3-D Forming of Paper Materials

ERIC LINVILL

Doctoral Thesis no. 99, 2017

KTH Royal Institute of Technology

School of Engineering Sciences

Department of Solid Mechanics

SE-100 44, Stockholm, Sweden

TRITA HFL-0608

ISSN 1654-1472

ISRN KTH/HFL/R-17/03-SE

ISBN 978-91-7729-250-0

© Eric Linvill, 2017

Akademisk avhandling som med tillstånd av KTH i Stockholm framlägges till offentlig

granskning för avläggande av doktorsexamen fredagen den 10 februari kl. 14:00 i sal F3,

KTH, Lindstedtsvägen 26, Stockholm. Avhandlingen försvaras på engelska.

Abstract

Paper materials have a long history of use as a packaging material,

although traditional paper-based packaging is limited in its shape,

complexity, and design. In order to better understand the deformation

and failure mechanisms during 3-D forming, two experimental studies of

paper materials have been conducted. Furthermore, constitutive

modeling combined with explicit finite element modeling have been

validated against numerous experimental setups and utilized to develop

further understanding of 3-D forming processes.

Two experimental studies were necessary to further investigate and

model the 3-D formability of paper materials. The combined effect of

moisture and temperature on the uniaxial mechanical properties of paper

was investigated, providing new insights into how moisture and

temperature affect both the elastic and plastic properties of paper

materials. Furthermore, the in-plane, biaxial yield and failure surfaces

were experimentally investigated in both stress and strain space, which

gave an operating window for 3-D forming processes as well as input

parameters for the constitutive models.

The constitutive modeling of paper materials and explicit finite

element modeling were directed towards two 3-D forming processes:

deep drawing and hydroforming. The constitutive models were calibrated

and validated against simple (typically uniaxial) mechanical tests, and the

explicit finite element models (which utilize the developed constitutive

models) were validated against 3-D forming experiments. Hand-made

papers with fibers partially oxidized to dialcohol cellulose, which has

greater extensibility than typical paper materials, was furthermore

characterized, modeled, and 3-D formed as a demonstration of the

potential of modified paper fiber products for 3-D forming applications.

Keywords

3-D forming, finite element method, constitutive model, moisture,

temperature, biaxial, hydroforming, deep drawing.

Sammanfattning

Papper har länge framgångsrikt använts som förpackningsmaterial,

men traditionella pappers- och kartongförpackningar är begränsade i

form och design. Två experimentella studier har utförts för att få bättre

förståelse för deformations- och brottmekanismer under 3D formning.

Resultat från konstitutivmodellering i kombination med explicit finit

element modellering har validerats mot ett flertal experimentella

uppställningar och använts för att utveckla bättre förståelse för 3D

formningsprocesser.

Två experimentella studier var nödvändiga för att ytterligare

undersöka och modellera pappersmaterials 3D formbarhet. I den första

undersöktes den kombinerade effekten av fukt och temperatur på

pappers enaxliga mekaniska egenskaper, vilket gav nya insikter om hur

fukt och temperatur påverkar både de elastiska och de plastiska

egenskaperna hos papper. I den andra har biaxiella (i planet) flyt- och

brottytor undersökts experimentellt i både spännings- samt

töjningsrymden, vilket gav ett processfönster för 3D formningsmetoder

samt ingångsparametrar för de konstitutiva ekvationerna.

Konstitutiv modellering av pappersmateriel samt explicit finit element

modellering riktades mot två 3D formningsprocesser: djupdragning och

hydroformning. De konstitutiva modellerna kalibrerades och validerades

mot enkla (oftast enaxliga) mekaniska experiment, och explicita finita

elementmodeller (som utnyttjar de utvecklade konstitutiva modellerna)

validerades mot 3D formningsexperiment. Handark med fibrer delvis

oxiderade-reducerade till dialkohol cellulosa, som har större töjbarhet än

andra pappersmateriel, har dessutom karakteriserats, modellerats, samt

3D formats som en demonstation av potentialen hos modifierade

pappersfiberprodukter i 3D formning.

Nyckelord

3D formning, finita elementmetoden, konstitutiv modell, fukt,

temperatur, biaxiell, hydroformning, djupdragning.

Preface

This thesis was completed at the Department of Solid Mechanics,

School of Engineering Sciences, at KTH Royal Institute of Technology

under the supervision of Professor Sören Östlund. All research presented

as part of this thesis was conducted as a part of the VINN Excellence

Center BiMaC Innovation.

BiMaC Innovation was founded in 2007 as a ten-year multi-

disciplinary platform for addressing major problems in the forest-based

sector. The mission of BiMaC Innovation, according to the 2016 strategic

plan, is “to be a facilitator for renewal and innovation in the forest-based

sector” utilizing “fundamental knowledge from several disciplines”. One

major goal of BiMaC Innovation is the “development of manufacturing

techniques for complex shaped papers”, which is the main focus of this

work.

This thesis directly investigates two processes (deep-drawing and

hydroforming) through which geometrically complex paper-based

structures can be created. These 3-D forming methods are investigated

through the use of finite element methods. Finite element methods have

been instrumental in the improvement of similar 3-D (sheet) forming

methods for metals, so a similar approach has been utilized for paper-

based materials. In addition to the creation of finite element models,

understanding for the mechanical behavior of paper materials and

material models based on that understanding have also been developed in

order to improve the accuracy and completeness of the finite element

simulations.

The research has been funded by BiMaC Innovation, which (at the

date of publication) includes the following partners/funders: KTH Royal

Institute of Technology, BillerudKorsnäs, Holmen, Innventia, Perstorp,

Stora Enso, Tetra Pak Packaging Solutions, and Vinnova.

Stockholm, February 2017

Eric Linvill

List of appended papers

This doctoral thesis is based upon the following six scientific articles:

Paper A

E. Linvill, S. Östlund (2014) “The Combined Effects of Moisture and

Temperature on the Mechanical Response of Paper”, Experimental

Mechanics, Vol. 54(8), Pages 1329-1341.

Paper B

E. Linvill, S. Östlund (2016) “Biaxial In-Plane Yield and Failure of

Paperboard”, Nordic Pulp & Paper Research Journal, Vol. 31(4), Pages

659-667.

Paper C

M. Wallmeier, E. Linvill, M. Hauptmann, J.P. Majschak, S. Östlund

(2015) “Explicit FEM Analysis of the Deep Drawing of Paperboard”,

Mechanics of Materials, Vol. 89, Pages 202-215.

Paper D

E. Linvill, M. Wallmeier, S. Östlund (2017) “A Constitutive Model for

Paperboard Including Wrinkle Prediction and Post-Wrinkle Behavior

Applied to Deep Drawing”, Internal Report No. 606, Department of Solid

Mechanics, KTH Royal Institute of Technology, Stockholm.

Paper E

E. Linvill, S. Östlund (2016) “Parametric Study of Hydroforming of

Paper Materials Using the Explicit Finite Element Method with a

Moisture-dependent and Temperature-dependent Constitutive Model”,

Packaging Technology and Science, Vol. 29(3), Pages 145-160.

Paper F

E. Linvill, P. Larsson, S. Östlund (2017) “Advanced Three-

Dimensional Paper Structures: Mechanical Characterization and Forming

of Sheets Made from Modified Cellulose Fibers”, Internal Report No. 607,

Department of Solid Mechanics, KTH Royal Institute of Technology,

Stockholm.

In addition to the appended papers, the thesis work has also resulted in

the following publications and conference presentations.

E. Linvill (2015) “Box Compression Strength: A Crippling Approach”,

Packaging Technology and Science, 28(12):1027-1037.

E. Linvill, S. Östlund (2016, August) “Biaxial (In-Plane) Failure and

Yield of Paperboard” In S. Schabel (Chair), Progress in Paper Physics

Seminar. Symposium conducted in Darmstadt, Germany.

M. Wallmeier, E. Linvill, M. Hauptmann (2016, August) “The Effect of

Inhomogeneous Material Properties in Explicit Dynamic Simulation of

Paperboard Forming” In S. Schabel (Chair), Progress in Paper Physics

Seminar. Symposium conducted in Darmstadt, Germany.

E. Linvill, M. Wallmeier, S. Östlund (2015, October) “3-D Forming of

Paper and Paperboard: Parametric Studies through the Use of Explicit

Finite Element Simulations” In A. Isogai (Chair), Progress in Paper

Physics Conference. Symposium conducted in Tokyo, Japan.

E. Linvill, S. Östlund (2015, March) “Explicit Finite Element Simulation

of 3D Forming Processes for Paperboard” In J.P. Majschak (Chair), VVD

2015 Verarbetungsmaschinen und Verpackungstechnik. Symposium

conducted in Dresden, Germany.

E. Linvill, S. Östlund (2016, September) “3-D Formability of Handsheets

at Various Moisture and Temperature Conditions” In J. Pawlak (Chair),

Progress in Paper Physics Seminar. Symposium conducted in Raleigh,

North Carolina, USA.

M. Wallmeier, E. Linvill, S. Östlund (2016, September) “Experimental

and Numerical Examination of Deep-Drawn Paperboard Cups” In J.

Pawlak (Chair), Progress in Paper Physics Seminar. Symposium

conducted in Raleigh, North Carolina, USA.

E. Linvill, D.D. Tjahjanto, S. Östlund (2014, July) “The Effects of

Moisture and Temperature on the Mechanical Response of Cellulose-

Based Materials: From Paper to Pressboard” In J. Barton (Chair), 16th

International Conference on Experimental Mechanics, Symposium

conducted in Cambridge, UK.

Contribution to the papers

Paper A

E. Linvill designed the experimental method, conducted the testing,

post-processed the testing data, wrote the manuscript, and acted as

corresponding author for the manuscript.

Paper B

E. Linvill developed the experimental method, conducted the testing,

post-processed the testing data, wrote the manuscript, and acted as

corresponding author for the manuscript.

Paper C

E. Linvill developed the material model, implemented the material

model subroutine in LS-DYNA, and conducted the uniaxial and biaxial

validation of the material model. E. Linvill and M. Wallmeier shared

equally the work to create and execute the finite element model of the

deep-drawing process as well as the writing of the manuscript. M.

Wallmeier was the corresponding author for the manuscript.

Paper D

E. Linvill developed the material model, implemented the material

model subroutine in LS-DYNA, conducted the validation of the new

aspects of the material model, executed the finite element simulation of

the deep-drawing process, wrote the manuscript, and acted as

corresponding author for the manuscript.

Paper E

E. Linvill developed the material model, implemented the material

model subroutine in LS-DYNA, and created and executed the finite

element model of the hydroforming process. Additionally, E. Linvill

developed the experimental method, conducted the testing, post-

processed the testing data, wrote the manuscript, and acted as

corresponding author for the manuscript.

Paper F

E. Linvill developed the finite element models, developed the method

to obtain an in-plane failure surface from tensile test data, processed

experimental data, wrote the manuscript, and acted as corresponding

author for the manuscript. P. Larsson and E. Linvill shared equally the

work of mechanical testing, and P. Larsson conducted the chemical

modification and prepared the samples.

Acknowledgements

This work is the product of a very interesting, interdisciplinary,

international collaboration. I would like to thank my supervisor, Sören

Östlund, as well as BiMaC Innovation for setting up the framework for

this project and the collaborations therein; I have greatly enjoyed the

journey. Additionally, I would like to thank my “processing machines”

counterpart Malte Wallmeier for the assistance and company on a variety

of very interesting and fruitful projects. Furthermore, I would like to

thank my “fibre and polymer chemistry” collaborator Per Larsson for

expanding my horizons into the world of chemistry and for always

coming with ideas and energy (and, not to mention, such a fun material to

experiment with).

Beyond the framework, I would also like to thank Sören Östlund for so

many interesting philosophical and technical discussions. You provided

me with the freedom to explore certain areas within the framework

provided but kept me focused on the important goals when I began

venturing off track. I greatly appreciate your supervision and support

throughout this work.

Special thanks are extended to the industrial partners of BiMaC

Innovation as well as Vinnova for sponsoring this work as well as for

providing a practical perspective and outlet for this research.

Finally but in no way least, I would like to thank my family and friends

for their support and company throughout this work. I would like to

thank my parents, Beth and Tom, for encouraging me to chase my dreams

no matter where they might take me. I would like to thank my brothers,

Tyler and Bret (in order of age, not relative importance), for keeping a

humorous face and always cheering me up. Finally, I would like to thank

Camilla and Franzi for making the past four-and-a-half years very special

and memorable. There are many friends who also deserve my gratitude

for their support and company and will receive those thanks from me in

person.

Nomenclature

Abbreviations

3-D .................................................................................. Three Dimensional

CD .......................................................................... Cross-Machine Direction

MD ................................................................................... Machine Direction

ZD .................................................................. Through-Thickness Direction

FOB ......................................................................................... Free On Board

Latin Symbols

k ................. Size of the Yield Surface ...................................................... [-]

𝑚𝑐 .............. Moisture Content ................................................................ [-]

𝐶1 ................ 1st Coefficient of the Hyperbolic Tangent Function ..... [MPa]

𝐶2 ................ 2nd Coefficient of the Hyperbolic Tangent Function ......... [-]

𝐶3 ................ 3rd Coefficient of the Hyperbolic Tangent Function .... [MPa]

𝐸 ................. Elastic Modulus ............................................................. [MPa]

𝑀𝑚𝑜𝑖𝑠𝑡 ......... Mass of Moisture within Paper ......................................... [kg]

𝑀𝑡𝑜𝑡 ............ Total Mass of Paper with Moisture .................................. [kg]

Greek Symbols

𝜀 ................. Strain .................................................................................... [-]

ν ................. Poisson’s Ratio .................................................................... [-]

𝜎 ................. Normal Stress ................................................................ [MPa]

𝜏 ................. Shear Stress ................................................................... [MPa]

Subscripts

𝑐 Compressive 𝑓 Failure

ℎ Hinge 𝑡 Tensile

𝑦 Yield 𝐶𝐷 Cross-Machine Direction 𝑀𝐷 Machine Direction

Superscripts

𝑝 Plastic

Contents

1. Introduction ............................................................................... 1 1.1. Objectives ............................................................................................... 2 1.2. Structure ................................................................................................. 3

2. Paper ........................................................................................ 4 2.1. Kraft Fiber Handsheets ........................................................................... 5 2.2. Kraft Fiber Handsheets with Fibers Partially Oxidized to Dialcohol

Cellulose ......................................................................................................... 6 2.3. Paperboard ............................................................................................. 6 2.4. Paper versus Other Packaging Materials ............................................... 7

3. Experimental Mechanical Characterization ................................ 9 3.1. The Effects of Moisture and Temperature .............................................. 9 3.2. Biaxial Failure and Yield ....................................................................... 10

4. Material Models ....................................................................... 13 4.1. Orthotropic Elastic-Plastic Model .......................................................... 13 4.2. Wrinkling in the Continuum ................................................................... 18 4.3. Validation .............................................................................................. 20

5. 3-D Forming of Paper .............................................................. 21 5.1. Deep Drawing ....................................................................................... 22 5.2. Hydroforming ........................................................................................ 24

6. Models of 3-D Forming of Paper .............................................. 25 6.1. Deep Drawing ....................................................................................... 25 6.2. Hydroforming ........................................................................................ 26

7. Conclusions ............................................................................. 28

8. Recommendations for Future Work ......................................... 30

References .................................................................................. 32

INTRODUCTION | 1

1. Introduction

Paper materials, which are for the sake of this thesis defined as any

sheet structure which is produced from wood fibers prepared for

papermaking, are one of the oldest known man-made materials. Paper as

it is today has developed through the times as follows [1]:

105 AD: de-fibered bamboo was utilized during the Chinese

Han Dynasty to create a variety of papers, including sized,

coated, and dyed papers.

610 AD: beating processes were introduced and papermaking

(utilizing cotton fibers) began to spread to spread through

Asia and Europe.

14th Century: technological advances are made, including the

use of water-mill power.

16th Century: widespread use of water-mill power and

improvements in machinery improve the paper-making

process.

17th and 18th Century: large-scale production begins. Shortages

of cotton fibers force paper producers to start using tree-based

(cellulose) fibers.

19th Century: wood becomes the primary raw material for

papermaking, and industrialization motivates the creation of

larger machines which run at greater speeds.

20th Century: Paper machines become fully automated and

become even larger and faster than the previous century.

Paper materials have a multi-century-long history of use as a

packaging material. The first known commercial cardboard box was

produced in 1817 in England as a replacement for wooden crates and

boxes that were in use during that time [2]. With the invention of the

gable top milk carton in 1915 [3], the use of cardboard boxes to contain

Kellogg’s ® Corn Flakes ® in 1906 [4], and the development of

continuous production methods in 1956 and aseptic paper-based

2 | INTRODUCTION

packaging in 1969 by Tetra Pak [5], paper materials became increasingly

more common in western society, especially as a food packaging product.

The dawn of plastics and plastic food packaging started with the

patent of Polymeric Vinylidene Chloride in 1939 [6], which then led to the

development of Saran wrap in 1949 by Dow Chemical [7]. Plastic-based

packaging products continued to expand until the dawn of the 21st

century, at which point environmental and sustainability concerns

regarding plastic packaging began to gain widespread public attention. In

recent years, an increased focus has been placed on food packaging

products which have a cradle-to-grave environmental impact that is as

small as possible, and consumers around the world perceive paper

materials as significantly more environmentally friendly than plastics [8].

Plastics have the benefit of being easily formed into almost any complex

shape, while paper products have typically only been successfully formed

into geometrically simple shapes. In order to compete with plastic-based

packaging, paper products will need to address this significant limitation.

The subsequent sub-sections respectively present the overarching

objectives of this work as well as the structure of the thesis.

1.1. Objectives

This work is directed towards the understanding of three-dimensional

(3-D) forming processes (see Section 5 for a definition and description of

3-D forming processes for paper materials). This thesis aims to answer, at

least to some degree, the following three overarching research questions:

1. How 3-D formable are paper materials?

2. How can paper materials be improved for specific 3-D

forming processes?

3. How can specific 3-D forming processes for paper materials be

improved?

The answers to these questions were pursued through a scientific

approach with both experimental and numerical solid mechanics as the

primary tools. By combining experimental and numerical solid

mechanics, simulations of various complex 3-D forming processes can be

both modeled and validated. The validated simulations can then be

utilized in parametric studies to investigate the independent effects of

each parameter on each process, which can be difficult or nearly

impossible in an experimental framework.

INTRODUCTION | 3

1.2. Structure

The following sections of this thesis are organized in such a way as to

provide the reader with an overall picture and purpose of the work

presented in Papers A-F. Papers A-B are purely experimental studies,

whilst Papers C-F combine numerical and experimental solid mechanics.

After this introductory chapter, a chapter is provided which presents

paper as an engineering material and presents the types of paper utilized

through this work. The third chapter discusses the experimental

mechanical characterization of paper materials developed and conducted

throughout this work; these characterizations were necessary to develop

increased understanding of and model the deformation and failure of

paper materials during 3-D forming processes. The fourth chapter

describes the framework for the various material models that were

developed. The fifth chapter introduces the types of 3-D forming

processes for paper materials that have been of recent interest in the

literature and provides illustration and discussion of the 3-D forming

processes studied in this thesis. The sixth chapter presents the finite

element models that were developed for the 3-D forming processes as

well as some key results. The seventh and eighth chapters respectively

present the major conclusions from this work and provide suggestions on

possible future extensions of this work. Papers A-F are respectively

attached as appendices and contain a majority of the scientific work of

this thesis.

4 | PAPER

2. Paper

Modern paper is produced on large-scale paper machines which range

up to 600 meters long and have production speeds of up to 2.2 kilometers

of paper per minute. The nature of this highly dynamic process has an

effect on the final mechanical properties of a commercially produced

paper sheet. A paper sheet is typically defined in its three approximately

orthogonal material directions: the machine direction (MD), the cross-

machine direction (CD), and the through-thickness direction (ZD). An

illustrative example of these three directions and how they relate to the

paper machine that the paper was produced on is shown in Figure 1.

Figure 1: The three orthogonal directions in paper – displayed in the paper machine

The green rolls in Figure 1 pull the paper sheet through the paper

machine in MD (as indicated by the black arrows in Figure 1). Speed

differences in the rolls can be utilized to strain the paper in MD. Drying in

PAPER | 5

such a strained condition (i.e. restraint drying) can create an increased

stiffness and strength in MD.

Paper is a composite material composed of fibers (each of which is a

multi-layer composite material composed of cellulose, hemicellulose, and

lignin [9]), fines (a non-homogeneous collection of particles, including

bands, flakes, and ray cells [10]), and pores. However, the fibers

themselves are the primary load-carrying part of the structure. Tree-

based fibers typically have lengths from 1 to 5 mm, widths from 10 to 40

µm, wall thicknesses from 2 to 10 µm, and hollow oval cross-sections.

However, during the drying part of the papermaking process, the hollow

oval cross-sections of these tree fibers typically collapse. These fibers are

quasi-randomly distributed throughout the sheet structure, and most

fibers are oriented within the in-plane (MD-CD) directions. Within the

in-plane (MD-CD) directions, fibers tend to have a bias towards MD; this

orientation causes MD to typically be stiffer and stronger than CD.

In the ZD direction, the fiber-fiber joints are the primary tensile-load-

carrying structure. In ZD tension, these joints are much weaker and more

compliant than the stiffness of the fibers for in-plane loading. In ZD

compression, the material response is driven by compression of the

fibers. The fibers are much more compliant in ZD compression than they

are during in-plane loading. Generally, the stiffness of paper in the in-

plane (MD-CD) directions is typically at least a magnitude of ten times

greater than the stiffness in the out-of-plane directions [11], making paper

a highly anisotropic material. This significant difference between the in-

plane (MD-CD) and out-of-plane stiffness is a primary reason why an in-

plane plasticity model with out-of-plane elasticity is utilized in this work.

The subsequent sub-sections present the specific types of paper

materials utilized in this work, including kraft fiber handsheets, kraft

fiber handsheets with fibers partially oxidized to dialcohol cellulose, and

commercial paperboard.

2.1. Kraft Fiber Handsheets

In Papers A & E, handsheets produced from Celeste kraft pulp from

SCA (Svenska Cellulosa Aktiebolaget) were utilized. This pulp was beaten

at 160 kWh/t, and the handsheets were made with white water

recirculation on the Rapid Köthen laboratory sheet former in the

Department of Fibre and Polymer Technology at KTH Royal Institute of

Technology. This beating level in conjunction with white water

recirculation provided the best strain-at-break properties for this pulp.

High strain-at-break properties were desired in order to provide the best

possible 3-D formability. The same beaten pulp was utilized in the

investigation of the 3-D formability of chemically modified paper

materials as is described in the following section.

6 | PAPER

2.2. Kraft Fiber Handsheets with Fibers Partially Oxidized

to Dialcohol Cellulose

In Paper F, the same beaten kraft pulp was utilized as in Papers A & E.

Prior to the handsheet-making process, the fibers in the pulp were

partially oxidized to dialcohol cellulose [12]. The purpose of the

modification was to dramatically increase the strain-at-break and thus 3-

D formability of the handsheets.

2.3. Paperboard

Paperboard is a type of paper which often consists of multiple layers.

The utilization of multiple layers allows one to optimize the bending

strength, print surface, and appearance. A visual representation of one

type of paperboard construction is provided in Figure 2.

Figure 2: Multi-layer paperboard construction

Typically, the outer layers of paperboard, which are represented by the

brown, thinner layers in Figure 2, have stiffer fibers with better

appearance properties than the middle layer, which is represented by the

green, thicker layer in Figure 2. The middle layer is typically bulkier and

more compliant than the outer layers, because the middle layer has less

effect on the bending stiffness. Additionally, since the middle layer is

hidden from view, the middle layer typically does not have to meet any

visual requirements.

The experimental study of the biaxial in-plane yield and failure

surfaces in Paper B utilized paperboard, because the thicker grades of

PAPER | 7

paperboard are sufficiently thick for utilization with laser engraving. The

paperboard utilized in Paper B had a thickness of 0.7 mm.

Due to the effectiveness of paperboard as a packaging material and its

superior visual quality, Papers C & D focus on the deep-drawing of

paperboard. Trayforma 350 from Stora Enso was selected for both studies

of deep-drawing. Unpublished experimental experience at TU Dresden

has shown that Trayforma 350 performs well in the deep-drawing

process.

2.4. Paper versus Other Packaging Materials

Paper materials are typically not as 3-D formable as other traditional

packaging materials such as some types of plastic or aluminum. However,

paper materials have many advantages as compared to other packaging

materials, and a comparison between paperboard and other typical

packaging materials is provided in Table 1.

8 | PAPER

Table 1: Comparison between paperboard and other typical packaging materials (many values are estimated from the provided references)

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EXPERIMENTAL MECHANICAL CHARACTERIZATION | 9

3. Experimental Mechanical Characterization

The subsequent sub-sections present the experimental techniques

developed as well as key results obtained for the two experimental studies

conducted during this work: the combined effects of moisture and

temperature on the mechanical response of paper handsheets, and the

biaxial failure and yield of paperboard.

3.1. The Effects of Moisture and Temperature

The constitutive behavior of paper materials is known to be sensitive

to both moisture and temperature, and Paper A contains a review of

published papers on the effects of moisture and temperature.

The definition of the moisture content can vary, but a consistent

definition of moisture content has been utilized throughout this work as

follows:

𝑚𝑐 =𝑀𝑚𝑜𝑖𝑠𝑡𝑢𝑟𝑒

𝑀𝑡𝑜𝑡 (1)

where 𝑚𝑐 is the moisture content, 𝑀𝑚𝑜𝑖𝑠𝑡𝑢𝑟𝑒 is the mass of moisture

contained within the paperboard, and 𝑀𝑡𝑜𝑡 is the total mass of the

paperboard including the moisture.

A new experimental setup was created in Paper A to study the

combined effects of moisture and temperature on the uniaxial, tensile

mechanical response of paper. The same experimental setup was utilized

in Paper F to study the combined effects of moisture and temperature on

the mechanical response of chemically modified paper. The experimental

setup utilized pre-conditioning of samples at a given relative humidity

and heating of the samples with a heating lamp during the tensile test.

In Paper A, moisture and temperature was found to have uncoupled

effects on the elastic modulus, tangent modulus, hardening modulus,

strain-at-break, and tensile energy absorption. In Paper F, moisture and

temperature was found to have uncoupled effects on all parameters

studied (i.e. elastic modulus, tangent modulus, and yield strength). In this

10 | EXPERIMENTAL MECHANICAL CHARACTERIZATION

context, uncoupled effects means that only three (as opposed to four)

coefficients are required to calculate the material property at a given

moisture content and temperature (within the studied ranges). See

Papers A & F for further discussion of the definition of coupled or

uncoupled effects.

The following hyperbolic tangent function was utilized to fit the tensile

test data:

𝜎 = 𝐶1 tanh(𝐶2𝜀) + 𝐶3 𝜀 (2)

where 𝜎 is the stress, 𝐶1−3 are curve-fitting constants, and 𝜀 is the total

strain. This hyperbolic tangent function fit all experimental results in

Papers A & F with an R-squared value of greater than 0.98, and

utilization of this hyperbolic tangent function allowed for systematic,

simple, and unbiased determination of material properties and the effects

of moisture and temperature on those material properties.

3.2. Biaxial Failure and Yield

The biaxial failure and yield of paper materials was directly (i.e.

experimentally) studied in Paper B and indirectly (i.e. combining

experimental and numerical methods) in Paper F. This section, however,

presents complete in-plane failure surfaces at zero in-plane shear stress

for different materials and utilizes those surfaces to analyze the 3-D

formability of the respective materials.

The experimental biaxial study in Paper B determined, among many

other conclusions, that the shapes of the yield and failure surfaces are

similar. This similarity, along with the assumption of a maximum stress

yield surface (discussed in more detail in Section 4.1), allows for direct

estimation of the strain-based failure surface (i.e. forming limit diagram)

from the uniaxial results, as conducted in Paper F. For an orthotropic

material with a maximum stress yield surface and zero in-plane shear

stress, the strain-based failure surface can be assumed to be composed of

the four following linear equations:

𝜀𝐶𝐷 = −ν𝑀𝐷−𝐶𝐷𝜀𝑀𝐷 + 𝜀𝐶𝐷,𝑓,𝑡(1 − ν𝑀𝐷−𝐶𝐷2) (3)

𝜀𝐶𝐷 = −ν𝑀𝐷−𝐶𝐷𝜀𝑀𝐷 + 𝜀𝐶𝐷,𝑓,𝑐(1 − ν𝑀𝐷−𝐶𝐷2) (4)

𝜀𝐶𝐷 = −1

ν𝐶𝐷−𝑀𝐷𝜀𝑀𝐷 + 𝜀𝑀𝐷,𝑓,𝑡 (

1

ν𝐶𝐷−𝑀𝐷− ν𝐶𝐷−𝑀𝐷) (5)

𝜀𝐶𝐷 = −1

ν𝐶𝐷−𝑀𝐷𝜀𝑀𝐷 + 𝜀𝑀𝐷,𝑓,𝑐 (

1

ν𝐶𝐷−𝑀𝐷− ν𝐶𝐷−𝑀𝐷) (6)

where 𝜀𝐶𝐷 and 𝜀𝑀𝐷 are respectively the CD and MD strains, ν𝑀𝐷−𝐶𝐷 and

ν𝐶𝐷−𝑀𝐷 are the Poisson’s ratios during tensile loading respectively in CD

EXPERIMENTAL MECHANICAL CHARACTERIZATION | 11

and MD, 𝜀𝐶𝐷,𝑓,𝑡 and 𝜀𝐶𝐷,𝑓,𝑐 are respectively the tensile and compressive CD

strain-at-break, and 𝜀𝑀𝐷,𝑓,𝑡 and 𝜀𝑀𝐷,𝑓,𝑐 are respectively the tensile and

compressive MD strain-at-break. Strain-based failure surfaces generated

from (3)-(6) for four different materials (i.e. reference handsheets,

chemically modified handsheets, Trayforma commercial paperboard, and

Invercote G commercial paperboard) are shown in Figure 3 and Figure 4.

Interpretations of the strain-based failure surface for various 3-D forming

operations are provided in Figure 5 and Paper B. For reference, Figure 5

also includes the strain path for tensile loading in MD and CD, which are

respectively represented by blue and purple/magenta lines.

Figure 3: Estimated strain-based failure surfaces (zero shear strain) for different materials at 100 °C

12 | EXPERIMENTAL MECHANICAL CHARACTERIZATION

Figure 4: Estimated strain-based failure surfaces (zero shear strain) for different materials at 100 °C, zoomed in on materials without chemical modification

Figure 5: Interpretation of the strain-based failure surface for 3-D forming, where the red/brown area represents possible initiation of wrinkling

MATERIAL MODELS | 13

4. Material Models

Material models were developed in order to capture permanent

deformations during the simulated 3-D forming processes in Papers C-F.

The foundation of these models is based on the assumption of orthotropic

material directions with in-plane (MD-CD), linear, quadrant hardening.

Quadrant hardening is the name given to hardening which only affects

the parts of the yield surface that enter the quadrant of the current

normal stress state.

The material properties were assumed to be affected by temperature

and moisture based on the experimental method in Paper A and

experimental results in Papers A & F.

The subsequent sub-sections provide: an overview of the orthotropic

elastic-plastic model with in-plane (MD-CD), linear, quadrant hardening,

an overview of the inclusion of wrinkle development and compression in

a continuum method, and a presentation of the utilized validations of

these models.

4.1. Orthotropic Elastic-Plastic Model

A few key assumptions were utilized to derive the in-plane quadrant

hardening model:

1. Orthotropic material behavior

2. No coupling between in-plane (MD-CD) and out-of-plane

behavior

3. In-plane maximum stress yield surface with quadratic

dependency on in-plane (MD-CD) shear stress

4. In-plane (MD-CD) linear quadrant (anisotropic) hardening

5. Linear elastic out-of-plane behavior

Paper C presents and discusses these assumptions. Although many of the

assumptions listed above are common for paper materials, this section

14 | MATERIAL MODELS

introduces those parts that are unique to this model (i.e. quadrant

hardening and the yield surface).

Due to the extensive experimental scatter in the yield point, a simple

stress-based yield surface was chosen: an in-plane (MD-CD) maximum

normal stress yield surface with quadratic dependence on the in-plane

shear stress. This surface is supported by the stress-based yield surface

obtained in Paper B as well as by the stress-based failure surface obtained

by Suhling et al. [19] (which exhibited quadratic dependence on in-plane

shear stress). The selected stress-based yield surface is shown in Figure 6.

Bisections of the stress-based yield surface at constant in-plane shear

stresses are projected onto a contour map in Figure 6.

Figure 6: Yield surface for an in-plane (MD-CD) stress state

At zero in-plane shear stress, Figure 6 becomes the two-dimensional yield

surface with four sub-surfaces in Figure 7. The four subsurfaces can be

described by:

𝑘𝑖 =𝜎𝑖

𝜎𝑦𝑖

+ (𝜏

𝜏𝑦)

2

(7)

where 𝑘𝑖 begins at a value of unity and expands with the yield surface for

each independent sub-surface, 𝜏 is the in-plane (MD-CD) shear stress

state, 𝜏𝑦 is the in-plane (MD-CD) shear yield stress, and 𝜎𝑖 and 𝜎𝑦𝑖 are

defined in Table 2.

MATERIAL MODELS | 15

Figure 7: Yield surface for a value of in-plane (MD-CD) shear stress

Table 2: Definition of the normal stress terms in (7)

𝒊 𝝈𝒊 𝝈𝒚𝒊

1 Tensile MD Stress Initial Tensile MD Yield Stress

2 Tensile CD Stress Initial Tensile CD Yield Stress

3 Compressive MD Stress Initial Compressive MD Yield Stress

4 Compressive CD Stress Initial Compressive CD Yield Stress

The anisotropic (non-associative) hardening behavior utilized in this

model was supported by experimental evidence in Paper C, in which

purely tensile loading in MD or CD only caused changes in the yield

surface in MD or CD respectively. Since paper is a fibrous material, small-

deformation plastic straining in one direction would likely not cause

plastic straining in an orthotropic direction. Conflicting evidence for this

assumption was found by Borgqvist et al. [20] utilizing a different

experimental method. However, the experimental method utilized by

Borgqvist et al. [20] subjected the specimens to biaxial loads (i.e. not

purely tensile loads) during the pre-stress portion of the experiments due

to the test piece dimensions (i.e. 160 mm wide and 145 mm between the

clamps), which could have significant effect on the results.

Quadrant hardening was chosen as the anisotropic (non-associative)

hardening behavior, and quadrant hardening means that the expansion of

the yield surface only occurs for those parts of the yield surfaces lying in

the same quadrant as the normal stress state. For example, in bi-axial

16 | MATERIAL MODELS

tension, only sub-surfaces 1 and 2 in Figure 7 expand. Because the

hardening behavior utilizes a non-associative plastic flow potential, the

corners of the yield surface shown in Figure 6 and Figure 7 do not affect

the solution uniqueness as would an associative plastic flow potential.

One property of quadrant hardening is that any loading in purely MD

or CD only causes yielding (i.e. change of the yield surface) in the

respective direction. This property means that transitions between

quadrants are continuous.

Linear orthotropic hardening during inelastic loading is assumed. In

the first quadrant, this becomes:

𝑑𝜎1 = 𝐻𝑀𝐷,𝑡𝑑𝜀𝑀𝐷𝑝

(8)

𝑑𝜎2 = 𝐻𝐶𝐷,𝑡𝑑𝜀𝐶𝐷𝑝

(9)

𝑑𝜏 = 𝐻𝑀𝐷−𝐶𝐷𝑑𝜀𝑀𝐷−𝐶𝐷𝑝

(10)

where 𝑑𝜎 is the stress increment, 𝐻𝑀𝐷,𝑡 is the tensile MD hardening slope,

𝐻𝐶𝐷,𝑡 is the tensile CD hardening slope, 𝐻𝑀𝐷−𝐶𝐷 is the in-plane shear

hardening slope, and 𝑑𝜀𝑝 is the plastic strain increment. The definition of

hardening according to (8)-(10) in combination with Hooke’s Law and an

additive split of elastic and plastic strains allow the plastic strain

increment to be directly solved for as follows (note that this derivation is

identical to but presented slightly different than in Paper B):

𝑑𝜀𝑀𝐷𝑝

=(𝐶11−

𝐶122

𝐻𝐶𝐷,𝑡+𝐶22) 𝑑𝜀𝑀𝐷+(𝐶12−

𝐶22𝐶12𝐻𝐶𝐷,𝑡+𝐶22

) 𝑑𝜀𝐶𝐷

𝐻𝑀𝐷,𝑡+𝐶11− 𝐶12

2

𝐻𝐶𝐷,𝑡+𝐶22

(11)

𝑑𝜀𝐶𝐷𝑝

=(𝐶22−

𝐶122

𝐻𝑀𝐷,𝑡+𝐶11) 𝑑𝜀𝐶𝐷+(𝐶12−

𝐶11𝐶12𝐻𝑀𝐷,𝑡+𝐶11

) 𝑑𝜀𝑀𝐷

𝐻𝐶𝐷,𝑡+𝐶22− 𝐶12

2

𝐻𝑀𝐷,𝑡+𝐶11

(12)

𝑑𝜀𝑀𝐷−𝐶𝐷𝑝

=𝐶44

𝐻𝑀𝐷−𝐶𝐷+𝐶44𝑑𝜀𝑀𝐷−𝐶𝐷 (13)

where 𝐶𝑖𝑗 are the components of the elastic stiffness tensor and 𝑑𝜀 is the

total strain increment. Combining equations (11)-(13) with (8)-(10) allows

the stress increments to be directly calculated from the strain increment

during inelastic loading in the first quadrant. Note that the yield surface

is not utilized in the calculation of the stress increment. Therefore, the

stress increment is independent from the definition of the yield surface.

With the definition of the yield surface according to (7), the expansion

of the yield surface can be determined by:

MATERIAL MODELS | 17

𝑑𝑘1 =𝑑𝜎1

𝜎𝑦1

+2𝜏𝑑𝜏

𝜏𝑦2 (14)

𝑑𝑘2 =𝑑𝜎2

𝜎𝑦2

+2𝜏𝑑𝜏

𝜏𝑦2 (15)

where 𝑑𝑘 is the change in the size of the yield surface.

Although (8) and (9) are simply based on tensile tests, plastic

straining for in-plane shear (10) has a less trivial interpretation. For the

case of off-axis tensile tests, the tangent modulus in the off-axis direction

becomes a closed-form function of all three hardening moduli (i.e. 𝐻𝑀𝐷,𝑡,

𝐻𝐶𝐷,𝑡, and 𝐻𝑀𝐷−𝐶𝐷) as well as the elastic stiffness constants. The tangent

modulus is typically greatest near or at MD and least near or at CD. In

order to capture this behavior, the in-plane shear hardening slope should

be well defined. Utilizing the material properties from Paper C, one

example of a well-defined and two examples of poorly-defined in-plane

shear hardening slopes are illustrated in Figure 8. Therefore, the shear

hardening slope and (10) can be interpreted as functions for describing

how the hardening slope changes with off-axis angle.

Figure 8: Three examples of the tangent modulus for uniaxial loading in off-axis directions utilizing material properties from Paper C; the middle, green curve is well-defined, because the maximum tangent modulus occurs in MD and the minimum tangent

modulus occurs in CD

18 | MATERIAL MODELS

4.2. Wrinkling in the Continuum

Paper D introduces a constitutive model for wrinkled paperboard in

the continuum, including large deformation and size effects. Wrinkled

paperboard is assumed to have a response equal to that of a perfectly

plastic hinge (i.e. the moment in the hinge is constant during permanent

deformation), which was validated for three types of paperboard and a

variety of simple experiments. The reader is directed to Paper D to learn

the details of the model and the validation.

This section specifically examines the wrinkling point in order to

provide a physical interpretation of the hinge yield strength, which is the

material property that determines the mechanical response of wrinkled

paperboard. The expression for the wrinkling point is given by the one-

dimensional equation in the direction of minimum in-plane principal

stress:

𝜀ℎ2 =

𝜎ℎ,𝑦

2𝐸ℎ(1 + 𝑒−2𝜀ℎ√1 − 𝑒2𝜀ℎ + tan−1(1 − 𝑒2𝜀ℎ) − 𝑒−2𝜀ℎ) (16)

where 𝜀ℎ is the hinge strain (i.e. a negative number), 𝜎ℎ,𝑦 is the hinge yield

strength (i.e. the material property defining the mechanical response of

the wrinkles), and 𝐸ℎ is the elastic stiffness in the direction of wrinkle

compression. From this expression, the effect of the elastic and hinge

material properties on the wrinkle initiation strain can directly be solved,

and the result is shown in Figure 9.

Figure 9: Compressive wrinkle initiation strain as a function of material properties

MATERIAL MODELS | 19

As is clearly illustrated in Figure 9, lesser hinge yield strengths result

in wrinkling at lower levels of strain. Alternatively, greater hinge yield

strengths result in wrinkle initiation at greater compressive strains.

Therefore, the hinge yield strength provides information about the

resistance to initiation and subsequent compression of wrinkles.

Generally, deep-drawn paperboard cups with greater number of

wrinkles have a better visual quality than the deep-drawn cups with lesser

number of wrinkles [21], thus implying that lower hinge yield strength

could be beneficial. However, paperboard is far from homogeneous, and

deviations in hinge yield strength would cause greater deviations in

wrinkle initiation strain at lesser hinge yield strengths than at greater

hinge yield strengths, as shown in Figure 9. An important quality

parameter for deep-drawn paperboard cups is the evenness of wrinkling

[21]. Therefore, the optimal hinge yield strength for deep-drawing would

fulfill two criteria: constant ratio (𝜎ℎ,𝑦 𝐸ℎ⁄ ) between hinge yield strength

and elastic modulus for all in-plane directions and smallest deviations of

that ratio.

The ratio of the hinge yield strength and elastic modulus of the three

paperboards utilized in Paper D to validate the plastic hinge assumption

are shown in Table 3.

Table 3: Ratios of hinge yield strength and elastic modulus for three types of paperboard in MD and CD.

Paperboard 𝝈𝑴𝑫,𝒉,𝒚

𝑬𝑴𝑫

𝝈𝑪𝑫,𝒉,𝒚

𝑬𝑪𝑫

% Difference

Trayforma Natura 350 0.0049 0.0040 23

Cupforma Natura 330 0.0020 0.0035 75

Invercote G 350 0.0031 0.0050 61

Historically, TU Dresden has had most success deep-drawing

Trayforma Natura 350. The greater success with Trayforma Natura 350

could be due to the fact that it has relatively small differences between the

MD and CD ratios between the hinge yield strength and the elastic

modulus, as is shown in Table 3, thus encouraging consistent and well-

distributed wrinkling.

20 | MATERIAL MODELS

4.3. Validation

In addition to presenting the material model, Papers C-F include also

validation of the utilized material models. These methods included:

Off-axis tensile testing (Paper C)

Tensile testing the anisotropic hardening (Paper C)

Three-point bending tests (Paper C)

Compressive testing of the plastic hinge assumption (Paper D)

Tensile testing of wrinkled paperboard (Paper D)

Off-axis tensile testing of wrinkled paperboard (Paper D)

Deep drawing

o Punch force (Papers C & D)

o Wrinkle number (Paper D)

o Springback and its asymmetry (Paper D)

Hydroforming (Papers E & F)

o Final forming depth (Papers E & F)

o Final forming volume (Paper E)

o Failure (Paper F)

The model was capable of matching well the experimental results with the

exception of predicting the bending response of paperboard in MD and

obtaining precise values for the 3-D forming processes. Validation of the

model in bending was carried out for a paperboard material. The

continuum model did not account for the composite structure of

paperboard which maximizes bending stiffness as illustrated in Figure 2,

which is why the model underestimated the bending stiffness. The

comparison between the simulated and experimental deep drawing and

hydroforming of paper materials was capable of showing that the model

captured the trends and approximate values for the respective processes.

The minor differences between simulations and experimental forming are

not surprising, because these 3-D forming processes consist of numerous

relatively unknown process parameters (e.g. coefficients of friction) and

tend to exhibit significant experimental variability. Therefore, the extent

to which these 3-D forming processes can be utilized to validate the

material model is somewhat limited.

3-D FORMING OF PAPER | 21

5. 3-D Forming of Paper

3-D forming of paper materials is defined as the production of

advanced, 3-D paper structures without utilizing traditional creasing and

folding techniques.

Traditional paper-based packaging products are mostly limited to

folded box structures or sacks. In other words, traditional paper-based

packaging tends to have a limited package shape. Especially when

compared to other packaging materials (e.g. plastics and aluminum

alloys), paper is limited in the extent to which it can be formed into

advanced, 3-D structures. The development of 3-D forming techniques

specifically designed for paper materials, however, could increase the

number of possible paper-based packages and thus increase the

competitiveness of paper-based packaging.

Vishtal [22] introduced a useful classification system for

distinguishing 3-D forming processes into two groups: sliding blank and

fixed blank processes. A sliding blank process is a process during which

the blank (i.e. the paper specimen) is allowed to slide in between the die

and blankholder. A fixed blank process is a process during which the

blank is held in place between the die and blankholder. Some types of 3-D

forming processes for paper materials are presented in Table 4.

Table 4: Types of 3-D forming processes for paper materials

Sliding Blank Fixed Blank

Deep Drawing Hydroforming

Stamping/Press Forming Blow/Vacuum Forming

Of the four processes shown in Table 4, deep drawing and hydroforming

were selected as the focus of this work. The deep-drawing process is

currently one of the most developed 3-D forming processes for traditional

paper materials and is even currently utilized in limited commercial

processes. Recent research and development of this process at TU

Dresden has realized a significant new potential for this process [23]-

[24], although improved understanding of the effects of material

22 | 3-D FORMING OF PAPER

properties and process parameters on quality measures for deep-drawn

paperboard cups (e.g. failure probability [25], springback, and wrinkle

distribution and number [21]) are necessary. Because experimental

determination of the effects on these quality measures has been difficult,

this work has studied these quality measures utilizing an explicit finite

element approach.

As opposed to the sliding blank processes in Table 4 for which

matching dies and blanks must be simultaneously produced for each

different 3-D packaging shape, the fixed blank processes in Table 4

require only one die to be produced for each shape. The paper blank in

blow/vacuum forming is directly subjected to an air pressure differential,

but the effectiveness of this pressure differential depends on the porosity

of the material. During hydroforming, on the other hand, a membrane

(e.g. rubber) is utilized to engage the pressure differential and transmit

the load from this pressure differential to the blank. Due to the porous

nature of paper materials and the simplicity of creating tooling for fixed

blank processes, the hydroforming process was also investigated in this

work.

The subsequent sub-sections respectively present and discuss the

deep-drawing and hydroforming processes.

5.1. Deep Drawing

The deep-drawing process is illustrated in Figure 10.

Figure 10: Sketch of the deep-drawing process

During the deep-drawing process, the blank is held between the

blankholder and die, and the punch travels downwards. As the punch

pulls the blank through the die, the blank slides between the blankholder

and the die. As the blank is drawn deeper into the die, the large

3-D FORMING OF PAPER | 23

compressive circumferential forces cause the blank to wrinkle. Precise

control over the force provided by the blankholder can be utilized to affect

the size and distribution of these wrinkles for paper materials. As defined

by Hauptmann and Majschak [23], a greater number of small wrinkles

(with a length scale on the range of hundreds of micrometers) are

preferable for maximizing visual quality.

The deep-drawing process can be utilized to create cup-like packages

such as the one shown in Figure 11.

Figure 11: A deep-drawn paperboard cup

The deep-drawn cup in Figure 11 has a diameter of 76 mm, a height of 46

mm, and consists of approximately 300 wrinkles around the

circumference of the cup. Since these wrinkles are so small, the overall

visual impression of the cup is not compromised by the wrinkles, as seen

in Figure 11.

24 | 3-D FORMING OF PAPER

5.2. Hydroforming

The hydroforming process is illustrated in Figure 12.

Figure 12: Sketch of the hydroforming process

During the hydroforming process, the blank is held in place between the

blankholder and die, and a pressure is applied within the balloon. This

pressure is then applied to the blank, forcing the blank to deform into the

shape of the die. The main purpose of this 3-D forming technique is to

create permanent (i.e. plastic) tensile in-plane (MD-CD) deformation in

the paper blank. Wrinkling is typically avoided during hydroforming for

both sealing and visual purposes. One could induce wrinkle formation in

the hydroforming process by allowing increased sliding of the blank into

the die, but this would also require precision control of the blankholder.

The hydroforming process can create complex double-curved surfaces,

such as the donut-shaped structure created with hydroforming shown in

Figure 13, which has a width of 15 cm and a formed depth of 3 cm.

Figure 13: A donut-shaped paper structure created with hydroforming

MODELS OF 3-D FORMING OF PAPER | 25

6. Models of 3-D Forming of Paper

Simulations of deep-drawing and hydroforming of paper materials

were conducted utilizing LS-DYNA, a commercial explicit finite element

solver. The developed material models presented in Section 4 were

implemented into the LS-DYNA user-defined material subroutine. Due to

the approximately orthotropic nature of paper, quarter symmetry was

utilized in all 3-D forming models utilized in Papers C-F. The subsequent

sub-sections summarize the major findings of the parametric studies

from Papers C-D (i.e. studies of deep drawing) and Paper E (i.e. study of

hydroforming). Furthermore, visual comparisons between experimental

and simulated 3-D formed paper structures are also provided.

6.1. Deep Drawing

Paper C utilized explicit finite element simulations to investigate the

effects of various process parameters on the punch force up to the

wrinkling point. The punch force was investigated, because the punch

force is related to the probability of failure [25]. Friction was especially

identified as a very important parameter for controlling the probability of

failure during the deep-drawing process (e.g. reduced friction would

reduce the probability of failure during the process).

Paper D, which utilized the material model with wrinkling in the

continuum, allowed for explicit finite element simulation of the entire

deep-drawing process and thus the study of many other quality

requirements or validation measures (e.g. springback and number of

wrinkles) for the deep-drawing process. The effects of various material

and process parameters on three different quality measures (i.e.

probability of failure, ovality, and average springback) were numerically

studied, and the hinge yield strength (𝜎ℎ,𝑦) was identified as the only

material parameter which could simultaneously improve all three studied

quality measures. A visual comparison between the simulated and

experimental deep-drawn paperboard cups is shown in Figure 14.

26 | MODELS OF 3-D FORMING OF PAPER

Figure 14: Simulated (pink elements are wrinkled while white elements are non-wrinkled) and experimental deep-drawn paperboard cup

6.2. Hydroforming

Paper E utilized explicit finite element simulations to investigate the

effects of various process and material parameters on the final volume of

the formed shape. Both the blankholder force and relative humidity were

found to have negligible effect on the final formed volume. The

blankholder force had no significant effect, because little to no sliding of

the blank occurred during all simulations regardless of the blankholder

force. Changes in relative humidity, however, cause changes in the

stiffness, yield point, and tangent modulus of the paper. For some reason

(possibly only for the specific material utilized in Paper E), the effects of

the changed stiffness, yield point, and tangent modulus happened to

cancel in their combined effects on the increase in final formed volume.

For reasonable changes in elastic stiffness, the increase in final formed

volume showed an approximately linear positive relationship with

stiffness. However, for stiffness greater than 9 GPa (such as the doubled

elastic stiffness in Paper E), drastic reductions in the forming volume

were obtained.

Paper F utilized explicit finite element simulations to investigate the

3-D formability and assumed strain-based failure surface of handsheets

with fibers partially oxidized to dialcohol cellulose. A visual comparison

MODELS OF 3-D FORMING OF PAPER | 27

between the simulated and experimental hydroformed modified

handsheet is shown in Figure 15.

Figure 15: Simulated (color scale shows in-plane strains in one direction; blue denotes large tensile strains) and experimental hydroformed handsheet with fibers partially

oxidized to dialcohol cellulose

28 | CONCLUSIONS

7. Conclusions

The three overarching research questions investigated in this work

(from Section 1.1) can be split into five specific sub-questions that were

addressed through this work:

1. How 3-D formable are paper materials?

a. What is the strain-based failure surface (i.e. forming

limit diagram) for paper materials?

b. What are the effects of moisture and temperature on

3-D formability?

2. How can paper materials be improved for specific 3-D

forming processes?

a. Which material properties are important for

hydroforming and deep drawing of traditional paper

materials?

b. How can the 3-D formability of paper materials be

improved through chemical modification?

3. How can specific 3-D forming processes for paper materials be

improved?

a. Which process parameters are important for

hydroforming and deep drawing of traditional paper

products?

The strain-based failure surface (question 1a) for paperboard and

handsheets with fibers partially oxidized to dialcohol cellulose were

experimentally and numerically explored in Papers B & F. Paper B found

that a part of the strain-based failure surface could be represented by a

bilinear surface. Furthermore, Paper B shows the limited extent to which

traditional paper materials can be formed in biaxial tension (e.g.

hydroforming or embossing). However, Paper B does indicate an increase

of the strain-at-break in sliding blank processes during which

compressive in-plane forces appear (e.g. deep drawing or press forming).

Paper F, on the other hand, developed a method to estimate the strain-

CONCLUSIONS | 29

based failure surface based on uniaxial testing results and validated that

method utilizing the finite element method. The handsheets utilized in

Paper F had fibers that were partially oxidized to dialcohol cellulose, and

the achieved forming limit diagrams in Paper F show the possible

improvement of 3-D formability by means of chemical modification

(question 2b).

The effects of moisture and temperature on 3-D formability (question

1b) were experimentally and numerically explored in Papers A, & D-F. In

Paper A, an experimental method was developed to enable the systematic

study of the combined effects of moisture and temperature on the

mechanical response of paper. This experimental method was applied to

chemically modified handsheets in Paper F, although Paper F also

utilized dynamic mechanical thermal analysis to study the glass transition

temperature (as a function of humidity) as well as to study the viscous

part of the mechanical response. The mechanical testing results from

both papers contain insights into the effects of moisture and temperature

on the materials and provide constitutive parameters that were utilized

for explicit FEM simulations in Papers C-F. The parametric studies

utilizing explicit FEM simulations in Papers D & E further quantified the

effects of moisture and temperature on three quality parameters (i.e.

average springback, springback asymmetry, and final formed volume) for

3-D forming.

The effects of various material properties on the hydroforming and

deep drawing of paper materials (question 2a) were numerically explored

and quantified through parametric study in Papers C-E. Furthermore, the

ratio between the plastic hinge strength and elastic modulus was shown

in Section 4.2 to be important for the deep drawing of high-quality

paperboard cups.

The effects of various process parameters on hydroforming and deep

drawing of paper materials (question 3a) were numerically explored and

quantified through parametric study in Papers C-E.

30 | RECOMMENDATIONS FOR FUTURE WORK

8. Recommendations for Future Work

This thesis work has produced many insights into the three

overarching research questions (from Section 1.1 and further discussed in

Section 7), although significant additional improvement in understanding

is possible.

Although the explicit finite element models of the hydroforming and

deep-drawing processes in Papers C-F were successful in developing

knowledge about the respective processes, these models could be further

utilized for more complicated package shapes. For example, the deep

drawing of packages with concave base shapes [24] could be studied in an

explicit finite element framework, non-symmetric hydroforming shapes

could be investigated in further detail than was conducted by Huang and

Nygårds [26], or press-forming of pre-creased paperboard [27] could also

be studied with help of the existing simulation tools.

Paper F investigated the increase in 3-D formability due to one type of

chemical modification, but the strainability of paper materials can also be

improved through mechanical treatment [28]. Studying the 3-D

formability of paper sheets that have undergone such mechanical

treatments would help determine the viability of producing advanced 3-D

formed structures from those types of paper.

The constitutive model for combined wrinkled and non-wrinkled

paper in Paper D has been validated utilizing various simple, tensile tests

and through comparison with deep drawing experiments. However, 3-D

forming processes (e.g. deep drawing) subject the material to 3-D load

conditions (i.e. in-plane and out-of-plane loads). Development of

experiments to measure the mechanical response of wrinkled and non-

wrinkled paper in 3-D loading conditions could be utilized to study the

accuracy of the constitutive model and thus perhaps also the accuracy of

the simulations. Additionally, Groche and Huttel [29] suggested that,

similar to metals, the application of hydrostatic pressure could increase

the size of the strain-based failure surface for paper materials, although

further experimental validation of this hypothesis is necessary. This

hypothesis would likely indicate that the out-of-plane loads have a

RECOMMENDATIONS FOR FUTURE WORK | 31

significant impact on the in-plane mechanical response and thus

constitutive model. If this hypothesis is true, 3-D forming processes for

paper could be tailored to enhance in-plane strainability by increasing

hydrostatic pressure during the process.

This thesis assumes that paper materials are approximately

homogeneous, which is not true [30]. Including the effects of

inhomogeneity in the explicit finite element simulations could provide

stochastic information about 3-D forming processes, and initial

investigations and simulations have been conducted [31]. At the current

state-of-the-art, however, the assumption of homogeneity may be

adequate for studies of the non-stochastic aspects of 3-D forming.

32 | REFERENCES

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