Cellulose fibers come in a wide range of shapes and sizes. The heterogeneity of the fiber length, width, wall thickness, curl and external fibrillation is detrimental to the mechanical performance of products such as paper and paperboard. Although micro-mechanical models of these materials sometimes incorporate features of this heterogeneity, so far there is no standardized method of fully incorporating this.We examine a large number of industrial mechanical fiber pulps to determine what information such a standardized method would have to have. We find that the method must allow for both non-Gaussian distributions and dependence between the variables. We present a method of characterizing mechanical pulp under these conditions that views the individual fiber as outcome of a sampling process from a multivariate distribution function. The method is generally applicable to any dataset, even a non-Gaussian one with dependencies.Using a micro-mechanical model of a paper sheet the proposed method is compared with previously presented methods to study whether incorporating both a varying fiber size and dependencies is necessary to match the response of a sheet modeled with measured characterization data. The results demonstrate that micro-mechanical models of paper and paperboard should not neglect the influence of the dependence between the characteristic shape features of the fibers if the model is meant to match physical experiments.

    Cellulose fibers exhibit a wide range of shapes and sizes. This variation influences the mechanical performance of paper and paperboard by affecting the stress distribution inside the network and the degree of fiber-to-fiber bonding which is possible at a given density. However, the methods used to characterize the distribution of fiber sizes in the pulp neglect that the characteristic features of a fiber are generally not independent.

    Here, we resolve this shortcoming by fitting the fiber population to a multivariate distribution without enforcing normality or independence between the properties. The high-dimensional multivariate function is recast as a set of univariate distribution functions and a series of bivariate distributions connected by a canonical vine. 

    Using a micro-mechanical model of a paper sheet the influence of this improved characterization is investigated. Reasonable margins and a description of the dependency is shown to be superior to assuming independence even for perfectly preserved marginal distributions. This result demonstrates that micro-mechanical models of paper and paperboard cannot by assumption neglect the influence of the interdependence between the characteristic features of fibers. 

Double-roll compaction is a process to create extensible paper and paperboard suitable for replacing plastic in 3D forming applications. Understanding the macro- and micro-mechanisms governing the compaction process allows increasing the stretch potential while maintaining sufficient strength and bending stiffness. In this work, we approach the compaction process of paperboard with micro-mechanical methods featuring the unprecedented level of details otherwise inaccessible with currently available experimental tools. The loading scheme is based on experiments and continuum level simulations. The different levels of compaction and their continuous impact on the fibers’ geometry, void closures, and irreversible deformation of the fibers are thoroughly characterized. We find that the structural changes are concentrated in the fibers oriented within 30 degrees of the direction of compaction. The deformation accumulates primarily in the wall of the fibers in the form of irreversible strains. The spring-back effect beyond the compaction is negligible. For the first time, the role of normal and frictional fiber-to-fiber interactions in the compaction process is investigated and quantified. The frictional interaction between the fibers has a surprisingly low impact on the outcome of the compaction process, and the normal interaction between the fibers has a dominant response. The consequence of this finding is potentially limited impact of the surface modifications targeting the friction.

Large deformations under in-plane compression of paperboard appear in forming processes like hydroforming, pressforming and deep drawing, but the mechanisms of deformation have not been studied on a micromechanical level. A constrained in-plane compression test is presented. This test allows for in-plane compression, buckling, wrinkling and compaction. The constrained compression test is realized using a DEBEN CT-500 in-situ tester for laboratory microtomography and synchrotron microtomography. Experiments with five different materials spanning from laboratory handsheets to commercially available multi-layered paperboards are performed. Image processing is used to observe the local out-of-plane fiber orientation and compaction. A phenomenological investigation of the deformation behavior of these materials is presented. Delamination is found to be the primary mechanisms of failure in the multi-layered boards. Furthermore, a porous network structure, created by using long and minimally refined softwood fibers, is found to facilitate the formation of uniform wrinkles and compaction.

Cellulose fibers are a major industrial input, but due to their irregular shape and anisotropic material response, accurate material characterization is difficult. Single fiber tensile testing is the most popular way to estimate the material properties of individual fibers. However, such tests only determine the longitudinal modulus of the fiber. Here, we compare sheet testing, micromechanical testing, and nanoindentation as methods to extract the elastic material properties of the individual fibers. We show that nanoindentation can be used to determine both the longitudinal and the transverse elastic modulus using only two indentations, additionally enabling the measurement of fiber properties in-situ inside a sheet of paper where the complete process history is captured. For the longitudinal modulus, the accuracy is comparable for larger indents, but with an increase of scatter of unknown origin as the probe size is decreased using an atomic force microscopy tip.

Cellulose fibers are a major industrial input, but due to their irregular shape and anisotropic material response, accurate material characterization is difficult. Single fiber tensile testing is the most popular way to estimate the material properties of individual fibers. However, such tests can only be performed along the axis of the fiber and are associated with problems of enforcing restraints. Alternative indirect approaches, such as micro-mechanical modeling, can help but yield results that are not fully decoupled from the model assumptions. Here, we compare these methods with nanoindentation as a method to extract elastic material constants of the individual fibers. We show that both the longitudinal and the transverse elastic modulus can be determined, additionally enabling the measurement of fiber properties in-situ inside a sheet of paper such that the entire industrial process history is captured. The obtained longitudinal modulus is comparable to traditional methods for larger indents but with a strongly increased scatter as the size of the indentation is decreased further.

Investigating the compression properties of randomly ordered fiber networks experimentally is difficult which has resulted in ongoing disputes as to the mechanisms controlling the compression strength in such materials. In this work, we investigated compression properties of randomly oriented fiber networks with a special emphasis on cellulose products such as paperboard. We numerically reconstructed the conditions of the short span compression test widely used to quantify the compression strength of paperboard. We found that the phenomenological failure mode of such networks is elasto-plastic buckling. The x-shaped failure mode observed in physical experiments appears when test specimen restraints are included in the model. The most significant improvements to sheet strength can be obtained by improving the elastic properties while the strain to failure is increased most by an improvement of the plastic yield and hardening properties of individual fibers. Bond breaks were confirmed to have a smaller influence on the overall response. Fiber level microscopic buckling was investigated in depth, providing quantitative estimates of the fraction of mass likely to buckle at the microscopic level. The analysis indicated that only a low to moderate number of load carrying fibers can be expected to buckle. The inherent strength reserve in non-ordered fiber networks was investigated by introducing hinge mechanisms throughout the network, and the effect was shown to be small for a small to moderate number of hinges.

A computationally efficient method to study the in-plane and out-of-plane dimensional instability of thin paper sheets under the influence of moisture changes is presented. The method explicitly resolves the bonded and the free segments of fibers in the sheet, capturing the effect of anisotropic hygroexpansion at the fiber level. The method is verified against a volumetric model. The importance of longitudinal fiber hygroexpansion is demonstrated in spite of the absolute value of longitudinal hygroexpansion being an order of magnitude lower than the transverse hygroexpansion component. Finally, the method is used to demonstrate the formation of macroscopic sheet curl due to a moisture gradient in structurally uniform sheets in the absence of viscoelastic or plastic constitutive behavior and through-thickness residual stress profiles.