Micromechanics of Fiber Networks
Svetlana Borodulina
Doctoral thesis no. 97, 2016 KTH School of Engineering Sciences
Department of Solid Mechanics Royal Institute of Technology SE-100 44 Stockholm Sweden
Akademisk avhandling som med tillstånd av Kungliga Tekniska högskolan (KTH) i Stockholm framlägges till offentlig granskning för avläggande av teknologie doktorsexamen fredagen den 2 september 2016 kl. 10.00 i Kollegiesalen, Kungliga Tekniska högskolan, Brinellvägen 8, Stockholm.
Fakultetsopponent är Professor Pierre J.J. Dumont, Institut National des Sciences Appliquées de Lyon, France
TRITA HFL-0598 ISSN 1104-6813
ISRN KTH/HFL/R-16/12-SE ISBN 978-91-7595-994-8
You cannot open a book without learning something.
ii
Abstract
The current trends in papermaking involve, but are not limited to, maintaining the dry strength of paper material at a reduced cost. Since any small changes in the process affect several factors at once, it is difficult to relate the exact impact of these changes promptly. Hence, the detailed models of the network level of a dry sheet have to be studied extensively in order to attain the infinitesimal changes in the final product.
In Paper A, we have investigated a relation between micromechanical processes and the stress–strain curve of a dry fiber network during tensile loading. The impact of “non-traditional” bonding parameters, such as compliance of bonding regions, work of separation and the actual number of effective bonds, is discussed. We conclude that large local strains are the precursors of bond failures and not the other way around. We attributed the overestimated network strength, as compared to experimental curves, to the overestimated number of contacts and other contact parameters.
In Paper B, we studied the impact of the chemical composition of the fiber cell wall, as well as its geometrical properties, on the fiber mechanical properties using the three-dimensional model of a fiber with helical orientation of microfibrils at a range of different microfibril angles (MFA). We found that the shape of the fiber cross-section does not affect its tensile response significantly, as long as the area of the cross-section and the average MFA of the fiber are preserved.
In order to accurately characterize the fiber and bond properties inside the network, via statistical distributions, microtomography studies on the handsheets have been carried out. This work is divided into two parts: Paper C, which describes the methods of data acquisition and Paper D, where we discuss the extracted data. Here, all measurements were performed at a fiber level, providing data on the fiber width distribution, width-to-height ratio of isotropically oriented fibers and contact density. We confirm that the number of fiber-to-fiber contacts in three-dimensional isotropic networks is controlled by the fiber’s aspect ratio. In the last paper, we utilize data thus obtained in conjunction with fiber morphology data from Papers C and D to update the network generation algorithm in order to produce more realistic fiber networks. We also successfully verified the models with the help of experimental results from dry sheets tested under uniaxial tensile tests. Further on, we carry out numerical simulations on these networks to ascertain the influence of fiber and bond parameters on the network strength properties. We conclude, among other things, that it is sufficient to account for the average bond strength value with an acceptable number of samples to describe the dry network strength.
Keywords: Network simulation, Mechanical properties, Fibers, Fiber-to-fiber
bonds, Free fiber length, Number of contacts, Contact density, Paper properties, X-ray microtomography
iv
Sammanfattning
En pågående utveckling inom produktion av papper är att behålla styrkan hos materialet samtidigt som priset minskas. Eftersom varje ändring som görs i processen samtidigt påverkar flera egenskaper hos pappret är det svårt att snabbt avgöra exakt vilka effekter dessa ändringar medför. Det är därför viktigt att analysera torra fibernätverk med detaljerade datormodeller för att kunna avgöra vilka effekter små processändringar har på slutprodukten.
I Artikel A har vi undersökt sambanden mellan mikromekaniska processer och spännings-töjnings kurvan för ett torrt fibernätverk som utsätts för en draglast. Bidragen från ”icke-traditionella” bindningsparametrar, såsom bindningarnas komplians, separationsarbete och det faktiskta antalet (verksamma) bindningar diskuteras. Vi visar att de största lokala töjningarna indikerar var bindningsbrott kommer att ske och inte tvärtom. Att nätverkens styrka överskattas jämfört med experimentellt framtagna värden beror på att antalet bindningar har överskattats. I Artikel B studeras hur en fibers mekaniska egenskaper påverkas av cellväggens kemiska sammansättning och fiberns geometri. Detta görs med hjälp av en tredimensionell (3D) modell av en fiber med spiralorienterade fibriller med olika fibrilvinklar. Vi fann att fiberns tvärsnittsgeometri inte påverkade dragegenskaperna signifikant, förutsatt att tvärsnittsarean och fibrilvinkeln bevarades.
För att kunna karakterisera fiberegenskaper och antal bindningar (kontakter) mellan fibrer inuti nätverket, med fokus på deras statistiska fördelning använder vi datortomografiska bilder (beräknade från multipla röntgenprojektioner) av laboratorieproducerade pappersark. Arbetet delades upp i två delar, Artikel C som beskriver bildanalysmetoder och Artikel D som diskuterar själva analysen av data. Alla mätningar utfördes på fibernivå vilket gav information om bland annat fiberbreddernasfördelning, höjd-till-bredd kvoten hos isotropt orienterade fibrer, samt densiteten av kontakter i ett fibernätverk. Vi bekräftade att antalet bindningar mellan fibrerna i tredimensionella isotropa nätverk styrs av fibrernas höjd-till-bredd kvot.
I den sista artikeln används data från Artikel C och D för att uppdatera den algoritm som genererar ett fibernätverk för att på så sätt simulera mer realistiska nätverk. Vi verifierade modellerna med hjälp av data från enaxligt dragprovade ark. Därefter genomfördes numeriska simuleringar av dessa fibernärverk för att fastställa vilken inverkan fiber- och bindningsegenskaperna har på nätverksstyrkan. Vi visade, bland annat, att det räcker med att beakta medelbindningsstyrkan för att beskriva nätverkstyrkan hos torra nätverk, förutsatt att tillräckligt många prov genomförts.
Nyckelord: Nätverkssimulering, Mekaniska egenskaper, Fibrer, Fiberkontakter,
Fri fiberlängd, Antal bindningar, Kontaktdensitet, Pappersegenskaper, Röntgen- tomografi
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Preface
The work presented in this doctoral thesis has been carried out at the Department of Solid Mechanics, KTH Royal Institute of Technology, Stockholm, Sweden between November 2010 and March 2016 within the competence center BiMaC Innovation, whose financial support is gratefully acknowledged.
During these years so many people have helped me in many ways, and I wish it were possible to thank them all. First and foremost, I would like to thank Associate Professor Artem Kulachenko for being an enthusiastic and smart supervisor. Artem has always been friendly and encouraging, maintaining the delicate balance between support and challenge. He was also patient beyond forbearance. My thanks are extended to my second supervisor Docent Mikael Nygårds. Despite his tight schedule, Mikael always managed to find time for discussion or help me in the lab at Innventia AB, which is greatly appreciated. I selfishly hope that the collaboration between me, Artem and Mikael will continue. I express my deep gratitude to those who helped me with the collaborative work in this interdisciplinary field: Sylvain Galland at the Wallenberg Wood Science Center, Erik L.G. Wernersson at the Centre for Image Analysis, SLU, Uppsala and Joanna Hornatowska at Innventia AB. My sincere thanks also go to the co-authors of my articles Denny D. Tjahjanto, Hamid R. Motamedian, Cris L. Luengo Hendriks, Gunilla Borgerfors and the anonymous reviewers for their comments during the publications.
I cannot go without mentioning my former and present colleagues and friends at KTH Solid Mechanics for the enjoyable atmosphere. I would also like to thank the administrative staff for all their help during my postgraduate studies.
Finally, I am deeply indebted to my family, especially my twins Emelie & Valerie and my husband Elias. Their love and support provided me with the energy to attain my goals. My family in Belarus has also been an important and indispensable source of spiritual support.
Last but not least, I would like to thank you for reading my thesis. I hope you find it useful and please do not hesitate to report to me any mistakes you may find at svebor@kth.se.
Stockholm, April 2016
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List of appended papers
Paper A: Stress-strain curve of paper revisited
Borodulina, S., Kulachenko, A., Galland, S., Nygårds, M.
Nordic Pulp and Paper Research Journal, 2012, 27(2), 318-328.
Paper B: Constitutive modelling of a paper fiber in cyclic loading applications
Borodulina, S., Kulachenko, A., Tjahjanto, D. D.
Computational Materials Science, 2015, 110, 227-240.
Paper C: Characterisations of the fibre networks in paper using micro computed
tomography images
Wernersson, E. L. G., Borodulina, S., Kulachenko, A., Borgerfors, G.
Nordic Pulp and Paper Research Journal, 2014, 29(3), 318-328.
Paper D: Extracting fiber and network connectivity data using microtomography
images of paper
Borodulina, S., Wernersson, E. L. G., Kulachenko, A., Luengo Hendriks, C.L.
Nordic Pulp and Paper Research Journal, 2016, 31(3). Accepted manuscript.
Paper E: Effect of fiber and bond strength variations on the tensile stiffness and
strength of fiber networks
Borodulina, S., Motamedian, H.R., Kulachenko, A.,
Report 597, Department of Solid Mechanics, KTH Royal Institute of Technology Submitted for publication
ix
In addition to the appended paper, the work has resulted in the following publications in conference proceedings fully reviewed prior to publication:
Effects of variation in morphological properties of fibers and bond strength on the tensile stiffness and strength of fiber networks.
Borodulina, S., Motamedian, H.R., Kulachenko, A.
Euromech Colloquium 569 - Multiscale Modeling of Fibrous and Textile
Materials, 5–7 April 2016, Châtenay-Malabry, France.
Micromechanical behaviour of fiber networks. Borodulina, S., Kulachenko, A.
European Forest Product Doctoral Symposium, 21–23 August 2013, Helsinki, Finland.
Stress–strain curve of a fiber network.
Borodulina, S., Kulachenko, A., Nygårds, M. Proceedings of the 23rd International Congress of Theoretical and Applied Mechanics ICTAM, Beijing,
China, 19–24 August, 2012.
Influence of paperboard structure and processing conditions on forming of complex paperboard structures.
Östlund, M., Borodulina, S., Östlund, S. Packaging Technology and Science, 24:331-341, 2011.
3D-forming of double-curved paperboard structures for packaging applications.
Östlund, S., Östlund, M., Borodulina, S. Progress in Paper Physics Seminars (poster), Graz, Austria, 2011, pp. 323-325.
Digital analysis of deformed corrugated boxes using 3D speckles. Borodulina, S., Östlund, S., Trost, T.
Proceedings of the 17th IAPRI World Conference on Packaging, 12–15
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Contribution to the papers
The author’s contributions to the appended papers are as follows:
Paper A: Principal author. Extended Kulachenko’s previously developed fiber
network model for wet fibers further to dry fibers by incorporating the appropriate fiber bonding conditions and constitutive relations. Experimental work was conducted together with Galland. Both the principal author and Kulachenko contributed to the writing process. Nygårds contributed with comments.
Paper B: Principal author. Performed simulation work and results evaluation.
Constitutive modeling of a fiber was conducted together with D.D. Tjahjanto. Both the principal author and Kulachenko contributed to the writing process.
Paper C: Responsible for the idea and planning. Prepared the samples. Manually
marked the fibers on which the described methods were tested. Participated in interpreting the results. Wrote most of the introduction and contributed to the review. Wernersson developed the methods and wrote the major part of the manuscript. Kulachenko, Borgerfors contributed with comments.
Paper D: Principal author. Evaluation and interpretation of the results. Both
Wernersson and Kulachenko contributed to the writing process. Luengo Hendriks contributed with comments.
Paper E: Principal author. Evaluation and interpretation of the results.
Motamedian is responsible for the improved network definition and wrote part of the methods section. Both the principal author and Kulachenko contributed to the writing process.
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Abbreviations
2D Two-dimensional 3D Three-dimensional BS Bond strength CD Cross-machine direction CS Cross-section CT Computed tomographyFFL Free fiber length
FMA Fiber morphology analyzer
MD Machine direction
MFA Microfibril angle
NCP Number of contact points
RBA Relative bonded area
RCA Relative contact area
SEM Scanning electron microscope
SSC Stress–strain curve
TMP Thermo-mechanical pulp
Contents
Abstract ... ii
Sammanfattning ... iv
Preface ... vi
List of appended papers ... viii
Contribution to the papers ... x
Abbreviations ... xi
Introduction... 1
Mechanical response of paper sheets ... 3
Network characteristics ... 4
Fibers ... 5
Bonds ... 6
Results ... 7
Stress–strain curve (SSC) of paper... 7
Mechanical response of a pulp fiber ... 9
Fiber and network connectivity data ... 12
X-ray computed tomography... 13
Extracting fiber morphological properties from μCT ... 15
Influence of fiber morphology and bond strength ... 17
Conclusions... 20
Final remarks ... 22
Bibliography ... 25
Summary of the appended papers ... 30
Paper A
Paper B
Paper C
Paper D
Paper E
1
Introduction
Papermaking is a multidisciplinary technology with a strong market competition. Up to the present, since Fourdrinier in 1810, the paper machine speed has increased by a factor of 170 [1], resulting in a production speed of up to 100 km/h. This makes paper one of the fastest continuously produced materials; and papermaking — one of the most technologically advanced industries. Despite a general expectation of a “paperless” society, based on rapid developments in the IT world (such as an increasing number of electronic reading devices and e-publications, digital advertisements via Facebook and Twitter [2], etc.) and the growing transition to efficient energy usage while maintaining economic profit [3, 4], paper still remains ubiquitous. It would be extraordinary if, one day, we stopped reading printed texts or did not make any contact with a paper-based material, whether it is a takeaway coffee cup or a grocery package. At the same time, the environmental perspective plays a key role, with increasing demand for a rational choice of raw materials, recycling in production and preparatory testing — all being a driving force for development. Yet, we have reached the point where the changes in the papermaking process are infinitesimal, mainly dominated by the product development.
The structure of paper is essentially a non-unique network of cellulose fibers (Fig. 1). Such random networks are also found in other modern materials, while mostly at the nanoscale, for example in biofabricated bacterial cellulose [5], collagen fibrillar networks [6], buckypaper made of carbon nanotubes [7], not to mention (rather conventional nowadays) cellulose nanofiber networks [8].
2
Fig. 1. A scanning electron microscope (SEM) image of a paper surface (courtesy of Jim Ekstrom): left – copy paper, right – Post-it paper (partly showing the sticky part with glue particles on the top of the picture).
The stochastic nature of paper can be distinguished from other materials by its manufacturing process [9]. To simplify, it starts from a wet suspension of pulp fibers and finishes with a dry porous fiber network structure. Depending on the application, these pores can be filled by other constituents like fines (small fractions of pulp less than 0.2 mm)1, microparticle fillers (clay, calcium carbonate, talc, etc.), coatings, binders or pigments [10]. Paper constituents are subjected to a complex loading history during its manufacturing process, with varying tension applied at continuously varying moisture content.
During drying, the network of fibers is held by fiber-to-fiber bonds. Despite decades of research, there is no consensus about the nature of these bonds. What is common in all the suggested mechanisms is that one or several factors are intertwined when two pulp fibers are in contact. A good review of the available theories can be found in [11, 12], with some recent remarks on the bonding between the fibers in [13–16].
Additionally, the sheet formation process causes structural anisotropy of paper [17]. Paper is known to have two principal in-plane directions called the machine- (MD) and cross-machine direction (CD). Finally, this process-dependent
1 Primary fines may contain materials naturally present in the pulp, such as ray cells or pith parts,
3
disorder is combined with large variations in the properties of fibers that originate from plant sources, thereby creating the very sophisticated, but entirely necessary structure we call paper.
The work presented in this thesis is a compilation of five scientific articles, named as Papers A to E, where the objective of each of the subsequent articles has been driven by the results of the previous work. The logical sequence of this thesis is as follows. In Paper A we identify the importance of various factors, namely, fibers, bonding, and paper structure on the stress–strain curve of paper. Since each of these parameters is on a different length-scale, we had to look at them separately. This was done in Papers B–E, respectively. In Paper B we investigate the influence of geometrical and material parameters on the mechanical response of the pulp fiber. In Papers C and D we extract information about bonds, which allows us to understand the connectivity properties of fibers in a network. Finally, we examine the importance of variation in fiber and bond properties on the mechanical properties of paper in Paper E. At all times, the focus was not only on developing tools that could be used to answer specific questions that arose in the course of this project but also on making them amenable for more general use.
Mechanical response of paper sheets
Over the past few decades, the structural characterization of paper has been performed mostly experimentally and was aimed at relating the properties of the major constituents (fibers and bonds) to the paper sheet properties by a variety of methods [18]. With the development of new testing techniques in the 1950s, it became possible to perform testing on single fibers [19], but the strength of bonds is generally more complicated to measure due to small bonded areas and the process of fiber-to-fiber bond manufacturing. There has also been a debate about the meaning of the bond strength [11] and the reported values [20]. At the same time, the role of disorder in the mechanical properties of paper is indeed abstract and it can neither be measured nor attributed experimentally. One of the solutions to overcome these difficulties would be to treat paper as a continuum or a
fiber-4
reinforced composite. However, the composite approach is unrealistic, as paper does not have a distinct “matrix” per se and the interaction between the fibers is complicated due to the presence of bonds.
On the other hand, describing paper as a continuum does not answer the key questions associated with the disordered structure: whether paper properties are governed by the randomness in fiber distribution, fiber micromechanical ultrastructure, the bond strength, sheet inhomogeneity, or a combination of these. The importance of these parameters in the paper should be resolved in another context, namely by performing numerical simulations of the network structure, which is also a better alternative in terms of efficiency and performance. for the From the material design perspective, numerical methods in network modeling provide far more possibilities in assessing valuable information, especially in those cases where experiments are not tractable.
The mechanical response of paper sheets to external in-plane tension is usually described by the load–elongation curve, often presented as a stress–strain curve (SSC). This term is slightly inaccurate because there exists uncertainty about which value to use for the thickness (also called caliper) of a paper sheet as its surface is uneven; and also because stresses and strains are not constant within the specimen. The elimination of thickness in the equation for density leads to unification of the “basis weight” term – also grammage, which is defined as mass per unit area, usually expressed in units of grams per square meter. We shall nevertheless use the term SSC interchangeably with load–elongation curve.
Network characteristics
We considered a three-dimensional network of fibers (Fig. 2). The network was created with the help of a deposition technique described in [21]. Each fiber is represented as a series of Timoshenko quadratic beam elements with a tubular (Papers A and B) or rectangular (Papers D and E) cross-section. The beam element has three translational and three rotational degrees of freedom at each node. The following parameters can be varied during construction of the network:
5
Fig. 2. A typical simulated fiber network with isotropic in-plane orientation and boundary conditions for tensile force-controlled loading (Paper B).
fiber length, fiber width, fiber wall thickness, fiber width-to-height (WH) ratio, fiber curl, sheet basis weight, network thickness and other material properties of the fibers. All the listed parameters can be varied according to a specified distribution law. We accounted for large deflections, rotations, and strains. A typical simulated fiber network had isotropic in-plane fiber orientation (Papers A, B, and E).
Fibers
It is known that fibers dried under compressive strains or having high fibril angles exhibit stronger non-linear behavior with a distinct hardening region as compared to untreated wood fibers [22, 23]. In this respect, drying inside the sheet affects the fibril angles, reduces the elastic modulus of the fiber and promotes a distinct non-linear hardening response by introducing compressive strains.We have used two different strategies to describe the non-linear stress–strain behavior of a single fiber.
In Papers A and E, we characterized fiber with the help of bilinear isotropic hardening plasticity (see Fig. 3 (left)). This formulation does not include the effects of the changed microfibril angle, which was accounted for in the second fiber model, presented schematically in Fig. 3 (right) (for details, refer to Paper B).
6
Fig. 3. The fiber material model used in the network simulations. (left) Stress–strain curve for bilinear isotropic hardening plasticity model (Papers A and E), where σy is the yield stress, E is
Young’s modulus and Etan is the tangent modulus. (right) Flow stress diagram for anisotropic–
kinematic hardening plasticity fiber model during cyclic loading (Paper B), where σy0 is the initial
yield stress, Ht and Hc are the kinematic hardening moduli in loading and unloading, respectively.
Bonds
Fiber-to-fiber bonds2 were modeled by point-wise contacts with debonding and subsequent friction [24, 25]. The failure of the bonds was captured with a cohesive zone model to specifically represent bond breakage and bond separation during network loading [26]. A mixed mode I and II failure was modeled in which the bond separation depends on both normal and tangential contact forces. The schematic representation of the adapted bilinear cohesive zone model for the tangent direction is shown in Fig. 4 (left). In both the normal and tangential directions, denoted by n and t subscripts, respectively, the initial linear loading continues until bond strength (Fmax) is reached at displacements un and ut, where the latter is the effective displacement for both tangential directions
2 2
1 2
t t t
u = u +u ; followed by a linear softening behavior until failure with critical displacement (u and nc u ). In this formulation, tc utc−ut is the separation distance that reflects the bond damage in the tangential direction, which is 15% greater
2 The terms “bonds” and “contacts” are used interchangeably in this work. Other notations, for
7
Fig. 4 (left) Schematic definition of the bond failure model for the tangential direction in terms of traction-separation diagram with linear softening (Papers A and E). Unloading subsequent to damage is assumed to be linear towards zero force. (right) Local separation displacements for two fibers in contact.
than the distance (ut) at bond strength (the same is true for the normal direction also). A contact was considered to be fractured when it reaches Fmax and
separated/debonded when it attains its critical displacement. When a bond fails, the contact between the corresponding fibers is described with a frictional contact. The dry friction turned out to have a negligible effect on the strength of the dry sheets.
Results
Stress–strain curve (SSC) of paper
In Paper A, we investigated the relation between micromechanical processes and the SSC of a dry fiber network during tensile loading. A typical SSC of paper from a standard tensile test is approximately linear at small strains, with Young’s modulus defining the initial slope (also referred to as tensile stiffness herein); also, it has a distinct plastic region prior to the failure (Fig. 5). When the specimen fails, the end point of the SSC defines the strain at break (also referred to as stretch) and tensile strength (or simply strength).
8
Fig. 5 Typical stress–strain curve of paper in tension with the defined methodology.
Fibers are the principal structural elements of paper [27] and it is recognized that the inelastic features of the hardening behavior of paper are due to the fibers and, surprisingly, not the bonds [28]. This conclusion was justifiably fascinating, as the natural explanation of dissipation during plasticity was attributed to the breakage of fiber-to-fiber bonds.
Here, we confirmed that the non-linear response of the network has its origin in the fibers. We compared simulations with and without applying bilinear plasticity for the fibers. These two networks attained almost the same strength, as shown in Fig. 6 (left). This figure demonstrates that the hardening behavior of the network is mainly controlled by the fibers and not by the bonds. Assigning a linear elastic material model for fibers resulted in almost 20% fewer fiber–fiber bonds completely separated prior to network rupture, as compared to the case when a plastic behavior of fibers was assumed, as shown in Fig. 6 (right).
Another important result from Paper A is that the large local strains (that depend on the initial details of the network structure — local fiber orientations, the number of bonds and density) are the precursors of bond failures and not the other way around. We compared the strain fields for the networks with weak bonds and no debonding at the same level of the global strain in Fig. 7. At this stage, the network with bond breakage allowed is just about to fail, but still attains
9
Fig. 6 (left) Stress–strain curves for different fiber material models utilized on fiber level in 10 x 4 mm2, 27 g/m2 network. (right) The total amount of fractured (solid line) and separated (dashed line) bonds for networks with different fiber material models. The legend is the same on both graphs: a. bilinear isotropic hardening plasticity; b. linear elastic fiber material model (Paper A).
Fig. 7. Strain field in the 10 x 4 mm2, 27 g/m2network, calculated prior to failure at a global strain of 0.7%, for the cases with (left) weak bonds (bond strength in normal and tangential directions being 8.5 mN and 1.4 mN respectively), (right) no debonding (Paper A).
the strain concentrations of the same size and magnitude as in the network with no debonding.
Mechanical response of a pulp fiber
If the inelastic response of paper is controlled by the fibers, the natural question is what controls the inelastic behavior of the fibers themselves.
Let us consider the fiber architecture in detail. A natural wood fiber consists of several cell-wall layers and has a helical internal structure, shown in Fig. 8.
10
Fig. 8. A schematic representation of wood fiber, redrawn from [29]. The following layers are shown: P - primary cell wall, S1, S2, and S3 - secondary cell walls. The angle of microfibril orientation with respect to the fiber axis in S2 is depicted as the microfibril angle (MFA).
Cellulose is the most important constituent of the cell walls and is responsible for the load-bearing capacity of the fiber. The thickest of the layers is S2, comprising approximately 70–80% of the fiber volume [30–32]. The structural stiffness of fiber can mainly be characterized by the properties of cellulose microfibrils embedded in a polymeric matrix of the S2 layer. The upper estimation for the stiffness of crystalline cellulose is 134 GPa [33], which may be compared with the elastic stiffness of ceramics or engineering alloys.
The ultrastructure of a fiber includes microfibrils that are wound around the axis of the fiber and form a helical assembly in bundles. It is certain that the key feature is the orientation angle of the microfibrils, known as the microfibril angle (MFA) measured with respect to the fiber axis. It is known that this orientation angle governs the mechanical properties of fiber [31, 32, 34–36]. To prepare pulp for papermaking, wood is fractionated and separated into single fibers by means of mechanical or chemical treatments, called pulping process. This modifies the cell structure irreversibly, including the MFA [37], as well as the composition of the cell walls. Thereby, fiber initially damaged after pulping may have another structure as shown in Fig. 9.
11
Fig. 9 (left) SEM image of thermo-mechanical (TMP) pulp fiber from pine, reproduced with the permission of the authors [38]. Bar 10 μm. (right) SEM image of a refined pulp showing disruption of the S2 layer due to cyclic loading reproduced with permission from Springer, Blackie Academic & Professional, ©1995 [39].
Fig. 10. Schematic representation of a 3D finite-element pulp fiber model utilized in Paper B. It shows fibrils orientation, microfibril angle MFA, introduced coordinate system and elliptic cross-section (mimicking a partly collapsed fiber).
In Paper B we concentrated on developing a three-dimensional pulp fiber model consisting of an S2 layer that accounts for the orientation of microfibrils, depicted in Fig. 9, and the chemical composition of the fiber cell wall. The schematic representation of the simulated fiber is shown in Fig. 10.
We considered chemical (bleached kraft) and thermomechanical pulp (TMP) fibers by assuming different fractions of cellulose microfibrils and surrounding matrix. Since MFA has a direct impact on the elastic modulus of the fiber, the effect is different for the kraft/TMP fiber types due to different amounts of lignin and disordered cellulose regions that also have low elastic moduli [40]. The effect
12
of the initial MFA on the mechanical properties of the two considered fiber types were incorporated into a constitutive model (Fig. 3 (right)). This model is capable of describing the complex response to the loading history that fibers undergo during paper manufacturing, simulated by several cycles of loading and unloading as presented in Fig. 11.
Fig. 11. Stress–strain curve comparison of the constitutive model of a fiber (hatched line) with simulated results (solid line) for TMP fiber with initial microfibril angle of 43º (left) and 21º (right). The details and material constants are presented in Paper B.
Fiber and network connectivity data
Based on the numerical results in Paper A, paper strength is largely affected by the number of fiber-to-fiber bonds. The predictability of the micromechanical models of paper relies heavily on this fundamental paper property. One way to describe the network connectivity is to address the statistics of its contacts. Many researchers have recently turned to approximation of the number of fiber-to-fiber bonds as being mainly an extension of van Wyk's theory for textile fibers [41], starting with the estimation of the spatial density of fiber-to-fiber contacts in a random fiber assembly [42], prediction of the number of contacts in a planar fiber network [43] and numerical evolution of the number of contacts under compressive loads [44, 45].
In the numerical models of fiber networks, the number of fiber-to-fiber bonds can either be used as an input parameter for network generation [46] or calculated from the generated network [47], giving the free fiber segment length in the
13
ranges of 15–39 μm [17, 42, 48–50] or even 375–530 μm [51]. However, since little experimental knowledge on the fiber-to-fiber bonds was available, this created a need for direct measurements of the number of bonds in paper.
X-ray computed tomography
Non-destructive X-ray computed tomography at micrometer resolution (μCT) [52] is acknowledged to have potential in describing the internal characteristics of fiber network structures. A series of two-dimensional μCT images, also called tomograms, provide the X-ray absorption coefficient of the tested material. An example of a paper tomogram is shown in Fig. 12. A set of consecutive μCT images is collected by 180-degree rotation of the sample maps three-dimensional coordinates with X-ray absorption. The basics of the geometrical optics of the projections and the reconstruction techniques can be found in the handbooks [53, 54]. Utilizing image processing of tomograms of the fiber network enables us to distinguish fibers from the air at a resolution of less than 1 μm. However, automatic processing has not yet made a breakthrough due to fundamental reasons, considering that there is no difference in the absorption coefficient between two fibers that are close to each other and two fibers that are actually bonded.
This is mainly because natural fibers, and particularly wood fibers, are difficult to outline or segment in the images due to their plant origin and varied morphology. Prior to pulping, wood fibers can be assumed to be hollow tubes or, in other words, have lumens. However, papermaking may disrupt the structure of the fiber, for example, by collapsing cell walls so that the lumen disappears or by grinding the fibers and, thus, introducing fines. Since three-dimensional μCT images only contain the absorption coefficients, all the direct measurements on the fibers require a segmentation of individual fibers subsequent to image acquisition.
14 Fig. 12 Tomogram of a handsheet; the scale bar is 100 μm.
There exist quite a few automatic methods for wood fiber segmentation that may be applicable to paper samples, as in [50, 55–58], to mention a few. However, none of the current image analysis methods can be used to segment all individual wood fibers in μCT images of paper in an obvious way. This means that there is no straight path from imaging to statistical measurements of the fiber network structure. Eventually, we aimed for fully automatic methods. However, a qualitative verification would be required to establish it for all situations. Hence, we have chosen a semi-automatic two-step approach: firstly, a manual fiber seeding, which requires minimal user interaction and secondly, completely automatic methods for actual measurements on the identified fibers.
In Paper C, we present a semi-automatic methodology for extracting important paper properties from 3D μCT images, such as the free fiber length (FFL), as well as fiber-to-fiber relative contact areas (RCA), and others. An example of one of the extracted network samples is presented in Fig. 13.
Based on the developed algorithms, we analyze the geometrical structure of the fiber network and its connectivity in Paper D. This is a large-scale (regarding the number of fibers) study to investigate the structural characteristics of paper networks, including estimation of fiber cross-sectional shapes, distances between fiber-to-fiber contacts, and relative contact areas, and to discuss the
15
Fig. 13. (left) Extracted 3D representation of one of the scanned samples with a size of 2mm × 2mm and thickness, t=44.9µm. (right) 3D visualization of the demarcated fibers from the test sample corresponding to the left figure. In the visualization, each fiber is defined by a fiber axis and the median of its inner and outer radii and shown as hollow cylinders of constant radii, discussed in Paper C.
representativeness of the studied samples. The results presented in this paper are compared with the existing statistics on the number of contacts [17, 41, 59]. Furthermore, we propose a methodology for the density of contacts in 3D fiber networks. The results include geometrical characterization of fiber networks in terms of distributions and/or mean values with the intention of a direct data transfer to fiber network mechanical modeling and data verification.
Extracting fiber morphological properties from μCT
In order to measure basic pulp morphologies, fiber morphology analyzers (FMAs) applied to a highly diluted fiber suspension are frequently used [60]. However, FMAs are applied to the swollen fibers. Consequently, the reported fiber wall thickness and fiber width are larger than those in the final sheet. Furthermore, these tools do not report the fiber WH ratio, which is an important factor that can be either inherited from or affected by the papermaking process. Hence, the primary focus of the developed methodology, presented in Papers C and D, was the extraction of the complementary information about cross-section properties in the dry sheet from μCT. The difference between the width data measured from the fiber suspension and those calculated from μCT is demonstrated in Fig. 14.
16
Fig. 14. Distribution of fiber widths measured by fiber morphology analyzer (FMA) [60] and computed from μCT images (Papers D and E).
We estimated that the transverse cross-sectional shrinkage of the unbleached softwood sulfate pulp from the completely wet state to the dry condition in the handsheet was about 20%, as estimated by comparing the fiber morphology data in the wet state with microtomography data acquired from dry sheets. The result corresponds well with the expected transversal shrinkage of the fiber cross-section, which is at a level of 15–40% [17, 27].
Since 2D models of the network structure are sufficient in many applications, we would like to couple our measurements to planar networks as well. Based on the extracted data from μCT, we compare our measurements on the number of bonds in paper for 3D networks with currently available knowledge for 2D networks [17], shown in Fig. 15.
Since the calculated values of NCPi correspond well with the number of contacts
predicted by the fiber aspect ratio (NCPpred), Fig. 15 signifies that almost a
complete association between the calculated (3D) and theoretical predictions (2D) has been reached. This implies that the number of fiber-to-fiber contacts in 3D networks is controlled by the geometrical parameters of the fibers, assuming isotropic orientation of the fibers. We suggest that this result be used as a rule of thumb for verification purposes when generating fiber networks.
17
Fig. 15 Theoretical prediction of the number of contact points NCPpred, estimated as the ratio of
fiber length to its width (fiber aspect ratio) plotted against the calculated NCPi acquired from µCT
results. The agreement between two quantities (NCPpred and NCPi) is found by the linear fit with
the calculated Bravais–Pearson's correlation coefficient (r) as 0.9530 ≤ r ≤ 0.9575 (95% confidence level) (Paper D).
Influence of fiber morphology and bond strength
The current trends in papermaking are about reducing the costs through the optimal use of raw materials [17]. To achieve these goals there are several possibilities. One of them involves a competent selection of the appropriate pulp fibers in terms of their morphological properties, such as fiber length, width, wall thickness, etc. Very often, the pulps are compared using length-weighted average values, since it does not significantly depend on the fraction of fines, and correlates better with paper properties [61]. Nevertheless, the morphological properties are subject to significant variations.
Another possible strategy for reducing the cost of paper products, while maintaining the basic properties on the prescribed level, is reducing a number of fibers in combination with improving the bond strength. The available strategies for increasing the bond strength, such as the addition of strengthening chemicals and mechanical refining of fibers, affect a number of important factors, including network density and fiber morphology. Therefore, it is difficult to single out the effect brought by the increased bonding strength in interpreting the results of physical experiments. The development of robust micromechanical tools for measuring and manipulating the fibers has enabled researchers to conduct extensive studies and retrieve reliable bond strength distributions [62]. It was
pred
Fiber length NCP
Fiber width =
18
possible to study how the bond strength responds to certain modifications, which affect not only the average value but also the distribution of the bond strength. The questions we answer in Paper E are directly linked to both of these possibilities which are related to a reduction in raw material usage. We reflect upon whether it is appropriate to use the mean values of fiber properties as compared to accounting for the details of variations within the fibers, and investigate how the details of the bond strength distribution affect the fiber network strength and strain to failure.
We used the extracted fiber data for model verification for both fiber network reconstruction and mechanical simulation. The verification is performed on the handsheet information from Paper A. Most of the unknown factors, such as the exact shape of the dry fiber cross-sections, the number of contacts and network thickness, were eliminated. The results for the verification networks are presented in Fig. 16.
Fig. 16 Stress–strain curve of the experimental laboratory sheets (Paper A), compared with the reference network with constant bond strength (Paper E).
As can be seen in Fig. 16, all the parameters (tensile stiffness, strength, and stretch) are well captured with the chosen fiber and bond constitutive properties, presented in Paper E. Another verification parameter is the ratio of normal to tangential bond strength, kept constant at a value of 0.25 in all the numerical tests. The contact parameters in the normal direction influence the results marginally (Fig. 17). This is true even when the normal bond strength is varied by a factor of 10 (originally used in Paper A), with respect to the corresponding
19
parameters in the tangential direction. This proves that the overwhelming majority of the fibers fail in nearly pure shear mode in the considered cases.
Fig. 17 Stress–strain curve of the networks with constant bond strength from Papers A and E, with different ratios between normal and tangential bond strengths.
When fiber-to-fiber bond strength follows some distribution, for example as presented in Fig. 18 (left), it affects the mean values of strength and stiffness of the network of fibers (Fig. 18 (right)). However, these results suggest that comparing the average bond strength while having a sufficient number of samples is an adequate strategy for characterization of the mean values of strength and stiffness of the network of fibers, only if the bond strength distribution is symmetric.
Fig. 18 (left) Distributions for tangential bond strength utilized in the simulation based on experimentally measured bond strength values of Marais et al. (2014). A constant normal to the tangential ratio for bond strength is assumed to be 0.25 in all the numerical tests. (right) Effect of bond strength distribution on the average network properties, shown for the networks of 27 g/m2. The relative difference in the average properties is calculated with respect to the “Constant BS” values.
20
Meanwhile, the variability in length-weighted fiber shape factor and fiber length have a major effect on mean values of strength and stiffness of the networks (Fig. 19). In particular, the network composed of fibers having the average length-weighted fiber shape factor constant had over 20% reduced strength and stiffness. This was partly due to the altered number of contacts, but mostly due to the increased bending deformation modes in the fibers.
Fig. 19 (left) Effect of the distribution of fiber properties on the average network properties (i.e. strength, stiffness, strain at failure (stretch), network thickness and number of contacts) shown for fiber networks of 10 mm x 4 mm and 27 g/m2 in (left) and 65 g/m2 in (right). The relative difference in the average properties is calculated with respect to the “Original pulp” values (Paper E).
Hence, in the variation in fiber properties, foremost in fiber length and fiber shape factors, should be given separate attention in benchmarking the fiber characterization data.
Conclusions
The network model used in this study is probably the first three-dimensional model that is capable of simulating the fracture process of paper by accounting for both nonlinearities at the fiber level and bond failures. With the help of the model, we have identified the effect of various factors on the stress–strain curve of paper, including the contribution from bonds, fibers and the paper structure itself. Additionally, we have examined the importance of variation in fiber and bond properties on the mechanical properties of paper.
21
We have shown that the original strain inhomogeneities due to the structure are transferred to the local bond failure dynamics. This accelerates the bond fracture in the regions of strain concentrations. The width of the strain concentration regions is at a millimeter scale and obviously depends on the initial details of the network structure, such as local fiber orientations and bond density. Again, in three-dimensional networks, the bond density is dependent on the fiber aspect ratio (the fiber’s width-to-height ratio). Hence, bond fractures are interdependent with fiber properties.
In addition, the complexity of the structural response of the three-dimensional layered fiber structure can be summarized into two factors:
• Geometrical – through the impact of the helical orientation of the microfibrils and,
• Material – through the inclusion of strong cellulose into the soft matrix in the fiber cell walls.
We have investigated the influence of both of these factors on the mechanical response of the pulp fiber. To sum up, in the elastic region, the change in the microfibril orientation upon axial straining is mainly a geometrical effect and is independent of the material properties of the fiber. During plastic deformations, the change in microfibril orientation is accelerated and makes it material-dependent. To cover these features, an anisotropic–kinematic hardening plasticity model is proposed for a fiber.
Using the data from the non-destructive X-ray tomography experiment, we have extracted and analyzed the connectivity properties of fibers of low-grammage handsheets with isotropic fiber orientation. The methods have been developed in order to obtain network characteristics, such as free fiber length, the number of contacts, spatial positions of the fibers and their orientations. All of these parameters are used to describe the initial details of the network geometry. The main advantage of the presented semi-automated approach is the possibility to extract the information from several measurements without a complete segmentation of the fiber network. Moreover, the developed algorithms work equally well for the complex fiber geometries – collapsed (without the lumen) and
22
non-collapsed fibers – and for the fibers with pores. We have calculated the fiber cross-section shrinkage factor to be 20%, which is a reasonable addition to the conventionally used Fiber Morphology Analyzer (FMA) data, measured on water-swollen fibers. This shrinkage factor has been used for FMA data correction on the reconstructed networks.
Having a model that uses verified input parameters gives us a reliable tool to investigate the impact of each independent parameter of paper strength. In turn, this makes process-dependent decision-making much easier.
Final remarks
This doctoral thesis combines experiments and numerical simulation of paper as a network of fibers, as well as comparison with experimental data, obtained both by myself and from the literature.
To summarize, paper is a natural and recyclable material. It is also a multifunctional material that can be cut, folded, perforated, crumpled, etc. and has a potential for broad applicability. The emerging novel applications for paper material include printed electronics, fluid control devices, microelectromechanical systems, etc. [64]. The reader interested in contemporary paper applications is recommended to visit Paper & Packaging – How Life Unfolds™ [65] for inspiration.
Understanding the impact of various factors on the specific paper properties is the key to product development. Since all this work is done towards industrial applications, a natural step for future work is revising the failure criteria for the anisotropic fiber networks in multiaxial loading conditions. This would give insights on the effect of fiber and bond variations on the strength of a wide range of paper products. It would be valuable to advance to denser samples larger than the fiber length, including contact statistics from µCT. That requires optimization of the presented image analysis methods, striving for fully automatic fiber segmentation. Finally, considering the fiber network under compression is a topical issue for many packaging materials. Addressing this problem would
23
require extending the formulation to describe local buckling and compressive failure of the fiber segments.
25
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Summary of the appended papers
Paper A: Stress–strain curve of paper revisited
We have investigated a relation between micromechanical processes and the stress–strain curve of a dry fiber network during tensile loading. By using a detailed particle-level simulation tool, we investigate, among other things, the impact of “non-traditional” bonding parameters, such as compliance of bonding regions, work of separation and the actual number of effective bonds. This is probably the first three-dimensional model that is capable of simulating the fracture process of paper accounting for non-linearities at the fiber level and bond failures. The failure behavior of the network considered in the study could be changed significantly by relatively small changes in bond strength, as compared to the scatter in bonding data found in the literature. We have identified that compliance of the bonding regions has a significant impact on network strength. By comparing networks with weak and strong bonds, we concluded that large local strains are the precursors of bond failures and not the other way around.
Paper B: Constitutive modeling of paper fiber in cyclic loading applications
In this paper, we investigate the influence of geometrical and material parameters on the mechanical response of the pulp fibers used in paper manufacturing. We developed a three-dimensional finite element model of the fiber, which accounts for microfibril orientation of cellulose fibril, and the presence of lignin in the secondary cell wall. The results showed that the change in the microfibril orientation upon axial straining is mainly a geometrical effect, and is independent of the material properties of the fiber, as long as the deformations are elastic. Plastic strain accelerates the change in microfibril orientation and thus makes it material-dependent. The results also showed that the elastic modulus of the fiber has a non-linear dependency on the microfibril angle, with the elastic modulus being more sensitive to the change of microfibril angle around small initial values of microfibril angles. Based on numerical results acquired from a 3D fiber model supported by available experimental evidence, we propose an anisotropic– kinematic hardening plasticity model for a fiber within a beam framework. The
31
proposed fiber model is capable of reproducing the main features of the cyclic tensile response of a pulp fiber, such as stiffening due to changing microfibril angle. The constitutive model of the fiber was implemented in a finite element model of the fiber network. By using the fiber network model, we estimated the level of strain that fiber segments accumulate before the typical failure strain of the entire network is reached.
Paper C: Characterizations of fiber networks in paper using microcomputed
tomography images
In this work, we present a methodology for microstructure characterization of individual fibers, as well as paper. The methodology is based on three-dimensional computed tomography images of paper at micrometer resolution. The first step of the method consists of a graphical user interface (GUI), designed to minimize the amount of manual labor. To manually identify a fiber from a 2x2 mm2 paper sheet takes about one minute with this GUI. Then several algorithms are available to analyze the image data automatically, guided by the user input. With this approach it is possible to measure several characteristic properties without complete segmentation of the individual fibers. The methodology includes a method to calculate the contact areas between fibers even in extreme cases of severely deformed fibers, which are naturally present in paper. Among the measurable properties are also estimators for the free fiber lengths and fiber wall thickness.
Paper D: Extracting fiber and network connectivity data using microtomography
images of paper
In this work, we apply image analysis methods based on microcomputed tomography (µCT) methods, developed in Paper C, to extract the parameters that characterize the structure and bonding parameters in the fiber network of paper. The scaling and variational properties of the µCT images are examined by analyzing the paper structural properties of two 1 × 1mm2 test pieces, which have been cut out from a low-grammage handsheet. We demonstrate the applicability of the methods for extracting the free fiber length, fiber cross-sectional data, the distances between the fibers, and the number of fiber-to-fiber bonds, which are the
32
key properties required for the adequate representation of the network in numerical models. We compare the extracted connectivity data with the earlier reported analytical estimations and conclude that the number of contacts in three-dimensional networks is controlled by the fiber aspect ratio. In addition, we compare the cross-sectional data with those measured by the fiber morphology characterization tools and estimate the fiber shrinkage from completely wet to dry state to be nearly 20%.
Paper E: Effect of fiber and bond strength variations on the tensile stiffness and
strength of fiber networks
As the fiber and bond characterization tools sophisticates, the information from the fiber scale becomes richer. It is used for benchmarking of different types of fibers by the paper- and packaging industry. In this work, we addressed a question about the effect of variability in the fiber and fiber bond properties on the average stiffness and strength of the fiber networks. We used a fiber-scale numerical model and reconstruction algorithm to address this question. The approach was verified using the experimental sheets having the fiber data acquired by fiber morphology analyzer and corrected by microtomography analysis of fibers in these sheets. We concluded, among other things, that it is sufficient to account for the average bond strength value with acceptable number of samples to describe dry network strength as long as the bond strength distribution remains symmetric. We also found that using the length-weighted average for fiber shape factor and fiber length data neglects the important contribution from the distribution in these properties on the mechanical properties of the sheets.
