# Characterisation of time-dependent, statistical failure of cellulose fibre networks

### Abstract

Cellulosic materials have special advantages for transport packaging, because of their light-weight and recyclable natures and also relatively high specific strength. The strength of such materials is normally evaluated by applying monotonically increasing, quasi-static displacement (or load). However, in real circumstances, the material is subjected to far more complex loading histories, such as creep, fatigue, and random loading. Failures under such circumstances are, not only time-dependent, but also notoriously variable. For example, the coefficient of variation for creep lifetime reaches or even exceeds 100%. The objective of this study is to develop a method to characterise both time-dependent and statistical natures of failures of cellulosic materials. We have used a general formulation of time-dependent, statistical failure, originally proposed by Coleman (J Appl Phys 29(6):968–983, 1958). We have identified three material parameters: (1) characteristic strength, representing short term strength, (2) brittleness parameter (or durability), and (3) Weibull shape parameter related to long-term reliability. These parameters were determined by special protocols of creep and constant loading-rate (CLR) tests for a series of containerboards. Results have shown that these two test methods yield comparable values for the materials parameters. This implies the possibility of replacing extremely time-consuming creep tests with the more time-efficient CLR tests. Comparing the cellulose fibre networks with fibres and composites used for advanced structural applications, we have found that they are very competitive in both reliability and durability aspects with Kevlar and glass-fibre composites.

### Graphical abstract

## Keywords

Time-dependent failure Statistical failure Fibre network Strength Creep Characterisation Durability Reliability## Introduction

Material strength is traditionally evaluated by critical stress at which the material fails under monotonically increasing displacement (or load) conditions. However, it is known that materials often fail even at a much lower stress level, if it is subjected to stresses over a prolonged period, e.g., creep and fatigue conditions. Material strength is generally time-dependent or loading-history dependent. An interesting question may be “Does short-term strength predict long-term strength?”.

Figure 1 shows an example of creep failure of corrugated boxes in compression, tested by Nyman (2004). (The work is based on his PhD work, and the authors received the data as his courtesy).

Prior to creep tests, ordinary compression tests (Box Compression Test, BCT) were performed for Box A and Box B, both of which are made of B-flute with the same size and configuration. Results showed that Box B was stronger (2.92 kN) than Box A (2.44 kN). However, creep strength, as measured by time-to-collapse (lifetime), showed the opposite: Box A had a longer lifetime than Box B (Fig. 1a). In other words, Box A is more durable than Box B, even though the “ordinary” strength is lower. This poses a fundamental question of the relevance of typical static strength to longer-term strength properties. Another important aspect of long-term performance is variability. Each data point in Fig. 1a is, in fact, an average of 10 measurements of creep lifetime. Its scatter is plotted at each load level in Fig. 1b. The variations of lifetimes among the boxes were extremely large, and the coefficient of variation (COV) varied from 34 to 77% for the different load levels, although the measurements were done under a *nominally constant* environmental condition.

Similar large scatters of box creep lifetime were also reported in the literature from early days (e.g., Kellicutt and Landt 1951; Stott 2017; Moody and Skidmore 1966; Koning and Stern 1977; Popil and Hojjatie 2010). This large uncertainty of lifetime is a source of overdesign (i.e., taking higher safety factor) of boxes which are subjected to creep or long-term loading. Therefore, an important question is how to evaluate this multi-faceted nature of long-term strength, particularly the *durability* aspect, and also the *uncertainty* (or conversely, certainty or *reliability*) aspect, in a general framework.

In the literature, time-dependent failure has been dealt with in the areas of fatigue strength and creep strength (e.g., Murakami and Endo 1994; Wilshire 2002) for many years. Because of its complexity of underlying mechanisms, the approaches are largely empirical and phenomenological, but a few important empirical laws has been found (e.g., Miner 1945; Hashin and Rotem 1978; Monkman and Grant 1956), which are widely used for organising fatigue and creep strength data and also provided insights for the later development of damage evolution models [e.g., Curtin and Scher 1997]. The basic limitation of these approaches is, however, that they are mainly deterministic, and thus there is no systematic treatment of the huge scatter inherent to fatigue and creep strength. In the area of cellulosic materials, the subject has been actively investigated as creep failure of container box under constant or cyclic humidity conditions (Kellicutt and Landt 1951; Stott 2017; Moody and Skidmore 1966; Koning and Stern 1977; Bronkhorst 1997; Kirkpatrick and Ganzenmuller 1997; Popil and Hojjatie 2010). Although enormous scatters of creep lifetime data were recognised in the early literature, systematic treatment of time-dependent, statistical failure has been still in infancy for cellulosic materials (e.g., for review, Coffin 2011).

The first rigorous treatment of time-dependent, stochastic failure is, probably, Coleman’s model (Coleman 1958). He considered a fibre subjected to a general loading history (not only creep and fatigue), and obtained an expression for lifetime distribution based on three postulates: (1) weakest-link scaling, (2) damage evolution laws (exponential or power laws), and (3) an algebraic form of probabilistic failure condition. In the cases of creep-type loading history and constant loading rate (CLR) history, the model predicts Weibull distributions for lifetime and strength distributions, respectively. Although Coleman’s model is for “fibre” (a chain of breakable elements), Phoenix and coworkers extended Coleman’s approach to a system of fibre bundles (a parallel arrangement of breakable elements) as a model for uniaxially-reinforced fibre-polymer composites (Phoenix 1978; Phoenix and Tierney 1983; Tierney 1982; Newman and Phoenix 2001; Mahesh and Phoenix 2004). The lifetime distributions of this system depend on the assumed load-sharing rules. In the case of the local-load sharing rule (a more brittle system), it shows a non-Weibull distribution. The weakest-link-scaling behaviour appears only after the system size grows sufficiently large, unlike the fibre case. Christensen and Miyano (2006, 2007) took a different approach by considering the time-dependent growth of a single crack and treating critical stress as a random variable. Interestingly, the resulted expression for the relation between creep lifetime distribution and strength distribution was very similar to that obtained by Coleman.

In order to investigate how the time-dependent, statistical failure characteristics of fibre, as formulated by Coleman, is translated into the behavior of fibre network, the authors used a central-force, triangular lattice network and performed Monte-Carlo simulations of creep failure (Mattsson and Uesaka 2015, 2017). We found that the weakest-link scaling asymptotically appears with increasing the system size, as the fibre bundle model also showed (e.g., Newman and Phoenix 2001). Interestingly, the damage evolution law defined for fibre was preserved even in the fibre network level. The creep lifetime showed a distribution slightly deviated from Weibull distribution. We found that the resulted distribution is the same double-exponential form distribution, called DLB-type distribution, which was initially found for static strength of random fuse models (Duxbury et al. 1987) and later fibre bundle models (Wu and Leath 1999). However, it is important to note that, within a typical, experimentally-acceptable probability range (e.g., 0.01–0.99), the non-Weibull feature is subtle, thus very difficult to detect in a statistically significant manner. In other words, Weibull distribution approximately holds for lifetime distribution of fibre networks.

In conclusion, although the model proposed by Coleman for fibre is phenomenological, the basic relationship describing time-dependent, statistical failure is still preserved on the fibre network level. The only precaution is that the size dependency of the material parameters is different from that for typical Weibull distribution and the prediction of the lower tail of the distribution is always conservative because of the slight non-Weibull feature.

## Theoretical background

*f*(

*t*) is load at time

*t*, and \(S_{c}\), \(\rho\), and \(\beta\) are material parameters, as we will describe shortly. This formula was originally derived by Coleman (1958), later generalised by Phoenix (1978), and also re-derived by Curtin and Scher (1997) based on a damage-evolution model. The most important feature of this model is that one can take into account any loading history, such as creep, constant-rate loading, fatigue, random loading, etc. to determine lifetime distributions.

As seen in these expressions (Eqs. 2, 4), the statistical failure responses in two different time scales (creep and static loading tests) are completely determined by the materials parameters, \(S_{c}\), \(\rho\), and \(\beta\).

The parameter \(S_{c}\), called characteristic strength, essentially represents short-term strength, i.e. it is the creep load at which approx. 63% of the samples fail within one unit time (second) in creep. (See Eq. 2. Note that \(1 - exp(1)=0.63\)).

*f*(

*t*), the higher the \(\rho\), the higher the rate of damage growth, i.e. the system becomes more brittle. Another meaning comes from its molecular interpretation given by Phoenix et al. (1988):

*kT*is its thermal energy with

*k*, Boltzmann constant, and

*T*, absolute temperature. As the potential barrier becomes comparatively higher than the thermal energy (the higher the \(\rho\)), less atoms go over the barrier and thus the system becomes more stable (more durable). Conversely, if temperature increases, then \(\rho\) decreases and the material becomes more ductile (or less brittle).

The parameter \(\beta\) is the Weibull exponent of lifetime distribution (Eq. 2): a higher value means less variation of lifetime, i.e., less uncertainty and thus more reliable. Therefore, from the material characterisation view point, it is the material property representing reliability of long-term strength (creep lifetime).

Among these parameters, characteristic strength, \(S_{c}\), is the closest to a routinely measured property (static strength). Both durability, \(\rho\), and reliability, \(\beta\), aspects, are, however, largely overlooked in the characterisation of strength properties of cellulosic materials.

## Experimental characterisations

### Materials and methods

As typical examples of cellulose fibre networks, we have used commercial containerboard samples (liner and fluting used in for example corrugated boxes) of varying quality and basis weight. These samples were collected from board mills located in Sweden, Germany, Austria, and Czech Republic. From these samples, test specimens were cut into the size of 105mm × 25mm (the actual testing area between the clambs was \(61\,\mathrm{mm} \times 25\,\mathrm{mm}\)), both in the machine and the cross-machine directions, and conditioned at \(23\,^\circ\)C and 50% relative humidity. Additionally, all samples were preconditioned at lower relative humidity in order to achieve the same equilibrium moisture content (TAPPI 2013). Prior to testing, samples were rigorously randomised by generating random numbers to obtain statistically homogeneous samples sets for the material parameter estimates. (See also the section “Characterisation of the material parameters, \(S_{c}\), \(\rho\), and \(\beta\)”). These specimens were subjected to both compression creep tests and constant loading rate (CLR) compression tests, as described below.

### Creep failure tests in compression

A series of compression creep tests have been performed under different applied loads. The time to failure are determined when the creep rate surpassed a certain threshold value. Typical creep failure curves are shown in Fig. 2a. We have used a long-span compression tester (LCT) with finger supports, which was built at Rise Bioeconomy (formerly Swedish Forest Products Research Laboratories, STFI). Testing procedures were detailed in Mattsson and Uesaka (2013). Applied loads have been varied on 3–4 levels, and at each load level 50–100 samples have been tested.

### Constant loading rate (CLR) tests in compression

More detailed descriptions on the experimental procedures are available upon request.

### Numerical determination of the parameters

*fitnlm*(MATLAB 2017) in the MATLAB environment. The starting values (initial guess) for nonlinear fitting can be obtained by the Weibull plots based on Eqs. 8 and 9.

The estimation errors for these three parameters are also based on the above solution. Typical estimation errors, as expressed as relative standard errors, were very small, in the range of 0.03% for characteristic strength, and 2–6% for \(\rho\) and \(\beta\). However, during the course of this study, we have found large variations of these material parameters, from batch to batch and from position to position, for nominally the same sample (e.g., linerboard samples with the same basis weight, produced from the same mill, from the same board machine, and even at the same position across the machine). Accordingly, we have developed and perfected, progressively, a random sampling procedure over the same period. Unfortunately, this has caused some constraints to the sample-to-sample comparisons, particularly the comparison of results between creep tests and CLR tests, since the latter were conducted at a later phase of the entire project period of 5 years.

## Results and discussion

### Time dependent failure/Variability

First, in order to illustrate the nature of statistical failure, we have performed creep and constant loading rate (CLR) tests in compression by using the same fluting material randomly chosen. Figure 3 shows the examples of creep curves at a given load, and force-displacement curves of CLR tests at a given loading rate. The difference in the stochastic nature of the failure is apparent between these loading histories: the variation of time-to-failure in creep (lifetime) is much greater than the variation of force at failure in CLR tests.

Figure 6a shows the frequency distribution of strength obtained by CLR tests at the same loading rate. The data is typically centered around the mean with a very small scatter, unlike the lifetime distributions. The corresponding data are plotted in a Weibull format in Fig. 6b, which shows, again, Weibull distribution (a straight line in the Weibull plot).

*m*. Coleman (1958) and lately Christensen and coworkers (2009) derived the relationship between the Weibull exponents for lifetime, \(m_{creep}\), and for strength, \(m_{strength}\):

In the case of CLR tests, the model predicts that the strength distributions plotted in a Weibull format are horizontally shifted to the right with increasing loading rates. Figure 7a indeed showed such trend. It also predicts that taking the mean strength values from Fig. 7a and plotting against the corresponding loading rates yield a linear relationship in the log-log plot. The result (Fig. 7b) precisely showed such relationship.

### Characterisation of the material parameters, \(S_{c}\), \(\rho\), and \(\beta\)

Nevertheless, results showed some interesting characteristics of cellulosic materials. The brittleness parameter, \(\rho\), varied from about 20 to 60, in spite of the fact that the samples consist of essentially the same polymer compositions (cellulose, hemicellulose and some residual lignin). In our previous studies (Mattsson and Uesaka 2015, 2017) based on Monte Carlo simulations of time-dependent, statistical failures, the parameter \(\rho\) is determined by the element-level (or molecular level) properties, including temperature effects (Phoenix and Tierney 1983), see Eq. 7, rather than the network structures. It would be interesting to see how \(\rho\) varies with chemical compositions or moisture content.

Another parameter, reliability, \(\beta\), also varied considerably from 0.4 to over 1.0. This parameter is affected by both fibre and network disorders (Mattsson and Uesaka 2015, 2017). The parameter \(\beta\) of the network is directly proportional to the one for the constituent fibres (Mattsson and Uesaka 2017). Therefore, any non-uniformity of fibre will affect the \(\beta\) parameter of network. Our previous study also discussed the types of disorders that affect \(\beta\) the most. Among the factors, both geometrical disorders (random network structures) and also fibre stiffness variations had significant impacts on the \(\beta\) parameter of the fibre network.

Although, in this study, we were not able to compare, directly, the material parameter values obtained from these creep and CLR tests, a few data are available in the literature for carbon-based fibres, glass fibres, Kevlar fibres, and their composites. Results showed excellent agreements of \(\rho\) and \(\beta\) values measured by the creep tests and the strength tests (constant displacement-rate (CDR) tests (Christensen 2009)).

Lastly, it might be worth to note the CDR tests. Equation 4 requires the constant loading-rate history in order to determine the three material parameters. However, since many of the brittle and quasi-brittle materials show nearly a linear stress-strain response up to failure, one may be tempted to use the CDR tests, instead of the CLR tests. We have compared these two test modes for a limited number of samples for cellulosic materials (containerboards). CDR tests tended to exhibit a rather gradual (or ductile) failure processes, particularly at slow loading (or displacement) rates, say, 1 N/s. This made the peak-value determination very difficult, as compared with the CLR tests, and thus we were unable to use the CDR tests for the estimation of the material parameters. By considering this uncertainty of the CDR tests, we have determined the upper and lower bounds of the peak values, and compared with the mean strength values and distributions from the CLR tests. The results showed that the CLR results are indeed well bounded by the CDR results. However, in order to justify the use of CDR tests, one still needs a more systematic comparison with the CLR tests.

### Comparisons of cellulosic fibre networks with advanced composite materials

## Conclusions

Strength properties of cellulosic materials have been investigated for many decades. However, the aspects of time-dependent, statistical failures, which are commonly seen in fatigue, creep failure, or, more generally, end-use failures of structural members, are still poorly understood. The approaches to handle such properties have been also largely empirical. We have used a general framework of time-dependent, statistical failures, which was originally proposed by Coleman (1958). This theory is phenomenological, but it has been examined and tested intensively by both theoretical analyses and Monte-Carlo simulations. We have demonstrated, experimentally, that this model can be applied also to cellulosic materials, by using creep tests and constant loading rate (CLR) tests. These tests can determine, independently, the three materials parameters, i.e., (1) characteristic strength, \(S_{c}\), (2) brittleness parameter (or durability parameter), \(\rho\), and (3) Weibull shape parameter (or long-term reliability parameter), \(\beta\). Results showed that these two test methods provide comparable values for the material parameters, in spite of the large difference in the measurement time-scale and the differences in equipment geometries. This suggests that, by replacing the traditional creep tests with the CLR tests, one can drastically shorten the testing time for determining the material parameters, say, from months to a few days. Lastly we have also compared the cellulose fibre materials (containerboards) with fibres and composites used for advanced composite structures. We have shown that the cellulosic materials are very competitive in terms of reliability and possess durability comparable with Kelvar and glass fibre composites.

## Notes

### Acknowledgments

The financial support provided by the KK-Foundation in Sweden is greatly appreciated. The authors wish to thank Staffan Nyström at Mid Sweden University for the help and assistance for setting up the experiments. Rickard Boman at SCA R&D is also greatly acknowledged for all help with experimental matters. We acknowledge the valuable discussions, experimental supports, and enthusiasms received from SCA R&D in Sundsvall, and BillerudKorsnäs at Gruvön mill.

### Funding

The funding was provided by Stiftelsen för Kunskaps- och Kompetensutveckling.

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