specimens.

Note that the equations holds for both MD and CD. It should however be noted that the Young’s modulus at the lowest rates deviates from this relation, the strain rates below 10⁻³ s⁻¹ were therefore not included in the liner fitting of modulus data.

When the free-laid plies were tested in the same manner and similar trends were seen, as in Figure 10. It was however observed that there were more scatter between the measurements. This was believed to be caused by the splitting procedure. First of all, in the Fortuna splitter, the plies are damaged as bonds are broken during splitting, which lower the strength and modulus. Second, the splitting can also be uneven and generating variation in thickness. Here, all samples were assumed to have the average thickness of the splitted sheet. Although, the plies had different strength and stiffness ranges, they followed the same trend a suggested in Eqs. (1) and (2). However, at rates higher than 1 s⁻¹ the ply properties were higher than the predictions. This is when the testing is no longer quasi-static, and when the machine is close to its maximum performance. To clarify the behaviour at high rates the strength was also evaluated from short tensile tests.

Figure 10:

Failure strength and Young’s modulus for the free-laid top, middle, and bottom plies measured by long tensile tests.

3.3 Short tensile tests

The short tensile specimens, see pictures of tested samples in Figure 11, were a factor 10 shorter than the long specimens, therefore it was possible to achieve higher strain rates for these specimens. In Figure 12, the failure strength for the short tensile tests have been plotted, the results confirm the findings from the long specimens. For strain rates in the range 10⁻⁴ <  ε ˙  < 10 the rate effect can be predicted by Eq. (1). The increase of ultimate strength and Young’s modulus observed in Figure 11, could not be observed for the short specimens for rates above 1 s⁻¹. The increase seen in Figure 11 was hence judged to be an effect of the machine settings. There are however deviations from the proposed equations, especially for the free-laid plies. It should however be recalled that these have been splitted by the Fortuna. The splitting can cause damage of the plies, and scatter and deviations are currently believed be caused by the splitting procedure. This is confirmed by the observations that Eqs. (1) and (2) holds very well for the whole paperboards in Figures 10 and 11.

Figure 11:

Failure in short MD specimens and CD specimens. The target crosshead velocities are indicated in the figure.

Figure 12:

Failure strength evaluated from short specimens for the paperboard and its free-laid plies.

4 Discussion and conclusion

Tensile tests of paperboard and its corresponding free-laid plies have been performed using crosshead speeds in the range 1–24,000 mm/min. Hence strain rates in the range 10⁻⁴–3 s⁻¹ were achieved according to the ISO 1924-3 standard, using L = 50 and 5 mm long specimens. A log-linear relation was observed where the strength increased 9 % and Young’s modulus 7 % per decade, it confirmed the rule of thumb that increasing the loading rate a factor 10 increases the strength and stiffness about 10 %. This effect was independent of MD or CD loading, neither did the fiber orientation nor ply type affect the behavior. This although the plies had different pulp compositions. The fact that the strength and stiffness increase with loading rate would indicate that the fibers become more dominating than the bonds at increasing rate. It should also be mentioned that the failure stress and Young’s modulus had the same behavior. When the loading rate was high, the fibers will not have time to deform within the network nor be torn out of the network. Neither will the failure crack have time to follow the weakest crack path across specimen. Consequently, the crack paths became straighter at high speeds.

Although an electro-mechanical testing machine was used it was difficult to reach crosshead velocities above 24,000 mm/min. These speeds are somewhat lower than the loading rates that has been reported for creasing operations. To compensate for the limited crosshead speed, also short tensile tests were used. By these strain rates that were 10 times higher could be measured. It was confirmed that for strength and stiffness a linear-log relationship could be achieved for rates up to 10 s⁻¹. During industrial converting it would hence be fair to assume that the fiber properties have even larger impact than indicated by lab measurements, the presented increase of strength properties can then be used to do prediction of material behavior at high loading rates. At high rates, there should be less plastic deformation, but more of both in-plane cracking and delamination compared to at low rates. One can assume that if the fibers have less time to deform in the network it will lead to a localization of damage, which will also lower the strain at break. Since CD samples have more fibers perpendicular to the loading direction this of lowered strain at break should be more dominating in these, since the fibers will have less time to bend to comply with the deformation.