eth-6907-02

Multiscale poroelastic model:bridging the gap from cellular to macroscopic scale

Ahmad Rafsanjani AbbasiDISS. ETH NO. 20821

DISS. ETH NO. 20821

Multiscale poroelastic model:bridging the gap from cellular

to macroscopic scale

A dissertation submitted to

ETH Zurich

for the degree of

Doctor of Sciences

presented by

Ahmad Rafsanjani Abbasi

Master of Science in Mechanical Engineering (2009)Iran University of Science and Technology

born January 18, 1984citizen of Iran

accepted on the recommendation of

Prof. Dr. Hans Jürgen Herrmann, examinerProf. Dr. Jan Carmeliet, co-examiner

Prof. Dr. Dominique Derome, co-examinerProf. Dr. Lambertus Johannes Sluys, co-examiner

2013

Cover: Cellular structure of Norway spruce softwood. Picture by MasaruAbuku and design by Ehsan Rafsanjani Abbasi.

Copyright © by Ahmad Rafsanjani Abbasi

To my lovely wife, Shima

Abstract

Many biological and engineering materials are essentially porous or cellular, afeature which provides them with a low density, a high strength and a hightoughness. The deformation of cellular materials in response to environmentalstimuli such as changes in relative humidity is of practical interest to evaluatethe durability of materials in different working conditions. In this thesis,the hygro-mechanical behavior of hierarchical cellular materials is investigatedusing a multiscale computational framework. Attention is focused on softwoodsbut the proposed model is general and can be applied to other cellularmaterials. In wood, the interaction of the moisture and mechanical behavior isbest observed in swelling. The complicated hierarchical architecture of woodintroduces a strong geometric anisotropy which is reflected in the anisotropyof its mechanical and swelling behavior. A two-step computational upscalingmethod is utilized to devise a finite element model for the estimation of swellingbehavior of softwoods. Starting from the cellular scale which represents theunderlying structure of the growth ring scale, an efficient scheme is developedfor the estimation of the hygro-elastic properties of periodic honeycombs asa model for the cellular structure of wood. Predicted results are found to becomparable to experimental data at both cellular scale and growth ring level.A poromechanical approach is also presented as an alternative formulationfor the estimation of the effective swelling coefficients of cellular materials.The computational approach proposed in this thesis provides a predictivetool for revealing the structure-property relations of biological and engineeringcellular materials and can also be used for the design of new functional cellularmaterials with tailorable swelling properties.

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Zusammenfassung

Viele biologische und technische Materialien haben eine poröse oder zellenar-tige Struktur, eine Eigenschaft die ihr eine niedrige Dichte, grosse Festigkeitund hohe Zähigkeit verleiht. Die durch Umweltfaktoren so z.B. die relativeFeuchte verursachte Deformation von zellenartigen Materialien ist für dieDauerhaftigkeit dieser Materialien unter verschiedenen Anwendungsbedin-gungen von grossem Nutzen. In der vorliegenden Arbeit wird das hygro-mechanische Verhalten von hierarchisch strukturierten zellulären Materialienunter Zuhilfenahme eines rechnergestützten und über mehrere Grössenskalenerstreckten Programms untersucht. Das Hauptaugenmerk ist auf Weichholzgesetzt aber das vorgeschlagene Modell ist auch auf andere zelluläre Materialienanwendbar. Im Falle von Holz zeigt sich die Wechselwirkung zwischenFeuchte und mechanischem Verhalten am besten am Phänomen des Quellens.Die komplexe hierarchische Struktur von Holz bewirkt eine bedeutsamegeometrische Anisotropie, die sich im mechanischen und Schwellverhaltenniederschlägt. Eine rechnergestützte zweistufige Hochskalierung wurde zurAufstellung eines Finite-Elementen Modells für das Quellverhalten von Weich-holz verwendet. Beginnend mit der zellulären Skala und dann weiterführendmit einer periodischen Wabenstruktur der Jahresringe wird ein leistungsfähigesModell entwickelt, um die hygro-elastischen Eigenschaften der Holzstrukturabzuschätzen. Vorhergesagte Resultate stimmen mit experimentellen Datenin beiden Skalen überein. Als Alternative wird auch eine poromechanischeFormulierung zur Abschätzung der effektiven Quellkoeffizienten von zellulärenMaterialien präsentiert. Der rechnergestützte Ansatz, der hier vorgestellt wird,ist als eine Vorhersage der Wechselwirkung zwischen Struktur und Eigenschaftvon biologischen und technischen Materialien zu verstehen. Des Weiteren,können hiermit neue funktionale zelluläre Materialien mit zugeschnittenemQuellverhalten entworfen werden.

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Résumé

Beaucoup de matériaux biologiques et d’ingénierie sont essentiellement poreuxou alvéolaires, caractéristique qui leur donne une faible densité, une résistanceélevée et une grande endurance. La déformation des matériaux alvéolairesen réponse à des stimuli environnementaux tels que les variations d’humiditérelative est d’un intérêt pratique pour évaluer la durabilité des matériaux sousdifférentes conditions de service. Dans cette thése, le comportement hygro-mécanique des matériaux alvéolaires hiérarchiques est étudié en utilisant uncadre computationnel multi-échelle. Nous portons notre attention sur lesbois de résineux, mais le modèle proposé est général et peut être appliquéeà d’autres matériaux alvéolaires. Pour le bois, l’interaction de l’humiditéet du comportement mécanique s’observe par le gonflement. L’architecturehiérarchique du bois présente une forte anisotropie géométrique qui se reflètedans l’anisotropie de son comportement mécanique et de gonflement. Uneméthode d’homogénéisation computationnelle sur deux échelles est utiliséepour concevoir un modlèle par éléments finis estimant le gonflement dubois de résineux. A partir de l’échelle cellulaire qui est la structure sous-jacente des cernes de crossance un système efficace est développé pourl’estimation des propriétés hygro-élastiques de structures en nid d’abeillespériodiques, utilisé comme modèle pour la structure alvéolaire du bois. Lesrésultats sont comparables aux données expérimentales, tant au niveau del’échelle cellulaire et de celui des cernes de croissance. Cette thèse présenteaussi une deuxiéme formulation de l’estimation des coefficients effectifs degonflement des matériaux alvéolaires basée sur l’approche poromécanique.L’approche computationnelle proposée dans cette thèse fournit un outilprédictif révélant les relations structure-propriétés mécaniques des matériauxalvéolaires naturels et d’ingénierie, approche qui pourrait également êtreutilisée pour la conception de matériaux alvéolaires innovants avec despropriétés de gonflement adaptables.

vii

Acknowledgment

I would like to express my gratitude for many who have supported me duringmy studies at ETH Zurich. First of all, I am grateful for the opportunityto have worked with and learned from my promoter Hans Herrmann. I havebeen strongly affected by his excitement for scientific research, and diversity ofacademic interests. Similarly, I am indebted to Jan Carmeliet who has providedconstant support and guidance, encouraging me to pursue my academic andcareer dreams. I am also thankful to Dominique Derome, who has freely sharedher time, thoughts, and enthusiasm for scientific discovery. She has guided methrough the academic challenges which have inspired my best work, and hasconsistently supported all of my academic and professional endeavors. Also,I would like to thank Bert Sluys for being in my examination committee,critically reading my manuscript and providing fruitful comments.

I am also very grateful for all the support I got from other researchers. Iwould like to thank all my colleagues in Sinergia project: Alessandra Patera,Martin Dressler, Christian Lanvermann, Francois Gaignat and Stephan Hering.I would like to thank Massaro Abuku and Stephan Carl for providing ESEMexperimental data on the swelling of wood at the growth ring level. I thankAlessandra Patera and Michele Griffa for providing X-ray tomography dataof wood at the cellular scale. I would like to thank Peter Niemz who hasa deep knowledge in wood physics and has welcomed discussion about thevalidation of the model with experiment. I would like to thank ChristianLanvermann for sharing his experimental data on swelling of wood. I want tothank Falk Wittel for fruitful discussion on the mechanics of cellular materialsand providing useful comments on my work. I would like to thank RobertGuyer for his continuous interest in my research. I thank all my colleaguesat Laboratory for Building Science and Technology and Wood Laboratory atEMPA for creating a nice working environment during last three years. I thankKarim Ghazi Wakili for German translation of the abstract and Dominique

ix

Derome for French translation. I would like to thank Margrit Conradin andMartina Koch for their administrative support. I thank my brother, Ehsanwho kindly designed the cover of this dissertation. Furthermore, I thank allmy friends and peers at EMPA, EAWAG and ETH Zurich, from Iran and allaround the globe, whose friendship made me feel at home away from home.

This research was funded by Swiss National Science Foundation in theframework of the Sinergia project “Multiscale analysis of coupled mechanicaland moisture behavior of wood ” under Grant number 125184. The projectwas also supported by Swiss Federal Laboratories for Materials Science andTechnology (EMPA). Their supports are greatly acknowledged.

I owe special thanks to my parents and my family for their decades of love,encouragement and support. I am also grateful to my kind parents-in-law whohave supported me as well. Finally, my lovely wife, Shima, deserves credit formy successes every bit as much as I do. She has stood by my side throughmy entire graduate education, making untold sacrifices graciously and withoutcomplaint. Shima, these words are for you.


November 2012

Contents

Contents xiii

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Sinergia Project: cooperative approach . . . . . . . . . . . . . . . 21.3 Objectives of the research . . . . . . . . . . . . . . . . . . . . . . . 41.4 Scope and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 State of the art 72.1 Softwood, a natural cellular material . . . . . . . . . . . . . . . . 8

2.1.1 The structure of softwood . . . . . . . . . . . . . . . . . . 82.1.2 Anisotropic swelling behavior of softwood . . . . . . . . . 11

2.2 Micromechanical modeling of softwood . . . . . . . . . . . . . . . 132.2.1 Cellular models . . . . . . . . . . . . . . . . . . . . . . . . 132.2.2 Hierarchical models . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Needs for further research . . . . . . . . . . . . . . . . . . . . . . . 18

3 Computational upscaling of cellular materials 213.1 Computational upscaling method . . . . . . . . . . . . . . . . . . 21

3.1.1 Strain averaging theorem . . . . . . . . . . . . . . . . . . . 233.1.2 Hill-Mandel condition . . . . . . . . . . . . . . . . . . . . . 243.1.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . 25

3.2 Finite element implementation . . . . . . . . . . . . . . . . . . . . 273.2.1 Stress and strain averaging . . . . . . . . . . . . . . . . . . 283.2.2 Master node technique . . . . . . . . . . . . . . . . . . . . 293.2.3 Hygroelastic constitutive equations . . . . . . . . . . . . . 31

3.3 Upscaling model for cellular materials . . . . . . . . . . . . . . . 323.3.1 Symmetry conditions of honeycombs . . . . . . . . . . . . 333.3.2 Effective swelling coefficients: hygro-elastic model . . . . 34

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3.4 Poromechanics approach . . . . . . . . . . . . . . . . . . . . . . . 343.4.1 Poroelastic constitutive equations . . . . . . . . . . . . . 35

Poroelastic material properties . . . . . . . . . . . . . . . 363.4.2 Effective Swelling of ideal microstructures . . . . . . . . . 36

Single-porosity material . . . . . . . . . . . . . . . . . . . 37Two-scale double-porosity material . . . . . . . . . . . . . 38

3.4.3 Effective swelling coefficients: poroelastic model . . . . . 413.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4 Hygroelastic behavior of honeycombs 454.1 Verification of the computational model . . . . . . . . . . . . . . 46

Shape angle . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Cell wall thickness . . . . . . . . . . . . . . . . . . . . . . . 48

4.2 Honeycombs with multi-layered cell walls . . . . . . . . . . . . . 494.2.1 Numerical results . . . . . . . . . . . . . . . . . . . . . . . 51

Hygro-expansion coefficients . . . . . . . . . . . . . . . . . 51Elastic and shear moduli . . . . . . . . . . . . . . . . . . . 52Anisotropy ratio . . . . . . . . . . . . . . . . . . . . . . . . 53Shape angle . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.3 Remarks on swelling of wood cells . . . . . . . . . . . . . . . . . . 564.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5 The role of eccentricity on swelling anisotropy of cellularmaterials 615.1 Periodic honeycombs . . . . . . . . . . . . . . . . . . . . . . . . . . 635.2 Real cellular structure of softwood . . . . . . . . . . . . . . . . . 645.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.3.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . 67Material properties of the cell wall . . . . . . . . . . . . . 69

5.3.2 Swelling behavior of periodic honeycombs . . . . . . . . . 69Symmetric honeycombs . . . . . . . . . . . . . . . . . . . . 69Eccentric honeycombs . . . . . . . . . . . . . . . . . . . . . 71

5.3.3 Hygro-elastic behavior of real cellular structure . . . . . 72Apparent swelling coefficients . . . . . . . . . . . . . . . . 72Apparent elastic moduli . . . . . . . . . . . . . . . . . . . 73Comparison with experiment . . . . . . . . . . . . . . . . 76

5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6 Poromechanics model for swelling of cellular solids 816.1 Moisture induced swelling in softwood . . . . . . . . . . . . . . . 816.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.2.1 Transverse tangent poroelastic properties . . . . . . . . . 856.2.2 Swelling coefficients . . . . . . . . . . . . . . . . . . . . . . 876.2.3 Shape angle . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.2.4 Comparing poroelasticity and hygroelasticity . . . . . . . 91

6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

7 Multi-scale modeling of hierarchical cellular materials 957.1 Hierarchical multiscale model . . . . . . . . . . . . . . . . . . . . 95

7.1.1 Hygro-elastic behavior of a growth ring . . . . . . . . . . 96Upscaling procedure . . . . . . . . . . . . . . . . . . . . . . 99Numerical results . . . . . . . . . . . . . . . . . . . . . . . 100

7.1.2 Free swelling of softwood . . . . . . . . . . . . . . . . . . . 1057.2 FE2 Multiscale method . . . . . . . . . . . . . . . . . . . . . . . . 114

7.2.1 Numerical results . . . . . . . . . . . . . . . . . . . . . . . 114A cellular material under uniaxial tensile load . . . . . . 115Pure bending of a cellular beam . . . . . . . . . . . . . . 115

7.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

8 Hygroelastic behavior of microstructured materials:from honeycombs to auxetics 1218.1 Hygro-elastic properties . . . . . . . . . . . . . . . . . . . . . . . . 122

8.1.1 Conventional vs. re-entrant honeycombs . . . . . . . . . 122Elastic moduli . . . . . . . . . . . . . . . . . . . . . . . . . 123Poisson’s ratios . . . . . . . . . . . . . . . . . . . . . . . . . 124Hygro-expansion coefficients . . . . . . . . . . . . . . . . . 125

8.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

9 Conclusion and Outlook 129Future perspectives . . . . . . . . . . . . . . . . . . . . . . 130

Appendix 133

Bibliography 137

Curriculum vitae 145

Chapter 1

Introduction

1.1 Background

Many biological and engineering materials show a hierarchical structure inwhich a cellular microstructure is embedded. Wood, bamboo, trabecularbone and honeybee combs are examples of natural hierarchical materials withcellular architecture (Lakes, 1993). The cellular nature of these hierarchicalmaterials plays a significant role in their effective macroscopic behavior.Understanding the hygro-mechanical behavior of cellular materials is crucialfor optimal material design. Most engineered honeycombed materials usessimple isotropic materials. The design of such materials could benefit fromthe study of the multiscale structure of natural cellular materials, with all thesophistication found at the micro-scale. In hierarchical materials, the origin oftheir specific macroscopic behavior lies at their microscale. It has been widelyrecognized that the morphology and the properties of the microscopic buildingblocks play fundamental roles on the behavior observed at the macroscale.From this insight, an endeavor emerges towards finding the structure-propertyrelations of multi-level materials which links the macroscopic properties to themechanics of their underlying microstructures. A successful outcome to suchproject would provide an alternative route to phenomenological models forpredicting the macroscopic material response and would open the doors forthe design of new functional materials and structures.

The prediction of the material response to external loads has retained theinterests of scientists and engineers for many decades. In addition to

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CHAPTER 1. INTRODUCTION

mechanical loading, environmental stimuli such as changes in relative humidityor temperature can also cause deformation in the material. We focus hereon the moisture induced swelling. Such swelling of microporous media isimportant for a huge variety of engineering application: for instance forsequestration of carbon dioxide in coal, for storage of hydrogen in metal-organic frameworks, for the purification of water, etc. Also, moisture inducedswelling is of practical interest when one strives for the dimensional stabilityof hygroscopic materials. We may also include in this list moisture activatedshape memory materials in e.g. biomedical applications. As a final point tokeep in mind, sorption in microporous materials may also induce undesiredinternal effects such as a decrease in permeability or cracking due to arestraining of the deformations.

Wood is a hygroscopic material with cellular microstructure which deformsin response to changes in environmental humidity. For this reasons, it isselected as the perfect model material for this thesis. The exposure ofwood elements to humidity generates differential strains inside the material,which can generate high stresses and eventually cracking. In practice, themoisture induced deformation of wood plays a significant role in degradationof the material over the time. The complex hierarchical architecture of woodis an obstacle for understanding the swelling behavior of wood with pureempirical methods which are indeed time consuming and expensive. Thus,computational multiscale methods are chosen as the preferred tool which isemployed for revealing the structure-property relations of such hierarchicalmaterials.

1.2 Sinergia Project: cooperative approach

This PhD project is part of SNF sinergia project “Coupled mechanicaland moisture behavior of wood ” which is introduced briefly in this section.Knowledge of mechanical and moisture properties of wood are crucial, sincethey determine to a great extent the durability and failure of wood elementsexposed to varying mechanical and environmental conditions. For predictivepurposes, computational models are used to forecast the relation betweenthe applied loads (mechanical and/or moisture) and the respective hygro-thermal and mechanical response. These computational models are mostlybased on continuum mechanics, where wood is assumed to behave as a linear(visco)elastic orthotropic (plastic) continuum. However, when we realize that

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1.2. SINERGIA PROJECT: COOPERATIVE APPROACH

the particular behavior of wood is driven by phenomena occurring at thecellular level, it is obvious that a continuum approach is a simplification ofthe real behavior. Macroscopic mechanical and moisture properties of woodhighly depend on lower level features such as cell geometry and orientation,angle of the cellulose fibrils in the cell wall, which vary between growth rings.These features span the whole range of spatial scales, from nanoscopic (cellwall fibril aggregates), over microscopic (cell wall layers, cell geometry), andmesoscopic (growth ring of early- and latewood) to the macroscopic scale(wood). Therefore, macroscopic material properties, such as elasticity modulusand moisture sorption isotherms (or moisture capacity), should be consideredas apparent material parameters, which incorporate actual physical constantssuch as the properties of the different chemical constituents of wood, andtheir spatial arrangement in nano- and microscale geometry of cells and ingrowth rings composed of late- and earlywood. The relationship betweenthe macroscopic apparent properties and the microscopic features is not fullyunderstood to date. Furthermore, mechanical properties and moisture capacityare interrelated, and such interaction is often not consistently taken intoaccount in continuum models. As a consequence, the available continuummodels have a limited range of physical validity. In cases where microscopicdetails are important, a multiscale modeling approach is more appropriatethan a continuum modeling approach, in order to understand the relativeimportance of microscopic features on the overall mechanical performance ofwood. Multiscale models are basically a hierarchy of submodels, which describethe coupled mechanical and moisture behavior at different spatial scales in sucha way that the submodels are interconnected. The development of such modelsis a challenging task because of the following reasons:

High resolution determination of the geometry of wood at the nano- andmicroscale, which is a basis for multiscale modeling, depends on the resolutionand the availability of advanced visualization techniques.

Measurement of mechanical and moisture properties at lower scales is difficultto carry out, and, alt-hough important technical improvements were maderecently, few experimental techniques are yet available.

Geometry computer modeling of a natural material in 3D at the micro- andnanoscale is not trivial.

Although upscaling of elastic behavior has become a well-known procedure,upscaling of time-dependent phenomena like damage and creep remains nottrivial.

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The objective of this research program is to investigate the relation betweenmacroscopic mechanical and moisture properties of wood and its microscopicstructure using a multiscale approach. Coniferous (softwood) species is themain type of wood used in building and civil engineering applications andpresents a simpler structure than deciduous (hardwood) species. Thus, twosoftwood species, spruce (Picea Abies) and pine (Pinus silvestris), will beconsidered as each species has unique geometry and cell dimensions anddifferent density. In terms of scope, the range of moisture loading is to fullhygro-scopic capacity (no liquid) and mechanical loading is studied to the edgeof failure (e.g. to the appearance of microdamage). We will address wood fromthe macroscale of a few centimeters (10−2 m) to the nanoscale (10−8-10−6m)which still includes subcellular features such as cellulose fibril aggregates.Expected contributions are improvements in the micro-meso-macro modelingand upscaling of a complex porous material, validation of the models atdifferent scales by advanced experimental techniques, development of advancedexperimental approaches for the combined study of moisture and mechanicalloading and improvement of a macromodel to include nonlinear, hysteretic,creep and damage phenomena. The combination of such advancements willallow to clarify the roles of the nano- and microfeatures of wood on itsmacroscale moisture and mechanical behavior, which is crucial to ensurethe durability of wood components exposed to varying environmental andmechanical loads.

1.3 Objectives of the research

The main objective of the research presented in this thesis is to develop amultiscale computational framework to bridge the mechanism of anisotropicswelling from microscale to macroscale for hierarchical cellular materials. Thespecific objectives are as follows:

To develop a multiscale computational upscaling framework for the predictionof the effective swelling behavior of cellular materials;

To study the moisture induced deformation mechanism and anisotropy inswelling behavior of cellular materials with respect to their microstructure;

To present a poroelastic description for the swelling behavior of cellularmaterials based on their poroelastic properties;

To investigate the contribution of morphology and material properties of cell

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1.4. SCOPE AND OUTLINE

wall constituents on the swelling behavior of softwood in order to understandthe material design principles in natural cellular materials.

1.4 Scope and outline

The research presented in this dissertation falls within the category ofcomputational solid mechanics. More specifically, the static equilibrium of amacroscopically homogeneous continuum and microscopically heterogeneoussolid subjected to mechanical loading and moisture is investigated. It isassumed that the moisture field is fully specified, i.e. we consider only moisturein equilibrium condition, and thus the transport of the moisture is not studied.The derivation is restricted to small deformations in two dimensions. Dynamiceffects and time dependent behavior such as visco-elasticity are not takeninto account. Furthermore, continuum mechanics is assumed to hold at alllength scales involved. In this thesis, multiscale refers to a framework inwhich multiple length scales are treated separately and there is an exchangeof information between these length scales. Softwoods, especially Norwayspruce, are of primary commercial importance in Switzerland. Thus attentionis restricted to softwoods in general and, where applicable, to Norway sprucein particular. In this thesis, the hysteresis and the mechano-sorptive effectsare not considered and the behavior is fully reversible.

This thesis is composed of nine chapters. The remainder of this thesis isstructured as follows:

In chapter 2, the microstructure of softwood is explained and the state of theart in micromechanical modeling of softwood is reviewed.

In chapter 3, the numerical models that are used in the subsequent chapters arediscussed. The essential ingredients of the computational upscaling method areexplained and the application of the method to the prediction of hygroelasticproperties of cellular materials with honeycomb microstructures is presented.Furthermore, based on a poromechanics approach, an alternative method ispresented for estimating the effective swelling properties of cellular materials.

In chapter 4, the effective hygroelastic behavior of honeycombs composed ofmulti-layered cell walls is investigated. Numerical simulations are conductedand the influence of different geometrical parameters on the development ofanisotropy in the swelling behavior of honeycombs is studied.

In chapter 5, the role of eccentricity on swelling anisotropy of cellular materials

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is studied. Synthetic honeycombs are generated based one the microstructuralmorphology of softwood. The effect of eccentricity in the geometry of theperiodic honeycombs on the swelling behavior is examined.

In chapter 6, the poromechanical approach developed in chapter 3 is appliedto softwoods and the effective swelling behavior of earlywood and latewoodtissues is estimated.

In chapter 7, a two-scale computational hierarchical upscaling scheme ispresented for the prediction of the anisotropic swelling behavior of softwood.The treatment of mechanical loads in a two-scale upscaling model for cellularmaterials is illustrated for simple load cases with comparison to referencesolutions.

In chapter 8, the computational model developed for estimating the swellingbehavior of honeycombs is extended to study another group of cellular solids,called re-entrant auxetics, which shows an unusual behavior, i.e. negativePoisson’s ratio. The hygroelastic properties of re-entrant auxetics are simulatedand the results are compared with the response of honeycombs.

Chapter 9 summarizes the main conclusions of this thesis and presentsrecommendations for future research.

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Chapter 2

State of the art

Cellular solids are present in both man-made and natural materials withremarkable properties (Gibson and Ashby, 1997). Cellular materials organizethe matter into an interconnected system of beams and plates usually ina much lesser density than the bulk material. Physical properties suchas stiffness, strength, thermo or hygro-expansion, thermal conductivity anddiffusivity of the bulk material can be changed by several orders of magnitudewhen organized into a cellular structure. The mechanical behavior of cellularmaterials can be distinguished by axial or bending dominated structure. Anaxial-dominated material can be exceptionally stiff and strong for a givendensity while a bending dominated structure is much more compliant andsoft and can absorb a lot of energy when compressed (Ashby, 2013). Whilesignificant research has been carried out on the mechanical properties of cellularsolids (Gibson and Ashby, 1997), knowledge concerning the swelling behaviorof cellular materials is limited. Ashby (2006) pointed out that the thermalexpansion coefficient of a cellular material is the same as that of the solid fromwhich it is made. The above conclusion does not apply to cellular solids whichare made of anisotropic cell walls. In case of natural cellular materials, thepresence of structures with different material architecture (e.g. geometricalheterogeneity, multi-layered cell walls, etc.) results in a complex swellingbehavior which is not yet understood completely. The full understandingof these materials with multiscale structure opens the possibility to designcellular hierarchical materials with tailored elastic and swelling properties. Inthis thesis, we focus on one of the natural cellular materials, wood, and use itas an inspiration for more advanced analysis.

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CHAPTER 2. STATE OF THE ART

This chapter presents first the hierarchical structure of softwood, as a naturalcellular material. The anisotropic swelling behavior of softwood is analyzed toillustrate the range of swelling coefficients offered by alternative configurationsof wood tissues at cellular scale. Then, the modeling strategies that have beenused for the mechanical and the swelling behavior of softwood as a hierarchicalcellular material are reviewed. Finally, based on this overview, the needs forfurther research are pointed out, which are addressed in this work.

2.1 Softwood, a natural cellular material

2.1.1 The structure of softwood

Wood is a natural cellular material with a hierarchical microstructure.Coniferous trees, e.g. cedars, Douglas firs, pines, spruces, and yews, are referredto as softwoods. Wood features and behavior are described in terms of its threemain orthotropic directions: radial, tangential and longitudinal directions. Theradial direction is normal to the circular growth patterns, from the pith to thebark. The tangential direction is tangent to the growth rings. The longitudinaldirection is perpendicular to the cross section of the wood and along the stem.

In temperate climate regions, tree growth occurs in the warmer part of theyear and is modulated by the environmental conditions and the biologicalneeds of the tree. As a consequence, there are some variations in the densityof cells within wood: the cells that form in the spring and early summer,called earlywood, are larger and have thinner walls than those that form laterin the season, named latewood, giving rise to the well-known annual growthrings in trees. A tree trunk is composed of millions of individual woodycells. These cells differ in size and shape, depending upon their physiologicalrole in the tree, most of them being many times longer than their width.Softwoods are made up of two types of cells: tracheids, which have a roughlyhoneycomb-like structure, and parenchyma, which make up the rays andhave a box-like structure (see Figure 2.1). The bulk of the cells (85-95%)is highly elongated tracheids that provide both structural support and act as aconduction path for fluids through small openings, called bordered pits, alongtheir sides (Dinwoodie, 1981). In softwoods, the rays make up 5-12% of thewood.

The cell wall material is akin to a multi-layer fibre-reinforced composite, asrepresented in Figure 2.2. The cell walls in wood are made up of a primary

8

2.1. SOFTWOOD, A NATURAL CELLULAR MATERIAL

layer (P), with cellulose fibrils randomly distributed in the plane of the layer,and three secondary layers, S1, S2 and S3, with cellulose fibrils helically woundin varying patterns in each of the three layers. A layer called middle lamellajoins the cells together. The S2 layer accounts for most of the thickness ofthe cell wall; in Norway spruce (Pices abies), which is studied in this thesis inmore detail, it makes up about 80% of the cell wall, while the primary layeraccounts for about 3%, the S1 layer 10% and the S3 layer 4% (Fengel and Stoll,1973). The composition of the cell wall varies through the four layers with thehighest fraction of lignin is in the primary layer and the highest fraction ofcellulose is in the S2 layer (Dinwoodie, 1981) as shown in Figure 2.2a. In theS1 layer, the cellulose microfibrils are wound roughly circumferentially aroundthe cell wall, nearly perpendicular to the longitudinal axis of the cells. TheS2 layer is composed of roughly 45% cellulose, 35% hemicellulose and 20%

wood

growth ring

tracheidray (parenchyma)

growthring

latewood

earlywood

cells

cell wall layers

S2 material

pit

S3

S2

S1

fibril aggregate

ML+P

CML

MFA matrix

L

TR

Figure 2.1: Schematic representation of the hierarchical structure of wood: growthrings, earlywood and latewood cells, cell wall layers and the microfibril angle withinthe S2 layer.

9


lignin (Dinwoodie, 1981), with the cellulose fibrils wound at a slight angle(the microfibrillar angle), typically 10 − 30 to the vertical (Dinwoodie, 1981).The orientation of the cellulose microfibrils in the S3 layer is 70. Some studiesreport a cross-fibrillar structure, with alternating left- and right-handed helices,while others find a single handedness of the helices, with different handednessin different cells from the same tree (Donaldson and Xu, 2005). Within the

(a) wood cell

(c) cellulose microfibrils

condensed lignin

hemicellulose (xylan)and non-condensed lignin

crystallinecellulose

amorphouscellulose

(b) S2 material

S3

S2

S1

P

ML

C HC L

volume fraction

25 m

100 nm 3-4 nm

glucomannan

Figure 2.2: Schematic representation of the hierarchical structure of wood cellwall. (a) Cell wall composition and arrangement where ML refers to middle lamella,P primary cell wall, S1, S2, S3 layers of the secondary cell wall, C cellulose, HChemicellulose, L lignin. (b) Schematic representation of a hypothetical model for thelenticular network of S2 material as well as the arrangement of matrix componentsfrom hemicelluloses and lignin (Salmen, 2004) (c) Idealized cellulose microfibrilshowing one of the suggested configurations of the crystalline and amorphous regions.

10

2.1. SOFTWOOD, A NATURAL CELLULAR MATERIAL

cell wall layers, cellulose crystals align to form microfibrils with diameter ofabout 3-4 nm (see Figure 2.2b). The microfibrils have both crystalline andnoncrystalline regions that merge together (see Figure 2.2c). The cellulosemicrofibrils themselves are aligned and bound together into fibril aggregatesor macrofibrils, roughly 10-25 nm diameter, by a matrix of hemicellulose andlignin. The complicated microstructural architecture of wood, which secrets areyet to be fully unearthed, is known to introduce a strong geometric anisotropyat the cell wall and the cellular scale which is reflected in the anisotropy of itsmechanical and swelling behavior.

2.1.2 Anisotropic swelling behavior of softwood

Although swelling originates at the cell wall level and lower, the anisotropicnature of wood swelling can be found both at the cell wall and at the cellularlevel. Derome et al. (2011) documented the swelling/shrinkage of a smallbundles of isolated earlywood (porosity of 78%) and latewood (porosity of50%) cellular tissues of Norway spruce softwood via high resolution synchrotronradiation phase-contrast X-ray tomographic microscopy. The samples wereexposed to controlled steps of the ambient relative humidity and the affineregistration technique was used for identification of the orthotropic swellingstrains. The resulting swelling strains in radial and tangential directions arereported as a function of equilibrium moisture content in Figure 2.3. For

(a) earlywood (b) latewood

Figure 2.3: Radial and tangential swelling/shrinkage strains versus moisture contentof (a) earlywood and (b) latewood. Strains are relative to initial state at equilibriumwith 25% RH (Derome et al., 2012).

11


softwood(tangential)

softwood(radial)

earlywood(tangential)

earlywood(radial)

latewood (radial)

latewood (tangential)

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.5

0.3

0.1

Swelling-density map of softwoods

relative density (-)

Sw

elli

ng

co

effic

ien

t (%

/%)

Figure 2.4: The swelling-density map of softwood and isolated cellular tissues ofearlywood and latewood for different softwood species based on a review of literatureby the author. The swelling coefficients are reported in %strain per %moisture contentwhich is briefly denoted (%/%).

latewood and earlywood, swelling and shrinkage strains were found to undergosimilar behavior in tangential direction. In latewood, the radial strain is similarto tangential direction while in earlywood, the radial strain is less than athird of the tangential strain. Furthermore, at macroscopic scale, wood is wellknown to shrink and swell anisotropically in the hygroscopic range. For Norwayspruce, reported values for dimensional changes within the hygroscopic rangeare typically in the order of 0.5% longitudinally, 4% radially and 6% alongthe tangential direction. More specifically, swelling coefficients (β = ε/∆m, %strain per % moisture content increase) over the 0 to 12% MC range, go from0.33 to 0.40, with a mean of 0.37, tangentially and from 0.17 to 0.22, with amean value of 0.19, radially.

The variation in the hierarchical microstructure of softwoods at different lengthscales gives rise to a remarkably wide range of swelling properties, illustrated inFigure 2.4. The data used in this figure are collected from different sources ofliterature (Vintila, 1939; Yao, 1969; Quirk, 1984; Watanabe et al., 1998; Pangand Herritsch, 2005; Derome et al., 2012). This figure shows the radial andtangential swelling coefficients (%strain/%moisture content) against relativedensity (density of wood normalized by that of the cell wall ρcw =1500 kg/m3

) for earlywood and latewood tissues as well as the wood at timber scale of

12

2.2. MICROMECHANICAL MODELING OF SOFTWOOD

different softwood species. The equilibrium moisture content (MC) at fibersaturation point is assumed to be 30% MC for all softwood species. Theswelling coefficient of earlywood in radial direction is as low as 0.05, whilethat of latewood is 0.5. The tangential swelling coefficient of earlywood andlatewood spans from 0.15 to 0.5. This chart clearly shows how the combinationof earlywood and latewood cellular tissues provides a hybrid material, i.e. bulkwood, with anisotropic swelling properties.

2.2 Micromechanical modeling of softwood

A better understanding and predictability of the interactions between themechanical and moisture behavior of wood and its dependency on the cellulararchitecture is needed, for example, to assess the durability of wood elementsexposed to varying mechanical and environmental loading. In this section,the theoretical and micromechanical models for prediction of hygromechanicalbehavior of wood, specifically those models used for softwood, are reviewed.These models can be divided into two main groups, which correspond typicallyto specific scale levels: cellular models and hierarchical multi-scale models.In cellular models, the honeycomb-like microstructure of wood is analyzed.Hierarchical models cover several orders of magnitude, from the cell wallstructure, to the structure of wood cells, to the timber scale.

2.2.1 Cellular models

Cellular models provide a framework for the analysis of the mechanicalbehavior of softwood at the mesoscale. They allow predicting the elasticproperties of softwood, taking into account the mesostructure of softwood,as a cellular structure, including shapes and properties of the cells. In aclassical work, Price (1928) put forward a description of the differences betweentransverse and longitudinal elastic properties of wood based on the tubularstructure of wood cells as shown in Figure 2.5a. Gillis (1972) extended the workof Price by showing that the cellular structure of wood can be approximatedas a honeycomb with prismatic cells. Gillis observed that the characteristiccommon to an array of wood cells is a large number of triple points, i.e. pointsof junction of three double walls. The structural model adopted in his analysisis a triple junction and for simplification it is assumed to be symmetric asillustrated in Figure 2.5b. Taking into account the competition between two

13


TL

R

(a)

T

L

R

(b)

Figure 2.5: Cellular models for wood (a) array of tubular cells proposed by Price(1928) (b) a symmetric triple junction used by Gillis (1972) as a basic element in theapproximation to the tracheid geometry.

deformation modes, stretching and bending, he found that the bending effectsenhance the deformation of the structure which causes the overall responseof the element to be anisotropic. Furthermore, he showed that the elasticconstants of wood are strongly dependent on the structural arrangement ofthe wood cells. Easterling et al. (1982) studied the mechanical behavior ofwood cells based on a regular honeycomb model. They showed that the elasticmoduli, the strength and the anisotropy of wood can be determined, in part,by the properties of the cell wall and, in part, by the dimensions and shape ofthe cells themselves. The irregular symmetric honeycomb models proposedby Gibson and Ashby (1997) have been widely used for the prediction ofelastic moduli of softwood. Kahle and Woodhouse (1994) extended the Gibsonand Ashby model to a periodic tessellation of non-symmetric hexagons. Theyshowed that, with respect to the macroscopic elastic stiffness, spruce softwoodcan be regarded as an orthotropic continuum. The elastic properties aredetermined by the geometric configuration of the honeycomb, together withthe intrinsic material properties of the cell wall. In this model, the cell-to-cell variations in growth and also the larger-scale modulation of cell propertiesin the annual growth rings are taken into account. Local elastic propertieswere deduced from the geometry of each structural element, and an averagingprocedure was devised for aggregating these into macroscopic elastic constants.Watanabe et al. (1999) approximated softwood cells as honeycombs where cellwalls can be bent and stretched. Watanabe et al. (2000) investigated theeffect of varying cell wall thickness approximating honeycomb cell walls astapered beams. Moden and Berglund (2008) developed a two-phase annual

14


ring model based on fixed densities for earlywood and latewood which includesboth cell wall bending and stretching as deformation mechanisms. Compositemechanics theories are then used to calculate the macroscopic elastic modulifrom the earlywood and latewood moduli. Hassel et al. (2009) proposed amicromechanical model for spruce softwood based on a hexagonal cell model.They showed that the functional density gradient growth rings can explain thelow transverse shear modulus of spruce. In summary, cellular models, whichare mainly based on the Gibson and Ashby theoretical honeycomb model, arecapable to estimate the elastic properties of softwood. The accuracy of thesemodels have been improved in several studies, as mentioned above, by takinginto account the non-uniformity of cellular structure, the variation in thicknessof the cell walls, the functionally graded arrangement of wood cells within thegrowth rings and adjustment of the elastic properties of the cell wall and theshape of the honeycomb unit cells. However, a main limitation is general tothese approaches as the transverse (tangential to radial) anisotropy in swellingproperties of wood cannot be explained with cellular models.

2.2.2 Hierarchical models

A more complex group of models are the multiscale or hierarchical modelswhich span several orders of magnitude, from the sub cell wall level tothe macroscopic scale. Harrington et al. (1998) presented a two-stageanalytic homogenization scheme to determine the equivalent orthotropic elasticconstants for the cell-wall layers. This model was based on the assumptionsthat the softwood cell wall can be considered as a heterogeneous continuum atthe nanostructural level. The classic laminate theory is then used to determinethe equivalent properties of the wall. Astley et al. (1998) used the resultsof the above model as input data for a finite element cellular model. Theyassumed that each closed cell of the cellular array contains seven compositelaminated layers, which are treated as a thick laminated shell using the classicallaminate theory. The finite element model was realized using thick compositeshell elements. They analyzed the relationship between the macroscopic elasticproperties of softwood and the local cell characteristics such as cell size, wallthickness and microfibril angle.

Hofstetter et al. (2005) developed a continuum micromechanics multi-scalemodel based on a three-step analytical homogenization scheme for softwoodfrom a length scale of several tens of nanometers to a length scale of severalmillimeters. The interaction of the elementary components is considered in

15


three successive homogenization steps. At the lowest scale, hemicellulose,lignin, and water are intimately mixed, and the stiffness of the polymericmatrix is estimated using a self-consistent scheme with inclusions of sphericalshape. The elastic behavior of the cell wall material, composed of cylindri-cal fiber-like aggregates of crystalline cellulose and of amorphous celluloseembedded in a contiguous polymer matrix is estimated by a Mori-Tanakascheme. Finally, softwood is modeled by representing the cell lumens ascylindrical pores which are embedded in a previously determined matrix,built up of the cell wall material of the last homogenization step usingagain a Mori-Tanaka scheme. Further, Hofstetter et al. (2007) presented ahomogenization scheme containing three homogenization steps, two based oncontinuum micromechanics and one on the unit cell method. Recently, Baderet al. (2011) extended this micromechanics framework to study the poroelasticrole of water in the cell walls of softwoods. Following the poro-micromechanicsof multi-phase materials, they estimated the effective poroelastic propertiesof softwood from a hierarchical set of matrix-inclusion problems at differentlength scales. In summary, continuum micromechanics models allow topredict the effective mechanical behavior of a micro-heterogeneous materialfrom microstructural characteristics. In particular, these models includecompositional (components and their amount), morphological (orientationand distribution), and mechanical information (stiffness and interaction ofcomponents) about the material. However, due to the fact that the cellularstructure of softwood is simplified to cylindrical pores, such models are notsuitable to distinguish between the transverse properties of wood in radial andtangential directions. Furthermore, the anisotropic swelling in the transverseplane of wood cannot be explained with these models.

Neagu and Gamstedt (2007) developed an analytical modeling approach forthe prediction of the hygro-elastic response of wood cells. A wood cell wasidealized as a multi-layered hollow cylinder made of orthotropic material withhelical orientation. The hygroelastic response of the layered assembly due toaxisymmetric loading and moisture content changes was obtained by solvingthe corresponding boundary value problem of elasticity. This analysis wascombined with an analytical ultra-structural homogenization method, usedto link the hygro-elastic properties of the constituent wood polymers to theproperties of each layer. It was found that, when the wood cells are constrainednot to twist, they are showing a stiffer response than those which are allowedto twist under uniaxial loading. It was also shown that the ultrastructure,i.e. the microfibril angle, controls the hygro-expansion in the same way

16


as it affects the compliance of the cell wall. Marklund and Varna (2009a)proposed a similar concentric cylinder model for the prediction of the elasticproperties of softwood at several length scales. In this model, the wood cellwas modeled as a three concentric cylinder assembly with the lumen in themiddle followed by the S3, S2 and S1 layers. Marklund and Varna (2009b)extended their concentric cylinder model to include the free hygro-expansion oforthotropic phase materials considering several length scales. Using propertiesof the three main wood polymers, cellulose, hemicellulose and lignin, thelongitudinal and transverse hygro-expansion coefficients for the microfibrilunit cell were obtained depending on the microfibril angle in the S2 layer.They found that a homogenization procedure replacing the S1, S2 and S3layers with one single layer does not influence the results significantly for lowmicrofibril angles. The combination of ultrastructural homogenization withthe concentric cylinder model provides a potentially fruitful method to studyeffects of ultra-structural morphology on mechanical properties of softwoodcells. However, more realistic models of wood including the irregular andmore realistic geometry are indispensable for a better and more quantitativeunderstanding of the hygro-elastic behavior of wood cells.

Persson (2000) proposed several finite element models including the irregularhexagonal cellular structure and the real microstructure of wood in which thecell walls are considered as a multi-layered structure. Using this approach,the stiffness and swelling properties of the growth rings in softwoods wereestimated. Harrington (2002) proposed a comprehensive hierarchical set ofmodels for hygro-elastic properties of softwood using analytical and numericalhomogenization techniques across nano-structural, the cell wall and the cellularscales. Qing and Mishnaevsky (2009b) developed a computational modelfor the micromechanical analysis of hygro-elastic and shrinkage properties ofsoftwood. Using this 3D hierarchical computational model, the influence ofmoisture, density and microstructure of latewood on the stiffness propertieswas investigated. The elastic properties of cell sub-layers have been determinedusing unit cell models as for fiber reinforced composites. In this model, themoisture effect is represented as an equivalent temperature induced effect,taking into account the moisture-dependent changes of the elastic propertiesof the cell wall layers. Qing and Mishnaevsky (2009a) also developed ahierarchical computational model for stiffness of softwood, which takes intoaccount the structure of wood at several scales from cellular structure andmulti-layered cell walls to composite-like structures of the wall layers. Atthe mesoscale, the softwood cell is presented as a 3D hexagon-shape-tube with

17


multi-layered walls. The layers in the softwood cell are considered as compositereinforced by cellulose microfibrils. The elastic properties of the layers aredetermined with the Halpin-Tsai equations and introduced into the mesoscalefinite element cellular model. This model is used to study the influence ofthe microstructure, including MFA, the thickness of the cell wall, the shape ofthe cell cross-section and wood density, on the elastic properties of softwood.The above computational models are mainly focused on the prediction of theeffective properties of softwoods and the anisotropy of cellular materials isnot studied in details. The swelling behavior of softwoods is studied by theseauthors. Most of the authors(Persson, 2000; Harrington, 2002) used strainaveraging for calculation of effective swelling properties which is not efficient.The proposed model of Qing and Mishnaevsky (2009b) does not distinguishbetween swelling behavior in radial and tangential direction.

2.3 Needs for further research

This literature study shows an important scientific interest in micromechanicsmodeling of wood as a natural cellular material. Although numerous studieshave been done concerning this topic, there still remains a large domain tobe investigated. Despite the considerable efforts devoted to the study of themechanical response of cellular materials, their swelling behavior has not beenstudied to the same extent. The hierarchical architecture of natural materialsintroduces a large number of parameters which cannot be fully determinedeven by exhaustive experiments. Furthermore, the origin of the anisotropicswelling behavior of natural cellular materials cannot be understood exceptwith an appropriate modeling tool which takes into account the respective roleof geometry and the material properties of the cell walls. Revealing the originof the complex swelling behavior of natural cellular materials can be moreimportant than its prediction. In this case, an appropriate computationalupscaling model is needed. The development of such models can incite usto explore the capacity of cellular materials to undergo anisotropic moisture-induced deformation from their underlying microstructures.

As most models in literature (Persson, 2000; Harrington, 2002; Qing andMishnaevsky, 2009b), this work also starts with assuming the thermalexpansion analogy as a basis for modeling the moisture-induced swellingbehavior of cellular materials in which the hygro-expansion behavior of thecell wall material is independent of its elastic properties. The intention of

18

2.3. NEEDS FOR FURTHER RESEARCH

this thesis is to add new aspects to the existing hygro-expansion models basedon a new poromechanics approach. In this setting, the swelling propertiesand the dependence of stiffness on moisture content are modeled as a coupledphenomenon which is physically more relevant since both processes originatefrom the same physical phenomenon of fluid-solid interactions at lower scales.

Cellular materials have the potential to make a key contribution to the devel-opment of new structural and functional materials. It is therefore of interestto explore the range of their attainable macroscopic properties (Lakes, 1996).The hygro-expansion of cellular materials has been much less investigated.Understanding the structure-property relations for hygro-elastic behavior ofcellular solids is an open field of research which is pursued at end of this thesis.

19

Chapter 3

Computational upscaling ofcellular materials

3.1 Computational upscaling method

Computational upscaling is a numerical multiscale method to determine theapparent or effective properties of a heterogeneous material starting from theunderlying microstructure based on the appropriate construction and solutionof the boundary value problem (BVP) at the micro level. In most cases, themicrostructure of real inhomogeneous materials (e.g. wood see Figure 2.1) arehighly complex. As a consequence, exact expressions for effective propertiescannot be given and approximations have to be introduced. Typically, theseapproximations are based on the assumption that the heterogeneous materialis statistically homogeneous. This implies that sufficiently large volumeelements selected at random positions within the sample have statisticallyequivalent phase arrangements and give rise to the same averaged materialproperties which are referred to as the overall or effective properties. A properrepresentative volume element (RVE) is a sub-volume of the material that isof sufficient size which contains all information necessary for describing thebehavior of the heterogeneous material. The concept of RVE is illustratedschematically in Figure 3.1. In the context of the principle of separation ofscales, the maximum microscopic length scale lm within the RVE should bemuch smaller than the characteristic size lM of the RVE which represents themacroscopic sample:

lm ≪ lM , (3.1)

21

CHAPTER 3. COMPUTATIONAL UPSCALING

sizeRVE

pro

pert

y

Figure 3.1: The concept of RVE.

The geometrical RVE can be defined by requiring it to be statisticallyrepresentative of the geometry of the microstructure being independent of thephysical property to be studied. On the other hand, the physical RVE is definedbased on the requirement that the overall responses with respect to some givenphysical behavior are independent of the actual position and orientation ofthe RVE and/or of the boundary conditions applied to it (Hill, 1963). Thesize of such physical RVE depends on both the physical property consideredand on the geometrical parameters of its microstructure. Unfortunately, forhierarchical cellular materials such as wood, none of the above situations ismet so that the applied boundary conditions have always an influence onthe predicted properties by any numerical method. A possible solution isperiodic model materials which are used to approximate the behavior of actual,non-periodic inhomogeneous materials. Computational upscaling of periodicmaterials involves studying unit cells i.e. volume elements with periodicarrangement that tile space by translation (Böhm et al., 2009).

The computational upscaling procedure is represented schematically in Figure3.2. The micro-macro transition procedure outlined here is deformation driven,i.e. for a point in a macroscopic medium, the macroscopic strain εM is imposedas a boundary condition on the external boundaries Γm of the microscopic unitcell Ωm. At the microscopic scale, the boundary value problem is solved and themacroscopic stress σM is obtained as the volume average of the microscopicstress over the microscopic sample. Also, a stress driven procedure, wherea local macroscopic stress is given to obtain the deformation, is possible.However, such a procedure does not fit into the standard displacement-basedfinite element framework, which is commonly employed for the solution of

22

3.1. COMPUTATIONAL UPSCALING METHOD

aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa a a a a a a a a a

u

t

t

u

t

t

m

M

M

micro BVP

CM

unit cell

Figure 3.2: Multi-scale modeling of cellular solids with computational upscalingscheme. The boundaries Γt and Γu are related to the displacement boundaryconditions and the mechanical loading, respectively. The macroscopic stiffness tensoris denoted CM .

macroscopic boundary value problems. At the microscale (denoted subscriptm), in the absence of inertial and body forces, the static equilibrium for theunit cell reads:

∇ ⋅σm = 0 ∶ (3.2)

where ∇ is the symmetric gradient operator. At the macro level (denotedsubscriptM), the mechanical equilibrium has a structure identical to the microlevel:

∇ ⋅σM = 0. (3.3)

The rest of this chapter presents the essential ingredients of the computationalupscaling method and its application to the prediction of the swelling behaviorof cellular materials.

3.1.1 Strain averaging theorem

The macroscopic strain tensor at any arbitrary point xM of the macroscopiccontinuum is achieved by volume averaging of the microscopic strain tensor εmover the domain Ωm:

23


εM =1

Ωm∫

Ωm

εmdΩ, (3.4)

The RHS of Equation (3.4) can be elaborated as follows:

εM =1

Ωm∫

Ωm

1

2(umi,j + u

mj,i)dΩ =

1

Ωm∫

Γm

1

2(umi nj + u

mj ni)dΓ, (3.5)

where the definition of the strain tensor is used in the first equality and theGauss theorem is employed to convert a domain integral into a surface integralin the second equality. In this equation, n denotes the unit outward normalto Γm and um denotes the microscopic displacement field. The position ofthe subscript m is switched to superscript when the index notation is used forconvenience of reading.

The displacement field at a location xm within the unit cell can be decomposedinto a macroscopic linear displacement contribution and a displacementfluctuation:

um(xm) = εM ⋅ xm + um(xm), (3.6)

where the displacement fluctuation field um represents the fine scale deviationswith respect to the average fields as a result of the heterogeneities within theunit cell. By substituting the Equation (3.6) into Equation (3.5), we obtain:

εM = εM +1

Ωm∫

Γm

1

2(um

i nj + umj ni)dΓ. (3.7)

Therefore, to fulfill the strain averaging theorem given in Equation (3.4), thedisplacement fluctuations field must satisfy the following condition:

∫Γm

1

2(um

i nj + umj ni)dΓ = 0. (3.8)

3.1.2 Hill-Mandel condition

The micro-to-macro transition is essentially based on the Hill-Mandel conditionalso referred to as macro-homogeneity principle (Hill, 1963; Suquet, 1985)which guarantees the equivalence between the energetically and mechanicallydefined effective elastic properties:

24

3.1. COMPUTATIONAL UPSCALING METHOD

σM ∶ εM =1

Ωm∫

Ωm

σm ∶ εmdΩ (3.9)

where the symbol (∶) denotes the double contraction operator of two secondorder tensors which is defined as:

A ∶ B = AijBij (3.10)

Using the Gauss theorem, the RHS of Equation (3.9) can be further elaboratedas:

σM ∶ εM =1

Ωm∫

Γm

tm ⋅ umdΓ (3.11)

where tm = σm ⋅n is the microscopic traction vector. Inserting Equation (3.6)into the RHS of Equation (3.11) yields:

σM ∶ εM = (1

Ωm∫

Γm

tm ⊗ xmdΓ) ∶ εM +1

Ωm∫

Γm

tm ⋅ umdΓ (3.12)

where the symbol ⊗ denotes the dyadic product. If the microscopic fluctuationfield does not contribute to the microscopic work, i.e. the displacementfluctuation field um is such that the second term of the RHS of Equation (3.12)vanishes, we obtain:

σM =1

Ωm∫

Γm

tm ⊗ xmdΓ (3.13)

As a result, the macroscopic stress tensor is defined as the volume average ofthe microscopic stress tensor:

σM =1

Ωm∫

Ωm

σmdΩ (3.14)

3.1.3 Boundary conditions

The macro-homogeneity principle given in Equation (3.9) is the basis fordifferent types of boundary conditions that can be imposed at the micro level.This condition is satisfied by three different types of boundary conditions (BC)for random media:

25


Uniform displacement BC (Dirichlet, kinematic, KUBC): This conditionprescribes a linear mapping between the displacement and the position of apoint on the RVE boundary:

um(x) = εM ⋅ x ∀x ∈ Γm. (3.15)

Uniform traction BC (Neumann, static, SUBC): This boundary conditionapplies a uniform traction on the unit cell boundaries:

tm(x) = σM ⋅ n ∀x ∈ Γm. (3.16)

The traction boundary conditions are not appropriate for the deformationdriven procedure to be pursued in the current computational homogenizationscheme. They were presented here for the sake of completeness.

Uniform displacement-traction BC (orthogonal mixed, MUBC): In thecase of MUBC, different combinations of a priori prescribed boundary vectorsare possible but have to fulfill the following condition:

(tm(x) −σM ⋅ n) ⋅ (um(x) − εM ⋅ x) = 0. ∀x ∈ Γm (3.17)

Periodicity compatible mixed uniform boundary condition (PMUBC) is aspecific set of mixed uniform boundary conditions that avoids prescribingnonzero boundary tractions. PMUBC is proposed by Pahr and Zysset(2008) for obtaining the apparent elastic tensor of cellular materials. Theseboundary conditions give exactly the same effective elastic properties asperiodic boundary conditions (PBC) if they are applied to a periodic andorthotropic micro-structured material. The PBC will be explained in thefollowing of this section. The PMUBC offer an attractive option for evaluatingestimates of the macroscopic properties of periodic and non-periodic volumeelements.

In addition to the above three groups of boundary conditions, in the caseof periodic microstructures, the periodic boundary conditions (PBC) canbe utilized and are known to yield a more accurate estimation of apparentmacroscopic properties of heterogeneous materials (van der Sluis et al., 2000;Terada et al., 2000).

26

3.2. FINITE ELEMENT IMPLEMENTATION

Periodic boundary conditions (PBC): This boundary condition statesthat the displacement fluctuation on periodic boundaries is periodic and thetraction is anti-periodic:

um(x+m) = um(x−m), tm(x+m) = −tm(x−m). (3.18)

where x+m and x−m refer to the positions of two points on periodic boundarypairs Γ+m and Γ−m (see Figure 3.4). In this case, the implementation of periodicboundary conditions leads to the following relations:

um(x+m) − um(x−m) = εM ⋅ (x+m − x−m). (3.19)

By substituting the first Equation of (3.18) into Equation (3.8), it can beeasily verified that PBC satisfies the strain averaging theorem. Furthermore,substituting Equation (3.18) into Equation (3.12) reveals that the microscopicfluctuation field um does not contribute to the work on the boundary Γm:

∫Γm

tm ⋅ umdΓ = 0. (3.20)

In this thesis, the PBC is used to predict the effective properties of periodicunit cells and PMUBC is used for non-periodic volume elements. The otherboundary conditions are presented for completeness. The choice of boundaryconditions affects the results of computational homogenization method andconsequently the homogenized properties. In Figure 3.3, the von Mises stresscontour of a honeycomb unit cell under shear loading is shown for three typesof boundary conditions, namely KUBC, PBC and PMUBC. As can be seenin this figure, the KUBC prediction is different from PBC and PMUBC. Asmentioned above, the results of PMUBC and PBC are very similar to eachother.

3.2 Finite element implementation

In this section, the computational aspects of the upscaling method arepresented. Several possibilities are available to compute the overall stiffnessproperties from the finite element results such as stress and strain averagingand the master node technique (Demiray et al., 2006; Pahr and Rammerstorfer,2006). The macroscopic tangent stiffness of the heterogeneous medium isdefined as :

27


(a) KUBC (b) PBC (c) PMUBC

Figure 3.3: von Mises stress contour for a honeycomb unit cell in shear loadingunder (a) uniform displacement boundary conditions, KUBC, (b) periodic boundaryconditions, PBC; (c) periodicity compatible mixed uniform boundary conditions,PMUBC. The PBC and PMUBC exhibit very similar behavior.

δσM =CMδεM . (3.21)

The derivation of CM using the standard stress and strain averaging methodand the master node technique is explained in the following.

3.2.1 Stress and strain averaging

Stress and strain averaging is the simplest way to compute the apparent elasticproperties but the procedure is computationally expensive. The macroscopicstresses and strains can be computed numerically in a classical manner fromthe Equations (3.14) and (3.4), respectively:

σM =1

Ωm∫

Ωm

σmdΩ =1

Ωm∑N

σmipdΩip, (3.22)

εM =1

Ωm∫

Ωm

εmdΩ =1

Ωm∑N

εmipdΩip, (3.23)

where the subscript ip denotes the integration point; N is the number of theintegration points in the unit cell domain and Ωip is the volume correspondingto an integration point. The coefficients of the macroscopic stiffness tensorCMijkl are found as averages of the local fields from the solution of successive

elasticity problems. The volume element is loaded, in turn, by one of the

28


n n

12

34

m, 2

m, 2

m,1m,1

+

+

-

+-

-

11

Figure 3.4: Periodic unit cell. The three corner nodes, q = 1,2,4, are considered asmaster nodes. The periodic boundary pairs, (Γ+m,1,Γ

−m,1) and (Γ+m,2,Γ

−m,2), and the

outward normals to boundary pair 1, (n+1,n−1), are shown in this figure.

components of the macroscopic strain field εM while the other components areset to zero.

3.2.2 Master node technique

An alternative approach to compute the stiffness tensor is the master nodetechnique. In this method, the consistent macroscopic stiffness matrix isderived from the total stiffness matrix of the unit cell by reducing the latterto the relation between forces acting on the retained vertices of the unit celland the displacements of these vertices. Thus, only a few resultant forcesand displacements have to be read and post-processed from the finite elementresults. Considering a two-dimensional periodic unit cell, the master nodes1, 2 and 4 are respectively located at the left-bottom, the right-bottom andthe left-top corners. The displacement of the node located at corner 3 isfully described by the displacement of corner 2 and corner 4 according toEquation (3.19). Therefore displacement at corner 3 is not an independentquantity and consequently does not appear in the equations. Following theperiodically fluctuating displacement boundary conditions which is given inEquation (3.19), the displacement vectors of the corner nodes 1, 2 and 4 arefully prescribed by:

uq = εM ⋅ xq q = 1,2,4, (3.24)

29


where the node corresponding to corner 1 is fixed to eliminate rigid bodytranslations. By putting Xp = [x1 x2 x4] and up = [u1 u2 u4],Equation (3.24) can be written in matrix form:

up =XTp εM . (3.25)

In the case of periodic boundary conditions, the macroscopic stress tensorwhich is defined by Equation (3.13) can be simplified as follows (Kouznetsovaet al., 2001):

σM =1

Ωm∫

Γm

tm ⊗ xmdΓ =1

Ωm∑

q=1,2,4

fq ⊗ xq, (3.26)

Written in matrix-vector notation, the macroscopic stress vector reads:

σM =1

Ωm[x1 x2 x4]

⎡⎢⎢⎢⎢⎢⎣

f1f2f4

⎤⎥⎥⎥⎥⎥⎦

=1

ΩmXpfp. (3.27)

For extraction of the macroscopic stiffness, the total system of equations forthe unit cell is rearranged as:

[Kdd Kdp

Kpd Kpp] [δud

δup] = [

0δfp

] . (3.28)

with Kdd,Kdp,Kpd and Kpp partitions of the stiffness matrix, while δup andδfp refer respectively to the incremental displacements and external forces ofthe prescribed retained vertices and δud to the incremental displacements ofthe dependent nodes. By condensing out the dependent degrees of freedomfrom the system, the reduced stiffness matrix KM is obtained as:

KMδup = δfp, KM =Kpp −KpdK−1ddKdp. (3.29)

Following Equation (3.27) and using Equation (3.29), the variation of themacroscopic stress is given by:

δσM =1

ΩmXpδfp =

1

ΩmXpKMδup. (3.30)

30


Inserting the variation of the displacement δup = XTp δεM which is given by

Equation (3.25) into Equation (3.30), one obtains:

δσM =1

ΩmXpKMXT

p δεM , (3.31)

from which the consistent tangent stiffness can be derived as:

CM =1

ΩmXpKMXT

p . (3.32)

This means that CM is obtained from the position vector of master nodes Xp

and the reduced stiffness matrix KM .

3.2.3 Hygroelastic constitutive equations

In the computational upscaling framework, the constitutive equation of thematerial at the macroscale is not required. At the microscale, the behaviorof the constituents has to be determined prior to simulation, e.g. fromphenomenological models, computational homogenization schemes from lowermaterial scales or from experiments. In the case of linear constituents at themicro-scale (subscript m), the hygroelastic constitutive equation of a linearlyelastic solid with moisture induced swelling is:

σm =Cm(εm −βm∆m), (3.33)

where Cm is the microscopic elasticity tensor, ∆m is the change in moisturecontent by mass from the initial state and βm is the microscopic second ordertensor of swelling coefficients. As a result of microscopic linear constituents,the constitutive equation at the macro level is also linear and has a structureidentical to the micro level constitutive equation:

σM =CM(εM −βM∆m). (3.34)

where βM is the macroscopic second order tensor of swelling coefficients.

31


h

32

1

-s

+s~

2

θ

l

(a) (b)

t

1

2

~

3

-s~

1+s

~

1

+s~

3

-s~

2

Figure 3.5: (a) The centerline of a periodic arrangement of honeycombs. Ahoneycomb unit cell is shown with bold lines and geometrical parameters, namelyheight of vertical walls h, length of inclined walls l, the shape angle θ and cell wallthickness t are defined. (b) A quarter of honeycomb RVE discretized with finiteelements. The positions of three master nodes (q = 1,2,3) and the correspondinglocal coordinate system, sp, centered on each master node are shown in this figure.

3.3 Upscaling model for cellular materials

Computational upscaling methods are well suited for studying the hygro-mechanical behavior of cellular materials. A cellular material can be conceivedas a material made of an interconnected network of solid struts or plates whichform the edges and faces of cells (Gibson and Ashby, 1997). Thus, a two-dimensional cellular material can be considered as an array of polygons whichpack to fill a plane area. A perfect example is the hexagonal cells of the bee.For this reason such microstrutures are called honeycombs. In this section, theupscaling method introduced in this chapter is applied to cellular materialsand honeycombs are selected for this purpose. The conventional treatmentof the mechanics of honeycombs is the approach proposed by Gibson andAshby (1997). They provided several relations for the mechanical propertiesof honeycombs and other cellular solids. However, thermal or hygro-expansion(swelling coefficient) of cellular materials are not considered in these model.Figure 3.2a shows a periodic arrangement of honeycombs. In this figure,a honeycomb unit cell is shown with bold lines and the three geometricalparameters, namely height of vertical walls h, length of inclined walls l andthe shape angle θ are displayed.

32

3.3. UPSCALING MODEL FOR CELLULAR MATERIALS

3.3.1 Symmetry conditions of honeycombs

In many periodic microstructures, the configuration of the constituents issymmetric. The symmetry conditions are very useful for describing simplemicrostructures and tend to give rise to small unit cells. The point symmetryfor a two-dimensional unit cell is defined by the invariance of the materialproperties and kinematic constraints to rotations of 180. Ohno et al. (2001)developed a homogenization model for periodic solids in which the constituentsare configured point symmetrically with respect to the body center of eachunit cell. For this class of periodic solids, the field of perturbed displacementsatisfies the point-symmetry with respect to all boundary facet centers as wellas to the body center of a unit cell. Flores and De Souza Neto (2010) extendedthis model to account for other boundary conditions such as SUBC andKUBC. Point symmetry has been used to analyze efficiently the homogenizedproperties of honeycombs (Asada et al., 2009; Flores et al., 2012). For aperiodic honeycomb unit cell, the point symmetry of displacement fluctuationum with respect to points 1, 2 and 3 as indicated in Figure 3.2b may be recastinto the following constraint relation (Weissenbek et al., 1994):

x(−sq) + x(+sq) = 2xq, (3.35)

where sq denotes a local coordinate system centered on a point of symmetryand xq is the position vector of this point (see Figure 3.5). As a consequenceof point symmetry, the displacement fluctuations of the points of symmetryvanish (Ohno et al., 2001):

uq = 0, q = 1,2,3. (3.36)

These symmetry points are selected as master nodes in the analysis ofhoneycombs. As a consequence of zero displacement fluctuation given byEquation (3.36), and using the displacement decomposition relation given inEquation (3.6), the displacement vectors of the master nodes can be writtenas follows:

uq(xq) = εM ⋅ xq. (3.37)

The macroscopic stress tensor of honeycombs can be calculated from thefollowing relation:

33


σM =1

Ωm∑

q=1,2,3

fq ⊗ xq. (3.38)

The effective stiffness tensor of honeycombs can be computed based on themaster node technique, as presented in Section 3.2.2:

CM =1

ΩmXpKMXT

p , (3.39)

where Xp = [x1 x2 x3] and KM is the reduced stiffness matrix of thehoneycomb unit cell.

3.3.2 Effective swelling coefficients: hygro-elastic model

The macroscopic swelling coefficients βM can be calculated by adding anauxiliary load case that constrains all displacements normal to the unit cellboundaries, i.e. εM = 0, and then applying a unit moisture content increment,i.e. ∆m = 1. This allows evaluating the macroscopic swelling stress tensorσM from Equation (3.38). Inserting σM in the macroscopic constitutiveEquation (3.34), the macroscopic swelling coefficients can be readily obtained:

βM = −C−1MσM . (3.40)

These are the basic elements needed for the computational homogenization ofswelling coefficients of cellular solids. The above procedure is based on the hy-groelasticity where the eigenstrains developed in the material are proportionalto the change of the moisture content. An alternative thermodynamicallysound method based on poromechanics approach is presented in the nextsection.

3.4 Poromechanics approach

In this section, the poromechanics approach is described for determining thehygro-mechanical behavior of a general anisotropic porous medium. Thegeneral expression for the macroscopically observable free swelling strain isderived from the poroelastic constitutive equations and is used to predict theswelling behavior of cellular materials.

34

3.4. POROMECHANICS APPROACH

3.4.1 Poroelastic constitutive equations

In poromechanics, the stress and pore pressure are jointly connected to thevariation in fluid content and the strain of a porous medium. The incrementalconstitutive equations of a saturated porous material with respect to thetangent properties read (Biot, 1941; Coussy, 2004):

dσ =Cdε − bdp, (3.41)

dmf

ρf= bdε +

dp

M, (3.42)

where σ is the stress, ε is the strain, b is the second-order Biot coefficienttensor and M is the Biot modulus. Combining relations (3.41) and (3.42), weobtain:

dσ =Cudε − bMdmf

ρf, (3.43)

where Cu =C+Mb⊗b is the undrained tangent stiffness tensor of the porousmaterial. Accordingly, in a stress-free adsorption experiment, i.e. dσ = 0, therelation which links the change in fluid mass content to the swelling induceddeformation (superscript sw) is:

dεsw = SubMdmf

ρf, (3.44)

where Su = (Cu)−1 is the undrained compliance tensor. The last equationis formally close to the relation which is obtained for stress-free expansivereactions in chemo-elastic concrete (Lemarchand et al., 2005). When thematerial properties are constants, this relation yields a linear dependencebetween the fluid mass change and the free swelling strain which appearsfrequently in macroscopic swelling models. In this model, the swelling strainis related to a dimensionless mass increase of the fluid by means of a linearoperator, i.e. the tensor of swelling coefficients:

β = SubM. (3.45)

We also remind that Equation (3.43) and Equation (3.33) are of similar formand Equation (3.45) gives a physical interpretation of the swelling coefficient:

35


it depends on the compliance, Biot coefficient and Biot modulus. The Biotmodulus M can be interpreted as the inverse of moisture capacity of thematerial. This relation shows that the swelling coefficient increases for a lessstiff material (less restraint) and in a material with higher fluid-solid interaction(increasing b). The swelling coefficient decreases with increasing moisturecapacity (decreasing M).

Poroelastic material properties

An expression for the poroelastic material coefficients of an anisotropic porousmaterial, bij andM , can be formulated based on two fundamental assumptions:micro-homogeneity and micro-isotropy of the solid matrix (Cheng, 1997; Chengand Abousleiman, 2008). The micro-homogeneity assumes that the solidmatrix of the porous material is homogeneous at the pore scale while thematerial can be heterogeneous at the macroscopic scale. The micro-isotropyis based on the assumption that the solid constituent of the porous mediumis isotropic at the pore level and that the material anisotropy is of structuralorigin, mainly resulting from pore shape and orientation. By adopting theseassumptions, the poroelastic constants, written in index notation, can bedefined as follows:

bij = δij −Cijkk

Ks, (3.46)

1

M=

1

Ks[(1 −

Ciijj

9Ks) − φ0 (1 −

Ks

Kf)] , (3.47)

where δij denotes the Kronecker delta. For saturated condition, the Lagrangianporosity φ which refers the current porous volume to the initial volume isrelated to the fluid (water) mass content by mf = ρfφ. The bulk modulus ofthe solid matrix and the fluid are noted Ks and Kf , respectively. CombiningEquations (3.46) and (3.47) with Equation (3.45) shows that the swellingcoefficient tensor, β, is dependent on the stiffness tensor, the porosity andthe bulk modulus of the solid and fluid.

3.4.2 Effective Swelling of ideal microstructures

In this section, the derivation presented in Sec. 3.4.2 for prediction of the effec-tive swelling coefficients of a porous medium is applied to ideal microstructured

36


porous solids with single porosity and double porosity materials composed ofisotropic constituents. The aim is to examine how the presence of porositiesat different hierarchical levels influences the effective swelling coefficients ofa porous medium. Again, in choosing this specific case, we are inspiredby wood showing a cellular structure, where the cell walls are porous andsorption of water molecules induces swelling deformation. The application ofthe computational poromechanical approach to a cellular porous material withanisotropic cell walls is then investigated.

Single-porosity material

We consider a single porosity medium with spherical pores and porosity φ.The porosity is totally filled by the liquid, i.e. we consider the materialis in saturated conditions. The change in moisture content is thus totallyrelated to the change in porosity, which induces swelling and is restrained bythe solid skeleton. The solid skeleton is isotropic with a bulk modulus Ks

and shear modulus µs. The effective material properties associated to singleporosity material are denoted with superscript I. In this case, the macroscopicconstitutive equations (superscript M) are:

dσM =KIdεM − bIdp, (3.48)

dφ = bIdεM +dp

M I, (3.49)

where σM here stands for the mean stress and εM is the volumetric strain.The parameters KI , bI and M I correspond respectively to the effective bulkmodulus, Biot coefficient and Biot modulus of the porous medium with singleporosity. The poroelastic properties given in Equations (3.46) and (3.47), fora single porosity material reduces to (Coussy, 2004):

bI = 1 −KI

Ks,

1

M I=bI − φ0

Ks. (3.50)

where φ0 is the initial porosity of the porous medium. We obtain expressionsfor the macroscopic poroelastic constants of Equations (3.48) and (3.49) usingMori-Tanaka homogenization scheme. These results are used to estimatethe effective swelling coefficient of the single-pore material system accordingto Equation (3.45). The Mori-Tanaka estimation (superscript mt) for

37


elastic properties of a porous material with spherical pores and porosity φ0

reads (Dormieux et al., 2006):

Kmt=Ks

4(1 − φ0)

3φ0Ks + 4µs, µmt

= µs(1 − φ0)(9Ks + 8µs)

9Ks(1 +23φ0) + 8µs(1 +

32φ0)

. (3.51)

Following Equation (3.45), the isotropic swelling coefficient of the porousmedium is:

βIM =1

3(

bIM I

KI + bI2M I

) . (3.52)

Using Equation (3.50) and assuming that KI = Kmt, the swelling coefficientof a single-pore medium reads:

βIM =1

3(

3Ks + 4µs3Ks + 4µsφ0

) (3.53)

This expression shows that the swelling coefficient depends on the bulk andshear moduli of the solid and the porosity. A similar relation to (3.53) ispresented by Lemarchand et al. (2005) for modeling the expansive reactionsin chemoelastic concrete. Knowing Ks, µs and φ, the swelling coefficient ofa single porosity material can be determined. It can be verified that, for anincompressible skeleton (i.e. Ks →∞), the swelling coefficient is βIM = 1/3.

Two-scale double-porosity material

In this section, we consider a two-scale double-porosity material with an idealmicrostructure composed of spherical pores at both scales. The porosity ofthe material manifests itself at different scales, which are separated by at leastone order of magnitude in length. This simplified microstructural model ispresented in Fig. 3.6. The homogenized linear poroelastic constitutive behaviorat the macroscopic scale, for the double porosity medium with isotropic solidskeleton may be written as (Dormieux et al., 2006):

dσM =KIIdεM − bII1 dp1 − bII2 dp2, (3.54)

38


p

p1

2

m

p

p

I

II

Figure 3.6: Schematic of a double porosity material.

dφ1 = bII1 dε

M+dp1

M II11

+dp2

M II12

, (3.55)

dφ2 = bII2 dε

M+dp1

M II21

+dp2

M II22

, (3.56)

where KII is recognized as the overall drained bulk modulus of the doubleporosity material. The macroscopic Biot coefficients, bII1 and bII2 , are associatedwith the micro-porosity of the porous matrix and the macropore space,respectively. The pore pressure p1 is the uniform pressure prevailing in themicroporosity of the porous matrix and the uniform pressure in the macroporespace is p2 ≠ p1. The Biot moduli are denoted as M II

ij . The normalized porevolume φ0

1 located inside the porous matrix Ωm is obtained by integration:

φ01 =

1

Ω∫

Ωm

φ0dΩ = (1 − ϕ02)φ

0, (3.57)

where φ0 = ΩIp/Ωm is the initial porosity of the porous matrix and ϕ0

2 = ΩIIp /Ω

is the volume fraction of macropores. Inspired by the behavior of wood, weassume that the liquid in the macropores is not exposed to an external pressureand that the pores are large enough such that pore pressures like capillarypressures can be neglected, i.e. p2 ≈ 0. In this situation, combining the firsttwo constitutive equations (3.54 and 3.55) and using Equation (3.57), aftersome manipulation, we obtain the isotropic swelling coefficient of the double-porosity medium at macroscopic scale:

39


βIIM =1

3

⎛

⎝

(1 − ϕ02)b

II1 M

II11

KII + bII12M II

11

⎞

⎠, (3.58)

which relates the macroscopic strain to the change of fluid content in themicropores of the porous matrix:

dεsw = βIIMdφ = βIIM

dmf

ρf, (3.59)

where the moisture content is related to filling of the second pore system.The macroscopic poroelastic constants of double-porosity material (KII , bII1

and M II11 ) are readily determined from the poroelastic constants of the porous

matrix (Dormieux et al., 2006). The successive use of relation (3.51) at bothscales yields the overall drained bulk modulus of the medium:

KII=

4Ksµs(9Ks + 8µs)(1 − φ0)(1 − ϕ0

2)

(9Ks + 8µs)(4Ks + 3µsφ0) + 3Ks(9Ks + 8µs + 6(Ks + µs)φ0)ϕ02

(3.60)

The Biot coefficient, bII1 can be estimated as (Dormieux et al., 2006):

bII1 = (1 − ϕ02)b

IAIIm , (3.61)

where AIIm is the average volumetric strain concentration coefficient of the

porous matrix in the double-porosity material. With Mori-Tanaka approxima-tion at both scales, this coefficient reads (Ulm et al., 2004):

AIIm =

(9Ks + 8µs)(4µs + 3Ksφ0)

(9Ks + 8µs)(4Ks + 3µsφ0) + 3Ks(9Ks + 8µs + 6(Ks + µs)φ0)ϕ02

. (3.62)

The effective Biot modulus M II11 is defined as Dormieux et al. (2006):

1

M II11

=bI ((1 − ϕ0

2)bI − bII1 )

KI+

1 − ϕ02

M I(3.63)

Inserting the resulting constants in the swelling relation (3.58) yields themacroscopic swelling coefficient of the double-porosity material:

40


βIIM =1

3(

3Ks + 4µs3Ks + 4µsφ0

) (3.64)

It is noteworthy that this relation is identical with the prediction of a singlepore system as presented in Equation 3.53, i.e βIIM = βIM . This shows that,when the solid phase is isotropic, the swelling of a porous material is the sameas the swelling of the matrix from which it is made. Here, with a simple idealmicrostructure, we showed that, in a two scale double porosity material withisotropic solid skeleton, if the macropores are not pressurized, the macroscopicswelling is governed by the change of the porosity of the micropores.

3.4.3 Effective swelling coefficients: poroelastic model

Cellular materials can be considered as two-scale double porosity materials aspresented in Sec. 3.4.2 with the exception that, in general, the cell walls ofmany natural cellular materials (e.g. wood) are anisotropic and the geometryof the pores is not simple. Therefore, given the complexity of the system, acomputational approach is needed. The poroelastic material properties of theanisotropic cell walls can be defined with relations (3.46) and (3.47). Theconstitutive equations of a cellular material with porous cell walls have astructure similar to those presented for double-porosity materials (Dormieuxet al., 2006) but with anisotropic solid skeleton. Now we are looking at theswelling behavior of a cellular material with anisotropic porous cell walls.Assuming p1 = p and p2 = 0, the constitutive equations read:

dσM =CMdεM − bMdp (3.65)

dmMf

ρf= bMdεM +

dp

MM(3.66)

Our goal is to link the macroscopic (superscript M) tangent poroelasticproperties, CM, bM and MM , to the microscopic ones, i.e. C, Ks and Kf .Then based on a poromechanics approach, the macroscopic swelling coefficientsare determined. We consider a honeycomb unit cell consisting of anisotropicporoelastic cell walls with periodic boundary conditions. The geometricalparameters of the honeycomb unit cell are the same as those defined in Fig. 3.5.The stiffness (compliance) tensor of honeycombs, CM (SM ), can be computed

41


with the computational upscaling procedure as described earlier in this chapterusing Equation (3.39). The effective poroelastic properties of honeycombs, bM

and MM , can be calculated computationally for a macroscopically constrainedunit cell, i.e. εM = 0, on which a pore pressure perturbation ∂p is imposed.The macroscopic Biot coefficient tensor, bM , is obtained from the macroscopicstress in the unit cell computed by Equation (3.38) through the state equation(3.65) as:

bM = −∂σM

∂p. (3.67)

The macroscopic Biot modulus,MM , can be computed in the same experimentfrom Equation (3.66):

MM = ρf∂p

∂mMf

, (3.68)

where the macroscopic fluid mass content is defined as:

mMf =

1

Ω∫

Ωm

mfdΩ. (3.69)

Finally, the macroscopic swelling coefficient of the cellular material can beexpressed in terms of macroscopic poroelastic properties:

βM = ρrSuMbMMM . (3.70)

3.5 Summary

In this chapter, the essential ingredients of the computational homogenizationmethod have been presented for the computation of the effective stiffness andswelling coefficients of heterogeneous cellular materials. The effective swellingcoefficients of honeycombs are determined based on two strategies. The firstapproach is based on the analogy between thermal expansion and moistureinduced swelling. In this method, the effective swelling coefficient is determinedfrom the swelling stress built up in a constrained unit cell and the effectivestiffness tensor of the medium. The essential material input parameters for

42

3.5. SUMMARY

this model are the stiffness and swelling coefficients of the cell walls of thecellular material.

An alternative approach for the estimation of the swelling properties is basedon poromechanics. Following the constitutive equations of poroelasticity, theeffective swelling coefficients of the cellular materials are determined as a linearoperator which relates the fluid mass change and the free swelling strain. Theinput material parameters in the poromechanical approach are the elasticproperties of the cell wall, the bulk modulus of the matrix from which thecell walls are made and the porosity of the cell walls material. The effectiveswelling coefficient based on the poromechanics approach is dependent onthe effective undrained stiffness, the Biot coefficient, the Biot modulus andthe relative density of the cellular material. The proposed model is appliedto an ideal porous medium. Interestingly, with a simple homogenizationscheme, we showed that, in a two-scale double-porosity material with isotropichomogeneous solid constituents, when the macropores are not pressurized, themacroscopic swelling coefficient is identical to the swelling coefficient of theporous matrix. The swelling behavior of cellular materials is studied as anextended case of two-scale double-porosity materials. In the next chapters,both methods will be used on comprehensive computational examples andthe resulting anisotropy in swelling behavior of the cellular materials will beexamined.

43

Chapter 4

Hygroelastic behavior ofhoneycombs

The anisotropy in swelling behavior of cellular materials is intriguing andyet to be fully understood. In this chapter, the hygro-elastic behavior oftwo-dimensional periodic honeycombs is investigated using the computationalmicro-mechanics approach presented in the previous chapter1. In response toan external stimulus, such as a variation in relative humidity or temperature,eigenstrains develop in cellular solids despite the absence of externally appliedforces due to the geometrical constraints embedded in the hierarchical architec-ture of the material. Hygric eigenstrains occur due to moisture induced swellingor shrinkage. For cellular materials composed of homogeneous isotropic cellwalls, the effective hygric or thermal expansion coefficient is the same as thatof the solid from which it is made (Ashby, 2006). However, the cell walls inmany natural cellular materials such as honeybee combs and wood cells exhibita complex multi-layered organization that can result in anisotropic properties.Figure 4.1a shows the macroscopic configuration of the natural honeybeecombs. The cell walls of honeybee combs are composed of a multi-layered cellwall, which enhances the structural integrity of the whole structure (Zhanget al., 2010). In wood, the stiff inextensible cellulose microfibrils constrainthe deformation of the cell wall during swelling (Burgert et al., 2007). Theorientation of microfibril angle (MFA) in the different cell wall layers (S1,S2 and S3) also modifies the swelling of the cell wall which might result inanisotropic swelling behavior at macroscopic scale.

1based on Rafsanjani et al. (2013)

45

CHAPTER 4. HONEYCOMBS WITH MULTI-LAYERED CELL WALLS

300 m2mm

(a) (b)

Figure 4.1: (a) Natural honeybee combs (Wikipedia) (b) cross section of a triplejunction showing the multi-layered cell walls in honeybee combs (Zhang et al., 2010).

In this chapter, the hygro-expansion of honeycombs composed of multi-layered cell walls is investigated to explain the anisotropic swelling behaviorof cellular materials. Before presenting the main results, the computationalmodel is verified against the analytical models available in literature fordifferent geometrical parameters. Then, inspired from wood cell wall, a simpleconfiguration is selected in which the swelling of the core of the cell wall isrestrained with two non-swelling sheath layers. Numerical experiments areconducted to examine the combined effects of the restraining sheaths on theswelling of the core material at the cell wall level. At the cellular scale, theinfluence of the shape angle as an important geometrical parameter of thehoneycombs on the macroscopic swelling behavior is investigated.

4.1 Verification of the computational model

In order to establish the overall validity of the work, limiting cases areconsidered and compared to analytical solutions available in the literature.For this purpose, two analytical models developed by Gibson and Ashby forhoneycombs with isotropic cell walls are considered (Gibson et al., 1982; Gibsonand Ashby, 1997). In the first model, named flexure model, the elastic responseof the honeycomb is primarily caused by the bending of the cell walls which aremodeled as beams. The second model, here referred to as general model, is ageneralized model which includes shear deformation and axial extension of thebeams. For a full description of these models and mathematical derivations,

46

4.1. VERIFICATION OF THE COMPUTATIONAL MODEL

we refer to Gibson and Ashby (1997). Table 4.1 summarizes the solutions ofthe flexure and general models for the elastic moduli of honeycombs with equalcell wall lengths h = l for two extreme cases when θ → 0 (brick arrangement)and θ → 30 (regular hexagon). These relations are expressed in terms ofthe thickness ratio α = t/l. The geometrical parameters of the honeycombsare presented in Figure 3.5. The computational results are presented fora honeycomb with homogeneous isotropic cell walls with elastic moduli Ec

and Poisson’s ratio νc = 0.3. Since the theoretical models are based on beamformulations which are generally in the state of the plane stress, plane stresselements are used in the computational model.

Shape angle

The normalized effective in-plane elastic and shear moduli of honeycombs asa function of the shape angle θ are computed and are shown in Fig. 4.2.The elastic moduli are normalized to the stiffness of the core material Ec.The cell walls have equal lengths h = l and the thickness is selected to beα = t/l = 0.1. The simulation results are compared with the flexure modeland the general model of Gibson and Ashby (1997) which includes axial andshear deformations. The elastic modulus E1 decreases with the increase ofthe shape angle while E2 and G12 increase. Regular honeycombs (h = l andθ = 30) show isotropic behavior, i.e. E1 = E2. For loading in the direction(1), the flexure model is not suitable for calculation of the elastic modulus E1

in honeycombs with the brick arrangement, i.e θ = 0. In this model, whenθ → 0 the elastic modulus in direction (1) tends to infinity, i.e. E1 →∞, which

Table 4.1: Normalized elastic and shear moduli of honeycombs with equal cell walllength, i.e. h = l, as a function of thickness parameter, α = t/l, when θ → 0 (brickarrangement) and θ → 30 (regular hexagons) based on flexure model (Gibson et al.,1982) and general model including axial and shear deformations (Gibson and Ashby,1997).

Moduli Shape angle Flexure model General modelE1/Ec θ → 0 ∞ αE2/Ec θ → 0 α3 α3/[1 + α2(2.4 + 1.5νs)]G12/Ec θ → 0 α3/3 α3/[3 + α2(7.2 + 4.5νs)]

E1/Ec = E2/Ec θ → 30 4α3/√

3 4α3/(√

3[1 + α2(5.4 + 1.5νs)])

G12/Ec θ → 30 α3/√

3 α3/(√

3[1 + α2(3.3 + 1.75νs)])

47


0 5 1 0 1 5 2 0 2 5 3 01 0 - 4

1 0 - 3

1 0 - 2

1 0 - 1

1 0 0

norm

alized

elas

tic an

d she

ar mo

duli (

-)

s h a p e a n g l e , ( °)

E 1 / E c E 2 / E c G 1 2 / E cc o m p u t a t i o n a l m o d e l g e n e r a l m o d e l f l e x u r e m o d e l

∞

Figure 4.2: The normalized effective in-plane elastic and shear moduli of single-layered honeycombs as a function of the shape angle, θ computed with the proposedcomputational model and compared with the flexure model (Gibson et al., 1982) andthe general model of Gibson and Ashby (1997). The geometrical parameters areselected as h = l and α = t/l = 0.1.

is not physical. In the general model, axial and shear deformations are takeninto account. As shown in this figure, the influence of these variations on E2

and G12 is negligible. Masters and Evans (1996) showed that, for loading indirection 1, the dominant deformation mechanism in regular honeycombs isflexure while for the honeycombs with brick arrangement, the deformation isin axial mode. The agreement between the simulation results and the generalmodel which includes axial and shear deformations is excellent. The advantageof the present computational model is that it can be extended to study thebehavior of honeycombs composed of anisotropic or multi-layered cell wallmaterial and the irregular cellular structures such as eccentric honeycombswhich will be discussed in the next chapter.

Cell wall thickness

To examine the influence of the cell wall thickness on the elastic properties, theeffective elastic and shear moduli of honeycombs with homogeneous cell walls

48

4.2. HONEYCOMBS WITH MULTI-LAYERED CELL WALLS

are computed for brick arrangement, i.e. θ = 0. The results are plotted as afunction of the thickness parameter, α in Fig. 4.3. The cell walls have equallengths, i.e. h = l = 1.0, and the elastic modulus of the cell walls is Ec. Since forθ = 0, the flexure model is not valid, the finite element results are comparedonly with the general model (Gibson and Ashby, 1997). The elastic moduli ofhoneycombs increase with the increase of the cell wall thickness. At large cellwall thicknesses, there is a slight difference between the computational and thetheoretical models. Again, the simulation results exhibit very good agreementwith the general model which confirms the validity of the proposed model forcomputation of effective elastic properties.

0 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0 0 . 2 51 0 - 5

1 0 - 4

1 0 - 3

1 0 - 2

1 0 - 1

E 1 / E c E 2 / E c G 1 2 / E c c o m p u t a t i o n a l m o d e l g e n e r a l m o d e l

norm

alized

elas

tic an

d she

ar mo

duli (

-)

t h i c k n e s s r a t i o , α= t / l ( - )

Figure 4.3: The effective in-plane elastic and shear moduli of honeycombs withisotropic cell walls for θ = 0 as a function of the thickness ratio α. The solid lines arecalculated with the proposed computational model and are compared with generalmodel of Gibson and Ashby (1997) which includes the axial and shear deformations.The selected geometrical parameters are h = l and θ = 0.

4.2 Honeycombs with multi-layered cell walls

In this section, the anisotropy of the hygro-elastic properties of honeycombswith multi-layered cell walls is investigated. We consider a periodic honeycombunit cell with multi-layered cell walls. The geometrical parameters of the

49


2 3

-s~

11

0.5l

0.5h

t

θ

Es

Ec,

c

,s

,c

2

1

h

l

θ

(a) (b)

+s~

1

-s~

2

+s~

2

-s~

3

+s~

3

t

Figure 4.4: (a) Schematic of honeycomb structure and its geometrical parameters,(b) discretized quarter of a multi-layered honeycomb unit cell.

honeycomb are shown in Fig. 4.4a. The height of the vertical walls is h,the length of the inclined walls is l and the shape angle is θ. The cell wallof thickness t is composed of three layers: a swelling core (denoted c) andtwo non-swelling sheaths (denoted s). The thickness of each sheath layer isδ. Realizing the large number of parameters involved here, we focus on aparticular subset of parameters. The elastic modulus and Poisson’s ratio ofthe core (Ec, νc) and the sheath layers (Es, νs) are linear elastic and assumedto be isotropic. The hygro-expansion coefficients of the cell wall core and thesheath layer are βc and βs = 0, respectively. The constitutive equation of thecell wall materials with moisture induced swelling can be written as:

σmij = Cmijkl(ε

mkl − β

mkl∆m), (4.1)

where σmij , εmij , C

mijkl, β

mij are respectively the stress, strain, stiffness and swelling

coefficients tensors at cell wall level (superscript m). The change in moisturecontent by mass from the initial state is denoted by ∆m. The constitutiveequation at the macroscopic level has identical structure:

σMij = CMijkl(ε

Mkl − β

Mkl ∆m), (4.2)

where σMij , εMij , CMijkl, βMij are corresponding to the macroscopic stress,

strain, stiffness and swelling coefficients (superscript M). The macroscopichygro-elastic properties are obtained from the computational upscaling of thehoneycomb unit cells as explained in Section 3.2.2.

50


The proposed computational upscaling approach has been implemented inthe finite element package ABAQUS (Rising Sun Mills, Providence, RI,USA) and used to predict the effective properties of the honeycombs. Theimplementation of the computational model in ABAQUS is presented schemat-ically in Appendix. We focus on the effective material properties arisingin problems of hygro-elasticity, specifically the elastic stiffness and hygro-expansion coefficients. The hygro-elasticity is an analog to thermo-elasticitywith moisture content playing the role of temperature. In all simulations, themesh density is increased until the results converge satisfactorily. The reducedhoneycomb unit cells are meshed using generalized plane strain elements(CPEG4) as shown in Figure 4.4b. The elastic properties of the cell wallcore are Ec = 1 GPa and νc = 0.3. The elastic modulus of the sheaths Es

is considered as a parameter in the simulations. The Poisson’s ratio of thesheaths is assumed to be νs = 0.3.

4.2.1 Numerical results

Hygro-expansion coefficients

Fig. 4.5 shows the normalized effective in-plane hygro-expansion (swelling)coefficients of multi-layered honeycombs, β1/βc and β2/βc, as a function ofthe elastic moduli ratio of the sheath to the core material Es/Ec. The shearcomponent of the effective hygro-expansion tensor is found to be almost zero,i.e. β12 ≃ 0, and is not shown in the results. The geometrical parameters ofthe honeycombs are h = l = 1.0, t = 0.1 and θ = 15. Two cases are consideredand the simulations are conducted for two sheath thicknesses δ = 0.1t andδ = 0.2t. It is found that the restraining effect of the sheath layers considerablyreduces the effective hygro-expansion coefficients as the elastic ratio Es/Ec

increases. The sheath layers act as corsets. With increasing stiffness of thesheath layers, the restraining effect and thus the reduction of effective hygro-expansion coefficients becomes more important. This reduction is larger forthicker sheath layers. As the stiffness of the sheath layers increases, the swellingbehavior of the honeycombs becomes more anisotropic.

The induced anisotropy stems from less expansion in direction (1) than indirection (2). The restraining effect of the sheath layers is more pronouncedwhen Es/Ec < 10. The presence of thin sheath layers with thickness δ = 0.1t andelastic modulus Es/Ec = 10 reduces the hygro-expansion of the honeycombs toβ1 = 0.25 βc in direction (1) and to β2 = 0.59 βc in direction (2). The reduction

51


1 5 1 0 1 5 2 0 2 5 3 00 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0hy

gro-ex

pans

ion, β

/βc

e l a s t i c r a t i o , E s / E c

β 1 / β c , δ/t = 0 . 1 β 2 / β c , δ/t = 0 . 1 β 1 / β c , δ/t = 0 . 2 β 2 / β c , δ/t = 0 . 2

Figure 4.5: The effective in-plane hygro-expansion coefficients of three-layeredhoneycombs, β1 and β2, normalized to the hygro-expansion coefficient of the coreβc as a function of sheath to core elastic ratio Es/Ec with geometrical parametersθ = 15, h = l and α = t/l = 0.1 for two sheath thicknesses δ/t = 0.1 and δ/t = 0.2.

of the hygro-expansion due to the thicker sheath layers with thickness δ = 0.2tand elastic modulus Es/Ec = 10 is even more. In this case, the hygro-expansioncoefficients are β1 = 0.11βc and β2 = 0.39βc.

Elastic and shear moduli

The effective elastic and shear moduli of multi-layered honeycombs as afunction of the sheath to core elastic ratio Es/Ec are shown in Fig. 4.6.The honeycomb geometry, the configuration of the layers and the materialproperties are the same as in the previous simulation. As expected, the elasticand shear moduli of honeycombs increase with the increase of the sheaththickness. The elastic response of honeycombs is strongly anisotropic andthe effective elastic modulus E1 is bigger than E2 by roughly one order ofmagnitude.

52


1 5 1 0 1 5 2 0 2 5 3 00 . 1

1

1 0

1 0 0ela

stic Yo

ung’s

and s

hear

modu

li (MP

a)

e l a s t i c r a t i o , E s / E c

δ/t = 0.1 E 1 E 2 G 1 2δ/t = 0.2 E 1 E 2 G 1 2

Figure 4.6: The effective elastic and shear moduli of three-layered honeycombs asa function of sheath to core elastic ratio Es/Ec with geometrical parameters θ = 15,h = l and α = t/l = 0.1 for two thicknesses of the sheath layers, i.e. δ/t = 0.1 andδ/t = 0.2.

Anisotropy ratio

Fig. 4.7 shows the hygro-expansion and elastic anisotropy ratios of three-layered honeycombs, respectively β2/β1 and E1/E2, as a function of sheathto core elastic ratio Es/Ec. These ratios correspond to the results presentedin Fig. 4.5 and Fig. 4.6. The anisotropy in elastic moduli is not influencedsignificantly with the increase of the elastic ratio Es/Ec and it is almostconstant. We may conclude that not the multi-layered configuration, butthe geometry of honeycombs mainly determines the elastic anisotropy of thematerial. By contrast, the anisotropy in hygro-expansion coefficients notablyincreases with the increase of the elastic ratio Es/Ec. The hygro-expansionratio β2/β1 shows almost a linear relationship with Es/Ec. The simulationresults show very similar trends for both thin and thick sheath layers.

53


1 5 1 0 1 5 2 0 2 5 3 00

3

6

9

1 2An

isotro

py ra

tio, β

2/β1 a

nd E 1/E

2 (-)

e l a s t i c r a t i o , E s / E c ( - )

E 1 / E 2 β 2 / β 1δ/ t = 0.1 δ/ t = 0.2

Figure 4.7: The hygro-expansion and elastic anisotropy ratios of three-layeredhoneycombs, β2/β1 and E1/E2, as a function of sheath to core elastic ratio Es/Ec

with geometrical parameters θ = 15, h = l and α = t/l = 0.1 for two sheath thicknessesδ/t = 0.1 and δ/t = 0.2.

Shape angle

The influence of the shape angle on the in-plane hygro-expansion propertiesof the honeycombs and the anisotropy ratio β2/β1 are shown in Fig. 4.8aand 4.8b, respectively. These results are presented for a honeycomb withgeometrical parameters h = l and α = t/l = 0.1 and for two sheath thicknessesδ/t = 0.1 and δ/t = 0.2. The elastic modulus of the core is Ec = 1GPaand the sheath to core elastic ratio is Es/Ec = 10. The elastic propertiesof these honeycombs and the corresponding anisotropy ratio are also displayedin Fig. 4.9a and 4.9b. It is found that honeycombs in the brick arrangement(θ = 0) show strong anisotropic swelling and mechanical behavior while theregular hexagon arrangement (θ = 30 and h = l) exhibits a completely isotropicbehavior in the plane. Therefore, irrespective of the anisotropy of the multi-layered cell wall, the effective hygro-elastic behavior of regular honeycombs isisotropic in the plane. However, the restraining sheaths modify the swellingproperties and the hygro-expansion coefficients decrease significantly.

As shown in Fig. 4.8b, the swelling coefficients of the honeycombs with

54


thicker sheath layer is smaller than those with thin sheath layers but theirhygro-expansion anisotropy ratio is bigger, specifically for small shape angles.Regarding the elastic properties, they show similar trends and the anisotropyratio for both configurations is almost the same. In order to determinethe effect of the orientation of the coordinates on the in-plane properties, theswelling coefficient and the elastic properties are calculated using the Eulertransformation for an orthotropic material. Figure 4.10 shows the polar plotsof the in-plane swelling coefficients, the elastic modulus and the shear modulusof the honeycombs with multilayered cell walls for h = l, α = t/l = 0.1 and fortwo sheath thicknesses δ/t = 0.1 and δ/t = 0.2. The results are presented forthree shape angles θ = 0,15,30. The first column of Figure 4.10 shows theswelling coefficient β/βc as a function of rotation of the frame. It can be seenthat for shape angles θ < 30, the smallest swelling coefficient occurs when thereis no rotation and the frame is in direction (1) and it reaches its maximum valuewhen the frame is rotated 90 which corresponds to direction (2). For regularhexagons the swelling coefficients are equal in all directions and the behavioris truly isotropic in the plane. The second column of Figure 4.10 displays thevariation of the elastic modulus with a rotation of the frame. When the frameis not rotated, the elastic modulus is maximum while for rotation of 90 itgoes to its minimum. The elastic modulus also behaves isotropically in the

0 5 1 0 1 5 2 0 2 5 3 00 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0 β 1 / β c , δ/ t = 0 . 1 β 2 / β c , δ/ t = 0 . 1 β 1 / β c , δ/ t = 0 . 2 β 2 / β c , δ/ t = 0 . 2

hygro

-expa

nsion

, β/β c

s h a p e a n g l e , θ ( °)

(a)

0 5 1 0 1 5 2 0 2 5 3 00

1

2

3

4

5

hygro

-expa

nsion

ratio

, β2/β

1

s h a p e a n g l e , θ ( °)

δ/t = 0.1 δ/t = 0.2

i n - p l a n e i s o t r o p i c b e h a v i o r

(b)

Figure 4.8: (a) The effective in-plane hygro-expansion coefficients of three-layeredhoneycombs, β1 and β2, normalized to the hygro-expansion coefficient of the core βcas a function of shape angle θ and (b) the corresponding hygro-expansion anisotropyratio β2/β1. The geometrical parameters are h = l and α = t/l = 0.1 and the sheath tocore elastic ratio Es/Ec = 10 for two sheath thicknesses δ/t = 0.1 and δ/t = 0.2.

55


0 5 1 0 1 5 2 0 2 5 3 00 . 1

1

1 0

1 0 0

δ/t = 0.1 E 1 E 2 G 1 2δ/t = 0.2 E 1 E 2 G 1 2

elastic

Youn

g’s an

d she

ar mo

duli (

MPa)

s h a p e a n g l e , θ ( °)

(a)

0 5 1 0 1 5 2 0 2 5 3 00 . 1

1

1 0

1 0 0

elastic

ratio

, E1/E

2

s h a p e a n g l e , θ ( °)

δ/t = 0.1 δ/t = 0.2

i n - p l a n e i s o t r o p i c b e h a v i o r

(b)

Figure 4.9: (a) The effective in-plane elastic and shear moduli of three-layeredhoneycombs, E1, E2 and G12, as a function of shape angle θ and (b) the correspondingelastic anisotropy ratio E1/E2. The geometrical parameters are h = l and α = t/l = 0.1and the sheath to core elastic ratio Es/Ec = 10 for two sheath thicknesses δ/t = 0.1and δ/t = 0.2.

plane for regular hexagons. An interesting observation is that for shape anglesθ < 30 the swelling coefficient is maximum when elastic modulus is minimumand vice versa. The influence of the frame orientation on the in-plane shearmodulus is also shown in the third column of Figure 4.10. It can be seenthat the shear modulus has a maximum at a rotation of 45. On the otherhand, either when there is no rotation or there is a rotation of 90 the shearmodulus is minimum. Also for regular hexagons the in-plane shear modulus isisotropic. In summary, these results show that by increasing the shape angle,the anisotropy of the in-plane swelling and the elastic properties of the multi-layered honeycombs decreases. The hygro-elastic behavior of regular hexagons(θ = 30 and h = l) is truly isotropic in the plane. It should be mentioned thatif we take into account the out of plane properties, the regular hexagons aretransversly isotropic.

4.3 Remarks on swelling of wood cells

The cell wall in wood are composed of a primary layer, with randomly orientedcellulose fibrils, and three secondary layers, S1, S2 and S3, with cellulose fibrilshelically wound in varying patterns in each of the three layers. Thus, the

56

4.3. REMARKS ON SWELLING OF WOOD CELLS

0.2 0.4 0.6 0.8 1

30

210

60

240

90

270

120

300

150

330

180 0

Swelling coefficient

θ=0°

0.2 0.4 0.6 0.8 1

30

210

60

240

90

270

120

300

150

330

180 0

θ=15°

0.2 0.4 0.6 0.8 1

30

210

60

240

90

270

120

300

150

330

180 0

θ=30°

0.5 1 1.5 2 2.5 3

30

210

60

240

90

270

120

300

150

330

180 0

Young’s modulus

θ=0°

0.5 1 1.5 2 2.5 3

30

210

60

240

90

270

120

300

150

330

180 0

θ=15°

0.5 1 1.5 2 2.5 3

30

210

60

240

90

270

120

300

150

330

180 0

θ=30°

0.2 0.4 0.6 0.8 1

30

210

60

240

90

270

120

300

150

330

180 0

Shear modulus

θ=0°

0.2 0.4 0.6 0.8 1

30

210

60

240

90

270

120

300

150

330

180 0

θ=15°

0.2 0.4 0.6 0.8 1

30

210

60

240

90

270

120

300

150

330

180 0

θ=30°

δ/t=0.1

δ/t=0.2

Figure 4.10: The polar plots for effective in-plane swelling coefficient, elastic Young’sand shear moduli of three-layered honeycombs with geometrical parameters h = l,α = t/l = 0.1 and thicknesses of the sheath layers δ/t = 0.1 and δ/t = 0.2 for threeshape angles θ = 0,15,30.

57


S2 layer which accounts for most of the thickness of the cell wall, drivesthe swelling of the cell wall material in transverse direction. In transversedirections, due to the circumferentially winding of the cellulose micro-fibrils,the S1 and S3 layers are much stiffer than the S2 layer. The in-extensibilityof the cellulose fibrils prevents the swelling of the S1 and S3 layers incircumferential direction of the cells and, as a result, they act as corsets whichcause swelling to be more important normal to the cell wall than along the cellwall direction. This behavior is also reported in the rare experimental studieson softwood species at the cell wall level (Nakato, 1958; Ishimaru and Iida,2001).

In this chapter, we showed that in a honeycomb with multi-layered cell walls,the difference in hygro-elastic properties of the cell wall layers results inmacroscopic anisotropic swelling properties in the transverse plane. In orderto explain the swelling anisotropy of the wood cells at the cellular scale, weexamine the hygro-elastic properties of the cell wall layers which are givenin literature based on the micro-mechanics models. For example, Neagu andGamstedt (2007) calculated the hygro-elastic properties of each cell wall layerwith a micro-mechanical approach taking into account the elastic propertiesand the volume fractions of the wood polymers in each layer. Transverselyisotropic elastic constants, hygro-expansion coefficients for S1, S2 and S3 andthe normalized thickness of each layer δ with respect to the cell wall thicknessare given in Table 4.2. In this table, (33) corresponds to the direction ofthe cellulose microfibrils and (11) refers to the transverse plane normal to thefibrils. Hypothetically, we can assume that the microfibril angle in S2 layer iszero and the orientation of the fibrils in S1 and S3 layers is perpendicular to S2layer. In this case, the S2 layer plays the role of the swelling core and either S1or S3 act as the restraining non-swelling sheath layers. In this configuration,the in-plane hygro-elastic properties of the S2 layer are βc = βS2

11 = 0.44 andEc = E

S211 = 6.08 GPa. The in-plane hygro-elastic properties of the non-swelling

Table 4.2: Hygroelastic properties of the wood cell wall layers along the cellulosefibril direction (33) and in the plane normal to the fibrils (11). These values arecalculated based on volume fractions and properties of wood polymers (Neagu andGamstedt, 2007).

layer δ ESi11 (GPa) ESi

33 (GPa) βSi11 βSi

33 ESi33/E

S211

S1 0.12 4.20 28.1 0.44 0.02 4.62S2 0.78 6.08 64.0 0.47 0.05 -S3 0.03 6.08 64.0 0.47 0.05 10.53

58

4.4. SUMMARY

sheath layers are given as βs = βS1,S333 = 0.02,0.05 and Es = E

S1,S333 = 28.1,64.0

GPa. The last column of the Table 4.2 shows the ratio of the elastic moduliof the S1 and S3 layers to the S2 layer in the plane of the wood cell which isEs/Ec = E

Si33/E

S211 ≃ 5 − 10. Now if we recall, the results of the three-layered

honeycomb model, and assuming δ = 0.03−0.12 for the sheath thickness, we canpredict the anisotropy of the honeycombs as presented in Section 4.2. Despitethe simplified assumptions on the detailed configuration of the cell wall, theabove reasoning justifies the anisotropic swelling behavior of the wood cellsbased on a simplified multi-layered model. The link between this simplifiedmodel and the anisotropic behavior of the wood could be made qualitativelyin this section. However, for an accurate quantitative estimation of hygro-elastic behavior of wood cells, the exact layout of the cell wall layers should betaken into account.

4.4 Summary

In this chapter, the effective hygro-elastic behavior of periodic honeycombsis investigated based on a micro-mechanics computational homogenizationscheme presented in the previous chapter. The overall validity of the model isestablished by comparison with analytical solutions available in literature forthe elastic properties of honeycombs. The analysis of honeycombs with multi-layered cell walls reveals that, for certain configurations of honeycombs, thenon-swelling sheath layers restrain the swelling of the core layer which resultsin anisotropic in-plane effective hygro-expansion coefficients. In contrast, thesheath layers have almost no influence on the elastic anisotropy ratio of thehoneycombs. The anisotropy in swelling behavior is found to increase whenthe sheath layers are much stiffer than the cell wall core material and thegeometry of the honeycombs approaches a brick-like arrangement. The layeredconfiguration of the cell walls leads to the anisotropic behavior both at the cellwall (evidently) and at the cellular scale and beyond.

59

Chapter 5

The role of eccentricity onswelling anisotropy of cellularmaterials

The anisotropic swelling behavior of honeycombs composed of multi-layeredcell walls is investigated in the previous chapter. We found that the swellinganisotropy of cellular materials can be attributed to two factors: (1) anisotropichygro-elastic properties of the cell wall material caused by the multi-layeredlayout and (2) shape or the geometrical configuration of the cellular structure.Many biological materials exhibit a structural hierarchy which can enhanceto a great extent the hygromechanical properties of the bulk material. Theinfluence of geometrical constraints embedded in the hierarchical architectureof the cellular materials on the swelling behavior is yet to be understood. Inthis chapter, we are looking at the role of geometrical irregularities on theswelling anisotropy of cellular materials using softwood as an example1. Inwood, we noticed that at the cell wall scale the varying winding pattern ofthe cellulose fibrils in different cell wall layers might constrain the swellingof the cell wall in the circumferential direction and allow more deformationin the thickness direction. At the cellular scale, which is the main focus ofthis chapter, the disorder in connectivity of the cell walls within the cellularstructure of wood is expected to constrain swelling in one direction and allow itin another direction due to possible locking mechanism of adjacent cells. Thegeometrical irregularities in natural cellular materials come from the natural


61

CHAPTER 5. THE ROLE OF ECCENTRICITY ON SWELLING ANISOTROPY

(a) (b)

100µm

R

T

L

earlywoodtracheid cells latewood

tracheid cells

bordered pitray cells

RT

L

growth ringboundary

earlywood latewood

Figure 5.1: (a) µCT X-ray tomography image of softwood at a growth ring boundaryshowing the cellular and sub-cellular features and (b) binarized scanning electronmicroscopic image of one growth ring of Norway spruce softwood.

variability associated to cell growth processes. For example, at the mesoscopiclevel, shown in Figure 5.1a, softwood consists mainly of longitudinal tracheidcells which are formed by repeated cell division of initial cells which formthe cambium at the xylem/phloem interface. Consequently, there is a strongdegree of cell alignment in the radial direction, whereas the cells are randomlystaggered in the tangential direction (Kahle and Woodhouse, 1994). As shownin Figure 5.1b, in cellular structure of wood there is a large number of triplejoints, i.e. points of juncture of three double walls (Gillis, 1972). In tangentialdirection, the periodicity of the cellular structure of wood depends on thearrangement of the cells present in contiguous radial rows. As shown inFigure 5.1b, across one growth ring, the thin-walled earlywood cells with largeinternal cavities, called lumen, gradually change to thick-walled latewood cellswith small-sized lumens.

In this chapter, we go one step further and investigate the role of eccentricityin the geometry of honeycomb unit cells and the influence of the swellinganisotropy of the cell walls on the macroscopic response of cellular solids. Thediscussion here is confined to the hygro-elastic behavior of two-dimensionalcellular solids. We aim at quantifying the respective contributions of theeccentricity, the periodicity of the microstructure and the anisotropy of thecell wall material in the macroscopic swelling properties of cellular materials.

62

5.1. PERIODIC HONEYCOMBS

5.1 Periodic honeycombs

Two types of periodic honeycombs are considered. Symmetric and non-symmetric honeycombs are selected for the present study according to theirsimilarities with softwood cells (see Figure 5.2). Since the longitudinaldimension of the wood cells is very large in comparison to the dimensionsin radial and tangential directions, the problem is reduced to the analysis ofthe 2D cross-section of the material which is in a state of generalized planestrain. Symmetric honeycombs are characterized by two parameters, namelythe aspect ratio η and the shape angle θ. The aspect ratio η is defined by theratio of the axes of the ellipse inscribed within the symmetric honeycomb:

η =

√1 + 4 ζ sinθ

2 ζ cosθ, (5.1)

where ζ = l/h is the ratio of the length of the inclined wall l to the vertical wallh. The thickness of the cell walls t is adjusted to achieve the desired materialporosity φ. A generic geometrical eccentricity is introduced to the symmetrichoneycombs by shifting the two middle joints along the radial direction.The resulting structure covers a wide range of cellular patterns in softwoods

(a) symmetric (b) antisymmetric (c) mirrored

D D

d

d

d

d R

T

L

θ

l

h

Figure 5.2: Above, representation of (a) symmetric, (b) antisymmetric and (c)mirrored configurations and below, examples of scanning electron microscopy imagesof Norway spruce softwood. The scale bars are 50µm. The reduced domains aremeshed and overlaid on each honeycomb. The eccentricity parameter is defined ase = d/D.

63


caused by growth interval in contiguous rows of cells. The three honeycombconfigurations used in this work and examples of scanning electron microscopyimages of Norway spruce softwood are shown in Figure 5.2. The presentedimages are acquired in low vacuummode using environmental scanning electronmicroscopy (ESEM, PHILIPS XL30-FEG) with beam intensity of 15kV2.

In non-symmetric honeycombs, the eccentricity parameter e = d/D is defined asthe ratio of the offset from the center line d to the distance between the centerline and the lateral walls D. If the joints are shifted in opposite directions,an antisymmetric honeycomb is generated and shifting the joints in the samedirection results in a unit cell with mirror symmetry, referred to as mirroredunit cell. Although such irregularity changes the length of some cell walls, itseffects on porosity is negligible. To generate the non-symmetric honeycombs, atthe first step, a symmetric honeycomb is generated with predefined shape angleand aspect ratio. Then, the position of the middle joints is varied to constructthe non-symmetric configurations for a given eccentricity. Both symmetric andnon-symmetric unit cells preserve periodicity but possess different symmetryconditions. Taking advantage of these symmetry conditions, we reduce theanalysis of the original unit cell domain into one quarter for the symmetricand antisymmetric unit cells and to one half for the mirrored honeycombs.In Figure 5.2, the reduced unit cells are meshed and are overlaid on eachhoneycomb configuration.

5.2 Real cellular structure of softwood

Periodic idealized configurations provide estimates of the effective propertiesof cellular solids but may not capture some details of a real microstructure. Tocomplement the study, we also use four real microstructures of wood extractedfrom different microscale computed tomographic (µCT) datasets for finiteelement simulations3. The phase contrast tomography method was used sincewood has a low X-ray attenuation and requires a low dose of deposited ionizingradiation to prevent damage. The scans are made with an effective spatialresolution of 0.70 µm. From these datasets volume elements with a side length

2The experiments are conducted by Dr. Masaru Abuku and Stefan Carl in the Laboratoryfor Building Science and Technology at EMPA, Dübendorf, Switzerland.

3The phase contrast synchrotron X-ray tomographic data were acquired at the TOMCATbeamline at the Swiss Light Source, Paul Scherrer Institut, Villigen, Switzerland and theiranalysis were performed at the Laboratory for Building Science and Technology, EMPA,Dübendorf, Switzerland.

64

5.2. REAL CELLULAR STRUCTURE OF SOFTWOOD

of 200-300 µm are cropped and analyzed. Further details on the experimentalset-up and the method used can be obtained from Derome et al. (2011). Themicrostructures of four samples taken at different places from different growthrings are characterized as displayed in Figure 5.3. The porosity, the averageaspect ratio of the lumen diameters and the average eccentricity are determinedfor each sample. The eccentricity of real structures is measured cell by cellmanually, in a way similar to that which is applied to the periodic eccentrichoneycombs with the relation e = d/D as shown in Fig. 5.2. For each cell, thedistance (d) between the middle tangential (vertical) wall to centerline of thecell is measured and divided by D which equals the half of the cell diameter inradial direction. However, it should be mentioned that the antisymmetric andmirrored configurations are not distinguishable in real cellular structures dueto the irregularities of their microstructure. The mean values of the parametersφ, η and e are listed in Table 5.1. The standard deviation of the e is also shownin the last column of this table.

(b) EW1, =0.80(a) LW1, =0.49

(d) EW2, =0.63(c) LW2, =0.53

50µm50µm

50µm 50µm

Figure 5.3: Slices of µCT datasets of real microstructures of softwood used for finiteelement simulations. (a) latewood, φ = 0.49, (b) earlywood, φ = 0.80, (c) latewood,φ = 0.53 and (d) earlywood, φ = 0.63,

65


porosity aspect ratio eccentricitysample φ mean(η) mean(e) std(e)LW1 0.49 1.05 0.53 0.31LW2 0.53 0.97 0.44 0.22EW1 0.68 0.95 0.43 0.28EW2 0.80 0.80 0.37 0.26

Table 5.1: Geometrical parameters of real structures of softwood tissues which arestudied in this work; φ is the porosity, η is the aspect ratio of the lumen diametersand e is the eccentricity parameter.

A powerful active contour technique based on minimal variance and level setsproposed by Chan and Vese (2001) is employed to detect the borders of thelumens on the µCT images of softwood. The basic idea of active contourmodels is to evolve a curve, subjected to constraints from a given image,in order to detect objects in that image. The employed method can detectobjects whose boundaries are not necessarily defined by the gradient of theimage, as in the classical active contour models, but are instead related toa particular segmentation of the image. The algorithm is implemented inMATLAB4. In the case of porous materials with anisotropic constituents, it isalso necessary to identify the material orientation within the solid phase. Thematerial orthogonal directions are aligned locally parallel (∥) and normal (⊥)to the cell walls. The third direction, normal to the plane is the longitudinal(L) direction. An elegant and easy way for the estimation of orthotropicmaterial directions is to use the gradient of the level sets. Once the levelsets are computed, the lumen borders and the material orientations can bereadily determined from zero level sets and the gradient of the level sets,respectively. The resulting material orientations are assigned to the finiteelement integration points. Figure 5.4 illustrates the implementation of theabove procedure for the earlywood sample EW1. In Figure 5.4a, a slice ofµCT dataset of earlywood is shown. The contour of the corresponding level setsegmentation is illustrated in Figure 5.4b in which the dark color representsthe positive values and the light color stands for negative values. The zerolevel set is shown with black solid lines. Before segmentation, sub-cellularfeatures like bordered pits are replaced by continuous cell walls. The resultinggeometry is converted into spline curves, imported into finite element programand meshed with four node elements. A part of the meshed sample is shown

4http://dmforge.itn.liu.se/lsmatlab/minimal_variance.html

66

http://dmforge.itn.liu.se/lsmatlab/minimal_variance.html

5.3. NUMERICAL RESULTS

(a)(b)

(d)(c)

50µm

R

T

L

ll

Figure 5.4: Characterization of a real cellular microstructure of softwood (a) oneslice of µCT dataset of earlywood, (b) level set segmentation, close-up of (c) finiteelement mesh and (d) orthogonal material directions.

in Figure 5.4c. The corresponding orthogonal material directions of this partare shown in Figure 5.4d.

5.3 Numerical results

5.3.1 Implementation

To assess the macroscopic swelling properties of cellular solids, the fi-nite element based computational scheme presented in Chapter 3 is em-ployed. All simulations are carried out using the finite element packageABAQUS/Standard (Rising Sun Mills, Providence, RI, USA) through itsPython scripting interface. Periodic boundary conditions (PBC) are usedfor synthetic honeycombs. In this case, the reduced honeycomb unit cellsare meshed using 2D four-node generalized plane strain elements (CPEG4)with linear elastic constitutive behavior. The mesh density is increased until

67


lR

lT

left

top

right

bottom

R

T

L

Figure 5.5: Notations of the faces (right, left, top, bottom) for PMUBC on a volumeelement of real microstructure with dimensions lR and lT .

the results converge satisfactorily. The total number of elements depends ongeometrical parameters of honeycomb. However, our simulations show thatthe convergence is achieved with a minimum of six elements in the thicknessdirection.

The real structure models are not periodic, thus the PBC cannot be used for theupscaling procedure. Therefore, a set of mixed boundary conditions denotedas periodicity compatible mixed uniform boundary conditions (PMUBC)proposed by Pahr and Zysset (2008) is used for this purpose. Figure 5.5 showsa volume element of softwood tissue and the definition of the boundary edges.The corresponding boundary conditions applied on the edges of the volumeelement are given in Table 5.2. These load cases are used for the calculation ofthe in-plane elastic properties and the swelling coefficients. The procedure forcalculation of the effective swelling coefficients is explained in Section 3.3.2.

Loadcase left right top bottom out of planeR u1 = 1/2εolR u1 = −1/2εolR u2 = 0 u2 = 0 u3 = 0T u1 = 0 u1 = 0 u2 = 1/2εolT u2 = −1/2εolT u3 = 0RT u2 = 1/2εolR u2 = −1/2εolR u1 = 1/2εolT u1 = −1/2εolT u3 = 0

Swelling u1 = 0 u1 = 0 u2 = 0 u2 = 0 u3 = 0

Table 5.2: Periodicity compatible mixed boundary conditions (PMUBC) applied onthe boundary faces of a volume element where symbol εo is the desired uniform strain;lR and lT are respectively the length of the volume element in radial and tangentialdirections.

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Material properties of the cell wall

To mimic the material properties of the cell wall which is a natural laminatecomposite, an overall orthotropic behavior is assumed. The elastic propertiesare selected from Persson (2000) and are listed in Table 5.4. The subscriptindices 1, 2 and 3 are equivalent to normal (⊥) parallel (∥) and longitudinal(L) directions. There is rare experimental information about the swellingproperties of the wood cell wall (Ishimaru and Iida, 2001). The swellingcoefficients of the cell wall (β∥ and β⊥) are assumed to be orthotropic andconsidered as parameters in this work. The swelling coefficients are inspiredfrom the behavior of wood where the cell walls swell much less along the wallthan in the thickness direction, i.e. β∥ ≤ β⊥.

E1 E2 E3 ν12 ν13 ν23 G12 G13 G23

4.36 7.02 33.20 0.403 0.023 0.112 1.18 1.65 4.38

Table 5.3: Orthotropic elastic properties of the cell wall (Persson, 2000). Theelastic and shear moduli are in GPa. The subscript indices 1, 2 and 3 are equivalentto normal(⊥), parallel(∥) and longitudinal(L) directions.

5.3.2 Swelling behavior of periodic honeycombs

Symmetric honeycombs

We first investigate the swelling of symmetric honeycombs (e = 0). Aparametric study is carried out to study the influence of the shape angle θon the macroscopic swelling properties for different porosities φ and aspectratios η. The iso-contour plots for radial (βR = βM11 ) and tangential (βT = βM22 )

swelling coefficients and the anisotropy ratio βT /βR as a function of the shapeangle θ and the cell wall swelling ratio β∥/β⊥ for three pairs of parameters(φ, η) are shown in Figure 5.6. In each case, the thickness of the cell walls isadjusted to keep the porosity constant. It is found that the influence of θ onthe effective swelling properties is more pronounced at high porosities. Theswelling coefficients in radial direction are almost independent from porosityalthough they decrease slightly with the increase of the porosity. The tangentialswelling coefficients decrease with θ specifically at high porosities. The swellinganisotropy ratio is almost not influenced by shape angle up to θ = 15 and thenit decreases more sharply for θ > 25 to reach values close to 1.0 at θ = 30. It is

69


found that the regular symmetric honeycombs (η = 1, θ = 30) show completelyisotropic swelling behavior in the plane regardless of the anisotropy of the cellwall material, βT /βR = 1. With increasing cell wall swelling ratio β∥/β⊥, theswelling coefficients βT and βR increase while the anisotropy in macroscopicswelling behavior βT /βR decreases. As expected, the swelling behavior forhoneycombs with isotropic cell walls (β∥ = β⊥) is isotropic.

θ()

β‖

/β⊥

βR

0.5

0.4

0.3

0.2

0.1

0 10 20 300

0.2

0.4

0.6

0.8

1

θ()

β‖

/β⊥

βT

0.5

0.4

0.3

0.20.1

0 10 20 300

0.2

0.4

0.6

0.8

1

θ()β‖

/β⊥

βT /βR

1.25

1.5

2

3

0 10 20 300

0.2

0.4

0.6

0.8

1

θ()

β‖

/β⊥

0.5

0.4

0.3

0.2

0.1

0 10 20 300

0.2

0.4

0.6

0.8

1

θ()

β‖

/β⊥

0.5

0.4

0.3

0.2

0 10 20 300

0.2

0.4

0.6

0.8

1

θ()

β‖

/β⊥

1.251.5

23

0 10 20 300

0.2

0.4

0.6

0.8

1

θ()

β‖

/β⊥

0.5

0.4

0.3

0.2

0 10 20 300

0.2

0.4

0.6

0.8

1

θ()

β‖

/β⊥

0.5

0.4

0.3

0.2

0 10 20 300

0.2

0.4

0.6

0.8

1

θ()

β‖

/β⊥

1

1.251.5

0 10 20 300

0.2

0.4

0.6

0.8

1

(a)

(b)

(c)

Figure 5.6: The iso-contour plots for radial and tangential swelling coeffi-cients (%strain/%moisture content) and the anisotropy ratio βT /βR of symmetrichoneycombs as a function of shape angle θ and the cell wall swelling ratio β∥/β⊥ forβ⊥ = 0.6 and (φ, η) equal to (a) (0.8,0.5), (b) (0.68,1.0), (c) (0.55,1.5). The inset in theupper left and right of each sub-figure shows the schematic of honeycomb geometryat θ = 0 and θ = 30 respectively.

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Eccentric honeycombs

The eccentric periodic honeycombs are analyzed and the effective swellingcoefficients for different eccentricity parameter e and the cell wall swellingratio β∥/β⊥ are computed. As illustrated in Figure 5.2, a negative eccentricitycorresponds to an antisymmetric honeycomb and a positive value representsa mirrored configuration. Thus, e = 0 indicates a symmetric honeycomb.

e(−)

β‖

/β⊥

βR

0.1

0.2

0.3

0.4

0.5

−0.6 −0.4 −0.2 0 0.2 0.4 0.60.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

e(−)

β‖

/β⊥

βT

0.2

0.3

0.4

0.5

−0.6 −0.4 −0.2 0 0.2 0.4 0.60.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

e(−)

β‖

/β⊥

βT /βR

3

2

1.5

1.25

−0.6 −0.4 −0.2 0 0.2 0.4 0.60.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

e(−)

β‖

/β⊥

0.2

0.3

0.4

0.5

0.1−0.6 −0.4 −0.2 0 0.2 0.4 0.6

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

e(−)

β‖

/β⊥

0.5

0.4

0.3

0.2

−0.6 −0.4 −0.2 0 0.2 0.4 0.60.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

e(−)

β‖

/β⊥

2

1.5

1.25

−0.6 −0.4 −0.2 0 0.2 0.4 0.60.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

e(−)

β‖

/β⊥

0.2

0.3

0.4

0.5

−0.6 −0.4 −0.2 0 0.2 0.4 0.60.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

e(−)

β‖

/β⊥

0.2

0.3

0.4

0.5

−0.6 −0.4 −0.2 0 0.2 0.4 0.60.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

e(−)

β‖

/β⊥

1.25

−0.6 −0.4 −0.2 0 0.2 0.4 0.60.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(a)

(b)

(c)

Figure 5.7: The iso-contour plots for radial and tangential swelling coeffi-cients (%strain/%moisture content) and the anisotropy ratio βT /βR as a function ofeccentricity parameter e and the cell wall swelling ratio β∥/β⊥ for β⊥ = 0.6, θ = 15 and(φ, η) equal to (a) (0.8,0.5), (b) (0.68,1.0), (c) (0.55,1.5). The inset in the upper left,middle and right of each sub-figure shows the schematic of antisymmetric, symmetricand mirrored honeycombs, respectively.

71


The results are presented in Figure 5.7 for three pairs of (φ, η) similarto those of the symmetric honeycombs at θ = 15. In radial direction,honeycombs are rather insensitive to the eccentricity and the effective swellingcoefficients are dependent only on the swelling properties of the cell wall.For low porosities, the radial swelling coefficients slightly increase at largeeccentricities. In contrast to what is seen in radial direction, the tangentialswelling is highly sensitive to eccentricity, specifically for low β∥/β⊥ ratios. Themaximum tangential swelling occurs for symmetric honeycomb (i.e. e = 0) and,surprisingly, both antisymmetric and mirrored honeycombs show very similaralthough not completely identical results. At low porosities, the influence ofeccentricity on the effective swelling properties decreases.

In general, the swelling anisotropy decreases with the increase of geometricaldisorder in the cellular structure. Consequently, the symmetric honeycombsshow the highest swelling anisotropy. This difference in behavior could beattributed to the hinge-like joints in symmetric honeycombs which providemore flexibility for the structure. For high eccentricity (e > 0.6), the alignmentof tangential walls of neighboring cells increases the rigidity of the middlejoints, thus significantly reducing the tangential swelling. The largest swellinganisotropy occurs in symmetric honeycombs which also have the smalleststiffness in tangential direction. Again we observe a different behavior forstiffness and swelling coefficients which according to Equation 3.70 showopposite trends. Our results show that the influence of eccentricity on swellinganisotropy is significantly diminished at lower porosities (φ ≃ 50%).

5.3.3 Hygro-elastic behavior of real cellular structure

Apparent swelling coefficients

Following the procedure described in section 5.2, finite element models offour different real cellular structures of wood are constructed. The apparentswelling coefficients of these four models as a function of porosity φ are shownin Figures 5.8a and 5.8b. For comparison, the results of symmetric andeccentric honeycombs are also provided at the same porosities, aspect ratiosand eccentricity as reported in Table 5.1. Since the results of antisymmetricand mirrored models are very similar, only the antisymmetric model resultsare included in these figures. At each porosity, three simulations are performedwhich correspond to the mean values of aspect ratio η and eccentricity e andplus/minus its standard deviation e±std(e). The capped lines show this range.

72


The corresponding swelling anisotropy is also shown in Figure 5.8c. It is foundthat, in all cases, the real structures show much less anisotropy in comparisonto the eccentric and symmetric honeycombs. Real structures swell more inradial direction than symmetric and eccentric honeycombs. On the other hand,the tangential swelling is smaller and consequently less anisotropy is observedfor real structures. The anisotropy in all models increases with porosity butanisotropy is much more apparent in ordered periodic structures than in realstructure models. This observation will be analyzed in more detail later. It isnoteworthy that the response of the eccentric honeycombs mostly falls betweenreal structures and symmetric models.

Apparent elastic moduli

The elastic moduli of the three models in radial and tangential directions, ER

and ET , are compared in Figure 5.9a and 5.9b. The radial and tangentialelastic moduli of periodic honeycombs show trends similar to those obtainedfor real structures. Similar to what was previously noted for swelling inradial direction, symmetric and eccentric honeycombs show very similar resultsfor the radial elastic moduli. The tangential elastic moduli of eccentrichoneycombs are closer to the results of real structure simulations. Thedegree of the stiffness anisotropy ER/ET found in Figure 5.9c shows that theanisotropy in elastic moduli increases with porosity. Real structures displayless anisotropy compared to symmetric and eccentric honeycombs. To betterunderstand the reduced swelling anisotropy observed in the real structure,the behavior of the earlywood sample with φ = 0.80 which is expected tohave the biggest swelling anisotropy is considered for further investigation.Figure 5.10a shows the contour of the free swelling induced internal stressesin the direction perpendicular to the cell wall σ⊥. A close-up view of atriple joint in the real microstructure is shown in Figure 5.10b. Results ofsymmetric and eccentric unit cells are also provided in Figure 5.10c and 5.10dfor comparison. In previous simulations, we observed that the swelling of thecell wall is more dominant in the direction which is normal to the cell wall andthe swelling along the cell wall is hindered due to the layered microstructureof the cell wall. We hypothesize that, in real structure model, there might besome internal restraining mechanism which prevents the structure to swellto the same extent as the periodic honeycombs. It can be seen that thereal structure model shows considerably higher compressive internal stressescompared to the symmetric and eccentric models. It is noteworthy that the

73


0.49 0.53 0.68 0.80

0.05

0.1

0.15

0.2

0.25

0.3

φ(−)

βR(%

/%)

real structuresymmetricantisymmetric

(a) radial

0.49 0.53 0.68 0.80

0.05

0.1

0.15

0.2

0.25

0.3

φ(−)

βT(%

/%)


(b) tangential

0.49 0.53 0.68 0.80

1

2

3

4

5

φ(−)

βT/βR(−

)


(c) anisotropy ratio

Figure 5.8: Swelling coefficients (%strain/%moisture content) in (a) radial and (b)tangential directions; and (c) βT /βR swelling anisotropy at different porosities φ forreal structure, symmetric and antisymmetric honeycombs for β⊥ = 0.6 and β∥/β⊥ = 0.1.The shape angle is assumed θ = 15 and the dimensions of the symmetric honeycmbsare (φ = 0.49, t/h = 0.364, ζ = l/h = 0.63), (φ = 0.53, t/h = 0.359, ζ = l/h = 0.70),(φ = 0.68, t/h = 0.237, ζ = l/h = 0.72) and (φ = 0.80, t/h = 0.153, ζ = l/h = 0.90). Thecapped lines show the range of modeling results for disordered honeycombs.

cell corners in real structures have round shapes and are thicker than theones of periodic models. Furthermore, the thickness of the cell wall in realstructures is not constant. These irregularities may have some influence onthe higher internal stresses observed in the simulated real structures. Thehighest stress concentrations are found at the centerline of the cell walls andin the corners throughout the domain. This compressive stress is generateddue to the mismatch between material orientations in adjacent cells. Thismismatch comes from the difference in the shape of the cells which leads to

74


0.49 0.53 0.68 0.80

0.5

1

1.5

2

2.5

3

φ(−)

ER(G

Pa)


(a) radial

0.49 0.53 0.68 0.80

0.5

1

1.5

2

2.5

3

φ(−)

ET(G

Pa)


(b) tangential

0.49 0.53 0.68 0.80

1

2

3

4

5

6

φ(−)

ER/E

T(−

)


(c) anisotropy ratio

Figure 5.9: Elastic moduli (GPa) in (a) radial and (b) tangential directions; and (c)ER/ET elastic anisotropy ratio at different porosities φ for real structure, symmetricand antisymmetric honeycombs. The shape angle is assumed θ = 15 and thedimensions of the symmetric honeycmbs are (φ = 0.49, t/h = 0.364, ζ = l/h = 0.63),(φ = 0.53, t/h = 0.359, ζ = l/h = 0.70), (φ = 0.68, t/h = 0.237, ζ = l/h = 0.72) and(φ = 0.80, t/h = 0.153, ζ = l/h = 0.90). The capped lines show the range of modelingresults for disordered honeycombs.

different material orientation in the place that two neighboring cells meet. Onthe other hand, the internal stresses in the cell wall of periodic configurationsare very small and the stress concentration is only at the triple connectionnotable. Therefore the internal stress which is generated inside the cell wallsmight be a possible reason for the reduced macroscopic swelling anisotropyof the real structure models. At the cellular scale, the irregular geometry ofthe cells may increase the hindered swelling at the cellular scale. Also, theboundary condition used for the analysis of the real structures (PMUBC) is

75


-15.0-12.5-10.0-8.5-7.0-5.5-4.0-2.0-0.80.0+8.0+2.0+4.0

(MPa)

(c) symmetric

(d) antisymmetric(a) real microstructure

(b) close-up

Figure 5.10: Contour of internal stresses induced in perpendicular direction σ⊥during free swelling for (a) the real microstructure of earlywood sample with φ = 0.80,(b) a close-up view of a triple joint, (c) symmetric and (d) antisymmetric periodichoneycombs. The stresses are reported for unit increment of moisture content.

not the same as for the periodic ones and with the increase of the irregularitiesin the real structure model the deviation from the periodic results increases.

Comparison with experiment

In this part, we compare the swelling coefficients of wood tissues at cellularscale with our simulations. The swelling strains are measured by high-resolution phase contrast synchrotron X-ray tomography experiment of isolatedsoftwood tissues exposed to varying relative humidity conditions and analyzedwith affine registration (Derome et al., 2011). Then, in a dynamic vaporsorption apparatus, the samples were subjected to the same relative humidityprotocol and the moisture content of the samples at each step of relativehumidity is obtained. The swelling coefficients are calculated from the slope ofthe linear regression of swelling strain versus moisture content. Figure 5.11aand 5.11b show the experimental and simulated swelling coefficients in radial(βR) and tangential (βT ) directions, respectively. The agreement is good,except for the sample with φ = 0.68 for which the results are not satisfactory. Inthis analysis, we do not take into account the mesoscopic features such as raysand sub-cellular units like bordered pits. Their possible influence on swelling

76


0.49 0.53 0.68 0.800

0.1

0.2

0.3

0.4

0.5

0.6

φ(−)

βR

(%/%

)

experimentmodel

(a) radial

0.49 0.53 0.68 0.800

0.1

0.2

0.3

0.4

0.5

0.6

φ(−)

βT

(%/%

)

experimentmodel

(b) tangential

Figure 5.11: Comparison of the experimental and modeling results for (a) radialswelling coefficient, βR and (b) tangential swelling coefficient, βT (%strain/%moisturecontent). The capped lines show the standard deviation for experimentaldata (Derome et al., 2011, 2012).

sample LW1 LW2 EW1 EW2φ 0.49 0.53 0.68 0.80β∥ 0.3 0.15 0.15 0.05β⊥ 0.70 0.70 0.80 0.80β∥/β⊥ 0.43 0.21 0.19 0.06

Table 5.4: Swelling coefficients of the cell wall (%strain/%moisture content) inparallel (∥) and normal (⊥) directions for different wood tissues with the porosity, φ,determined by FEM model of real structure.

behavior cannot be modeled with the current approach. The determinedswelling coefficients of the cell wall for all tissues are listed in Table 5.4. Theobtained results are suggesting that the cell walls swell to a much less extentalong the cell wall (∥) than normal to the thickness direction (⊥). This behavioris also reported in the rare experimental study on other softwood species atthe cell wall level (Ishimaru and Iida, 2001). Figure 5.12 shows the simulationresults of the internal stresses generated during free swelling in each sampleusing the values given in Table 5.4 as input for the swelling coefficients of thecell wall. As seen in the figure, the real structures show high compressive stressat the border and the corner of the cells.

77


-20.0-18.0-16.0-14.0-12.0-10.0-8.0-6.0-4.0-2.00.0

+2.0+4.0

(MPa)

Figure 5.12: Contour of internal stresses induced in perpendicular direction, σ⊥,during free swelling for a uniform moisture increment, ∆m = 1.0 for real structuremodels (a) earlywood, φ = 0.80, (b) latewood, φ = 0.49, (c) earlywood, φ = 0.68 and(d) latewood, φ = 0.53.

5.4 Discussion

In this chapter, we examined how the presence of eccentricity in periodichoneycombs influences the macroscopic swelling properties. A generic eccen-tricity parameter is introduced in symmetric honeycombs along the tangentialdirection. It seems that alignment of micro-structures in radial directionsreduce the radial swelling. In symmetric honeycombs, the tangential swellingis larger than in the eccentric ones. This difference in behavior could bedue to the presence of hinge-like joints which provide more flexibility tothe structure. At high eccentricity (e > 0.6), the alignment of tangentialwalls of neighboring cells increases the rigidity of the middle joints, thussignificantly reducing the tangential swelling. The largest swelling anisotropyoccurs in symmetric honeycombs which also have the smallest stiffness intangential direction. Our results show that the influence of eccentricity onswelling anisotropy is significantly diminished at lower porosities (φ ≃ 50%). Asurprising similarity is observed in the swelling map of the antisymmetric andthe mirrored honeycombs with the same eccentricity parameter which suggeststhat the eccentricity plays an influential role in swelling anisotropy of periodiccellular solids.

The results of periodic honeycombs are compared with the real cellularstructures of wood. The anisotropy in swelling properties of real structuremodels is less pronounced. It can be related to the high degree of geometricaldisorder in the real cellular structure of wood such as the non-uniformity in thecell wall thicknesses, size and shape of the cells and thicker cell corners. As a

78

5.4. DISCUSSION

results, the anisotropy ratio of the swelling coefficients and the elastic modulideceases in real structure models. Comparing periodic honeycombs with realstructures shows that, in radial direction, the elastic moduli of real structuresare slightly larger than periodic models while the difference in tangentialdirection is more notable. Furthermore, in the radial direction, the swellingcoefficient of real structures is larger than that of periodic structures. It shouldbe mentioned that in wood, the radial swelling may also be restrained by therays, a cellular feature which is not included in the present model. Furthermore,the analysis of internal stresses reveals that the periodic idealization of cellularsolids underestimates the influence of internal stresses which leads to anoverestimation of the swelling anisotropy.

It is noteworthy that the key factor in macroscopic anisotropic swellingbehavior of cellular (porous) solids is the anisotropy of the cell wall material.However, as demonstrated in the previous chapter, the perfect regularhoneycomb (η = 1, θ = 30) is a special case which exhibits transversely isotropicmechanical and swelling behavior irrespective of the anisotropy of the cellwall material. The specific geometry of the cellular materials (e.g. smallshape angles) may result in anisotropic elastic stiffness even if the cell wallmaterial is isotropic. But, regarding the swelling behavior, the situation isdifferent and the anisotropy of the cell wall is the necessary condition forobserving macroscopic anisotropic behavior. A unit area of material mayexpand anisotropically because of two reasons: it may be acted on by forceswhose distribution is anisotropic, or its ability to resist force may be anisotropic(Baskin, 2005). The swelling forces acting on the cell wall originate fromswelling pressure, which is hydrostatic and thus isotropic. Wood cell walls arereinforced anisotropically by cellulose microfibrils in a multilayer order whichprovides anisotropic resistance to swelling forces. This behavior results in adifference between the swelling coefficient of the cell wall which is parallel tothe cell wall and that which is normal to the cell wall or a ratio β∥/β⊥ < 1. Aspresented in the previous chapter, a main reason that β∥ is different from β⊥is the corset action of the S1 and S3 layers. Finally, we compared the resultsof our simulation with free swelling experiments. The swelling coefficients ofthe cell wall are adjusted to achieve the best match. This comparison revealsthat, in wood, the swelling ratio of the cell wall β∥/β⊥ decreases with porosityas reported in Table 5.4. Consequently, for thin cell walls, the ratio β∥/β⊥ issmaller than for thick cell walls. One possible explanation for this behavior isthe microfibril angle (MFA) of S2 layer which has been found to be consistentlysmaller in latewood cell walls than in those of earlywood (Barnett and Bonham,

79


2004), thus yielding a higher anisotropy ratio in thinner cell walls. Consideringthe mechanical and swelling behavior of different arrangements of periodic andnon-periodic cellular solids, it can be seen that elastic and swelling propertiesare strongly correlated.

5.5 Summary

In summary, the proposed models of symmetric and eccentric honeycombsprovide some bounds for the swelling anisotropy in periodic irregular honey-combs. The symmetric honeycombs show more swelling in tangential directionin comparison to eccentric models. In radial direction, the cell walls in allmodels are well aligned which leads to a less sensitive swelling behavior. Wefound that the swelling anisotropy in periodic symmetric honeycombs can belarger for low porosities while the presence of eccentricity in the structure maysignificantly decrease the swelling anisotropy. The results provided in this workcan be used for understanding the behavior of cellular biomaterials in responseto environmental stimuli. However, a coupling of the moisture and elasticbehavior may be useful for an even further understanding of such process.Such coupling is looked at in the next chapter.

80

Chapter 6

Poromechanics model forswelling of cellular solids

In the two previous chapters, we studied the swelling behavior of cellularmaterials using an uncoupled hygro-elastic approach. However, swelling whichoriginates from the moisture/material interactions at the lower scales, isstrongly coupled to the mechanical properties. Thus, we now complete ourstudy at the cellular scale by investigating these interactions. Poromechanicsis a fully coupled approach as already presented in section 3.4 and is used togo further in understanding the role of the geometry and material anisotropyon swelling behavior of cellular materials.

6.1 Moisture induced swelling in softwood

We apply a poromechanical framework to study the moisture-induced swellingof softwood, originating from the sub-cellular scale and influenced by thehierarchical structure of the material. As shown in Fig. 6.1, in wood, twomain groups of porosity can be distinguished: macropores and micropores.The macropores consist of cell lumens, large pores of typical diameter of afew to hundred microns. Within the lumens, water is present in liquid orvapor phase. Water in lumens does not participate in the moisture induceddeformation in wood (Bazant, 1985; Skaar, 1988). The second group of poresin wood are micropores which are found in the porous cell walls and range insize from 5nm down to the level of single water molecules bound to hydrophilic

81

CHAPTER 6. PROMECHANICS MODELING OF SWELLING BEHAVIOR

wood cellscell wall

lumen

micropore

θ

t

l

h

honeycomb RVE

L

T

R

latewood

earlywood

Figure 6.1: Center, schematic representation of wood cells, left, micropores withinthe cell wall and right, examples of earlywood and latewood geometries. Thehoneycomb unit cell and the geometrical parameters are shown in this figure.

sites of the wood polymers (Thygesen et al., 2010). In wood, sorption of watermolecules in the hygroscopic range occurs in the cell wall. Water is a polarmolecule that binds with hydrogen bonds to the hydroxyl sites of the polarmolecules. Except for crystalline cellulose, all wood polymers demonstrate,to different extent, an affinity for water. In the dry state, the cell wall hasa low porosity where water molecules can nevertheless find available sorptionsites. For further sorption to take place, i.e. above 2-3% moisture content, thecreation of new porosity by the displacement of molecules is required. Thus,sorption of water molecules in between the hydrophilic molecules pushes theconstituents apart, resulting in swelling (Derome et al., 2012). Therefore, thecell walls can be considered to be always fully saturated with water (Maloneyand Paulapuro, 1999). The apparent initial porosity of the cell wall φ0 can thenbe linked directly to the moisture content u. In wood science, moisture contentis defined as the mass of water in wood sample, over the mass of oven-driedsample:

u =mwet −mdry

mdry=

ρfφ0

ρs(1 − φ0)(6.1)

where mwet = mdry + ρrρfφ0Ω and mdry = ρrρs(1 − φ0)Ω are the masses ofthe sample in wet and dry conditions, respectively. The density of the solidphase and the fluid (water) are denoted as ρs and ρf , respectively. From thisrelation, the apparent initial porosity of the cell wall can be calculated as:

φ0 =ρsu

ρf + ρsu(6.2)

82


cell wall E1 E2 E3 ν12 ν13 ν23 G12 G13 G23

earlywood 4.36 7.02 33.20 0.403 0.023 0.112 1.18 1.65 4.38latewood 4.77 6.43 43.00 0.416 0.022 0.067 1.22 2.12 3.50

Table 6.1: Orthotropic elastic properties of the cell wall at u = 0.12 (Persson, 2000).All moduli are in GPa. The subscript indices 1, 2 and 3 are equivalent to normal (⊥),parallel (∥) and longitudinal (L) directions, respectively.

The moisture dependency of elastic properties of the cell wall is estimatedfrom the micromechanical analysis of the cell wall constituents. Table 6.1shows the elastic properties of the cell wall for earlywood and latewood tissuesat u = 0.12. The elastic properties of the cell wall in transverse plane (normal⊥ and parallel ∥ to the cell wall), longitudinal direction (L) and the shearmodulus in transverse plane are normalized to the elastic properties at u = 0.12and are shown as a function of moisture content in Figure 6.2. These valuesare obtained from the dependency of elastic properties of the S2 layer in thecell wall on moisture content (Qing and Mishnaevsky, 2009c). We observethat the influence of moisture on the transverse elastic moduli is greater thanthe shear modulus, and also we can see that the moisture has less impacton the longitudinal elastic modulus. The above material properties and theirdependency on moisture content are used as inputs in the following simulations.

6.2 Numerical results

The proposed computational poromechanical approach as presented in Sec-tion 3.4 has been implemented in the finite element package ABAQUS (RisingSun Mills, Providence, RI, USA) and used to predict the swelling behavior ofcellular tissues of softwood.

The growth rings of softwood can be considered bi-layered with almost 80%low density (called earlywood) and 20% high density (called latewood) cells.Thus, two honeycomb configurations are considered representing earlywoodand latewood tissues with relative density ρr = 0.19 and ρr = 0.44, respectively.The density of the cell wall material and water are ρs = 1500 kg/m3 andρf = 1000 kg/m3, respectively. The shape angle for both configurations isθ = 10. The aspect ratio of cells which is defined by Equation (5.1) is η = 0.8for earlywood and η = 1.5 for latewood. The elastic properties of the cell wallare assumed to be moisture dependent as explained in Sec. 6.1 and as shown

83


0 . 0 0 0 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0 0 . 2 50 . 4

0 . 6

0 . 8

1 . 0

1 . 2

1 . 4

1 . 6

norm

alized

stiffn

ess (

-)

m o i s t u r e c o n t e n t , u ( - )

t r a n s v e r s e ( ⊥, ||) l o n g i t u d i n a l ( L ) s h e a r

Figure 6.2: Normalized stiffness of the cell wall respect to the elastic propertiesat u=0.12 as a function of moisture content obtained from homogenized materialproperties of the S2 cell wall layer (Qing and Mishnaevsky, 2009c).

in Figure 6.2. The bulk modulus of the cell wall material is assumed to be notmoisture-dependent and is selected to be Ks = 80 GPa. Water is consideredto be incompressible (Kf = ∞). Equations (3.46) and (3.47) are used todetermine the Biot coefficient and the Biot modulus of the cell wall. These twoparameters depend on the bulk modulus of the cell wall, porosity and stiffnessof the cell wall. In all cases, the porosity of the cell wall is defined accordingto Equation (6.2) as a function of moisture content. Since the longitudinaldimensions of the wood cells are very large in comparison to the dimensions inradial and tangential directions, the problem is reduced to the two dimensionalanalysis of the cross-section of the material which is in a state of generalizedplane strain. In the next sections, first the effective poroelastic propertiesof honeycomb unit cells are computed. The effective swelling coefficientsare calculated based on the calculated poroelastic properties and then theinfluence of the shape angle on poroelastic properties and swelling behavior isinvestigated. At the end, the swelling properties which are predicted by theporoelastic approach are compared to the hygroelastic model which is explainedin the previous two chapters.

84


6.2.1 Transverse tangent poroelastic properties

The effective Biot coefficients of honeycombs in radial (bR = bM11) andtangential (bT = bM22) directions are computed and plotted against the moisturecontent in Fig. 6.3. As shown in this figure, the Biot coefficients in radial andtangential directions are almost constant and they are only slightly influencedby the moisture dependency of elastic properties. Earlywood exhibits ananisotropic behavior and its Biot coefficient is notably larger in radial directionthan tangential direction. On the other hand, in latewood, the Biot coefficientsin radial and tangential directions are very close to each other. The Biotcoefficients in latewood are larger than ones in earlywood indicating that if thehoneycomb unit cell is constrained and the cell walls are pressurized by thefluid water inside them, more pressure is transmitted from the fluid within thecell wall to the boundaries of the unit cell in latewood in comparison to whatoccurs in earlywood. This can be explained by the fact that, more bulky wallswill lead to higher stresses when the porosity is pressurized. Furthermore,in earlywood, the pressure transferred from the water within the cell wallsonto the radial boundaries of a constrained honeycomb unit cell is largerthan onto the tangential ones. In comparison to other porous materials suchas sandstones and rocks where b = 0.6 − 0.8, the Biot coefficients predictedfor softwood are much smaller. The reason can be due to the presence ofmacropores (lumens) which do not contribute in transmission of the pressure.

The effective Biot modulus MM for earlywood and latewood honeycombs areillustrated in Fig. 6.4. The Biot modulus can be interpreted as the inverseof the moisture capacity of the material. As can be seen in this figure, thedependence of elastic properties on moisture content has an important impacton the effective Biot modulus. The Biot modulus shows a similar trend towhat is seen in the elastic modulus. It exhibits a maximum at low moisturecontent and afterwards it decreases smoothly due to stiffness softening. TheBiot modulus of latewood is smaller than earlywood; this relationship is due tothe more bulky walls of the latewood and its higher moisture storage capacity.Due to the softening of the solid material, moisture can easily enter the porosityat higher moisture contents.

The effective elastic moduli of earlywood and latewood cells in radial (ER =

1/SM1111) and tangential (ET = 1/SM

2222) directions as a function of moisturecontent are presented in Fig. 6.5. The macroscopic elastic properties are highlydependent on moisture content. For earlywood, the elastic modulus in radialdirection is much larger than the tangential one. The elastic properties of

85


0 . 0 0 . 1 0 . 2 0 . 30 . 0 0

0 . 0 5

0 . 1 0

0 . 1 5

0 . 2 0

T

R

T

R

TRe a r l y w o o d

Rl a t e w o o d

Biot c

oeffic

ients,

b R and

b T (-)


T

Figure 6.3: Effective Biot coefficients of honeycombs in radial and tangentialdirections for earlywood with ρr = 0.19, η = 0.8, θ = 10 and latewood with ρr = 0.44,η = 1.5, θ = 10.

0 . 0 0 . 1 0 . 2 0 . 31 0

2 0

3 0

4 0

5 0

6 0

l a t e w o o d

e a r l y w o o d

Biot m

odulu

s, M M

(GPa

)


Figure 6.4: Effective Biot modulus of honeycombs for earlywood with ρr = 0.19,η = 0.8, θ = 10 and latewood with ρr = 0.44, η = 1.5, θ = 10

86


0 . 0 0 . 1 0 . 2 0 . 30 . 0

0 . 5

1 . 0

1 . 5

2 . 0

T

R

T

R

T

T l a t e w o o d

e a r l y w o o d

elastic

mod

ulus,

E R and

E T (GPa

)


R

R

Figure 6.5: Effective elastic modulus of earlywood with ρr = 0.19, η = 0.8, θ = 10

and latewood with ρr = 0.44, η = 1.5, θ = 10 in radial and tangential directions.

thick walled latewood cells are notably larger than earlywood. The anisotropyof elastic properties in latewood is very small and the elastic moduli in radialand tangential directions are close to each other. The variation of the tangentialelastic modulus of earlywood with moisture content is not distinguishable inthis figure because it is much smaller in magnitude. The above poroelasticproperties complement the prediction of macroscopic swelling coefficients basedon the poromechanics model as formulated by Equation (3.70) and will bediscussed in the next section.

6.2.2 Swelling coefficients

Once the poroelastic properties, CMijkl (S

Mijkl), b

Mij and MM are obtained, we

can calculate the effective swelling coefficients of honeycomb structures usingEquation (3.70). In wood science, the swelling coefficients are defined basedon the definition of moisture content according to Equation (6.1):

εswij = β∗iju (6.3)

where β∗ij can be expressed in terms of βMij calculated from Equation (3.70):

87


0 . 0 0 . 1 0 . 2 0 . 30 . 0

0 . 2

0 . 4

0 . 6

0 . 8 T

R

T

R

T l a t e w o o d

T

e a r l y w o o d

swelli

ng co

efficie

nts,

R and

T (-)


R

R

Figure 6.6: Effective swelling coefficients of earlywood with ρr = 0.19, η = 0.8, θ = 10

and latewood with ρr = 0.44, η = 1.5, θ = 10 in radial and tangential directions.

β∗ij = βMij

ρsρf

(1 − φ0) (6.4)

Fig. 6.6 shows the effective swelling coefficients of earlywood and latewoodin radial (βR = β∗11) and tangential (βT = β∗22) directions as a function ofmoisture content. It is found that the swelling coefficients in tangentialdirection are larger than in radial direction and the T/R swelling ratio isconsiderably higher in earlywood. This conclusion is in line with the swellingmodel based on the hygroelasticity which is presented in Chapter 4 but,as mentioned before, the results presented here stem from the poroelasticbehavior of the cell wall. In this model, the swelling coefficients of the cellwall material are not inserted as an input to the model and the anisotropy inmacroscopic swelling behavior results from the poroelastic interactions withinthe cell wall. The macroscopic swelling coefficients are almost constant and notstrongly dependent on moisture content while the poroelastic properties such asstiffness and Biot modulus are highly moisture dependent. Consequently, theswelling strain is a linear function of moisture content which is also observed inextensive experimental studies (Skaar, 1988). This poroelastic behavior can beexplained considering the moisture dependent behavior of effective complianceand Biot modulus. The compliance increases with moisture content while the

88


Biot modulus decreases and they both compensate the influence of each otheraccording to Equation (3.70). Furthermore, it is notable that, based on ourresults, the poroelastic model is capable to predict the degree of anisotropy inswelling behavior of wood cells at tissue level. Previous experimental resultson swelling behavior of wood also show similar trends (Derome et al., 2011).The earlywood cells show more anisotropy in swelling behavior while thethick walled latewood cells swell almost isotropically. It is noteworthy thatthe honeycomb configuration considerably affects the macroscopic swellingproperties. As a result, the swelling behavior of low density earlywood andthick-walled latewood cells is different. In the next section, the effect of thegeometry of earlywood and latewood cells on the macroscopic properties isinvestigated by focusing on the influence of the shape angle.

6.2.3 Shape angle

A parametric study is carried out to investigate the influence of the shapeangle of the honeycomb unit cells on the effective poroelastic properties and theswelling coefficients at two densities which are corresponding to earlywood andlatewood cells. Figure 6.7 shows the effective Biot coefficients of honeycombs inradial (bR) and tangential (bT ) directions for two relative densities representingearlywood (ρr = 0.19) and latewood (ρr = 0.44) as a function of the shape angleθ. The aspect ratio for both unit cells is η = 1.0 and the results are presentedat moisture content u = 0.12. The effective Biot modulus of earlywood andlatewood honeycombs MM is shown in Fig. 6.8.

The effective radial (ER) and tangential (ET ) elastic moduli are shown inFig 6.9. The effective swelling coefficients in radial (βR) and tangential (βT )directions are presented in Fig 6.10. It is found that the anisotropy in Biotcoefficients, elastic moduli and swelling coefficients is more pronounced at smallshape angles, while increasing the shape angle results in isotropic behavior atθ = 30 in the plane. By contrast, the effective Biot moduli are found tobe almost independent on the shape angle. Fig. 6.7 shows that bT is moresensitive to shape angle than bR. Similar behavior is also predicted for swellingcoefficients. As shown in Fig. 6.10, βT is more sensitive to shape angle thanβR. This may be explained by the fact that the sensitivity of elastic moduli onshape angle is also more pronounced in radial direction. Fig. 6.9 shows thatER decreases with shape angle while ET is almost constant. This behavior canbe interpreted by the deformation of honeycombs. For small shape angles, i.e.θ → 0, the inclined walls (see figure 6.1) are aligned and they can bear the

89


0 5 1 0 1 5 2 0 2 5 3 00 . 0 0

0 . 0 5

0 . 1 0

0 . 1 5

0 . 2 0T

R

T

R

l a t e w o o d

e a r l y w o o d

Biot c

oeffic

ients,

b R and

b T (-)

s h a p e a n g l e , ( °)

T

R

R

T

Figure 6.7: Effective Biot coefficients of earlywood (ρr = 0.19) and latewood (ρr =0.44) honeycombs unit cells in radial and tangential directions with η = 1.0 for differentshape angles θ at moisture content u = 0.12.

0 5 1 0 1 5 2 0 2 5 3 00

1 0

2 0

3 0

4 0

5 0

6 0

l a t e w o o d

e a r l y w o o d

Biot m

odulu

s, M

(GPa

)

s h a p e a n g l e , ( °)

Figure 6.8: Effective Biot modulus of earlywood (ρr = 0.19) and latewood (ρr = 0.44)honeycombs unit cells with η = 1.0 for different shape angles, θ at moisture contentu = 0.12.

90


0 1 0 2 0 3 00 . 0

0 . 5

1 . 0

1 . 5

2 . 0

2 . 5T

R

T

R

Rl a t e w o o d

e a r l y w o o d

elastic

mod

ulus,

E R and

E T (GPa

)

s h a p e a n g l e , ( °)

TR

T

Figure 6.9: Effective elastic moduli of earlywood (ρr = 0.19) and latewood (ρr = 0.44)honeycombs in radial and tangential directions with η = 1.0 for different shape angles,θ at moisture content u = 0.12.

load axially. For large shape angles, i.e. θ → 30, the deformation mechanismis in flexural mode since the bending stiffness is smaller than the axial stiffness.

6.2.4 Comparing poroelasticity and hygroelasticity

In this section the two approaches which are presented in chapter 3 forprediction of swelling behavior of cellular materials are compared. For thispurpose, the earlywood and latewood unit cells which are used in section 6.2.1are considered. The effective swelling coefficients are calculated at moisturecontent u = 0.12. The parameters of the equivalent hygroelastic model areconstructed based on the poroelastic model. First, the Biot coefficients andthe Biot modulus of a general anisotropic porous medium are calculated fromEq. (3.47) and the results are inserted into Eq. (3.45) to find the effectiveswelling coefficients of the cell wall. Finally, the effective swelling coefficientof the two unit cells are calculated from the hygroelastic model according tothe procedure which is described in section 3.3.2. Figure 6.11 compares theeffective swelling coefficients of earlywood and latewood unit cells in radialand tangential directions which are estimated according to hygroelastic and

91


0 1 0 2 0 3 00 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0T

R

T

R

RT

T

swelli

ng co

efficie

nts,

R and

T (-)

s h a p e a n g l e , ( °)

e a r l y w o o d

l a t e w o o d

R

Figure 6.10: Effective swelling coefficients of earlywood (ρr = 0.19) and latewood(ρr = 0.44) honeycombs in radial and tangential directions with η = 1.0 for differentshape angles, θ at u = 0.12.

poroelastic models. Both models predict similar results for swelling coefficientsof earlywood and latewood cells in radial and tangential directions. The slightdifferences between these two approaches can be attributed to the differentloading conditions which are used for the estimation of the effective swellingbehavior. In the hygroelastic model, it is assumed that the moisture contentis uniform while in the poroelastic model the pore pressure is consideredto be uniform through the whole unit cell. As mentioned before, in thehygroelastic model the swelling behavior and the elastic properties at the cellwall are independent. On the other hand, the poromechanics approach linksthe swelling behavior to the elastic properties of the cell wall which enables usto have an educated guess about the swelling coefficients of the cell wall. Thiscomparison shows that the hygroelastic model can still predict the effectiveswelling behavior as well as the poroelastic model if we have good estimate forthe effective swelling coefficients of the cell wall. With this simple example weshowed, beside comparison with experimental results at different hierarchicallevels which will be discussed in the next chapter, that the poromechanicsapproach can be used as an alternative tool for the prediction of the swellingbehavior of cellular materials.

92

6.3. SUMMARY

L W ( β R ) L W ( β T ) E W ( β R ) E W ( β T )0 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

swelli

ng co

efficie

nts (%

/%) P o r o e l a s t i c

H y g r o e l a s t i c

Figure 6.11: Comparison between the estimation of the poroelastic and thehygroelastic models for the effective swelling coefficients (% strain/ % moisturecontent) of earlywood (ρr = 0.19) and latewood (ρr = 0.44) honeycombs in radialand tangential directions with η = 1.0 and θ = 10 at moisture content u = 0.12.

6.3 Summary

In this chapter, the swelling anisotropy in cellular materials is investigatedbased on a poromechanics approach. The general expression for the macro-scopically observable free swelling strain is derived by upscaling the constitutiveequations of a microscopic poroelastic medium. The poroelastic properties andswelling coefficients of the representative earlywood and latewood unit cellsare calculated and the evolution of anisotropy in poromechanical propertiesis investigated. The proposed model predicts the general trends accuratelyand gives a physical understanding of swelling phenomena which originatesfrom the interaction of the micro level pressure induced by moisture withinthe anisotropic porous cell walls. The anisotropy in swelling and mechanicalbehavior of earlywood cells is properly captured with this model. Based onthe proposed formulation, the increase of the compliance with moisture contentis compensated by the decrease of Biot modulus and as a result the swellingcoefficients are not dependent on the moisture content. Consequently, theswelling strain has an almost linear relationship with moisture content whichis widely observed in free swelling experiments. The regular honeycombs show

93


isotropic behavior in the plane and the anisotropy increases as the shape angledecreases. The proposed model provides a poromechanics description of theswelling phenomenon in porous materials which links the macroscopic swellingbehavior to the poroelastic properties and to the morphology of the cellularsolids.

94

Chapter 7

Multi-scale modeling ofhierarchical cellular materials

In this chapter, we study the hygro-mechanical behavior of hierarchical cellularmaterials using a multi-scale framework to bridge the gap from cellularmicrostructure to macroscopic scale. Many biological tissues, such as woodand bone, show a hierarchical structure which enhances their mechanicalbehavior. This enhancement in mechanical properties are believed to be dueto a functional adaptation of the structure at all levels of hierarchy (Fratzl andWeinkamer, 2007). In the previous chapters, the anisotropic swelling behaviorof cellular materials is investigated and the respective role of anisotropy ofthe cell wall material and the shape of the cells is studied. The complicatedarchitecture of wood, as explained in Section 2.1, introduces a strong geometricanisotropy which is also reflected in the anisotropy of its mechanical andswelling behavior at larger scales. A better understanding and predictability ofthe interactions between the mechanical and moisture behavior of wood acrossthe scales is needed, for example, to assess the durability of wood elementsexposed to varying mechanical and environmental loading.

7.1 Hierarchical multiscale model

In this section, we use a hierarchical multiscale model based on a computationalupscaling technique with application to the hygro-mechanical behavior of thegrowth rings in softwoods. The multiscale method used here is an uncoupled

95

CHAPTER 7. HIERARCHICAL CELLULAR MATERIALS

hygro-elastic model and the solution of the microscopic and the macroscopicproblems are carried out separately. In the rest of this section, a linearmacroscopic constitutive behavior according to relation (3.34) is assumed andthe macroscopic material properties are calculated from the computationalupscaling of the underlying cellular structure.

7.1.1 Hygro-elastic behavior of a growth ring

We investigate the distribution of transverse anisotropy in the swelling andmechanical behavior of softwood from the cellular scale to the growth ringlevel by means of a two-scale finite element-based model1. The growth ringbehavior is considered here as the macro-scale. The mechanical field at themacroscopic level (growth ring) is resolved through the incorporation of themicro-structural (cellular structure) response by the computational upscalingof different unit cells which represent the cellular structure of wood. Thegeometrical parameters of the unit cells are selected from a morphologicalanalysis of wood at the cellular scale. The proposed multiscale frameworkaims at achieving a more realistic characterization of the anisotropic swellingand mechanical behavior of wood and, in particular, at describing the latewoodand earlywood interaction caused by the material heterogeneity on the cellularscale. We consider a problem of infinitesimal small deformations with moistureinduced swelling. The growth rings in softwood are analyzed as a cellularporous solid within the framework of the computational upscaling. Since thelongitudinal dimensions of the wood cells are very long in comparison to thedimensions in radial and tangential directions, the problem can be reduced tothe analysis of a two dimensional cross-section of the material, which is in astate of generalized plane strain. In the regions far away from the center ofthe stem of the tree, growth rings are periodically arranged in radial directionwhich justifies an assumption on global periodicity of the growth rings. Atthe micro-scale, we assume local periodicity where, although wood cells havedifferent morphologies corresponding to different relative ring positions, eachcell repeats itself in its vicinity. The concept of local and global periodicityin softwood and the proposed two-scale upscaling scheme are illustrated inFigure 7.1.

At the microscopic scale, the underlying cellular structure of softwood isapproximated as honeycomb unit cells. The dimensions of the wood cells werecharacterized using a scanning electron microscopy (SEM) image of a single


96

7.1. HIERARCHICAL MULTISCALE MODEL

Γl

Γ

Γ

Γ

r

t

b

R

T

L

Microstructure (Honeycomb RVEs)

Macrostructure (Growth Ring)

Figure 7.1: Two-scale upscaling scheme and underlying RVEs corresponding todifferent elements.

growth ring of Norway spruce (Picea abies) on 10 continuous cell rows in theradial direction (Derome et al., 2012). A parametric function is fitted to eachdataset to define the geometry of the cellular structure across the growth ring.The lumen diameters, LR and LT , and the cell wall thickness, tR and tT , alongthe relative ring position and the corresponding fitted functions are presentedin Figures 7.2a and 7.2b respectively.

relative ring position(-)

lum

en d

iam

ete

r (

m)

0 0.2 0.4 0.6 0.8 10

20

40

60

80

LR

LT

(a)

0 0.2 0.4 0.6 0.8 10

2

4

6

8

t

t


cell

wall

thic

kness (

m

)

T

R

(b)

Figure 7.2: (a) Lumen diameter and (b) cell wall thickness in radial and tangentialdirections as a function of the relative ring position.

97


R

ΓrΓ

l

Γt

Γb

h

l

2tR

2tT

θ

3

2

1

T

L

-s~

3

3

+s~

3

-s~

1

2

1

-s~

2

+s~

2

+s~

2

(a) (b)

LT

LR

Figure 7.3: Honeycomb unit cell (a) geometrical parameters (b) discretized quarterof a honeycomb unit cell.

The geometrical parameters of the honeycomb unit cells, as shown in 7.3a, arethe length of the vertical wall h, the length of the inclined wall l and the shapeangle θ. These parameters are determined from the dimensions of the woodcell in combination with a shape angle. The relation between the measuredlumen diameters and the geometrical parameters of the honeycomb unit cellcan be expressed as:

LR = 2l cos θ − 2tR, LT = h + 2l sin θ − 2tT sec θ (7.1)

The cell wall properties are assumed to be orthotropic and the three orthogonalmaterial directions, as shown in Figure 7.3a, are oriented normal to the plane(1-axis), along the cell wall (2-axis) and normal to the cell wall thickness (3-axis). The material properties of the cell wall for earlywood and latewoodare provided in Table 7.1. The elastic properties are chosen from Persson(2000) and the swelling coefficients are selected from Nakato (1958). Thesevalues should be considered as the effective material properties of the wholecell wall. Material properties are the one of cell material at 12% moisturecontent. We quantify the shape angle parameter to range linearly from θ = 5

in earlywood to θ = 10 in transition zone and then it increases linearly upto θ = 20 in latewood. The density of each cell can be calculated from thevolume fraction of the cell wall in a unit cell multiplied by the cell wall density1460 kg/m3 (Derome et al., 2012).

98


× ×

××

Gauss point Finite element

Figure 7.4: Macroscopic finite element mesh for growth ring simulations.

Upscaling procedure

The two-scale hygro-mechanical framework has been implemented in the finiteelement package ABAQUS using user-defined material routines (ABAQUSInc., Providence, RI) and its Python scripting interface. According to thedistribution of earlywood and latewood in a specific growth ring, the macrodomain is discretized. Since the variation in geometrical parameters intangential direction is negligible, the macro domain is meshed only in radialdirection using standard four-node elements with finer resolution for latewood.Mesh sensitivity analysis is carried out to reach convergence. The finalmacroscopic mesh is shown in Figure 7.4. Here, the upscaling procedure iscarried out in two steps. First, at each integration point, based on the micro-structural geometrical distribution of wood cells along the relative growth ringposition, a honeycomb unit cell is generated. The reduced stiffness matrixof the unit cells at the retained master nodes is extracted and is used tocalculate the macroscopic stiffness tensor following the procedure explainedin Section 3.3. Then the swelling coefficients are calculated applying a uniformmoisture increment to a constrained unit cell, using Equation 3.40. Thisprocedure is explained in Section 3.3.2 in more details. Finally, the storedeffective properties are transferred to the macro-problem.

Table 7.1: Elastic properties of the cell wall (in GPa) from micromechanicalmodel (Persson, 2000) and swelling coefficients (% strain/%moisture content) fromexperiments (Nakato, 1958).

Material E11 E22 E33 G12 G13 G23 ν21 ν31 ν32Earlywood 33.20 7.02 4.36 4.38 1.65 1.18 0.112 0.023 0.403Latewood 43.00 6.43 4.77 3.50 2.12 1.22 0.067 0.022 0.416

β22 β33Cell wall Radial wall Tangential wall Radial wall Tangential wallEarlywood 0.10 0.15 0.45 0.45Latewood 0.40 0.35 0.50 0.60

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Numerical results

To better understand the swelling and mechanical behavior of softwoods,different parametric studies at the cellular scale are performed. Then, thetwo-scale model is utilized to study the behavior of a single growth ring. Thesimulated results are compared to experiments at both scales.

Unit cell We investigate the effects of the unit cell geometry and of the cellwall material anisotropy on the elastic and swelling properties of individualwood cells. We assigned earlywood and latewood material properties to thestart and end of the relative ring position respectively. Within the growth ringthe material properties are assumed to be linearly dependent on the cell wallthickness, due to the increasing thickness of the S2 layer. The effective swellingcoefficients in tangential (T) and radial (R) directions and the respective T/Ranisotropy ratios are shown in Figure 7.5 and compared with microscopicobservations of transverse swelling of isolated earlywood and latewood samplesas reported in Derome et al. (2011). The tangential swelling coefficients aregenerally greater than the radial ones, except in the region close to the end ofthe growth ring as presented in Figure 7.5a.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5


sw

elli

ng c

oeffic

ients

(%/%

)

tangential FEM

radial FEMtangential EXPradial EXP

(a)

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5


T/R

sw

elli

ng a

nis

otr

opy r

atio

FEM

EXP

(b)

Figure 7.5: (a) The homogenized tangential and radial swelling coefficients forhoneycomb unit cells and (b) the respective T/R swelling ratios and experimentalresults for Norway spruce (Derome et al., 2011).

The swelling coefficients of latewood cells are greater than of earlywood ones.The increase of swelling coefficients in radial direction is more significant thanin tangential direction. As shown in Figure 7.5b, the T/R swelling anisotropy

100


ratio in latewood is smaller than in earlywood and swelling in latewood isalmost isotropic, i.e. the T/R ratio equals to one. The swelling of earlywoodcells is on the contrary clearly anisotropic, with swelling ratios up to 4. Thisobservation is remarkable since the cell wall elastic properties of earlywood andlatewood cells do not differ significantly which means that the main influencecomes from the geometrical cellular parameters and the swelling coefficientsof the cell wall. We observe a good agreement between simulations andexperiments for both earlywood and latewood tissues. Figure 7.6a gives theelastic modulus in tangential and radial directions along the growth ring andexperiments of micro-tensile tests in the transverse plane of earlywood andlatewood parts of spruce (Farruggia and Perre, 2000). The elastic modulusin earlywood is very low and increases strongly in the latewood region. T/Relastic anisotropy ratios of honeycomb unit cells are depicted in Figure 7.6b.

0 0.2 0.4 0.6 0.8 110

-2

10-1

100

101


ela

stic m

odulu

s (

GP

a)

tangential FEM

radial FEM

tangential EXP

radial EXP

(a)

0 0.2 0.4 0.6 0.8 10

0.4

0.8

1.2

1.6

2


T/R

ela

stic a

nis

otr

opy ra

tio

FEM

EXP

(b)

Figure 7.6: (a) The homogenized tangential and radial elastic moduli for honeycombunit cells and (b) the respective T/R elastic ratios and experimental results forspruce (Farruggia and Perre, 2000).

In the earlywood region, the modulus of elasticity in radial direction is greaterthan in the tangential one, while the opposite is obtained in the latewoodregion. This observation can be explained considering the change of thegeometry of earlywood and latewood cells over the growth ring. As it canbe seen in Figure 7.2a, the radial lumen dimension LR gradually decreasesover the earlywood layer, followed by an even more pronounced decrease inthe latewood layer which provides considerably stiffer material properties intangential direction. The lumen dimension in the tangential direction LT isalmost constant throughout the growth ring, with a slight decrease in the

101


latewood layer. Therefore, less increase in radial elastic modulus is observed.In conclusion, our results show that the cell geometry and the orthotropicmaterial properties of the cell wall are responsible for the anisotropy in stiffnessand swelling properties of softwood cells. In particular, the different swellingcoefficients within the cell wall in transverse plane play an important role inthe anisotropic swelling of earlywood cells. These findings are in line with therare experimental data available at the cell wall level (Nakato, 1958; Ishimaruand Iida, 2001) and lead us in distinguishing different swelling mechanisms inearlywood and latewood.

Growth ring level The two-scale model is now used to calculate the effectiveproperties of a specific growth ring. The geometrical parameters of theunderlying unit cells are related to the relative ring position with the aid of thefunctions which are fitted to the measurements of the wood cell dimensions asdepicted in Figure 7.2. The scatter of the geometrical parameters in tangentialdirection is ignored. The resulting effective elastic properties and swellingcoefficients are presented and confronted with literature data in Table 7.2and 7.3, respectively. The result of the growth ring simulation indicates thatthe tangential swelling is greater than the radial one. The periodic boundarycondition which is imposed on the growth ring level includes the effect of theearlywood and latewood alternation for softwoods grown in the temperatezone. As a result, the strong bands of latewood in tangential direction forcethe weak bands of earlywood to swell tangentially to about the same extent asthe latewood.

Finally, we analyze whether the present multiscale model can predict theexperimental trend at the growth ring level. The swelling of a growth ringwas analyzed experimentally using environmental scanning electron microscopy

Table 7.2: Effective elastic properties of growth rings in Spruce (elastic moduli inGPa)

in-plane ρ (Kg/m3) ER ET GRT νTR

Two-scale model 443 0.757 0.460 0.030 0.34Bodig and Goodman (1973) 390 0.680 0.430 0.030 0.31Keunecke et al. (2008) 460 0.625 0.397 0.053 0.21out-of-plane ρ (Kg/m3) EL νRL νTL

Two-scale model 443 11.13 0.035 0.017Bodig and Goodman (1973) 390 10.80 0.022 0.019Keunecke et al. (2008) 460 12.80 0.018 0.014

102


Table 7.3: Effective swelling coefficients of the growth rings in Spruce (in %strain/%moisture content)

swelling coefficients ρ (Kg/m3) βR βT βLTwo-scale model 443 0.17 0.31 0.009Kollmann (1968) 375 0.19 0.37 -Fredriksson et al. (2010) 480 0.15 0.32 -

(ESEM), on images of a growth ring taken from a larger piece of wood, attwo different relative humidities, i.e. 32% and 64%RH2. Figure 7.7a and 7.7bshow the reference ESEM image at 32%RH. The measured displacements areanalyzed based on the affine transformation:

ua = (F − 1).x + c (7.2)

where F is the affine transformation matrix composed of F = S.L.R with S theshear matrix, L the scaling matrix and R the rotation matrix. This matrixis used to register the image and to determine the remaining displacements,i.e. the non-affine displacements. Assuming further that the non-affinedisplacements scale linearly with the moisture content variation, as is assumedfor the affine displacements, we obtain the non-affine displacement normalizedto the moisture content. Since it was not possible to determine the exactmoisture content of the sample in the ESEM chamber, the displacements werescaled to the maximum (negative) displacement and are presented in Figure 7.7along the relative ring position (0 marks the start of the growth ring and1 the end). According to this registration procedure, a non-zero non-affinedisplacement indicates non-homogenous cellular deformation (see Figure 7.7b).We observe that the highest displacements occur in radial direction in themiddle of the growth ring, i.e. in a transition zone between earlywood andlatewood, where the more important swelling of the stiffer latewood pushesradially the softer earlywood. The cell swelling behavior in radial directionalong the growth ring is thus significantly different between earlywood andlatewood while, in tangential direction, the behavior is almost homogeneousfor earlywood and latewood. The resulting experimental deformation fieldis compared to simulation results of a two-scale growth ring model andis presented in Figure 7.8. The free swelling simulation is followed bythe registration procedure as done in the processing of the ESEM images.

2The free swelling experiments are conducted by Dr. Masaru Abuku and Stefan Carl inthe Laboratory for Building Science and Technology at EMPA, Dübendorf, Switzerland.

103


(a) 32% RH

(b) 64% RH

(c) deformation field

100µm

Figure 7.7: ESEM images of radial-tangential cross section of Norway spruce at(a) 32% RH and (b) 64%RH; (c) deformation field between 32% and 64% relativehumidity.

The comparison gives us an insight about the interaction of the earlywoodand latewood tissues within the growth ring. A good agreement betweenexperiment and simulation is obtained. As can be seen, the maximumdeformation in the simulation occurs at the earlywood-latewood transitionfront towards the earlywood which is also observed in experiments.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−1

−0.8

−0.6

−0.4

−0.2

0

relative ring position(−)

nom

aliz

ed r

adia

l dis

plac

emen

t(−

)

experimentsimulation

Figure 7.8: Experimental and simulation results of normalized radial displacementduring free swelling between 32% RH and 64%RH.

104


7.1.2 Free swelling of softwood

We use the two-scale model to investigate the free swelling of a piece of wood atequilibrium moisture content. Figure 7.9 shows the schematic representation ofthe two-scale model used in this example. The macroscale is a section of woodin the RT plane which has several growth rings. In the macroscopic model,the boundary of each growth ring is represented by a spline curve. In eachgrowth ring, a distribution of the earlywood and latewood tissues is assignedto the relative ring position and the properties of each point are calculatedby computational upscaling of a honeycomb unit cell which represents theunderlying cellular microstructure. The relative ring position of each pointwithin a growth ring is calculated with respect to the boundaries of theneighboring rings. Within a growth ring, each point is located on a curvewhich can be represented by the average of the two growth ring boundariesweighted by the relative ring position of that point:

f(r, x) = rfi(x) + (1 − r)fi+1(x) (7.3)

where fi(x) and fi+1(x) are the spline curves which represent the boundariesof the growth ring i and r is the relative ring position of the point of interestin this growth ring. As mentioned before, the relative ring position r = 0points to the start of the growth ring (i.e. earlywood) and r = 1 marks theend (latewood). In order to consider the local anisotropy of the material, theorthotropic material orientations are needed. The tangential direction (T) isthe tangent to this curve and the radial direction (R) can be readily obtainedby calculating its normal. The macroscopic radial and tangential directionsare respectively represented with underlined letters R and T. At microscale,a complex honeycomb unit cell is used in which the cell wall layers are takeninto account. Each wall is composed of seven layers: three pairs of secondarylayers (S1, S2 and S3) and the compound middle lamella (CML).

The moisture induced deformation of a piece of wood in a free swellingexperiment at equilibrium moisture content is simulated using the proposedtwo-scale model3. Figure 7.10a shows the softwood sample which is consideredin this analysis. The swelling strains are obtained in three regions of this sample(right, middle and the left) for 9 growth rings as shown in Figure 7.10b. Thischoice is motivated by the fact that, due to edge effects, the distribution of theswelling strains in the middle part of the sample differs from the right and the

3The experimental part of this example is conducted by Christian Lanvermann in theInstitute for Building Materials at ETH Höngerberg, Switzerland.

105


growth ring boundaries

f

i+1f

i

0

1

r

R

T

CMLS3 S2 S1

(a) (b)

R

T

Figure 7.9: Schematic representation of the two-scale model for softwood. (a)growth ring boundaries and the definition of the local (R, T) and macroscopic (R, T)coordinates and (b) the honeycomb unit cell with seven layers.

(a)

R

T1

2

3

4

5

6

789

(b)

left middle right

Figure 7.10: (a) A piece of Norway spruce softwood and (b) the schematicrepresentation of the growth rings inside this sample which are experimentallycharacterized in three regions: right, middle and left.

106


left edges. The width of the right and the left regions is 17.6% of the total widthof the sample. The elastic properties of the cell wall layers which are tabulatedin Table 7.4 are transversely isotropic and are chosen from a micromechanicalmodel proposed by Neagu and Gamstedt (2007). The elastic properties of thecell wall layers are moisture dependent and they follow the curves which areshown in Figure 6.2 in the previous chapter. The swelling coefficients, thicknessand MFA of cell wall layers are presented in Table 7.5. The thickness ratio ofthe cell wall for each cell wall layer of Norway spruce was measured by Fengeland Stoll (1973). The micro-structural parameters of the wood cells (densitydistribution, radial and tangential lumen diameters, cell wall thickness andMFA) are measured along the radial direction for 9 continuous growth rings.They are plotted as a function of relative ring position of the correspondinggrowth ring in Figure 7.11. The remaining three geometrical parameters ofthe honeycomb unit cells (h, l and θ) are characterized using two relations ofEquation (7.1) and by equating the density of the unit cells to those of the woodcells as shown in Figure 7.11a. These parameters are plotted in Figure 7.12for each growth ring. The effective hygro-elastic properties of honeycomb unitcells are calculated for these 9 growth rings in the whole range of the relativering position and moisture content. These values are stored in a databasewhich is accessible at the macroscopic scale. Then, the macroscale simulationis conducted and the results are compared with experimental findings.

First, the mean values of the swelling strains in radial and tangential directionsfrom zero to 19.24% moisture content are computed for each region and alsofor the whole sample. Then, the swelling coefficients and the correspondingswelling anisotropy are calculated. As presented in Table 7.6, the simulationresults are in good agreement with the experiments and in both model andexperiment, the swelling anisotropy is βT /βR ≃ 2. It should be mentionedthat the swelling strains measured in this experiment are slightly lower thanthose values which are reported in literature for Spruce wood (see Table7.3). Apossible reason for this difference can be the low density of the wood sampleswhich are used in this experiment.

Table 7.4: Elastic properties of the cell wall layers of Norway spruce. The elasticproperties are based on a micromechanical model presented by Neagu and Gamstedt(2007).

layer E11 = E22 E33 G12 G13 = G23 ν12 ν13 = ν23CML/S1 4.20 28.1 1.49 1.64 0.41 0.05S2/S3 6.08 64.0 2.14 2.54 0.42 0.03

107


Table 7.5: Swelling coefficients, MFA and the thickness of the cell wall layers whichare used in this simulation. The percentage of the thickness of the cell wall layer isselected from the measurements of Fengel and Stoll (1973). The MFA of S2 layer isselected from Figure 7.11e.

earlywood latewoodlayer β11 β22 MFA() % of t β11 β22 MFA() % of tCML 0.05 0.05 45 4.0 0.05 0.05 45 2.0S1 0.05 0.05 75 13.0 0.05 0.05 75 9.0S2 0.10 0.30 see Fig. 7.11e 79.0 0.30 0.40 see Fig. 7.11e 86.0S3 0.05 0.05 50 5.0 0.05 0.05 50 3.0

The computed strain fields (εR, εT and εRT ) are compared with experimentalresults at 19.24% moisture content in Figure 7.13. The experimental strainfields are obtained in the range 0-19.24% moisture content using Digital ImageCorrelations (DIC) technique. The prediction of the proposed two-scale modelis in good agreement with experiments. In radial direction (R), the alternationof earlywood and latewood cells results in a non-uniform strain field. The radialstrains εR are small in earlywood and gradually increase toward latewood. Onthe other hand, in tangential direction (T), the strain field εT is almost uniformexcept in the small regions which are close to the edges. The distribution ofthe shear strains εRT is also influenced by the edge effects. According to thesimulations, shear strains are very small in the middle of the sample and areonly significant near the edges. The experimental shear strains are larger thanthose in the simulation. However, the overall behavior is qualitatively closeto experiments. One possible explanation for low shear strains in simulationresults could be the choice of honeycomb unit cells which exhibit very smallshear strain during swelling due to their symmetric geometry. Furthermore,

Table 7.6: Simulation results for mean swelling strains (in %) in radial and tangentialdirections and the corresponding swelling coefficients and swelling anisotropy ratio atdifferent regions of the sample. The strains are measured from zero to 19.24% moisturecontent. The experimental results are given in parenthesis.

region left middle right totalmean(εR) 2.597 ( 2.905) 2.616 (2.674) 2.617 (2.982) 2.617 (2.790)mean(εT ) 5.291 (5.444) 5.426 (5.695) 5.291 (5.541) 5.349 (5.618)βR 0.135 ( 0.151) 0.136 (0.139) 0.136 (0.155) 0.136 (0.145)βT 0.275 (0.283) 0.282 (0.296) 0.275 (0.288) 0.278 (0.292)βT /βR 2.04 (1.87) 2.08 (2.13) 2.02 (1.85) 2.04 (2.01)

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0 1 2 3 4 5 6 7 8 90

500

1000(a) density profile

ρ (k

g/m

3 )

0 1 2 3 4 5 6 7 8 90

20

40

60(b) radial lumen diameter

L R (

μm)

0 1 2 3 4 5 6 7 8 920

30

40

50(c) tangential lumen diameter

L T (

μm)

0 1 2 3 4 5 6 7 8 90

2

4

6(d) cell wall thickness

t (μm

)

0 1 2 3 4 5 6 7 8 90

10

20

30(e) microfibril angle of the S2 layer

MF

A (° )

ring position ()

Figure 7.11: Experimental characterization of the microstructural parameters of9 continuous growth rings: (a) density ρ (kg/m3), (b) radial lumen diameter LR

(µm), (c) tangential lumen diameter LT (µm), (d) cell wall thickness t (µm) and (e)microfibril angle of S2 layer MFA ().

109


0 1 2 3 4 5 6 7 8 90

10

20

30(a) shape angle

θ (° )

0 1 2 3 4 5 6 7 8 920

30

40

50(b) length of vertical call walls

h (μ

m)

0 1 2 3 4 5 6 7 8 910

15

20

25(b) length of inclined cell walls

l (μm

)

ring position ()

Figure 7.12: Numerical characterization of the geometrical parameters of the unitcells corresponding to 9 continuous growth rings. These parameters are shape angleθ (), length of the vertical wall h (µm) and the length of the inclined wall l (µm).

the experimental results which are presented here are not direct measurements.The strain fields are extracted from processed data which might be influencedby processing errors.

In order to compare the strain fields quantitatively, the average strain profileof each growth ring is compared with experimental results at 19.24% moisturecontent as shown in Figure 7.13. For the computation of strain profiles,the radial (εR), tangential (εT ) and shear (εRT ) strains are averaged overthe same relative ring position r of each growth ring. Both simulation andexperiment show that the radial strain profile is highly non-uniform and variesfrom εR ≃ 2% in earlywood to εR ≃ 6% in latewood regions. It is noteworthythat the radial strain profile follows the density profile which is shown inFigure 7.11a. On the other hand, in tangential direction, the swelling strainis very uniform and in the whole sample is εT ≃ 6%. This uniformity comesfrom the similar swelling behavior of earlywood and latewood cellular tissues in

110


Simulation Experiment

radia

l str

ain

tangential str

ain

shear

str

ain

0.0100

0.0600

0.0225

0.0350

0.0475

0.0100

0.0600

0.0225

0.0350

0.0475

-0.030

0.030

-0.015

0.0

0.015

Figure 7.13: Comparison of the simulation and experimental results of the freeswelling strain field of a softwood sample for radial, tangential and shear strains fromzero to 19.24% moisture content.

tangential direction and the high stiffness of latewood cells which force the lowstiffness earlywood cells to swell to the same extent in tangential direction. Theexperimental shear strains are larger than simulated ones. Both experimentaland simulation results show that the strain profiles are smoother in the middleof the sample in comparison to the right and the left edges. This behavior is also

111


0 1 2 3 4 5 6 7 8 9-0.02

0

0.02

0.04

0.06

0.08

0.1

sw

elli

ng

str

ain

(le

ft)

Simulation

0 1 2 3 4 5 6 7 8 9-0.02

0

0.02

0.04

0.06

0.08

0.1Experiment

εR

εT

εRT

0 1 2 3 4 5 6 7 8 9-0.02

0

0.02

0.04

0.06

0.08

0.1

sw

elli

ng

str

ain

(m

idd

le)

0 1 2 3 4 5 6 7 8 9-0.02

0

0.02

0.04

0.06

0.08

0.1

0 1 2 3 4 5 6 7 8 9-0.02

0

0.02

0.04

0.06

0.08

0.1

normalized ring position

sw

elli

ng

str

ain

(rig

ht)

0 1 2 3 4 5 6 7 8 9-0.02

0

0.02

0.04

0.06

0.08

0.1

normalized ring position

Figure 7.14: Prediction of experimental and computational strain profiles of asoftwood sample during free swelling for radial, tangential and shear strains in threeregions from zero to 19.24% moisture content.

observed in simulations specifically in tangential and shear strains. However,due to simplifications made in our computational model, the results of thesimulations are notably smoother than experimental results. For example, themicrostructural parameters are measured partially in the radial direction. Inthe tangential direction, there is no variation in material properties and thegeometry of unit cells.

112


In summary, the proposed two scale model can efficiently predict the swellingbehavior of softwood taking into account the microstructural informationsuch as density profiles, geometrical parameters and microfibril angles. Manyfeatures of the swelling strain fields and the strain profiles within the growthrings are successfully captured by this model which justifies the accuracy of theproposed model. The observed behavior in this analysis is also qualitatively ingood agreement with other experimental investigations of free swelling behaviorof softwood at the growth ring level (Keunecke et al., 2012).

113


7.2 FE2 Multiscale method

In order to investigate the behavior of cellular materials under external me-chanical loads, a two-scale computational upscaling method is employed. Theessential ingredients of this multiscale technique are explained in Chapter 3 (seeFigure 3.2). It is sometimes referred to as FE2 in literature which representsa two-scale finite element based upscaling scheme (Kouznetsova et al., 2002).The procedure pursued in this method is briefly reviewed here. In the first step,the geometry of the macroscopic structure is discretized and the macroscopicexternal load is applied incrementally. The solution of the macroscopic problemis obtained by solving the non-linear system of equations in a standard iterativeprocedure. Based on the micro-structural morphology of the material underinvestigation, at each macroscopic integration point, a periodic unit cell isassigned. Once the macroscopic nodal displacement is available, the localmacroscopic strain εM at each macroscopic integration point is computed andapplied as the boundary condition to the microscopic boundary value problem.The boundary value problem at the unit cell level is solved using master nodetechnique and the averaged stresses are computed from the resulting forcesat the master nodes. The consistent tangent stiffness is obtained from thereduced stiffness matrix of the unit cell using Equation (3.32) according tothe procedure described in section 3.2.2. Then, the stress tensor at eachintegration point is transferred to the macroscopic problem from which theinternal macroscopic forces can be evaluated. The local macroscopic tangentstiffness matrices are assembled to construct the macroscopic stiffness matrix.At the end, the macroscopic displacement field is updated by solving themacroscopic system of equations. Once the convergence is achieved, themacroscopic simulation proceeds to the next time increment.

7.2.1 Numerical results

In this section, two examples are analyzed using the FE2 multiscale model.The deformation of a cellular solid in tension and the pure bending of acellular beam have been examined and the results are compared against thereference solutions based on the direct numerical simulation. The referencemodel is composed of a repeated group of cells while, in the FE2 model,periodic boundary conditions are assumed at the microscale. The proposedmethod is fully implemented in the finite element package ABAQUS (RisingSun Mills, Providence, RI, USA) and the interactions between macroscopic

114

7.2. FE2 MULTISCALE METHOD

and microscopic scales are carried out by user defined material subroutinesand Python scripting interface.

A cellular material under uniaxial tensile load

First, we consider a two-dimensional rectangular section of a cellular materialwhich is assumed to be in the state of the generalized plane strain. The sampleis subjected to a uniaxial tensile load applied to the right edge. The dimensionof the cellular section is 3 × 1.73 mm2. The macroscopic model is discretizedwith four-node generalized plane strain elements which has four integrationpoints. Two macro mesh size of 1 × 1 and 2 × 2 elements are considered whichcorrespond respectively to the reference models with 2 × 2 and 4 × 4 numberof cells. The microscopic unit cell is a regular honeycomb with geometricalparameters h = l, t = 0.1h and θ = 30. The cell walls are modeled as an elasto-plastic material with Young’s modulus E = 7GPa, Poisson’s ratio ν = 0.33 andan initial yield stress of σ0 = 15MPa. A uniform displacement of u1 = 0.03mmis applied on the right edge and the responses of both models are obtained.The the force-displacement diagram of FE2 multiscale method are shown inFigure 7.15 and compared to the reference solutions. Both the referencemodel and the FE2 models are in good agreement specifically within the linearregime. In the transition to the plastic regime, the reference solution shows aslightly stiffer behavior. This difference originates from the assumption of theperiodicity of the microstructure in the FE2 model at the unit cell level whichneglects the influence of edge effects in the reference model. Furthermore, byincreasing the number of macro elements, the difference between the results ofthe FE2 model and the reference solution decreases.

Pure bending of a cellular beam

In the second example, we consider the pure bending of a beam which iscomposed of a cellular material. The cellular beam is fixed at one end anda macroscopic rotation is prescribed at the other end through the rotationof a reference node on a rigid bar that is attached to the end of the sample.The macroscopic rotation Θ and the reaction bending moment of the beamare equal to those of the reference node, which are given by the finite elementcalculations at each strain increment. The geometry of the reference modeland the macroscopically prescribed rotation at the end of the beam are shownin Figure 7.16 where H and L are the height and the length of the beam,

115


(a)

(b)

1.73 mm

3.0

mm

0.03mm

1.73 mm

3.0

mm

0.03mm

Figure 7.15: Comparison of FE2 model with reference model for a sample underuniaxial tensile load. (a) 2 × 2 and (b) 4 × 4 unit cells.

respectively. The reference model consists of 2 × 16 cells and the aspect ratioof the beam is H/L =

√3/8. The respective two scale model is composed of

eight macroscopic generalized plane strain elements. Eight-noded quadrilateralelements with reduced integration (each element has four integration points)are selected for this analysis since they show good results for bending problems.In this setting, a periodic unit cell is assigned to every integration point in theFE2 model. At the microscale, a regular honeycomb unit cell (h = l and θ =30) with isotropic homogeneous cell walls is considered. Again, the cell wallsare modeled as an elasto-plastic material with Young’s modulus E = 7GPa,Poisson’s ratio ν = 0.33 and an initial yield stress of σ0 = 15MPa.

In Figure 7.17, the results of the two-scale model (FE2) compared with the

116

7.2. FE2 MULTISCALE METHOD

H

L

Figure 7.16: The reference model for a macroscopically bended beam. The rotationΘ is applied at te right end of the beam. The aspect ratio of the beam is H/L =

√3/8

and it includes 2 × 16 honeycomb cells.

reference solution. In this figure, the average bending moment of the cellularbeams as a function of the applied macroscopic rotation Θ are shown. Forsmall rotations, the bending moment increases linearly with increasing Θ. Thisbehavior is followed by a nonlinear transition regime which corresponds tothe formation of localized plastic hinges. For small rotations, both modelsshow similar results but, with increasing the rotation, the FE2 model showsa stiffer behavior. This behavior can be due to the small number of cells

0 . 0 0 0 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0 0 . 2 50 . 0

0 . 1

0 . 2

0 . 3

0 . 4

Bend

ing m

omen

t (N.m

)

r o t a t i o n Θ( r a d )

2 × 1 6 F E 2

Figure 7.17: Comparison of the FE2 model with reference solution. The bendingmoment of a cellular beam as a function of the applied macroscopic rotation iscomputed. The reference solution is composed of 2× 16 unit cells and the FE2 modelconsists of 8 elements with 4 integration points.

117


von Mises (MPa)

+0.00+0.02+0.04+0.05+0.07+0.09+0.11+0.13+0.15+0.16+0.18+0.20+0.22

von Mises(MPa)

+0.0+ 1.7+ 3.3+ 5.0+ 6.6+ 8.3+10.0+11.7+13.3+15.0+16.7+18.0+20.0

honeycomb RVE Cellular beam

Figure 7.18: Stress distribution in cellular beam and two microscopic RVEs underpure bending at Θ = 0.25 rad based on the coupled multiscale model.

in the reference model. Consequently, the presence of the edge effects andthe local deformations limit the applicability of the FE2 model with periodicmicrostructure. The distributions of von Mises stress in the cellular beam andtwo underlying honeycomb unit cells are shown in Figure 7.18. The stressis lowest in the middle, i.e. at the neutral axis of the beam, and reaches toits maximum value in regions closest to the top and bottom boundaries, i.e.furthest from the neutral axis. At the microscale, the honeycomb unit cellwhich is located above the neutral axis of the beam is in tension and one thatis located below the neutral axis bears a compressive load. The highest stressconcentration occurs at the corners where the plastic hinges are created. Herewe showed that the macroscopic mechanical behavior of cellular materials canbe investigated using a two-scale multiscale method.

7.3 Summary

In this chapter, the hygro-mechanical behavior of softwood is investigated usinga two-scale model based on a hierarchical computational upscaling technique.In this analysis, both material anisotropy of the cell walls and the geometryof the cellular structure are taken into account. The transverse anisotropy inmechanical and swelling properties of softwoods are computed and comparedwith experiments. Simulation results are in good agreement with experimentswhen keeping in mind that the complex structure of natural materials suchas wood introduces a superposition of different influences which makes thequantitative prediction of the effective behavior difficult. In the second partof this chapter, the deformation of cellular materials under mechanical loads

118

7.3. SUMMARY

are investigated through simple load cases using the FE2 multiscale method inwhich the macroscopic problem is solved simultaneously with the microscopicequilibrium at the integration point level for all loading steps.

119

Chapter 8

Hygroelastic behavior ofmicrostructured materials:from honeycombs to auxetics

In previous chapters, we studied the hygro-elastic behavior of cellular solidswith honeycomb microstructure taking into account various settings. Theframework proposed for honeycomb materials can be extended to study theswelling behavior of other cellular solids. Playing with geometry as shown inFigure 8.1, we can extend the domain of the study from conventional to re-entrant honeycombs. We focus here on a specific cellular materials showingoverall negative Poisson’s ratios, called auxetics. The re-entrant mechanismoccurs when an applied tension to a side of a material leads to an expansionof the adjacent sides. The unusual behavior of auxetic materials is of practicalinterest in packaging, shock absorption and acoustic damping. They are knownalso to be present in natural materials with a cellular microstructure such ascancellous bone (Williams and Lewis, 1982). An extensive overview of theperformance and applications of auxetic materials can be found in the reviewpaper by Evans and Alderson (2000). Many materials properties of auxeticscan be enhanced as a result of having a negative Poisson’s ratio. In this chapter,we aim to explore how the methodology developed in this work can be used toinvestigate the mechanical and swelling behavior of a different class of cellularmaterials.

121

CHAPTER 8. SWELLING OF HONEYCOMBS AND AUXETICS

shape angle

aspect ra

tio

=/

l

h

0 90-90

0.5

l

h

hl

t

t

(a)

(b)

(c)

=

1.0

1.5

2

1sin

Figure 8.1: Geometrical parameters of honeycomb unit cells (a) θ > 0 and (b)θ < 0, (c) ζ-θ domain with examples of variation of the geometry with shape angleand aspect ratio.

8.1 Hygro-elastic properties

In this section, we investigate the mechanical and hygro-expansion propertiesof re-entrant honeycombs. Figure 8.2a shows a network of periodic re-entranthoneycombs. The geometry of re-entrant honeycombs is very similar tohoneycombs except that the shape angle is negative. The unit cell of re-entranthoneycombs poses point symmetry. Therefore, as depicted in Figure 8.2, onlya quarter of the unit cell is needed to be analyzed. This implies that the samecomputational model presented in Chapter 3 can be used for calculating theeffective properties of re-entrant honeycombs.

8.1.1 Conventional vs. re-entrant honeycombs

We compare the hygro-elastic behavior of conventional and re-entrant honey-combs in the following through a series of numerical simulations. Three caseswith different length to height ratios are considered where ζ = l/h = 0.5,1.0,2.0.In all simulations, the thickness of the cell wall is α = t/h = 0.1. The shape angleranges from θ = −60 to θ = 60. A shape angle θ < 0 refers to a re-entrantstructure and θ > 0 is representative of a conventional honeycomb geometry.As mentioned in chapter 4, θ = 0 corresponds to brick arrangement. For re-entrant honeycombs, only feasible shape angles are considered. The smallest

122

8.1. HYGRO-ELASTIC PROPERTIES

(b)

l

h

θ32

1

-s~

2

+s~

2 +s~

3

+s~

1-s

~

1

-s~

3

(a)

t

1

2

(c)

=0.5 =1 =2

Figure 8.2: (a) A periodic arrangement of re-entrant honeycombs. A unit cell isshown with bold lines and four geometrical parameters, namely height of verticalwalls, h, length of inclined walls, l, the shape angle, θ and thickness t are defined.(b) A quarter of re-entrant honeycomb unit cell discretized with finite elements. Thepositions of three master nodes (p = 1,2,3) and the corresponding local coordinatesystem, sp, centered on each master node are shown in this figure. (c) Schematic ofthree re-entrant honeycombs with ζ = l/h = 0.5,1 and 2 considered in this work.

possible shape angle for re-entrant honeycombs is θmin = −arcsin 1/2ζ. Thecell walls are isotropic with the elastic modulus Ec = 1GPa and Poisson’s ratioof νc = 0.3. The swelling behavior of the cell wall is assumed to be stronglyanisotropic with β∥/β⊥ = 0.1. In the following, we look at the effective elasticmoduli, Poisson’s ratios and hygro-expansion coefficients.

Elastic moduli

The in-plane elastic moduli E1 and E2 are calculated and the results areshown in Figure 8.3. The results are normalized by the elastic modulus ofthe cell wall Ec. It can be seen that, for a brick-like structure (θ = 0), the

123


elastic modulus E1 is maximum. As mentioned in Chapter 4, the dominantdeformation mechanism for a brick-like structure is axial deformation. Theelastic Young’s modulus E1 of re-entrant honeycombs (θ < 0) is generallylarger than of conventional honeycombs (θ > 0). The elastic modulus E1 ofboth conventional and re-entrant honeycombs decreases with increasing thelength to height ratio ζ. The elastic modulus E2 shows a different behavior,exhibiting a minimum at θ = 0 for low ζ ratios. For higher ζ ratios, the elasticmodulus E2 continuously decreases with decreasing the shape angle. Thesestructures are generally anisotropic. In the conventional honeycombs, furtherincrease of θ results in structures where E1 < E2.

- 6 0 - 3 0 0 3 0 6 01 E - 5

1 E - 4

1 E - 3

0 . 0 1

0 . 1

E 1 ζ= l / h = 0 . 5 E 1 ζ= l / h = 1 E 1 ζ= l / h = 2 E 2 ζ= l / h = 0 . 5 E 2 ζ= l / h = 1 E 2 ζ= l / h = 2no

rmaliz

ed el

astic

modu

li (-)

s h a p e a n g l e , θ (°)

Figure 8.3: Normalized in-plane elastic moduli E1 and E2 for ζ = l/h = 0.5,1,2 andα = t/h = 0.1 as a function of shape angle θ.

Poisson’s ratios

The two in-plane Poisson’s ratios ν12 and ν21 of re-entrant and conventionalhoneycomb structures are shown in Figure 8.4. This figure shows that ν12

goes to very large negative values for re-entrant auxetics. On the otherhand, in conventional honeycombs, ν12 reaches very large positive valuesspecifically for larger ζ. To be more specific, for auxetics, the Poisson’s ratioν12 reaches significant negative values close to a shape angle of θ ≃ −5 while for

124

8.1. HYGRO-ELASTIC PROPERTIES

conventional honeycombs it goes to very high positive values at θ ≃ 5. Thisbehavior was also observed in other homogenization based models proposedfor 2D lattices (Gonella and Ruzzene, 2008; Dos Reis and Ganghoffer, 2012).Increasing ζ results in the increase of the magnitude of ν12. On the otherhand, ν21 shows almost similar behavior for all three values of ζ. As shownin Figure 8.4b, the Poisson’s ratio ν21 increases with the increase of the shapeangle specially when θ > 30. The Poisson’s ratios of regular honeycombs areequal to one, i.e. ν12 = ν21 = 1 and those of brick-like structures are zero, i.e.ν12 = ν21 = 0.

- 6 0 - 3 0 0 3 0 6 0

- 2 0

- 1 0

0

1 0

2 0 ζ = l / h = 0 . 5 ζ = l / h = 1 ζ = l / h = 2

Poiss

on’s r

atio,

υ 12 (-)

s h a p e a n g l e , θ°

(a)

- 6 0 - 3 0 0 3 0 6 0- 2

0

2

4

6 ζ = l / h = 0 . 5 ζ = l / h = 1 ζ = l / h = 2

Poiss

on’s r

atio,

υ 21 (-)

s h a p e a n g l e , θ°

(b)

Figure 8.4: In-plane Poisson’s ratio of auxetics and honeycombs for different ζ = l/has a function of shape angle θ. (a) ν12 and (b) ν21.

Hygro-expansion coefficients

The normalized effective in-plane hygro-expansion (swelling) coefficients ofre-entrant and conventional honeycombs, β1/β⊥ and β2/β⊥, are shown as afunction of the shape angle θ in Fig. 8.5. The shear component of the effectivehygro-expansion tensor is found to be almost zero, i.e. β12 ≃ 0, and is notshown in the results. It is found that the effective hygro-expansion coefficientof conventional honeycombs (θ > 0) in direction 2 is less than β⊥, i.e. β2/β⊥ < 1.For re-entrant honeycombs (θ < 0) however, we observe that β2/β⊥ is muchlarger than 1 for high ζ and, with decreasing ζ, it becomes even smaller thanβ1/β⊥. The high hygro-expansion coefficient in direction (2) for high ζ ratiocan be explained by the low elastic modulus of such structures. The hygro-expansion coefficient in direction (1) is less sensitive to ζ and for all three

125


configurations, similar results are obtained with normalized values less than 1.The swelling anisotropy ratio β2/β1 as a function of shape angle θ is plottedin Figure 8.6. The cellular structures exhibit large anisotropy when θ is small.In this region, the anisotropy increases with increasing ζ. The anisotropy inswelling behavior of conventional honeycombs decreases with the increase ofthe shape angle. Further increase of θ > 30 results in honeycomb structureswhere β1 > β2 and, as mentioned above, E1 < E2 (see Figure 8.3). It canbe seen that, the highest anisotropy ratio occurs at θ ≃ 15 for conventionalhoneycombs. However, it should be mentioned that the swelling coefficientβ2 in these honeycombs is very small in comparison to re-entrant structuresspecifically for large ζ. For honeycombs with θ ⪰ 30, the hygro-expansionanisotropy ratio is very low, β2/β1 ≃ 0.1 which implies that β1 is much largerthan β2. Therefore, the swelling behavior of the resulting structure is highlyanisotropic.

- 6 0 - 3 0 0 3 0 6 0- 1

0

1

2

3 β

1/ β

⊥, ζ= l / h = 0 . 5

β1/ β

⊥, ζ= l / h = 1

β1/ β

⊥, ζ= l / h = 2

β2/ β

⊥, ζ= l / h = 0 . 5

β2/ β

⊥, ζ= l / h = 1

β2/ β

⊥, ζ= l / h = 2

hygro

expa

nsion

, β1/β ⊥

and β

2/β ⊥

s h a p e a n g l e , θ ( °)

Figure 8.5: Normalized in-plane hygro-expansion coefficient β1/β⊥ and β2/β⊥ ofauxetics and honeycombs for different ζ = l/h as a function of shape angle θ. Allsimulations are conducted for β∥/β⊥ = 0.1.

126

8.2. SUMMARY

- 6 0 - 3 0 0 3 0 6 00 . 0 1

0 . 1

1

1 0hy

gro-ex

pans

ion an

isotro

py, β

2/β1

s h a p e a n g l e , θ ( °)

ζ = l / h = 0 . 5 ζ = l / h = 1 ζ = l / h = 2

Figure 8.6: Hygro-expansion anisotropy ratio β2/β1 of auxetics and honeycombs fordifferent ζ = l/h as a function of shape angle θ. All simulations are conducted forβ∥/β⊥ = 0.1.

8.2 Summary

In this chapter, the computational framework developed for estimating thehygro-elastic behavior of conventional honeycombs is extended to study a groupof cellular materials which exhibit an unusual behavior: negative Poisson’sratio. The motivation of this study was to understand how this novel behaviorchanges the swelling properties. A set of numerical simulations are carriedout and the swelling behavior of re-entrant auxetics is investigated and theobtained results are compared to normal honeycombs as their conventionalcounterparts. We found that the variation of β2 of re-entrant auxetics is muchbigger than for conventional honeycombs with respect to different length toheight ratios ζ. The swelling behavior of re-entrant honeycombs in the otherdirection is not affected by varying the length to height ratio ζ. Re-entrantstructures may exhibit very large swelling anisotropy. Furthermore, it is shownthat the geometrical parameters of re-entrant and conventional honeycombscan be designed to obtain structures with a wide range of swelling behavior.This knowledge increases the potentials of auxetic materials to make a keycontribution to the development of new structural and functional materials.

127

Chapter 9

Conclusion and Outlook

Understanding the behavior of cellular solids is an essential step for revealingthe complex relations between structures and properties of many biologicaland engineering materials. The objective of the present research was todevelop a multi-scale computational strategy for studying the hygro-elasticbehavior of cellular materials to bridge the gap from microscopic cellularstructure to macroscopic scale. This thesis mainly focuses on the descriptionof the anisotropic swelling behavior of softwoods at the cellular scale from amechanical perspective. The method however is general and can be extendedto study the hygric and thermal expansion of other cellular materials. Thisknowledge provides a predictive tool for analyzing existing materials and isalso a source for the design of bio-inspired materials where material designprinciples from nature are investigated in order to provide the conceptualfoundation necessary for the development of new materials. The mainconclusions of the present work are summarized in this section.

Computational advances:

Development of a computational framework for predicting the effective hygro-elastic behavior of periodic honeycombs based on master node technique; Inthe proposed model the smallest possible unit cell for honeycombs is generatedtaking into account the point symmetry conditions.

Computational upscaling of swelling behavior of cellular material based on aporomechanical approach; A physically relevant description for free swelling ofcellular materials is derived based on poroelastic constitutive equations. It isshown that the effective swelling coefficient of a cellular materials is dependent

129

CHAPTER 9. CONCLUSION AND OUTLOOK

on the elastic properties of the cell wall, the bulk modulus of the matrix fromwhich the cell walls are made and the porosity of the cell walls material.

Advances in understanding the behavior of generic cellular materials:

Investigation of anisotropy in effective hygro-elastic properties of cellular ma-terials including different geometrical parameters and identifying the respectiverole of shape of the cells and the mechanical properties of the cell wall on thedevelopment of anisotropy in cellular materials;

Understanding the effective swelling behavior of auxetics in comparisonto honeycombs; The computational framework developed for estimating theeffective hygro-elastic behavior of honeycombs is extended to study re-entrantauxetics, a cellular material with negative Poisson’s ratio. It is shown that,the hygro-expansion coefficients of auxetics can be designed for a wider rangeof desirable properties in comparison to honeycombs.

Advances in understanding wood:

Understanding the swelling behavior of earlywood and latewood at thecellular scale; It is found that the orthotropic swelling properties of the cell wallin thin-walled earlywood cells produce anisotropic swelling behavior while, inthick latewood cells, this anisotropy vanishes.

Understanding the role of the eccentricity on the swelling anisotropyof periodic cellular models for wood; It is found that periodic symmetrichoneycombs provide the upper bound for anisotropic swelling ratio, whilepresence of eccentricity in arrangement of the cellular structure of softwoodsreduces the swelling anisotropy.

Upscaling the effective mechanical and swelling behavior of softwoods fromtheir underlying cellular structure taking into account the anisotropy of the cellwall material; A two-scale hierarchical multiscale model is devised to estimatethe effective swelling and mechanical behavior of softwoods. The proposedapproach provides the ability to consider the complex microstructure whenpredicting the effective mechanical and swelling properties of softwood.

Future perspectives

Based on this thesis, a number of directions for further research can beformulated. These issues are discussed here:

There is a need to include large deformations in modeling the swelling

130

behavior of cellular materials; This work is restricted to small deformations,while for a better understanding of swelling deformations of softwoods andother natural cellular materials, large deformations have to be considered.

Development of analytical models for the prediction of the effective swellingbehavior of cellular solids; For estimating the effective swelling behavior ofhoneycombs, we used the finite element method to solve the microscopicboundary value problem. Alternatively, the elastic properties of honeycombsare successfully modeled with analytical cellular models (Gibson and Ashby,1997). Similar analytical derivations for swelling properties result in con-siderable reduction of the computational times in multi-scale models andallow simulating larger systems taking into account morphological parameters.However, further development and optimization of multiscale computationalapproach are necessary for studying coupled problems.

Investigating the size effects in swelling behavior of cellular materialsusing second-order computational homogenization methods; The second-ordertheories (e.g. micropolar elasticity) are used in literature for modeling themechanical behavior of natural cellular materials, e.g. in bone (Lakes, 1995).Similar developments could be expected for modeling the swelling behaviorof cellular materials using second-order computational homogenization meth-ods (Kouznetsova et al., 2002).

Incorporating computational upscaling methods for design of innovativefunctional cellular materials with desirable swelling properties;

Further understanding of the origin of swelling behavior by downscaling tomolecular scale; For example, molecular dynamics simulation of the interactionof wood polymers with water might incite us about the improvement ofmacroscopic swelling models.

Including hysteresis and collapse of cellular materials in swelling models.

131

Appendix

In this appendix, the implementation of the homogenization procedure fora periodic honeycomb unit cell is illustrated. The proposed computationalmodel, as presented in Chapter 3, is implemented in the finite element packageABAQUS using its Python scripting interface. For this purpose two codesare written which are used for pre- and post-processing in homogenizationprocedure. The preprocessor creates a quarter honeycomb unit cell withorthotropic cell walls. The unit cell is meshed with generalized plane strainelements and the periodic boundary conditions are imposed on its boundaries.In hygroelastic model, first a substructure analysis is performed. Then, auniform moisture content increment is imposed to a macroscopically restrainedunit cell. After solution of these two steps in ABAQUS, the post processing iscarried out. In the postprocessor, the macroscopic stiffness matrix is calculatedfrom the reduced stiffness matrix which is obtained from the substructure step.The resulting swelling stresses are calculated from the reaction force of masternodes in swelling static step. Once the macroscopic stiffness and swellingstresses are known, the effective swelling coefficients of the honeycomb unitcell are calculated from the hygroelastic model.

The poromechanics approach is also implemented in ABAQUS similar tohygroelastic model. First, the preprocessing is carried out and a unit cell is gen-erated and poroelastic properties of the cell walls are defined. The macroscopicstiffness of honeycombs is extracted from a substructure analysis. Afterwards,using the Soil step, a uniform pore pressure is applied to a macroscopicallyrestrained honeycomb unit cell and the macroscopic poroelastic propertiesand swelling coefficients are calculated in the postprocessor. The schematicimplementation of hygroelastic and poroelastic models are shown in Figures 1and 2, respectively.

133

APPENDIX . ABAQUS IMPLEMENTATION

ABAQUS/standard

Postprocessor.py

Read output.odb

Hygroelastic modelPreprocessor.py

Create geometry

Geometrical properties

(h, l, t, )

SketchSectionPartition

Mesh and BC

Structured mesh

Define master nodes

Apply PBC with Equation

Analysis steps

SubstructureRetained nodal DOF

Static

stepdefine

stepRestrain masternodesMoisture increment

Master node

Compute

Reduced stiffness matrix

Reaction forces

MacroscopicProperties

Elastic moduli

Swelling coefficients

32

1

-s~

2

+s~

2+s

~

3

+s~

1-s

~

1

-s~

3

Figure 1: Schematic of ABAQUS implementation of hygroelastic model.

ABAQUS/standard

Postprocessor.py

Read output.odb

Poroelastic modelPreprocessor.py

Create geometry

Geometrical properties

(h, l, t, )

SketchSectionPartition

Mesh and BC

Structured mesh

Define master nodes

Apply PBC with Equation

Analysis steps

SoilsStep

SubstructureRetained nodal DOF

stepdefine

stepRestrain masternodesPore pressure increment

Master node

Compute

Reduced stiffness matrix

Reaction forces

MacroscopicProperties

Elastic moduli

Biot Coefficients

Biot Modulus

Swelling coefficients

32

1

-s~

2

+s~

2+s

~

3

+s~

1-s

~

1

-s~

3

Figure 2: Schematic of ABAQUS implementation of poroelastic model.

134

Generalized plane strain elements in ABAQUS

The generalized plane strain theory used in Abaqus assumes that the model liesbetween two bounding planes, which may move as rigid bodies with respect toeach other, thus causing strain of the thickness direction fibers of the model. Itis assumed that the deformation of the model is independent of position withrespect to this thickness direction, so the relative motion of the two planescauses a direct strain of the thickness direction fibers only. This strain and itsfirst and second variations are defined as follows. Let P0(X0, Y0) be a fixedpoint in one of the bounding planes, as shown in Figure 3. The length ofthe fiber between P0 and its image in the other bounding plane is t0 + ∆uz, where t0 is the length of this fiber in the initial configuration and ∆uz isthe change in length of this fiber. ∆uz is the value of degree of freedom 3at the reference node of the element. The reference node should be the samefor all elements in any given connected region so that the bounding planes arethe same for that region. Different regions may have different reference nodes.Since the bounding planes are rigid, the length of a fiber at any other point inthe element is

t = t0 +∆uz +∆φx(y − Y0) −∆φy(x −X0) (1)

where ∆φx and ∆φy are the rotational degrees of freedom of the boundingplanes. In our analysis, these two values are set to zero and the boundingplanes are parallel.

+

+

Bounding planes

Conventional element node

Reference node

(x,y)

(X ,Y)0 0

y

x

Figure 3: Generalized plane strain element.

135

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Curriculum vitae


Date of birth 18 January 1984Place of birth Mashhad, IranNationality IranianMarital status MarriedE-mail address [email protected]

Education

12/2009 - 3/2013 Doctor of Sciences (Defence date: 26/11/2012)Swiss Federal Institute of Technology Zurich

09/2006-02/2009 Master of Science in Mechanical EngineeringIran University of Science and Technology

09/2002 - 09/2006 Bachelor of Science in Mechanical EngineeringFerdowsi University of Mashhad, Iran

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CURRICULUM VITAE

Research Experience

4/2013 - present Postdoctoral Researcher at EMPA, SwitzerlandLaboratory for Building Science and Technology

12/2009 - 3/2013 Research Assistant at EMPA, SwitzerlandLaboratory for Building Science and Technology

09/2006 - 07/2009 Research Assistant at IUST, IranIran University of Science and Technology (IUST)

Journal Publications

Rafsanjani, A., D. Derome and J. Carmeliet (2013). Micromechanicsinvestigation of hygro-elastic behavior of cellular materials with multi-layeredcell walls. Composite Structures 95: 607-611.

Rafsanjani, A., D. Derome, F. K. Wittel and J. Carmeliet (2012). Computa-tional up-scaling of anisotropic swelling and mechanical behavior of hierarchicalcellular materials. Composites Science and Technology 72 (6): 744-751.

Rafsanjani, A., D. Derome and J. Carmeliet (2012). The role of geometricaldisorder on swelling anisotropy of cellular solids. Mechanics of Materials 55:49-59.

Derome, D., A. Rafsanjani, A. Patera, R. Guyer and J. Carmeliet (2012).Hygromorphic behavior of cellular material: hysteretic swelling and shrinkageof wood probed by phase contrast X-ray tomography. Philosophical Magazine92 (28-30): 3680-3698.

Derome, D., A. Rafsanjani, S. Hering, M. Dressler, A. Patera, C. Lanver-mann, M. Sedighi-Gilani, F. K. Wittel, P. Niemz and J. Carmeliet (2013). "Therole of water in the behavior of wood." Journal of Building Physics 36 (4):398-421.

Rafsanjani, A., D. Derome, R. A. Guyer and J. Carmeliet (2013). Swellingof cellular solids: from conventional to re-entrant honeycombs, Applied PhysicsLetters, accepted.

Rafsanjani, A., D. Derome and J. Carmeliet (2013). A poromechanicsapproach to predict the swelling behavior of cellular materials, submitted.

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CURRICULUM VITAE

Refereed Conference Proceedings

Rafsanjani, A., D. Derome and J. Carmeliet, A poromechanics approachto predict the effective swelling behavior of cellular materials, 5th BIOTConference on Poromechanics. Vienna, Austria, July 10-12, 2013.

Rafsanjani, A., A. Patera, D. Derome and J. Carmeliet, Multiscale model-ing of anisotropic hygro-mechanical behavior of natural cellular composites,Mechanics of Nano, Micro and Macro Composite Structures, Politecnico diTorino, Turin, Italy, 18-20, June 2012.

Rafsanjani, A., D. Derome and J. Carmeliet, Effective poroelastic propertiesof softwood in relation to moisture induced swelling, 4th InternationalConference on Porous Media and Annual Meeting of the International Societyfor Porous Media, Purdue University, West Lafayette, USA, May 14-16, 2012.

Rafsanjani, A., D. Derome and J. Carmeliet, Multiscale computationalhomogenization for the hygro-mechanical analysis of growth rings in softwoods,COST Action FP0802, Experimental and Computational Micro-CharacterisationTechniques in Wood Mechanics, Vila Real, Portugal, April 27-28, 2011.

Accepted Research Proposals

Rafsanjani, A. (Principal Investigator), D. Derome, A. Patera, J. Carmeliet,K. Jefimovs, Three-dimensional measurement of moisture induced swelling ofwood cell wall material as a natural fiber-reinforced nano-composite usingnano-tomography, TOMCAT, Swiss Light Source, Paul Scherrer Institute,Villigen, Switzerland (November 2012).

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