Identifying the weak spots in packaging paper: local variations in grammage, fiber orientation and density and the resulting local strain and failure under load

In this study the local structural properties of four \(73\,\hbox{g}/\hbox{m}^{2}\) Kraft paper strips are linked to their strain and damage distributions during tensile testing. The paper is produced from unbleached softwood Kraft pulp. Figure 1 illustrates the principle to link local paper properties by showing four maps of the same paper specimen imaged in different ways: (a) a photograph, (b) an IR thermogram, (c) a \(\upbeta\)-radiograph. In the last analysis step each sample is split into 14–18 layers to measure the local fiber orientation, Fig. 1d shows a high resolution image of one of the split layers, the enlarged region shows that the local fiber orientation is visible. The dashed lines are indicating the region of interest (ROI) where all local paper parameters were determined in a resolution of \(1\,\hbox{mm}^2\).

The strips had dimensions of \(40\times 65\,\hbox{mm}^{2}\) including a region of interest (ROI), i.e. the region where the linking of local properties was conducted, with dimensions of \(35\times 60\,\hbox{mm}^{2}\). The ROI was marked identically on both sides of each specimen with four dots indicating the corners of the ROI. On the front side, where the strain map was determined with a photography during tensile testing and a subsequent DIC analysis, the marking was done with a felt pen. Furthermore, a random speckle pattern was printed (HP Color LaserJet 4700n) on the front side to enable an accurate DIC analysis. All relevant DIC hard- and software parameters are given in Online Resource 1. On the back side, where the IR thermogram was measured with an IRT during tensile testing simultaneously with the strain map, the landmark dots were marked with silver ink due to its high IR reflectance which made the dots clearly visible in the recorded thermograms. High local basis weight within the silver ink dots made them also clearly detectable in the measured \(\upbeta\)-radiographs.

In addition, the region where considerable damage occurred just prior to tensile rupture is highlighted with a circle in the IR thermogram in Fig. 1. The same region is also marked to the photograph, the \(\upbeta\)-radiograph and the split layer of the specimen. This particular example illustrates that the damage occurred in the region with low basis weight surrounded by regions with high basis weights in the straining direction, i.e. machine direction (MD). In order to study the relationships between local structure, local strain and local damage comprehensively, local basis weight, local thickness, local density and local FO were linked to local strain and local temperature increase (thermal energy dissipation) just prior to tensile rupture as shown in Fig. 2. Local basis weight was studied with a \(\upbeta\)-radiographic transmission method, local thickness with a twin laser profilometer (TLP) and local density by dividing the former by the latter. A sheet splitting method was in turn utilized to measure the local fiber orientation (FO). As already mentioned, local temperature increase (thermal energy dissipation) was measured with infrared thermography IRT and local strain with the combination of photography during tensile testing and digital image correlation DIC. In the following all these measurement methods as well as the linking procedure are presented more in detail.

The set-up for measuring strain and damage distributions is shown in Fig. 3. The strain map was determined on the front side of the tensile tester (Zwick/Roell Z010) with a \(10\,\hbox{kN}\) load cell. First the tensile test of the paper strip was recorded with a digital single-lens reflex (SLR) camera (Nikon D5100) combined with a macro lens (Nikon ED AF Micro Nikkor \(200\,\hbox{mm}\) 1:4 D). The frame rate and resolution of the recording were 25 Hz and \(37\,\upmu \hbox{m}/\hbox{pixel}\), respectively. The ROI within recorded photographs was then analyzed with a two-dimensional (2D) DIC (Eberl et al. 2006) by utilizing the printed random speckle pattern. A comprehensive review of the 2D DIC is given by Pan et al. (2009). The DIC analysis was conducted with a subset size of \(20 \times 20\) pixels and thus the strain map resolution became 0.74 mm/pixel. The step size was in turn 15 pixels (0.56 mm). Finally, the determined strain map was resized (MATLAB imresize function) to the size of the ROI just prior to rupture, i.e. \(35\times 61\,\hbox{mm}^{2}\).

Simultaneously with the measurement of the strain map on the frontside, temperature increase (thermal energy dissipation) map was recorded on the backside of the tensile tester. The principle of studying paper damage with IRT is described in detail by Yamauchi (2012). The key idea is that during tensile testing fiber–fiber bonds are starting to fail long before the actual macroscopic rupture of the paper. The strain energy stored in the fibers is dissipated as heat when the bond between fibers is failing. Thus the local heat distribution in the paper provides information on where fiber–fiber bonds in paper are failing. Thus we use the local temperature as an indicator for local failure of the paper or local damage in the bonding of the paper.

We used a long-wavelength (7.5–13 \(\upmu \hbox{m}\)) IR camera (Optris PI 450). The long-wavelength IR emittance was approximated to be 0.93 (Hyll 2012). The frame rate and resolution of the recording were in turn 80 Hz and 0.38 mm/pixel, respectively. The temperature increase map just prior to tensile rupture was determined from the recorded thermograms by resizing (MATLAB imresize function) the ROI within thermogram prior to straining (\(35\times 60\,\hbox{mm}^{2}\)) to the same size as the ROI within thermogram just prior to rupture (\(35\times 61\,\hbox{mm}^{2}\)) and then subtracting the unstrained ROI from the strained one.

The tensile test was performed in MD with a strain rate of \({40}\,\hbox{mm}/\hbox{min}\) under a standard climate of \(23\,^{\circ }\hbox{C}\) and 50% relative humidity. The tensile test was not performed until complete rupture of the strip but stopped just prior to/during rupture, i.e. at the strain level of 2.2% (size of the ROI \(35\times 61\,\hbox{mm}^{2}\)). The reason was because the measurement of local fiber orientation (sheet splitting method) is also a destructive method, and thus the local structural properties were measured after the measurement of strain and damage distribution.

The development of strain and temperature maps during tensile test was investigated through calculation of mean and variance over ROI (see Fig. 1). Every recorded strain map (25 Hz) and every fifth temperature map (16 Hz) was analyzed. For strain and temperature mean and variance were calculated.

The local basis weight was measured with a \(\upbeta\)-radiographic transmission method described in detail by Keller and Pawlak (2001). First the \(\upbeta\)-radiographic transmission through a paper strip and references with known basis weights was recorded on a storage phosphor screen in a Formex Betaformation Box (Science Imaging Scandinavia AB). Second the screen was scanned (Fujifilm BAS 1800-II) and the basis weight map within the ROI of the strip was calculated with a resolution of \(50\,\upmu \hbox{m}/\hbox{pixel}\) by using the references.

The local thickness was measured with a TLP described in detail by Keller et al. (2012). TLP determined the surface topography of the paper strip simultaneously on both sides with the aid of two lasers. The local thickness was then calculated with a resolution of 100 \(\upmu \hbox{m/pixel}\) by using the surface topographies. Due to the high local thickness within the silver ink dots, the dots were clearly visible in the thickness maps. Local density was calculated by dividing local grammage by local thickness.

The local FO was measured with a sheet splitting method described in detail by Hirn and Bauer (2007). As it is a destructive method, the determination of local FO was the last measurement step. Before starting the procedure the landmark dots were made visible by bunching holes on the corners of the ROI. The local FO measurement was started by splitting the paper strip in thickness direction (ZD) into thin layers and subsequently scanning (Nikon Super CoolScan 8000 ED) the layers with a resolution of \(13\,\upmu \hbox{m}/\hbox{pixel}\). Each scanned layer was then analyzed through image analysis. In other words, each scanned layer was divided into subsets of \(1\times 1\,\hbox{mm}^{2}\) followed by a determination of FO distribution within each subset. The FO map for all the layers in local \(1\times 1\,\hbox{mm}^{2}\) subsets was then determined by averaging the FO distributions of the layers in ZD and subsequently fitting orientation ellipses to the averaged distributions. The averaged FO distribution and fitted orientation ellipse of one subset are shown in Fig. 4c. The FO of a subset, i.e. FO vector, is defined with the aid of FO anisotropy and FO angle (\(\theta\)). FO anisotropy is the eccentricity (e) of the orientation ellipse calculated as follows:where a is the major semi-axis and b the minor semi-axis of the ellipse. FO angle \(\theta\) is in turn the angle between the major semi-axis of the ellipse and MD. The magnitude and angle of the FO vector are then defined as the eccentricity and FO angle, respectively.

As local FO angle and local FO anisotropy combined are needed to investigate the influence of local FO on local strain/damage, a variable called FO load carrying factor (FOLCF) is introduced. It links the local FO to the local ability of the fiber network to carry tensile load and is technically defined as the local probability density of fiber segments pointing in load direction (2r) within a subset. The FOLCF is calculated by using the polar equation of the orientation ellipse as follows:

In order to enable a quantitative pointwise correlation, the measured maps were aligned and rescaled to a resolution of \(1\times 1\,\hbox{mm}^2\) per pixel. This was done through a landmark based image registration procedure by utilizing the four landmark dots visible in the corners of each map and aligning the images using these landmarks. The maps were aligned using a shape preserving coordinate transform, this procedure is described in Hirn et al. (2008). At this point the density map was also calculated by dividing the aligned basis weight map with the aligned thickness map. The noise of the aligned maps was then reduced by the aid of Gaussian low pass filter with a kernel size of \(5\times 5\,\hbox{mm}\) pixels and a standard deviation of 1.0 (MATLAB imgaussfilt function).

The aligned dataset of one paper specimen is shown in Fig. 5. Please note that, as a result of the registration procedure described above, all the maps have a common coordinate system where each pixel represents a \(1\times 1\,\hbox{mm}^2\) subarea on the paper specimen. Furthermore, the size of the final data maps was \(32\times 57\,\hbox{mm}^{2}\), i.e. slightly smaller than the actual ROI just prior to rupture (\(35\times 61\,\hbox{mm}^{2}\)), due to removal of the landmark dots from the analysis. By extracting all the structural parameters, the strain and the temperature from all the data maps at one specific coordinate position, all the local properties of a specific subarea on the sample can be collected. In this way 7296 data points with local structural and local deformation/damage data are collected from the four paper specimen analyzed. One of these data points at the coordinate position \(\hbox{x}=20/\hbox{y}=10\) is indicated in Fig. 5, its according properties can be read from the data maps.

In total 48 local data values were detected as outliers using Grubbs’ test with a significance level of 0.05. The quantitative analysis of relations between strain map, damage map and structural maps was conducted through correlation, simple/multiple linear regression and analysis of variance (ANOVA) of the 7296 data points (outlier data values excluded) collected (Neter et al. 1996).

The investigated Kraft paper had a basis weight of \(73\,\hbox{g}/\hbox{m}^{2}\) and density of \({1041}\,\hbox{kg}/\hbox{m}^{3}\). The global mean FO paramaters were in turn as follows: FO anisotropy 0.71, FO angle \(22.7\,^{\circ }\) and FOLCF 1.13. The tensile stress at break was \({111\pm 3}\,\hbox{MPa}\) in MD and \({58\pm 2}\,\hbox{MPa}\) in CD. However, in the current study the tensile test in MD was stopped just prior to rupture at strain level of 2.2% and the corresponding tensile stress was \(100\pm 2\,\hbox{MPa}\) (see Fig. 6a). In the following the local properties of the investigated Kraft paper are discussed.

Figure 5 qualitatively demonstrates the relationships between the measured local structural properties, the measured local deformation and the measured local damage. It shows the local basis weight, thickness, density, fiber orientation (FOLCF), strain and temperature increase for one of the studied paper strips. Clear mesostructural inhomogeneities are visible and similar conclusions can be made as in the previous investigations described in “Introduction” section of this paper. In other words, both the local strain and local temperature seem to correlate to a certain extent with the local basis weight. Similarities can be also seen between the strain and the damage distribution maps. Furthermore, the other structural maps seem to correlate to a certain extent with the strain and the damage distribution maps as well. In order to investigate these causalities in a more detailed and systematic manner a quantitative statistical analysis is utilized.

Before entering to the statistical analysis of the local properties the development of strain and temperature maps during tensile test is discussed. Fig. 6 shows averaged (all four specimen included) stress, averaged temperature, variance of temperature and variance of strain as a function of paper strain. It can be seen, Fig. 6b, that the temperature decreased first in the elastic region of the stress–strain curve and then increased considerably after reaching the plastic region. This behavior is well in line with previous studies and can be explained with thermoelastic effect (Dumbleton et al. 1973; Hagman and Nygårds 2017). The increase of the temperature at 0.7% strain (red line) indicates the onset of fiber–fiber bond failure, which, as discussed above, is caused by the heat dissipation of the strain energy in the fibers when the bonds break. This corroborates with results from acoustic testing (Isaksson and Hägglund 2013) of paper during tensile testing where the onset failure of fiber–fiber bonds was also observed at a strain considerably below the macroscopic failure of the paper.

The variance of temperature describes the localization of damage in the sample, Fig. 6c, i.e. when regions with locally higher damage are forming. The variance of temperature decreased also first in the elastic region. According to Hagman and Nygårds (2017) this can be explained with reduced noise due to cooling of the strip. However, the variance of temperature started to increase only in the middle of the plastic region (blue line at 1.5% strain). In other words, the damage started to strongly localize after reaching the second half of the plastic region, well after the initiation of bond failure (red line). Interestingly, the variance of strain increased constantly over the whole strain range, Fig. 6d. This in turn means that the strain started to localize much earlier in comparison to damage. Taking into consideration that local strain and local temperature increase correlate, Fig. 7, it is plausible that the local damage was preceded by the high local strain. This is well in line with the findings of Borodulina et al. (2012) who concluded that high local strain is the precursor for local failure of fiber–fiber bonds. In other words, the data presented here suggests, that already at a low load regions with locally higher strain are emerging. After onset of bond failure at 0.7% strain the strain keeps on localizing, the damage initially is not showing increasing inhomogeneity. This is changing at 1.5% strain where also the damage is starting to localize and regions with locally concentrated bond failure are occurring. Catastrophic failure (sheet rupture) finally initiates within one of the regions where local failure has accumulated and reached a critical level. In low density papers, according to Krasnoshlyk et al. (2018b), the failure propagation in the fracture process zone ahead of the crack tip is seen as an increase in paper thickness and porosity. These increases are caused by rotation and twist of the bonded fibers and gradual breaking of the fiber–fiber bonds. Figure 1b shows the damage (temperature) distribution just prior to rupture of a sheet. The white circle shows a region with significantly increased temperature, local failure has started to heavily concentrate in this region, sheet rupture is about to initiate in this region.

Figure 7 shows a scatterplot describing the pointwise relationship between local temperature increase maps and local strain maps of the four paper strips just prior to rupture, i.e. at the endpoint of the stress–strain curve shown in Fig. 6. Each spot in this scatterplot is representing one physical \(1\times 1\,\hbox{mm}^2\) subarea on the sample and the respective temperature and strain values in this spot, like e.g. the green spot at coordinates \(\hbox{x}=20/\hbox{y}=10\) in Fig. 5e, f. It can be seen that a positive linear correlation exists between these two distributions (\(r = 0.65\)). By taking into account that the local temperature increase was preceded by the high local strain, it can be said by fitting a linear regression model and calculating a coefficient of determination (\(R^2\)) that the local strain explains 42% of the total variation in the local temperature increase (\(R^2=0.42\)). This also means that the remaining difference 58% are explained by other variables and local strain is not the only indicator for local failure. Hence local strain and the local temperature increase, although related, are partially different which supports the conclusion that local paper strain and local paper failure are related, but not governed by the exactly same parameters.

In the following the above analyzed local strain and temperature maps are linked to the corresponding paper structure.

We begin the analysis by studying the pointwise correlation between strain maps and structural maps of the four paper strips. Figure 8 shows scatterplots describing these relationships. Like described in “Linking procedure” section and illustrated in Fig. 5 each spot in a scatterplot represents one \(1\times 1\,\hbox{mm}^2\) subarea on a paper specimen, and its respective local properties.

First it can be observed that the existing correlations are negative, \(r<0\). Local strain of the paper tends to be higher for regions with less local grammage, lower local density and less fiber pointing in the direction of the load (FOLCF). The highest, yet still moderate, negative linear correlation exists between local strain and local basis weight (\(r=-0.45\)). Furthermore, the local basis weight explains 20% of the total variation in local strain (\(R^2=0.20\)). A relevant negative linear correlation exists also between local strain and local fiber orientation (FOLCF, \(r=-0.37\)). The local FOLCF is capable of explaining 14% of the total variation in local strain (\(R^2=0.14\)), i.e. it is only a slightly weaker predictor for local strain than local basis weight. This is a remarkable result, it suggests that local fiber orientation is about equally relevant for the local strain of paper under tensile load like local grammage. Interestingly enough, Praast and Göttsching (1997) concluded that the local orientation of fibers correlates with the orientation of flocs (high basis weight regions). This means that the orientation of flocs might influence on the local strain as well. A weak negative linear correlation is in turn present between local strain and local density (\(r=-0.18\)). The local density explains 3% of total variation in local strain (\(R^2=0.03\)). These results indicate, that the well known relations between the structural properties of paper, i.e. basis weight, fiber orientation and density, and paper stiffness described in “Introduction” section also apply to the local straining under tensile load. Finally, no correlation is present between local strain and local thickness (\(r=-0.04\) and \(R^2=0\)). This can be explained with the fact that density variation is independent of thickness variation (Dodson et al. 2001).

Dodson (1970) found earlier a significant correlation between local deformation (dilatation) and local basis weight of fibrous networks (see “Introduction” section). Furthermore, he predicted this relation by using a statistical theory of elasticity. In the following we utilize the same approach to predict the relation between local basis weight and local strain shown in Fig. 8a. The theoretical (th) relative strain (\({\widetilde{\delta }}_{th}\)) as a function of local grammage (\({\widetilde{g}}\)), i.e. theoretical regression line, was locally calculated with the following equation:where strain is denoted by \(\varepsilon\), local value by tilde and global mean value by bar. The theoretical correlation coefficient (\(r_{th}\)) was in turn calculated as follows:This allowed the theoretical prediction for scatter about the theoretical regression line by calculating the standard deviation (\(\sigma\)) of the scatter:In Fig. 9 the theoretical regression line as well as \(\pm 2\sigma _{th}\) theoretical limits for scatter are plotted together with the experimental data (see Fig. 8a). It can be seen that the theoretical predictions are in good agreement with the experimental relations. i.e. the slopes of experimental and theoretical regression lines are close to each other, 95% of the experimental data points fall within the error bounds and the difference between \(r_{th}=-0.59\) and \(r=-0.45\) is not large. These conclusions were also made by Dodson (1970) and his correlation coefficients \(r_{th}=-0.56\) and \(r=-0.52\) are similar to ours. Thus, the statistical theory of elasticity is suitable for predicting the relation between local strain and local grammage also in the case of Kraft paper.

Scatterplots describing the pointwise relationship between local damage (temperature increase maps) and local structural maps of the four paper strips are in turn shown in Fig. 10. Similar like for the local straining behavior, also local failure of fiber–fiber bonds (temperature increase) tends to occur more pronounced in paper regions with less local grammage, lower local density and less fiber pointing in the direction of the load (FOLCF). The strength of the negative linear correlation between local temperature increase and local basis weight (\(r=-0.36\)) is slightly weaker than the correlation between local strain and local basis weight. The local basis weight explains 13% of the total variation in local temperature increase (\(R^2=0.13\)). A weaker negative linear correlation is also present between local temperature increase and local fiber orientation (FOLCF, \(r=-0.17\)). The local FOLCF is capable of explaining only 3% of the total variation in local temperature increase (\(R^2=0.03\)). On the other hand, the linear negative correlation is at the same level between local temperature increase and local density (\(r=- 0.22\)) in comparison to the correlation between local strain and local density. The local density explains 5% of the total variation in local temperature increase (\(R^2=0.05\)). As in the case of local strain and local thickness, no correlation exists between local temperature increase and local thickness (\(r=0.05\) and \(R^2=0\)). In general the correlations of local paper structure to local bond failure, Fig. 10, are somewhat lower than to local strain, Fig. 8. It is unclear if this difference is significant or not.

So far simple linear regression model has been utilized to study the relationships between local strain, local temperature increase and structural maps. In the following the analysis is extended to multiple linear regression model. This means that the local strain and local temperature increase are modelled as a linear function of local basis weight, local thickness, local density and local FOLCF. The analysis revealed that the structural properties combined explain 39% of the total variation in local strain (\(R^2=0.39\)). In other words, the local basis weight, local thickness, local density and local FOLCF are locally able to predict the local strain to a degree of 39%. In the case of local temperature increase the structural properties combined explain in turn 25% of the total variation. This means that the structural properties predict the local strain better than the local failure of the material (temperature increase). The remaining differences of \(R^2\) to 1 is 0.61 and 0.75 for the local strain and the local temperature increase, respectively. These differences can be explained with other influence factors, i.e. other structural variables not measured, inaccuracies/noise in the measurements, data map registration errors, et cetera. Analyzing the intercorrelation between the paper structural parameters revealed low linear dependance, which is also confirmed by the significantly higher correlation of the multiple regression model than the individual correlations. This leads to the conclusion that the structural parameters paper formation (local grammage), local fiber orientation and local density are independently responsible for the weak spots contributing to the distribution of local strain and local bond failure in paper.

The final catastrophic failure, i.e. sheet rupture, is always initiating from regions with locally higher strain and bond failure observed already earlier, at lower load. In the final part of the analysis we are focusing on these more extreme regions of the paper, i.e. the regions with particularly high and low local strain and local fiber bond failure. In order to do that the amount of analyzed pixels was reduced as illustrated in Fig. 11. More and more pixels were removed from the middle local strain/temperature increase range and only pixels from the lowest and highest ends were included to the analysis. For example when 10% of the strain pixels were included, 5% of the pixels with highest strain and 5% of the pixels with lowest strain were chosen to the analysis. Pixelwise correlations between local structural properties, local strain and local temperature increase were then investigated similar like above, but utilizing the selected pixels only.

Figure 12 shows that the correlations and \(R^2\) values (simple linear regression models) between local strain and temperature increase and local structural properties generally increase with decreasing percentage of pixels. This means that the structural paper properties are better able to predict the local strain and local fiber bond failure in the more extreme regions of the paper. The linear correlation between local strain/temperature increase and local thickness differs from this behavior as no correlation started to occur even at low pixel percentages. This further illustrates the fact that the density variation is independent of thickness variation. It is interesting to note that the best predictors in descending order for the regions with pronounced local strain (solid lines) are: local basis weight, local fiber orientation (FOLCF) and local density (as in Fig. 8 where 100% of the regions included). However, for prediction of local failure the order (descending) is in turn following: local basis weight, local density and local fiber orientation (FOLCF) (as in Fig. 10 where 100% of the regions included).

Finally in Fig. 13 the local strain and local temperature increase are modeled as a linear combination of the local structural properties with only more and more extreme regions being considered (as explained in Fig. 11). It can be seen from Fig. 13 that the \(R^2\) value increases with the decreasing pixel amount in both cases. The local structural properties combined explain the total variation in local strain overall better in comparison to the local bond failure (temperature increase) of the paper. In the case of 2% of pixels the local structural properties combined are locally able to predict the local strain to a degree of 87% and the local temperature increase to a degree of 72%. The results suggest that for the most extreme regions on the paper (i.e. the regions with the highest- and lowest deformation), the local variation of the analyzed paper structural properties local grammage, local fiber orientation and local density is highly related to the local concentrations of paper strain and fiber–fiber bond failure.

The relations in the current study were found by investigating unbleached softwood Kraft paper. However, these relations might differ to various degrees depending on the paper grade. For example, the density of paper (Krasnoshlyk et al. 2018a) and addition of dry-strength agent (Isaksson and Hägglund 2013) influence on the strain and damage distribution in paper. Thus, in the future it would be interesting to investigate how well the findings of the current study apply to other paper grades.