An investigation of strength variations in paper containing a sharp notch
HÅKON NORDHAGEN¹, ØYVIND WEIBY GREGERSEN²
¹ PhD student, Norwegian University of Science and Technology, Dept. Chem. Engin, N-7491 Trondheim, Norway,
² Professor, Norwegian University of Science and Technology, Dept. Chem. Engin, N-7491 Trondheim, Norway,
ABSTRACT
We intend to investigate the hypothesis that it is the local material properties around a defect that determines the variation in individual critical strengths of notched paper.
Three paper grades (SC, newsprint, copy paper), 100mmx50mm samples, 20 samples of each grade, with an introduced edge cut were analyzed by a beta-formation apparatus (Fujifilm FORMEX/BAS-1800). Areas of different sizesand shapes at the crack-tip were then analyzed with respect to various basis weight statistics. Statistical parameters of the local basis weights like the mean, standard deviation and the minimum value were correlated to the strength of the individual paper samples. For copy paper, a highly significant correlation was found between strength and the local mean and minimum basis weight, but only for areas in a certain range (3mm²< area < 10mm²). Correlations were found to a much lesser degree in the newsprint and SC grade.
This method of studying paper strength might give novel information about small scale paper properties – not necessarily caught by the formation measure - that is important for fracture properties of paper. The method may also be developed to a tool for the paper maker to assess the severity of a detected defect.
INTRODUCTION
The tensile strength of paper has been subject to many studies and has been successfully correlated to the runnability in pressrooms[1]. Various models for paper strength, such as the model of e.g. Page[2] give a rough estimate of the mean tensile strength - but the parameters needed in the model, such as bond and fiber strength, are difficult to measure. Addressing runnability issues calls for a system approach, that is,one need to knowboth the distribution of the paper strengths as well as operating conditions such as web-tension variations and other dynamics. In the weak-link theory of paper one measure the distribution of tensile strengths and infer from this the probability of failure at loads closer to web operating conditions[3].
The weak-link theory[3] of paper strength describes the failure of paper through interacting local weak areas in the paper. That is, a loaded paper, sufficiently large, is viewed as a chain of smaller areas (critical clusters) of various strengths, and the paper fails macroscopically when the weakest of these areas fail.
This theory explains the fact that larger areas of paper have a lower tensile strength than small samples. It also explains some of the observations linking the formation of paper to the tensile strength distribution of paper – suggesting that micro formation (a scale of about 1mm) might be the most important length scale regarding the connection between formation and paper strength [4].
It also suggests a lower limit of the tensile strength in paper, but due to the large amount of samples needed, preferably several hundred in order to measure the tail properly, it is not a very practical approach.
Another methodfor estimating the paper strength is using the tools of fracture mechanics – it is from this viewpoint we choose to approach the runnability problem. This approach is basicallyanother version of the weak-link theory, but we are now reducing the problem by constructing a weakest-link, that is, a macroscopic defect (a cut) much larger than the size of the critical cluster.
Cuts, holes, shives and other defects normally - but seldom - found in paper, drastically reduce its strength. Defects originate from the production and later handling of the paper such as in the pressroom or during transport. Reducing the number of defects or putting a lower limit to the size of those passingthe hole-detection system found on many paper machines, can be an effective way to improve the runnability of paper.But macroscopic defects will always exist.
Although macroscopic defects in paper (>5-10mm) will not always explain web-break problems in e.g. a pressroom[5], theymay often be the source of problems - depending on the type and grade of the paper and the press and printing method used.
A large stress and strain concentration, with an accompanying damage zone, will build up around a defect when the paper is loaded[6]. Our hypothesis is that it is the local material properties around the crack-tip that determines, or at least will correlate with the strength of a notched paper (for a given notch length). This is, of course, only if the crack is of such size (> 3-5mm) that the fracture of the paper is initiated at the crack-tip – which is what we observe in all the experiments reported in this article.
It has been shown that by using a relatively simple non-linear elastic material model together with a critical J-integral criterion for the initiation of a break[7], strengths of large paper sizes (>1x0.5m) containing a cut are predicted reasonably well from small paper samples of the same grade[8]. Despite this, the industry has been very reluctant to adopt the fracture mechanics technique. One reason for this may be that the predicted strengths using this method are in many cases too large to be able to account for web breaks at operating conditions.
A more complicated model – incorporating for example 3D-effects such as out of plane buckling of paper and material model parameters for the high strain rates and moisture content often found in a pressroom situation is believed to bring the predicted strengths closer to the observed failure conditions.
Another approach to “justifying” fracture mechanics approach to the industry is to focus on the extreme statistics of the predictions. That is, because of the extremely rare phenomenon web-breaks represent, the predictions should report the most conservative strength estimate possible.
Few efforts have been made to explain or describe the strength variation in paper containing a defect[9].
A better understanding of the effects that govern the variation in strength of paper with a notch will be useful for the papermaker - both in order to assess the grade of the paper product and to report the most conservative estimate of the strength to the costumer. Especially in the process of measuring the fracture toughness of paper and details regarding the fracture mechanics of the system, it is of vital interest how local material variations will affect the fracture properties [10].
BACKGROUND
At the heart of fracture mechanics is the ability to predict, from experiments performed on small structures, the critical load and elongation of much larger structuresThis phenomenon is called transferability. The material's ability to withstand crack growth is often referred to as the fracture toughness – and is the energy released when the crack grows a small amount at the point of instability. For a recent thoroughreview and discussion see [11].
The fracture toughness, which may calculated from the maximum force needed to break a sample having a cut, may be considered as a material-model parameter[7]. The standard size of the small paper strips needed to measure the fracture toughness is normally 100x50mm with a 20mm sharp cut in the middle(SCAN-P 77:95). The paper is strained to break in a tensile testing apparatus, with the crack perpendicular to the pulling direction, and the force and elongation are determined when the crack starts to grow uncontrolled – so called critical force and critical elongation when there is a crack present. In Figure1thenormalized critical forces and elongations for 100 samples of 50x100mm SC paper (52g/m²) with a 20mm centre crack are plotted together with 100 tensile strengths (no pre-crack) of the same paper grade. All the strengths are normalized (divided by the mean value) in order to compare the two measures.
Figure1Normalized critical forces (tensile strengths) and elongations (stretch)for 100 tensile tests on 100x50mm SC-paper samples with 20mm center crack (*) and without (o). Also shown are the 95% confidence intervalswith crack (- -) and without a crack (-).
As can be seen from the plot, the range of values (95% confidence intervals are shown) is roughly within 10% of the mean normalized value, but the critical values confidence interval (--) shows a slightly larger span than the tensile strengths (-) interval. It is also observed that the spread of relative elongations arelarger than the spread of forces (different size of the rectangles inFigure 1). Although the details about the distribution of critical values will not be discussed in depth here, we have in Figure 2 presented a normal probability plot of the same normalized values (critical forces) as presented inFigure1. The distribution of critical forces with a crack (+) and without (o) do not seem to deviate much from the Gaussian (normal). A slight, but statistically significant (through F-test) difference in the standard deviation in critical values,reflecting the difference seen in Figure1, indicates that the two distributions are not similar.
It should be mentioned that in order to draw any conclusions about the type of distribution of strengths, many more samples are needed, but it is sufficient to mention that it is widely accepted that the distributions of tensile strengths should have a Weibull shape [4] and that for relatively uniform paper grades (high m-value = good formation), such as the SC grade,the Weibull distribution cannot be easily discerned from a normal distribution.
Figure2Normalpropability plot of 100 critical forces (seen in Figure1) with crack (+,-) and without (o,.-). Distribution types other than Gaussian would show up as a curvature in the plot.
However, this presentation is not about the distribution of strengths of paper, but about possible reasons for the wide range of strengths for seemingly identical paper strips containing a defect.
For tensile tests on paper strips without a defect, it has been observed that the failure zone passes through areas where the basis weightis below average [12].No method has yet, to the authors’ knowledge, been able to tell whether a strip of paper will break a below or above the average strength of aspecific type of paper. A simple experiment, with a non-surprising result, was performed in order to illustrate such a correlation: 24 samples of 52g/m² SC paper, exactly 50mm wide and 180mm long was weighted and stretched to failure (normal tensile test). In Figure 3the results are plotted and a correlation with a p-value, calculated from student t-distribution with 23 degrees of freedom, of 0.02 was found.
Figure 3 Tensile tests of 24 identical 180x50mm strips of SC paper individually weighted. A significant correlation was found between the weight of the sample and its tensile strength.
That is, if there were no correlation, the chance that the plotted values arose by pure chance is about 0.02 – ergo highly significant. Apart from this obvious correlation, we have found noliteraturedescribing the correlation of the strength of individual paper samples with any other physical property such as e.g. a formation measure of the sample or shape and position of the fracture zone – that varies from sample to sample. For paper samples with a crack the situation is quite different. The crack is expected to start growing at or near the crack tip - and it is therefore natural to search for a correlation between material properties in this region and the strength properties of the sample.
METHODS AND EXPERIMENTS
The papers to be investigated were first cut into A4 sheets and the region of interest (about 120x120mm), where the beta-formation was to be measured, were marked by 8 through-going holes (as shown in Figure4) in order to align the basis weight image with the position of the crack-tip at a later stage.
Figure4 A sketch of the paper to be scanned.
Two 120x50mm paper strips were then cut from each A4 sheet (12 for each grade) making sure that the 4 holes on each strip were outside the 100mm region that was to put between the clamps in the tensile test performed later. A 6mm edge cut in CD direction was made in each strip, and the sample was scanned - using transmitted light in a flatbed scanner (in the same resolution as the beta-formation scanner) for later identification of the cut in the beta-formation pictures.
The sharp cut was barely visible in the beta-formation pictures so the method of flatbed-scanning the paper strips after the edge crack had been made was found necessary.
All the digital image processing were done using the ImageJ software[13]. A plug-in script for ImageJ, called TurboReg[13], was used to align the scanned picture and the beta formation picture. An example of two aligned pictures of copy paper can be seen in Figure 5 where one easily can see the common details.
Figure5 Aligned pictures (copy paper) used for identifying the region around the crack-tip. The transmitted light scan seen to the left and the betaformation scan seen to the right. The region shown is about 8mm in length.
Another plug-in called Image Correlation1e[13], which calculates the correlation coefficient between two images, was used to control that the pictures were reasonably aligned. Even a small disturbance in the picture alignment would show up as a big change in the correlation coefficient. The copy and newsprint gradeachieved a higher correlation coefficient than the SC grade. This is probably caused by the high degree of calendering in the SC grade – making dense areas with much optical contact show up as regions of high light transmission.
Measurements and correlations
The physical properties resulting from the tensile tests, namely the maximum measured force (critical force) and the corresponding elongation (critical elongation), were the two strength parameters of interest. The functional relationship between these and the basis weight properties around the crack-tip were then investigated. The region of interest around the crack-tip, illustrated in the right picture in Figure 5, was varied both in shape and size. Figure 6 shows the regions where the basis weights were analyzed.
Figure 6The circles, with diameters of 2,4, 8 and 14 mm, show the region where basis weight information was captured. Light areas have lower basis weight than darker. The type of paper shown is SC.
The choice of areas for the circles shown in the pictures was based primarily on the fact that the fracture process zone, which is expected to be the place where a through going crack will start to grow, having the size of about 1-10mm [14].
Statistical analysis (see methods) of the different regions were all performed using the ImageJ software, which also provide effective batch processing of the images. Measures of the greyscale values that were investigated was the mean, mode, median, min/max value, standard deviation, and kurtosis.
Theimage intensities were corrected according to the simultaneously recordedMylar film weights with basis weights equivalent to 50, 100, 150 and 200 g/m².
Paper materials
Three paper grades, SC (super calendered) 52g/m², newsprint 46g/m², and Canon copy paper 78g/m², were investigated in this study. The SC grade contained a mix of TMP fibers and 10-15% long fiber kraft pulp. The copy paper was found to be made of a mix of various sorts (pine, aspen, birch) of bleached long and short fibers. Both the SC and copy grade contained fillers (20-30% of weight) while the newsprint did not contain any.
All paper grades were kept at 23 degrees Celsius and at 50% relative humidity for at least 48 hours before testing.
Apparatus
Tensile tests were performed using a L&W Tensile Tester with Fracture Toughness, and the raw data (elongation and force) was collected for each experiment using a RS-232 cable and a PC for logging. The tensile tests were performed at a strain-rate of 100%/min.
The acquisition of the beta formation images was done using the Fujifilm FORMEX/BAS-1800 system.The spatial resolution was 50 µm, with 16bit grey tone resolution. The flatbed scanner used was aEpson Reflection 4990 PHOTO scanner. Scanning was performed at a resolution of 508dpi – the same as the beta-formation scanned images.
In order to keep the sample flat while scanning, a plate made of clear transparent glass was put on top of the sample to be scanned.
Statistical Methods
We have used three methods, the Pearson, Spearman and the Kendall method [15] to test for correlations between our two physical variables X (critical strength and elongations) and the data from the beta-formation pictures Y. We also report the significance level, which should be less than 0.05 for statistical significance, for each observed correlation.
Pearson’s method is the usual measure of correlation.Pearson's r is a measure of association which varies from -1 to +1, with 0 indicating no relationship (random pairing of values) and 1 indicating perfect relationship. It is calculated from covariance of standardized variables:
Here cov(X,Y)is the covariance, defined as∑((x- mean(x))(y – mean(y)) ).
The significance of the correlation or the p-value (defined significant when <0.05) is computed using a Student's t distribution for a transformation of the correlation - which is exact when X and Y are normal (see [16] for the corr() function).
The Spearman and Kendall methodsare rank correlations and do not assume normal distributions of X and Y. They are also less sensitive to outliers, making them an important contribution in our experiments, where experimental errors can occur in many of the steps described above and causing outliers.
Frequently ρ is used to abbreviate the Spearman correlation coefficient. For pairs of values (here n=20) it is defined as