Paperboard packaging is made by processing board materials into sheets or rolls and shaping them through creasing, cutting, folding, and erecting. The conversion process generates residual moments at the folds that cause panel bulging. This study experimentally investigates how the bulging introduced during the converting processes influence the mechanical response of paperboard packages during point load testing within the elastic deformation range. The study shows that panel bulging may significantly affect packaging performance as-perceived strength and stiffness. Bulging, influenced by the board's basis weight, can affect the package performance even more than packaging stiffness. Point load tests in the elastic region were performed on empty packages (78 mm & times; 50 mm & times; 110 mm) with force applied at specific points along their long sides. The packages evaluated in this study were made of two identically processed materials of different grammages. The heavier material showed more pronounced bulging than the lighter one, leading to overlapping force-displacement curves for the packages, and to that, a lower force and stiffness may be measured at a certain indentation depth for the package of heavier material. This complicates material choice according to functional requirements. The results show that a highly bulged package might resemble one with less bulging of another material. According to the results, it is not certain that a higher grammage package shows a higher indentation force and stiffness than a lower grammage package when measured at a certain indentation. This indicates that optimizing the creasing and folding processes can be a way to enhance performance rather than simply increasing board weight. The study underscores the importance of controlling converting parameters, especially creasing and folding behavior. Well-performed creasing and folding gives a low residual momentum, little bulging, and a high stiffness and compression strength at point loading in the elastic region. Proper optimization can improve packaging performance and manual handling user-friendliness. Application: The results from this study are of significant importance for the package industry, especially in package design, as the results show that a lower grammage may give a higher stiffness, depending on loading position.

Delamination strength is an essential property for the creasing and folding operations of paperboard into boxes. Due to fixation during creasing, the paperboard suffers in-plane straining. In the present study, we aim to increase our understanding of how in-plane straining affects the delamination properties of paperboard. Samples of paperboard were first strained in in-plane tensile loading, both in the machine-direction and in the cross-direction. Afterward, the paperboard is loaded in the out-of-plane (ZD) direction. Three different grades of commercial paperboard from two major manufacturers were tested in a climate-controlled lab. The results showed similar results for all grades of paperboard, with the delamination strength and the out-of-plane stiffness decreasing virtually linearly with pre-straining. With about 5% plastic in-plane straining, the strength was reduced by about 20% and the stiffness decreased by more than 50% for all grades of paperboard. Normalizing the strength and the stiffness with their values without pre-straining reveals virtually the same relation for all grades of paperboard. If proven to be a general result, this will prove valuable in reducing the demand for experiments. Application: The results from this study are important for the understanding and modeling of creasing and folding of paperboard.

Micro-CT analysis of experimentally creased and folded multilayer cardboards reveals insights into how the material deformation due to the creasing and folding process of cardboard impact the material concerning delaminations and position of broke particles. Delaminations were found in various locations and varied in size from just under a tenth of a millimeter to up to four times the thickness of the cardboard. The particles varied in size, ranging from a few micrometers to slightly larger than the cardboard thickness. Characteristic dimensions for the creased and folded cardboard were measured for selected cross sections. The differences in characteristic dimensions for the cross sections among the samples were typically a few hundredths of a millimeter. There are differences between cross-sections that are a few hundredths of a millimeter apart.

In packaging, the structural integrity of paperboard under complex stresses is essential. This study examines the delamination behavior of paperboard under normal and shear loading modes, which are the base for developing mixed-mode models. Four commercial paperboard grades, sourced from fibers across three geographical regions, are investigated using Double Cantilever Beam (DCB) and Split Double Cantilever Beam (SDCB) tests, which are used to calibrate a cohesive zone model from the literature. This model introduces a shape parameter that governs the cohesive traction-separation relationship, consistent across both normal and shear modes. Experimental findings support this approach, demonstrating that, while the shape parameter remains mode-independent, its numerical value varies uniquely for each paperboard quality.

A phase-field ductile fracture formulation for orthotropic paperboard materials is proposed, based on an anisotropic, multi-surface elastoplastic model describing the in-plane behavior of paperboard. A variational statement for the finite-step elastoplastic problem is extended to include the variational description of Griffith-type brittle fracture by a phase-field gradient term. The interaction between plastic and fracture dissipation mechanisms is modeled by introducing a scalar modulation function, assuming plasticity driven damage growth. This function depends on a scalar measure of the plastic strain components in the material orthotropy frame. It modifies the fracture activation criterion in a non-variational fashion, resulting in a direction-dependent material strength against crack propagation. The model performance is assessed by comparing numerical simulations and experimental tests conducted in a climate-controlled laboratory.

This study presents a thermodynamically consistent continuum damage model for fiber-based materials that combines elastoplasticity and damage mechanisms to simulate the nonlinear mechanical behavior under in-plane loading. The anisotropic plastic response is characterized by a non-quadratic yield surface composed of six sub-surfaces, providing flexibility in defining plastic properties and accuracy in reproducing material response. The damage response is modeled based on detailed uniaxial monotonic and cyclic tension-loaded experiments conducted on specimens extracted from a paper sheet in various directions. To account for anisotropic damage, we propose a criterion consisting of three sub-surfaces representing tension damage in the in-plane material principal directions and shear direction, where the damage onset is determined through cyclic loading tests. The damage evolution employs a normalized fracture energy concept based on experimental observation, which accommodates an arbitrary uniaxial loading direction. To obtain a mesh-independent numerical solution, the model is regularized using the implicit gradient enhancement by utilizing the linear heat equation solver available in commercial finite-element software. The study provides insights into the damage behavior of fiber-based materials, which can exhibit a range of failure modes from brittle-like to ductile, and establishes relationships between different length measurements.

A structured approach to derive the water activity function of cellulose-based materials which describe the sorption behavior from dry fiber up to the fiber saturation point is proposed. The coupling between the sorption isotherm and the net isosteric heat of sorption is incorporated in a consistent way, which allows for coupled heat-diffusion problems to be solved. Three different models are derived, generalizations of the classic Oswin and Henderson isotherm models and a new model denoted LW. Experimental data from the literature is used to evaluate and compare model performance to the GAB model, and it is shown that generalizations of the classic Oswin, Henderson isotherms and the new LW model perform well.

A continuum damage framework is proposed for modelling the delamination process occurring in paperboard during mechanical loading. The main application of interest is line creasing and subsequent line folding used in package forming. To adequately capture creasing and folding, a continuum damage framework that uses a single isotropic damage variable that evolves with the plastic strains associated with out‐of‐plane shearing is used. The damage evolution is calibrated against folding experiments for a specific reference mesh, and a simple scaling strategy is proposed to reduce the inherent mesh dependency. To highlight the potential of the proposed model, an illustrative 3D example is considered where a paperboard sheet is creased by two plates and folded to resemble the corner of a package.

Compression of paperboard is a common procedure during industrial package forming and better knowledge of the material response is needed to avoid defective packages and waste. To go beyond current modelling approaches, experimental identification of mechanisms underlying the macroscopic stress–strain responses is needed. In this study, in-situ uniaxial compression of paperboard is studied through synchrotron tomography at high spatiotemporal resolutions. Both the microstructural evolution of the fibre network and the actual boundary conditions of the loading were quantified and analysed. At the microscale, the loading equipment plates were not perfectly flat resulting in an increasing sample-equipment contact area with loading. This is, however, shown to only have a small effect on the form of the macroscopic stress–strain curves. The evolution of 3D strain fields showed that strain accumulated close to the sample surfaces in the early part of the compression process, whereafter the main deformation zone shifted to the out-of-plane centre. Both fibre walls and pore volumes were observed to decrease during loading (and recover partly after unloading). Regarding the pore volume, the main reduction mechanism was seen to be closure of layers between fibres. Even if the total pore volume reduction was seen to be the dominant deformation mechanism in a second stage of compression, the volumetric change of fibre walls was non-negligible. Fibre wall compression is not commonly considered in theoretical treatments of paperboard compression, but this work suggests that the stored elastic energy could be a driver for the elastic recovery of the fibre network during unloading.

A cohesive interface model based on a master curve is proposed for the analysis of delamination in paperboard under various loading, unloading, and reloading conditions. The model is thermodynamically consistent and considers the effects of elasticity, plasticity, and damage. The proposed model is verified by comparing its predictions with experimental data obtained from multiple loading–unloading–reloading cycle experiments using a split double cantilever beam specimen. The results show that the model can predict the cyclic behavior of shear loading and provide insight into the damage evolution associated with different loading paths by analyzing the shear stress distribution in the fracture process zone. The model's calibration process requires monotonic normal and shear loading data but only cyclic normal loading data. Additionally, the model accounts for the paperboard's fiber–fiber friction and normal dilatation due to shear loading. In total, nine parameters are needed to calibrate the model.

Development of three-dimensional continuum models for paperboard is an active field and the need for reliable measurements to calibrate and validate such models is evident. An experimental device and protocol for cyclic out-of-plane loading is developed. This loading sequence is present during converting operations of paperboard. The experimental tests reveals that the commonly observed soft initial non-linear response during out-of-plane compression is a structural effect that stems from the surface roughness rather than being an inherent material behavior. A gluing procedure, used to perform cyclic out-of-plane loading, is mitigating the effect of the surface roughness. Several novel cyclic loading experiments are performed, alternating between compression and tension which indicates that fiber bonds are not recovered in compression after they have been broken through delamination. Measurements also show that the transition in compression and tension is continuous, hence the use of a switch function present in a number of constitutive continuum models for paperboard is deemed questionable.

This thesis focuses on understanding the mechanical behavior of fiber-based materials by utilizing various modeling approaches. Particular emphasis is placed on their structural variability, anisotropic properties, and damage behavior. Furthermore, the study explores moisture diffusion phenomena within these materials, leveraging machine learning techniques. The research employs a blend of multiscale modeling, experimental investigation, machine learning, and continuum modeling to enhance the predictive capabilities for modelling fiber-based materials. In Paper I, the work investigates the impact of stochastic variations in the structural properties of thin fiber networks on their mechanical performance. A multiscale approach that includes modeling, numerical simulation, and experimental measurements is proposed to assess this relationship. The research also considers the influence of drying conditions during production on fiber properties. The study finds that spatial variability in density has a significant impact on local strain fields, while fiber orientation angle with respect to drying restraints is a key influencer of the mechanical response. In Paper II , the research delves into the investigation of anisotropic properties and pressure sensitivity of fiber network materials. It draws a comparison between the Hoffman yield criterion and the Xia model, which are widely utilized for simulating the mechanical response in fiber-based materials. The study performs a detailed analysis of these models under bi-axial loading conditions, assessing their numerical stability and calibration flexibility. Further supporting the research community, the paper provides open-source access to the user material implementations of both models and introduces a calibration tool specifically for the Xia model, thereby promoting ease of usage and facilitating further research in this domain. In Paper III a novel thermodynamically consistent continuum damage model for fiber-based materials is introduced. Through the integration of elastoplasticity and damage mechanisms, the model employs non-quadratic surfaces comprised of multi sub-surfaces, augmented with an enhanced gradient damage approach. The model’s capability is demonstrated by predicting the nonlinear mechanical behavior under in-plane loading. This study provides valuable insights into the damage behavior of fiber-based materials, showcasing a range of failure modes from brittle-like to ductile. In Paper IV , the study examines moisture penetration in fiber-based materials and the resultant out-of-plane deformation, known as curl deformation, using a combination of traditional experiments, machine learning techniques, and continuum modeling. The paper compares the effectiveness

The anisotropic properties and pressure sensitivity are intrinsic features of the constitutive response of fiber network materials. Although advanced models have been developed to simulate the complex response of fibrous materials, the lack of comparative studies may lead to a dubiety regarding the selection of a suitable method. In this study, the pressure-sensitive Hoffman yield criterion and the Xia model are implemented for the plane stress case to simulate the mechanical response under a bi-axial loading state. The performance of both models is experimentally assessed by comparison to bi-axial tests on cruciform-shaped specimens loaded in different directions with respect to the material principal directions. The comparison with the experimentally measured forces shows the ability of the Hoffman model as well as the Xia model with shape parameter k≤2 to adequately predict the material response. However, this study demonstrates that the Xia model consistently presents a stiffer bi-axial response when k≥3 compared to the Hoffman model. This result highlights the importance of calibrating the shape parameter k for the Xia model using a bi-axial test, which can be a cumbersome task. Also, for the same tension-compression response, the Hill criterion as a special case of the Hoffman model presents a good ability to simulate the mechanical response of the material for bi-axial conditions. Furthermore, in terms of stability criteria, the Xia model is unconditionally convex while the convexity of the Hoffman model is a function of the orthotropic plastic matrix. This study not only assesses the prediction capabilities of the two models, but also gives an insight into the selection of an appropriate constitutive model for material characterization and simulation of fibrous materials. The UMAT implementations of both models which are not available in commercial software and the calibration tool of the Xia model are shared with open-source along with this work.

A better physical understanding of mesoscale and microscale mechanisms behind deformation and failure of paperboard material is important to optimize industrial packaging converting processes and decrease waste. In this study, these mechanisms were investigated using synchrotron X‐ray tomography during in situ continuous uniaxial tensile loading. High spatial and temporal data resolution enabled quantification of rapid changes in the material occurring before, during and after material failure. The evolution of 3D strain fields, fibre orientations and sample thickness showed that deformation and failure mechanisms differ significantly between samples tested in machine direction (MD), cross direction (CD) and 45° from the loading direction. In 45° and CD, gradual failure processes could be followed across several load steps. Immediately after failure, the in‐plane fracture region was significantly larger in both 45° and CD compared to MD. Both fracture characteristics and strain field distributions differed between the three material directions. Significant fibre reorientation was an active deformation mechanism in 45° already from the beginning of the loading, also present in CD after peak load but absent in MD. The MD‐dependent mechanisms interpreted and quantified at the scale of the fibre network in this study can help guide model development and likely have wider applicability to other paper‐based materials.

An experimental study to characterize properties controlling delamination of paperboard is presented. The normal and shear traction–separation laws are measured and evaluated using a double cantilever beam (DCB) and a split double cantilever beam (SCB) specimen. The DCB-experiments provides normal separation data in good agreement with results using alternative experimental techniques. From the measured data, both normal and shear fracture resistance data are obtained. A length parameter is introduced. The length parameter allows for the cohesive law to be obtained from a dimensionless master curve which is valid both for normal and shear loading. Taking advantage of the master curve, a mixed-mode potential is proposed. The mixed-mode potential is implemented as a user interface to a finite element code. As a final test, the experimental setups of the DCB and SCB specimens are simulated to validate the identified normal and shear properties.

A rate-dependent continuum model for paperboard is developed within a framework for finite strains and finite deformations. A multiplicative split of the deformation gradient into an elastic and an inelastic part is assumed. For the in-plane modes of deformation, viscoelasticity is introduced via a thermodynamically consistent generalization of the Maxwell formulation. The elastic transition between out-of-plane compression and out-of-plane tension is smooth, excluding the need for a switch function which is present in a number of existing paperboard models. The evolution of the inelastic part is modeled using two potential functions separating compression from shear and tension. To calibrate the material model, a set of experiments at different loading rates have been performed on single ply paperboard together with creep and relaxation tests for in-plane uniaxial tension. The model is validated by simulating two loading cases related to package forming, line-folding followed by subsequent force-relaxation and line-creasing during different operating velocities in conjunction with a creep study. (C) 2021 Elsevier Inc. All rights reserved.

To allow for large‐scale forming applications, such as converting paperboard into package containers, efficient and reliable numerical tools are needed. In finite element simulations of thin structures, elements including structural features are required to reduce the computational cost. Solid‐shell elements based on reduced integration with hourglass stabilization is an attractive choice. One advantage of this choice is the natural inclusion of the thickness, not present in standard degenerated shells, which is especially important for many problems involving contact. Furthermore, no restrictions are imposed on the constitutive models since the solid‐shell element does not require the plane stress condition to be enforced. In this work, a recently proposed efficient solid‐shell element is implemented together with a state‐of‐the‐art continuum model for paperboard. This approach is validated by comparing the obtained numerical results with experimental results for paperboard as well as with those found by using 3D continuum elements. To show the potential of this approach, a large‐scale forming simulation of paperboard is used as a proof of concept.

In this Master’s Thesis x-ray tomography was used during tensile experiments on paperboard to study delamination and cohesive failure. Digital volume correlation of the x-ray tomograph images enabled quantitative analysis of strain fields. Tensile experiments on different specimen geometries were also conducted to investigate how the geometry affected the response of the specimens during loading. By analysing the size effects and by using normalisation it was found that the behaviour of the material during tensile experiments was independent of the geometry. Using x-ray tomography images, a thickness increase was measured, all the way from loading start to sample failure. It was found that right before the failure strength, the material experienced a higher dilation compared to during the rest of the experiment. It was further found, using digital volume correlation, that the normal strains in the loading direction localised in parabolic zones with higher strains between the notches in the test sample. From the shear strain fields it was also noted that in close proximity to the failure strength, shear strains increased. The thickness increase right before failure was probably caused by delamination of the paperboard. However, even though delamination results in dilation of the sample it was proven, by performing tensile tests on pre-delaminated samples, that it does not affect the cohesive failure. This means that delamination does not cause in-plane failure. From the analysis it was instead observed that the in-plane failure occurs at the zones with higher strains in the loading direction. During this Master’s Thesis it was found that the combination of x-ray tomography and digital volume correlation is effective to gain more information about the internal structure and deformation of paperboard.

Analytical relations to describe experimentally measured traction-separation laws are often expressed in dimensionless quantities. The traction and the separation are commonly normalized using the cohesive strength and a length measure, respectively. The ratio between the fracture energy and the cohesive strength is often used as a length measure. An alternative length measure is the ratio between the cohesive strength and the maximum slope of the traction-separation law. A relation between these two length measures are established. To illustrate the implications on cohesive laws, three existing cohesive laws are rewritten using the alternative normalization. As a result it is shown that the number of unknown material parameters can be reduced. One of the derived dimensionless cohesive law is validated against experimental uniaxial tension and compression load -deformation data of different sample sizes and different quasi-brittle materials, i.e. concrete and paperboard. A good fit of the cohesive law is shown to all the investigated data. These findings indicate that the derived normalized cohesive law is independent of material directions, moisture contents and sample size. (C) 2017 Elsevier Ltd. All rights reserved.

In-plane cohesive failure of paperboard was characterized by short-span uniaxial tension tests. Six paperboards' qualities were experimentally investigated, from which cohesive stress–widening curves were extracted. A fracture energy was defined, expressed in the tensile strength and maximum slope of the cohesive stress–widening relation. Analytical cohesive relations were derived based on the tensile strength and maximum slope, utilizing the Morse potential for diatomic molecules. It was experimentally found that the maximum slope and fracture energy depend on the tensile strength. The ratio of the maximum slope to the elastic modulus (stable length) was shown to be independent of the tensile strength.