An Experimental Study of the Rate Dependencies of a. Nonwoven Paper Substrate in Tension using Constitutive. Relations

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2 An Experimental Study of the Rate Dependencies of a Nonwoven Paper Substrate in Tension using Constitutive Relations A thesis submitted to the Graduate School of the University of Cincinnati in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering In the School of Dynamic Systems of the College of Engineering & Applied Science January 2012 By: Mark Burchnall Bachelor of Science in Mechanical Engineering Purdue University, West Lafayette, IN December 2009 Committee: Richard Hamm Dr. Yijun Liu Dr. Kumar Vemaganti i

3 Abstract Paper is an everyday product used for various reasons by the consumer. This study focuses on a low basis weight nonwoven paper substrate used as toilet paper. Paper manufacturers are always trying to increase line speed to push out more paper at the same cost and the manufacturing process this research focuses on is the embossing process. This study will focus on determining the rate dependent inplane constitutive relations that define the mechanical behavior of the paper substrate in tension. Once the constitutive models are created they can be imported into a finite-element software package and used to study changes made to the embossing process. Experimental tests were run in order to isolate specific properties of the material. Uniaxial tension tests were run at 0.1 1/s, 4.5 1/s and 45 1/s total engineering strain rates in order to determine the rate dependent effects on the material. Stress relaxation tests were run at varying moisture contents and temperatures with the idea of determining the viscoelastic model and how moisture and temperature affect the material. Viscoelastic and viscoplastic behavior models were developed to characterize the rate dependencies in the elastic and plastic regions of the material. A generalized Maxwell model is used to model the viscoelastic region and a modified form of G sell s flow stress law was used in conjunction with the viscous based overstress theory (VBO) to define the viscoplastic region. The research done on this paper substrate details a method to define the rate dependent constitutive properties of any fiber network material through an experimental study. ii

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5 Acknowledgements I would first and foremost like to thank those who served on my thesis advisory committee: Dr. Yijun Liu, Dr. Kumar Vemaganti and Richard Hamm. All of you helped increased my knowledge and passion for engineering during my time at the University of Cincinnati. I am grateful for Dr. Yijun Liu (Professor School of Dynamic Systems) as he served as my academic advisor as well as recommending me for the UC Simulation Center. Dr. Kumar Vemaganti (Professor School of Dynamic Stystems) helped me with the research associated with this project and to which I am deeply grateful for. I am also extremely appreciative of the research guidance Richard Hamm (Procter and Gamble - Masters from Georgia Tech) gave to me and his passion for engineering and materials will forever be embedded in me. I am also extremely thankful for Procter & Gamble as they funded this research and other projects I completed during my time at UC. I would especially like to thank Fred Murrell for growing my knowledge of engineering and helping propel me to a strong start in my future career. I am extremely grateful for the guidance and support Don Bretl, Kelly Anderson and Bernie Rudd gave to me during my time with P&G. I would also like to specifically thank Casey Fox and Angela Smith for the help they provided in completing the experiments for this research. Most importantly I would like to thank my wife Mallory Burchnall for loving and supporting me unconditionally throughout my graduate studies. I would also like to express my gratitude to my parents John and Ruth Anne Burchnall for supporting me in everything I have ever done. iv

6 Table of Contents ABSTRACT... II ACKNOWLEDGEMENTS... IV TABLE OF CONTENTS... V LIST OF FIGURES... VIII LIST OF TABLES... XII 1 INTRODUCTION MANUFACTURING BACKGROUND DESCRIPTION OF PAPER MOTIVATION MODELING BACKGROUND OBJECTIVES THESIS STRUCTURE LITERATURE REVIEW POLYMER DEFINITION ELASTICITY HYPERELASTICITY VISCOELASTICITY MOISTURE DEPENDENCY TEMPERATURE DEPENDENCY PLASTICITY FAILURE LITERATURE PAPER MODELS v

7 3 EXPERIMENTAL METHODS TENSILE SPECIMEN PREPARATION MOISTURE-SORPTION ISOTHERMS UNIAXIAL TENSION AT VARYING STRAIN RATES OFF-AXIS UNIAXIAL TENSION CYCLIC LOADING AT VARYING STRAIN RATES STRESS RELAXATION AT VARYING TEMPERATURE AND HUMIDITY UNIAXIAL TENSION TO DETERMINE POISSON S RATIO TOTAL TEST PLAN MODELING EXPERIMENTAL DATA MOISTURE-SORPTION ISOTHERM MODELS Moisture-Sorption Isotherm Experimental Data Moisture-Sorption Isotherm Model Description TIME-TEMPERATURE-MOISTURE SUPERPOSITION Stress Relaxation Experimental Data Temperature Shifting Model Moisture Shifting Model STRAIN MAPPING Cyclic Tensile Experimental Data Strain Map Model Uniaxial Tensile Experimental Data VISCOELASTIC MODEL Stress Relaxation Master Curve Generalized Maxwell Model Instantaneous and Long Term Response vi

8 4.4.4 Viscoelastic Model Description PLASTICITY MODEL Plastic Experimental data Viscoplastic Model Description FAILURE ENVELOPE MODEL Failure Envelope Model Description POISSON S RATIO FLOW CHART USED TO OBTAIN VISCOELASTIC AND VISCOPLASTIC EQUATIONS DISCUSSION CONCLUSION SOURCES OF ERROR FUTURE WORK REFERENCES vii

9 List of Figures Figure Embossing Process... 4 Figure Sample SEM Photograph of Nonwoven Paper [Hagglund, 2006]... 7 Figure Sample SEM Photograph of Copy Paper [Makela, 2003]... 7 Figure Loading of Paper Sheet from a Hypothetical Emboss Feature Figure a) Linear Polymer b) Branched Polymer c) Crosslinked Polymer Figure Cellulose Molecular Structure [Biermann, Handbook] Figure a) Maxwell Model b) Kelvin-Voight Model Figure Generalized Maxwell Model Figure Diagram of Moisture-Isotherm Test Setup Figure Diagram of Sample for Off-Axis Uniaxial Tests Figure Sample Test Set-Up for determining Poisson s Ratio Figure Cropped Image of markings on sample pre and post color threshold analysis Figure Dynamic Experimental Data for Moisture Sorption Isotherm at 296K (23 C) Figure Moisture Sorption Isotherm Experimental Data for the paper substrate Figure Moisture Sorption Isotherm Model at 278 K (5 C), 288 K (15 C), 296 K (23 C) and 301 K (28 C) Figure Experimental Data taken at 23 C versus Kraft Sack paper data [Salmen, 1980] viii

10 Figure Experimental Stress Relaxation Curves in MD for varying Temperature and Moisture Contents Figure Experimental Stress Relaxation Curves in CD for varying Temperature and Moisture Contents Figure Stress Relaxation Master Curves in MD after Temperature Shifting for 4%, 6% and 8% Moisture Contents Figure Stress Relaxation Master Curves in CD after Temperature Shifting for 4%, 6% and 8% Moisture Content Figure WLF Temperature Shifting Model for MD Figure WLF Temperature Shifting Model for CD Figure WLF Temperature Shifting Models for MD and CD Figure Stress Relaxation Master Curve in MD at 8.25% Moisture Content after Temperature and Moisture Shifting Figure Stress Relaxation Master Curve in CD at 8.25% Moisture Content after Temperature and Moisture Shifting Figure WLF Moisture Shifting Model for MD and CD Figure Experimental Cyclic Stress-Strain Breakdown for MD Figure Experimental Cyclic Stress-Strain Breakdown for CD Figure Experimental Elastic/Plastic Strain versus Total Strain at 0.1 1/s and /s Strain Rates for MD Figure Experimental Elastic/Plastic Strain versus Total Strain at 0.1 1/s and /s Strain Rates for CD Figure Elastic and Plastic Strain Mapping Model for MD ix

11 Figure Elastic and Plastic Strain Mapping Model for CD Figure Experimental Data taken in Uniaxial Tension for MD at Strain Rates of 0.1 1/s, 4.5 1/s and 45 1/s Figure Experimental Data taken in Uniaxial Tension for CD at Strain Rates of 0.1 1/s, 4.5 1/s and 45 1/s Figure Stress Relaxation Master Curve at 8.25% MC for MD Figure Stress Relaxation Master Curve at 8.25% MC for CD Figure Maxwell Model fit to Stress Relaxation Master Curve for MD at 8.25% MC Figure Maxwell Model fit to Stress Relaxation Master Curve for CD at 8.25% MC Figure Instantaneous Response in MD Figure Long Term Response in MD Figure Instantaneous Response CD Figure Long Term Response CD Figure Viscoelastic Fits at Strain Rates of 0.1 1/s, 4.5 1/s and 45 1/s in the MD and CD Figure Viscoelastic Model at Strain Rates Equal to 0 (Long Term), 0.1, 4.5, 45 and Inf (Instant) 1/s for MD Figure Viscoelastic Model at Strain Rates Equal to 0 (Long Term), 0.1, 4.5, 45 and Inf (instant) 1/s for CD Figure Plastic Strain versus Stress Data for varying Strain Rates in the MD.. 85 Figure Plastic Strain versus Stress Data for varying Strain Rates in the CD x

12 Figure Fitting Modified G'Sell Flow Stress Law in MD Figure Fitting Modified G'Sell Flow Stress Law in CD Figure Viscoplastic Fits at Strain Rates of 0.1 1/s, 4.5 1/s and 45 1/s in the MD and CD Figure Viscoplastic Model in MD at Strain Rates Equal to 0 1/s (Long Term), 0.1 1/s, 4.5 1/s and 45 1/s Figure Viscoplastic Model in CD at Strain Rates Equal to 0 1/s (Long Term), 0.1 1/s, 4.5 1/s and 45 1/s Figure Failure Envelope Fit for Paper Substrate Using Tsai-Hill and Tsai-Wu Figure Rate Dependent Failure Model for MD and CD with rates 0.1 1/s, 4.5 1/s and 45 1/s Figure Strain in MD versus Strain in CD as strain increases in the MD Figure Flow chart of work done with teal, white, blue and black boxes representing experiments, in-between calculations, models and final models respectively xi

13 List of Tables Table Average Physical Values of Paper at Standard TAPPI conditions... 8 Table Relative Humidity Increments Depending on Temperature Table Uniaxial Test Information depending on Strain Rate Table List of all Tests done on the Paper Substrate Table GAB Fitted Constants for paper substrate Table M0, C and K values at each constant temperature isotherm Table Values of Constants for WLF Temperature Shifting for MD and CD Table Values of Constants for WLF Moisture Shifting Model for MD and CD. 60 Table Strain Mapping Constants for MD and CD Elastic/Plastic Strain Table Calculated Initial Stress for Stress Relaxation Master Curves in MD and CD Table Prony Series Fits for MD Table Prony Series Fits for CD Table Constants for Instantaneous and Long Term Stress Responses in terms of Total Strain for MD and CD Table Constants for Elastic Instantaneous Response in MD and CD Table Constants for Plastic Long Term Stress Response in MD and CD Table Constants for Modified G'Sell Flow Stress Equation for MD and CD Table Failure Envelope Constants for Tsai-Hill and Tsai-Wu Table Constants for Failure Envelope due to Strain Rate for MD and CD xii

14 1 Introduction 1.1 Manufacturing Background Low basis weight paper is massed produced across the world for a variety of different products. Paper sheets can be formed to make paper towels, facial tissue, toilet paper, and many other forms. The paper product dealt with in this study is a non-embossed toilet tissue that goes through many of the same manufacturing processes of the other types of nonwoven paper sheets. Paper sheets are produced by forming a network of pulp fibers, usually wood pulp is the main ingredient but is not limited to wood pulp. The papermaking process is generally completed by a four-step manufacturing process, which includes [Biermann, 1996]: 1) Forming 2) Draining 3) Pressing 4) Drying Not all papermaking operations follow these exact steps to produce paper but in the interest of this research these are the assumed processes. The forming section is where the pulp is laid into the network of fibers that will be the basis for the structure of the paper sheet. Wetting the pulp fibers, with the intention of facilitating bonding between the fibers, and then laying them out onto a belt is the basis of the forming process. When the wetted fibers are laid out on a belt, referred to as the wire, the basic structure of the paper is complete. The other three steps of the papermaking process deal with removing the moisture from 2

15 the paper but also contribute to the mechanical properties of the paper. For example the pressing process compresses the paper between two rolls in order to force out water in the paper sheet and to compress the fiber network into a denser structure. For most paper products there is another process after the four-papermaking steps. This area is called converting and can be composed of processes such as embossing, plying, printing and etc. The main focus of the converting process is to prepare the paper produce for consumer use. The process in which this research is focused on is the embossing process. Embossing is used to add a three-dimensional pattern to the paper sheet after the basic structure is formed. Some of the reasons to emboss a paper sheet are to make the paper product more appealing to the consumer, to make it easier to ply multiple pieces of paper together or to adjust how the product performs at its intended purpose. Most of the nonwoven paper products that a typical consumer uses in a day have some sort of embossing pattern on the paper sheet. There are many different methods in which embossing is completed but the most common one is the roll-on-roll method [Schift, 2006]. The roll-on-roll method consists of one steel roll with a pattern on its cylindrical surface that is pressed into a rubber-coated roll. As the rolls are spinning the paper sheet is moving through the roll contact area, referred to as the nip and can be seen in Figure 1.1-1, at extremely high speeds. The three-dimensional pattern is than imprinted onto the paper sheet in the nip region. For information on the dynamic forces found in the 3

16 rubber to steel roll contact region refer to Parish [Parish, 1958]. The steel roll in his research does not have an emboss pattern on it but the dynamics involved can be assumed to be about the same. Emboss Roll Paper Sheet Pressure Roll Figure Embossing Process With these rolls moving at high speeds, the paper is subjected to large stresses and strains at varying strain rates. Every nip has a different set of parameters that affect the emboss pattern that is imprinted on the paper sheet. A few of the parameters that will affect the emboss pattern are the pattern on the steel roll, the diameter of the steel roll, the diameter of the rubber roll, the type of material on the rubber roll, the length of the nip, etc. So it is important to understand how changes in the embossing process will affect the paper as it passes through the nip. The embossing process can be modeled using computational methods such as Finite- Element Analysis (FEA) but in order to do that a constitutive model must be in place for the paper. 4

17 1.2 Description of Paper The paper being used in this research is a type of non-embossed toilet tissue. Non-embossed paper was used as to obtain a better understanding of the material before it enters the embossing process. The embossing process will change the geometry of the sheet and making the material model more heavily dependent on geometry. A paper sheet is usually defined by three directions, those being: machine direction (MD), cross-machine direction (CD) and the into the paper direction (ZD). The machine direction is classified as such as it is the direction that the paper sheet is fed on the production line. The cross-machine direction is defined as the direction on-plane with the sheet perpendicular to the machine direction. Paper is formed into a network of pulp fibers by the manufacturing process described in the previous section. During the forming process, pulp fibers are spread out onto the wire to initially form the wet paper sheet but in doing this most fibers are aligned in the MD. This happens because the momentum of the sheet moving in that direction causes a majority of the fibers to reorient to the machine direction. Thus the mechanical properties of the paper substrate in the MD and CD directions cannot be considered identical. The fiber network for the paper sheet used in this research is created using wood pulp fibers, which is the most common type of pulp used in nonwoven paper products for consumer use. Although other types of pulp can be used to form paper, the paper substrate picked for this research uses exclusively wood pulp. Wood pulp consists of naturally occurring polymers cellulose, hemicellulose, and lignin 5

18 [Biermann, 1996]. These polymers contribute to the time-dependent mechanical response of the paper substrate. The strength of the fiber network is widely believed to be dependent on the hydrogen bonds that form between pulp fibers after the sheets have been dried. From this we can conclude that, moisture and temperature will have great influence on the mechanical response of the paper [Haslach, 2000]. The failure of these hydrogen bonds introduces permanent damage to the material and is what contributes to the inelastic response. Figure shows an example of a nonwoven material fiber structure under a scanning electron microscope (SEM) taken from research Hagglund did on low basis weight paper [Hagglund, 2006]. Figure also shows a SEM picture taken of copy paper for a material model study Makela did on it [Makela, 2003]. These give the reader an idea of the complex and almost random structure of fibers that make up these nonwoven materials. The paper used in this research was not photographed for this study. 6

19 Figure Sample SEM Photograph of Nonwoven Paper [Hagglund, 2006] Figure Sample SEM Photograph of Copy Paper [Makela, 2003] 7

20 Table lists a few properties that will help clarify what type of material this toilet tissue is. Basis weight is the measure of the weight of the paper over a unit area defined in-plane with the MD and CD directions. All of these values are approximate values taken from the many samples used in the experimental research for this paper at a temperature of 23 C and 50% relative humidity, TAPPI standard testing conditions. It is important to remember that these values will change with respect to temperature and humidity. Table Average Physical Values of Paper at Standard TAPPI conditions Property Value Density (gf/m 3 ) 107,895 Basis Weight (gf/m 2 ) 20.5 Thickness (mm) Motivation The main motivation for constructing a constitutive model for a paper sheet is to have the ability to determine how changes in the embossing process will affect the paper sheet before changes are made to the production line. A paper product manufacturer stands to cut costs by being able to optimize processes on the production line if they have a constitutive model of the paper sheet they are working with. As mentioned before, computational models using FEA can be used in conjunction with a constitutive model of the paper to determine the effect production changes will have before they are made. Costs can rapidly build up if the production line is continually being changed in order to produce paper with a 8

21 certain desired effect. This method not only applies to the embossing process but to every other mechanical process on the converting line. The geometry, line speed, and other nip characteristics of the emboss process can be optimized in order to prevent material failure, increase the effectiveness of the product, increase production rate and increase the quality of the emboss image. The main process improvement that is of interest to paper manufacturers is increasing the speed of the production line. The reason for the increase in line speed is the company can put out more product for the same amount of cost and manufacturing equipment. With the increase in line speed, the embossing rolls must also increase their rotational speeds. Thus increasing the rate at which strain is applied to the sheet when it goes through the nip. In order to study the effects of increasing the line speed of the paper production line, an experimental study of the rate dependent behavior of the paper substrate should be completed. This should be done by first determining what other internal and external forces affect the mechanical structure of paper. Embossing is permanent so the plasticity of the paper should be studied and determined if it is rate dependent or not. Manufacturing conditions, such as temperature and humidity, are not held constant and should be researched in order to see how they will affect the mechanical response of the paper. Each of these factors needs to be explored in all three directions of the paper, as the structure is different in each. The failure criteria for the paper substrate should also be determined to add the capability to understand what changes made to the embossing process will cause the substrate to fail. 9

22 The research will be done in accordance to the loading pattern of the paper sheets as it goes through embossing process. There will be tensile forces in the MD and CD directions and compressive forces in the ZD direction as the paper passes through the embossing roll, an example of the loading process within the nip can be seen in Figure The focus of this research will be on the tensile loading in the MD and CD directions in order to simplify some of the aspects that will need to be studied. It cannot be assumed that the paper sheet has the same mechanical response in the CD direction as the MD direction because of the fiber orientation in the fiber network, which means any experiment done needs to be able to model the mechanical behavior in both the MD and CD. Figure Loading of Paper Sheet from a Hypothetical Emboss Feature 10

23 1.4 Modeling Background A constitutive model, or material model, is defined as a model that determines the stress depending on different variables such as strain, strain rate, temperature, moisture, stress history, etc. Constitutive models are useful as the user can obtain information about the material that cannot be experimentally measured. Many different constitutive equations can be used to describe a material, but particular models are picked in order to explain a certain behavior of the material. In the example of paper, factors that may change the mechanical behavior of the paper could be fiber network structure, type of pulp used, different manufacturing processes, and many more. The resulting variation in the mechanical response should be incorporated into the constitutive relations. These models become even more useful when they are used by numerical solvers such as the finiteelement method (FEM). FEM is a numerical method that is most commonly used to solve solid mechanics type problems, but is not limited to solid mechanics and is often used to solve fluid mechanics type problems as well. This numerical method is used to determine field variables, such as displacement, stress, temperature, etc. The solution is completed by discretizing the model into different pieces, finite elements, and creating a continuous mesh. This research did not use the FEM or any other numerical method but the constitutive equations determined will be used in FE models of the embossing process. For more information on the finite-element method, look up Cook s work on FEM [Cook, 2001]. 11

24 1.5 Objectives The focus for the research done on this master s thesis is as follows: 1) Experimentally study the rate dependence of the toilet tissue paper substrate in the MD and CD directions. 2) Experimentally determine the moisture and temperature dependent effects of the mechanical response of the paper. 3) Experimentally determine the failure criteria for the paper substrate. 4) Experimentally study permanent deformation and determine if it is rate dependent or not. 5) Model these individual forces in order to come up with constitutive relations that hold true on the macro-model scale, meaning the fiber network and not down to the fiber or molecular levels. 1.6 Thesis Structure Chapter 1 aims to construe to the reader the goals and basic background of the research that was done in this paper. Chapter 2 focuses on providing the necessary background research that was done in order to formulate the constitutive equations found in this research. Chapter 3 details the experiments run for this research and the method in which they were run so that the reader could conceivably recreate the experimental data. Chapter 4 discusses the models used to represent the experimental data and how they relate to the experimental data obtained from testing. Chapter 5 presents the results and discusses the validity of the models used. Chapter 6 details ideas for future work in this subject area. 12

25 2 Literature Review 2.1 Polymer Definition In the previous chapter it is discussed that paper is made up of a network fiber polymers, notably cellulose, hemicelluloses and lignin. In order to understand how these specific polymers affect the mechanical structure it would be diligent to understand the basics of polymers. Polymers are generally defined as molecules, with a high molecular weight, that consist of a repeated structure of chains. The polymer chains are made up of monomers covalently bonded together. Polymers can occur naturally in our environment or can be produced synthetically, such as polyethylene and PVC. Cellulose, hemicellulose and lignin are all naturally occurring polymers derived from wood that our found in nonwoven paper. The mechanical characteristics of polymers, for the most part, are highly sensitive to the rate deformation (strain rate), the temperature, and the chemical nature of the environment (the presence of water, oxygen, organic solvents, etc.) [Callister, 2007]. For the purpose of this study, all environmental factors besides deformation rate, temperature and presence of water (moisture content) will be ignored. Polymers can be linear in nature, branched, cross-linked or a combination of the three which can be seen in more detail in Callister [Callister, 2007]. These different types of polymer structures can lead to differences in the mechanical response of each polymer. 13

26 a) b) c) Figure a) Linear Polymer b) Branched Polymer c) Crosslinked Polymer In order to study the polymer physics of cellulose, hemicellulose and lignin it is necessary to have a basic understanding of the molecular structure of each polymer. Cellulose is an organic polymer that is found in many different organic structures, most notably plants. The cellulose used in this research is found in wood pulp so it takes the form of Cellulose 1β [Nishiyama, 2002]. For most wood products cellulose is the most abundant material which also can be assumed for this paper substrate. The full molecular structure of cellulose can be seen in Figure Cellulose is considered to be a linear semi crystalline polymer, with wood cellulose being about 50-70% crystalline [Biermann, 1996]. Linear polymers usually are bonded by hydrogen bonds, which is consistent with the bonds seen in cellulose. Figure Cellulose Molecular Structure [Biermann, Handbook] Hemicellulose is another polymer found in wood pulp and according to [Scheller, 2010] its most important biological role is to strengthen the plant cell s wall by interacting 14

27 with cellulose and lignin. Hemicellulose varies depending on the wood pulp used to make the paper but can be considered to be predominantly a linear polymer with only slight branching and rarely crystalline. Again the linear polymers mean that there is a large presence of hydrogen bonds in the material structure. A few examples of hemicellulose are xlyloglucan, xylan, mannan, glucomannan and many more, whose molecular structures can be found in [Scheller, 2010]. Lignin is the last type of non-additive polymer found in wood pulp. Though lignin is not found in all types of wood pulp, lignin is thought of as a binder that holds the other wood fibers together. Lignin is considered to be a cross-linked polymer, which gives it a rubber type elastic response. Paper is then defined as a nonwoven composite of the three polymers, cellulose, hemicellulose and lignin, assuming no other material was added during the manufacturing process. The only other substances considered to be found in paper for this research besides wood pulp are air and water, which both contribute to the mechanical response. 2.2 Elasticity The theory of elasticity relates stress and strain to each other in a linear or nonlinear fashion. This relationship means that at every stress point, P, there is a defined relationship by one strain point regardless of the other conditions [Boresi, 2000]. A typical model for use as an elasticity model is Hooke s Law, equation (1). Hooke s Law relates stress, σ, to strain, ε, by the Young s modulus, E, which can be defined as change in stress over the change in strain. For a linear elastic problem Young s modulus is constant because the stress-strain relation doesn t change depending on the amount of strain. A nonlinear elastic problem can have a Young s modulus that is dependent on the strain, which usually 15

28 requires a more complex constitutive model like hyperelasticity (discussed later on). Steels usually display linear elastic behavior whereas rubbers and plastics tend to display nonlinear elastic responses. (1) Engineering stress, s, is defined as the force over the initial area even. True stress, σ, is defined as force over the deformed area and can be determined from the engineering stress by using equation (2). These equations only hold true for volume preserving solids, solids with a Poisson s Ratio of 0.5. For a more accurate definition of stress the Cauchy Stress Tensor, the 1 st Piola-Kirchhoff or the 2 nd Piola-Kirchoff stress tensors can be used to describe the stress response (more information can be found in Holzapfel) [Holzapfel, 2000]. True strain, ε, can be related to engineering strain, e, by equation (3) as well. (2) ( ) (3) ( ) A material becomes inelastic when a load is applied and does not return to its initial strain state. Inelasticity usually starts to occur at a certain stress which is typically referred to as the yield limit, σyl [Boresi, 2000]. After the yield limit then plastic strain, εp, and elastic strain, εe, are both part of the stress-strain relationship. Equation (4) shows the additive decomposition of total strain, εt, into the elastic and plastic parts. Whenever hysteresis starts to occur below the yield point and beyond the yield point during unloading/reloading, then the material is also known to not be perfectly elastic. Hysteresis is defined as when the unloading stress-strain relationship is below the loading curve, 16

29 indicating energy dissipation. This usually means that the material exhibits ratedependencies and should be described by a viscoelastic model and not a perfectly elastic model. (4) 2.3 Hyperelasticity Hyperelasticity is used to develop constitutive equations for materials that undergo a nonlinear elastic response under finite strains, also referred to as large strains. The constitutive equations are based on the concept of a strain energy function, ψ, which indicates the energy stored in the material as a function of deformation [Holzapfel, 2000]. Strain-energy functions are usually described in terms of their strain invariants that can be determined from the principle stretches, λ 1, λ 2 and λ 3. Principal stretches are the eigenvalues of the stretch tensor. In one dimension, the material stretch, λ, is related to the engineering strain, e, by equation (5). Hyperelastic models are by and large phenomenological, meaning that they are not dependent on the physical structure of the material and are more focused on describing the stress-strain response using a mathematical approach. That being said there are hyperelastic models that are physical in nature and are not always phenomenological. There are many different types of strainenergy functions that can be used to construct a hyperelastic model and can be found in Holzapfel [Holazapfel, 2000]. (5) 17

30 2.4 Viscoelasticity Viscoelasticity is used to describe materials that show rate-dependent behavior. Materials that exhibit rate-dependency tend to have two parts to their mechanical response, the viscous almost fluid like state and the elastic state. Viscoelasticity combines both of these aspects into one type of constitutive relation. A material is usually characterized as viscoelastic if a material shows stress relaxation, recovery and/or creep. Stress relaxation occurs when a material is strained to a constant strain point and left there for a certain amount of time and if the material exhibits stress relaxation the stress will start to decrease over time. Creep can be observed in a material when that material is stressed to constant stress point and held over time and if the material exhibits creep the strain will increase over time. There are two basic models that were developed to describe creep and stress relaxation using linear viscoelasticity. The first model is the Maxwell model and can be seen in Figure a) and the other model is the Kelvin-Voight Model which can be seen in Figure b) [Ward, 2004]. Both models have a viscous part that is described by a Newtonian dashpot, represented by viscosity value η, and an elastic part that is described by a Hookean spring, displayed as stiffness E. The Maxwell model is meant to describe stress relaxation whereas the Kelvin-Voight model is meant to describe creep. Since this research deals with describing the rate-dependencies of the material using stress relaxation data the Maxwell model will be focused on. Assuming a constant zero strain rate, the constitutive equation for the Maxwell model can be calculated in terms of initial stress, σ0, time, t, and time constant, τ, which can be seen in equation (6) [Nielsen, 1994]. 18

31 The time constant is a function of the viscosity of the dashpot, ηi, and stiffness of the spring, Ei (7). Figure a) Maxwell Model b) Kelvin-Voight Model (6) ( ) (7) A more complex model using multiple Maxwell components and an instantaneous elasticity component is called a Generalized Maxwell (also known as Maxwell Wiechert model [Shaw, 2005]) model, which can be seen in Figure This model is used assuming the initial viscous dashpot to be very low which allows the model to have a long term response, E [Ward, 2004]. The instantaneous response is the combined response of all of the springs in parallel. The reason for the use of this model is because most materials, polymers included, cannot have their relaxation described by just one exponential function. The Generalized Maxwell model fits up to N exponential functions to the relaxation, one function for every decade of data is used in this research but there is not a required number of variables. 19

32 Figure Generalized Maxwell Model The above model is linear by nature but most viscoelastic materials exhibit nonlinearities. Nonlinearities can be applied to the same model by allowing the springs in the Generalized Maxwell model to be nonlinear meaning the stiffness, Ei, varies with strain becoming, Ei(ε). 2.5 Moisture Dependency Two environmental factors that influence the mechanical response of a polymer are temperature and relative humidity. The combination of both of humidity and temperature determine the moisture level in the paper substrate. Hydrogen bonding between the polymers is directly affected by moisture and will contribute to the change in the mechanical response [Haslach, 2000]. It is believed that most of the interfiber bonds that bond the cellulose, hemicellulose and lignin fibers together are hydrogen bonds. According to Patterson, increasing relative humidity at constant temperature accelerates creeping in paper [Patterson, 2007]. Salmen and Back did research that shows that the elastic modulus of paper decreases in stiffness as moisture content increases [Salmen, 1980]. The elastic modulus of the fiber network decreases in stiffness because moisture weakens the 20

33 hydrogen bonds between the fibers. This is the same reason why creep accelerates under increasing moisture content. As we are interested in the rate-dependencies of paper, it must first be determined how moisture affects the paper substrate used in this study. The first step is to determine the amount of moisture in the paper sheet at all conditions, which can be calculated by equation (8). Moisture is dependent on humidity and temperature and is captured in the paper in the space between the fibers. The amount of moisture is then dependent on the fiber network structure of the paper substrate. Moisture sorption is a dynamic effect that can be broken into two different parts, adsorption and desorption. Adsorption being the act of adding moisture to the paper and desorption the act of removing moisture from the sheet. A moisture-sorption isotherm is a curve that is used to predict the moisture content over a range of relative humilities 0% (bone dry) to 100% (dew point) for a given temperature. The curve may change shapes and magnitudes as the temperature changes, so it is important to obtain multiple isotherms over the applicable range of temperatures. These curves can then be used to predict the moisture in the paper at certain conditions and have mathematical models fitted to the data. There are many methods for modeling moisture-sorption isotherms for paper. The main model that will be used is the Guggenheim, Anderson, and de Boer (GAB for short) [Parker, 2006] equation, which can be seen in equation (9). The GAB equation has the constants M 0, C and K which have physical meanings of the mono-layer moisture content, the Guggenheim constant, and a constant relating the substrate to its pure liquid states respectively. The other variables MC and a w stand for the moisture content of the paper (g/g) and the water activity which is the relative humidity/100%. The base form of the GAB model does not account for 21

34 temperature but by applying equations (10), (11) and (12) it can be accounted for [Parker, 2006]. That leaves six different coefficients (M 0, C, K, k 1, k 2 and k 3 ) to be fitted for the substrate in order to fully characterize the moisture sorption properties of the material. Equation (10) describes the mono-layer moisture contents dependency on temperature phenomenologically with M 0 and k 1 constants. Equation (11) and (12) have physical basis in that C and K are constants but k 2 and k 3 are based on molar sorption enthalpies of the substrate. The GAB model is assumed accurate for relative humilities of up to 95% and can be assumed to have the same amount of accuracy with the temperature adapted GAB model [Gitte, 2003]. There are other models out there such as the BET model [Parker, 2006], but was only found to fit data of 5-45% whereas the GAB was found to fit data up to 90%. Blahovec [Blahovec, 2008] promotes a generalized form of the GAB model in which the water activity portion of the equation is taken to higher orders for more accurate fits. The temperature adapted GAB model is used in this study to determine the moisture content of the substrate at different environmental conditions because it depends on temperature and fits the necessary range of humidity needed. (8) ( ) ( ) (9) ( )( ) (10) (11) (12) After the moisture isotherms are developed in order to determine how much moisture is present in the paper at any current environmental conditions, it is important to model 22

35 how the moisture affects the mechanical response of the substrate. This can be done by using a process called time-moisture superposition. Time-moisture superposition shifts curves for different mechanical responses, such as for the normalized stress, along the time axis to a defined reference moisture content. This shift factor, am, is determined by manually shifting stress versus log time curves horizontally to match the response at the reference moisture content. The shift factor, am, can be related to the moisture content by a model similar to the Williams-Landel-Ferry (WLF for short) equation (equation (16) seem on page 26) for timetemperature superposition [Williams, 1955]. The modified WLF equation for moisture content can be found in equation (13). M is the moisture content for the current mechanical response at time tm and M0 is the reference moisture content that the mechanical response is shifted to, for this study that reference moisture is the moisture content at standard TAPPI test conditions 23 C and 50% RH. D1 and D2 are the empirical constants used to fit the moisture content shift factors found from the results of the experimental study. Equation (14) shows that moisture content shift factor, am, shifts a mechanical response from the current moisture content to the reference moisture content by relating the times of the for identical responses, tm is used for the current moisture content and tmref is used for the reference moisture content. Ishisaka and Kawagoe used this adaptation of the WLF equation for moisture to determine a moisture content shift factor for epoxy [Ishisaka, 2004]. Chaleat also used the same moisture shifting equation to model the moisture shift factors for a starch blend [Chaleat, 2008]. (13) ( ) ( ) 23

36 (14) Moisture shifting can be used to obtain a master curve of the mechanical response at times where it cannot be measured. For example when a stress relaxation test is run it is impossible to instantaneously strain the material, so shifting is used to determine the relaxation response at times shorter than what can be practically measured at the reference state. The last thing that moisture will affect that has an impact on the mechanical response of the material is the glass transition temperature. The glass transition temperature, Tg, is the temperature in which an amorphous polymer (or the amorphous part of a semicrystalline polymer) changes material structure from a rubbery state to a brittle state. The glass transition temperature is less than the melting temperature, Tm, for polymers no matter the crystallinity. The glass transition temperature is important for the polymers because the WLF equation is meant to be used for temperatures above the glass transition temperature, even though there are ways around this [Williams, 1955]. According to Salmen and Back, the glass transition temperature decreases as moisture in the paper increases [Salmen, 1980]. Time-moisture superposition and time-temperature superposition will not hold if a transition temperature for the material is crossed. 2.6 Temperature Dependency As stated in the definitions of polymers, the mechanical condition of a polymer is highly dependent up on temperature and paper is no different. While discussing the moisture dependency of the paper substrate it was argued that the hydrogen bonds between the cellulose, hemicellulose and lignin were affected by moisture. The bonds are 24

37 also affected by temperature. An amorphous polymer or amorphous region of a semicrystalline polymer can undergo multiple state transitions due to thermal loading. Below the glass transition temperature the polymer is glassy and brittle but as temperature increases past the glass transition temperature the material becomes rubbery and then becomes almost a liquid state before the melting point is reached. Goring found that the dry glass transition temperature for cellulose was 230 C and between 165 C and 225 C for dry hemicellulose [Goring, 1963]. Whereas Blechschmidt performs an experiment on wood pulp and sees a glass transition temperature between 115 C and 145 C [Blechschmidt, 1986]. So depending on how the glass transition temperature is affected by moisture, for the most part the polymers are below their glass transition temperature and glassy. Since we know temperature affects the mechanical response of the paper substrate it could be concluded that the hydrogen bonds are weakening at higher temperatures and strengthening at lower temperatures. Considering moisture content held constant, an increase in temperature will lead to softening of the material and a decrease in temperature will lead to a stiffening of the substrate. Salmen [Salmen, 1980] completes an experimental study showing the modulus of elasticity softening as temperature decreases. Again the effect of temperature is considered to be dynamic and not take place as soon as the temperature changes as it takes time for the fiber network to acclimate to the surroundings. In order to model the effects of temperature on the paper substrate time-temperature superposition is used. Time-temperature superposition says that a mechanical response at one temperature can be related to another temperature purely by a time shift factor [Ward, 2004]. The time shift factor due to temperature, at, is related to the time at a reference 25

38 temperature, for this study 23 C as it is TAPPI standard, by equation (15). Shift factors are determined by plotting experimental data on a stress versus log time plot and manually shifted one by one to line up with the reference temperature response. There are multiple different models used to model the temperature dependence but the two most common models are the Williams, Landel, Ferry Equation and the Arrhenius equation. The WLF equation, which can be seen in equation (16), models the time shift factor due to temperature, at, from the temperature of the current response, T, and the reference temperature, T0. The two constants, C1 and C2, can be fit to the obtained experimental data for the time shifts due to temperature. The generalized form of the WLF equation, equation(16), relates the shift factor to a reference temperature, T0, whereas another version of the equation can be written in terms of the glass transition temperature, Tg, seen in equation (17) [Ward, 2004; Nielsen, 1994]. In 1955 Williams, Landel and Ferry defined a specific model of the WLF that was supposed to fit all amorphous polymers above the glass transition temperature [Williams, 1955]. So for this study the generalized model, equation (16), was used as the temperatures of importance are well below the dry glass transition temperature for the paper substrate discussed before. (15) (16) ( ) (17) ( ) The other model that is routinely used to model the time shift with respect to temperature is the Arrhenius equation. The Arrhenius equation can be seen in equation 26

39 (18) with the time shift factor, at. The shift factor is a function of the universal gas constant, activation energy, reference temperature and temperature shown by symbols R, Ea, T0 and T respectively. Ding uses the Arrhenius equation to shift factors to predict aging of paper [Ding, 2007]. Ishisaka uses the Arrhenius equation to model the dynamic viscoelasticity of epoxy [Ishisaka, 2004]. Ishisaka shows that the Arrhenius fits the shift factors better for lower temperatures than the WLF equation does. (18) ( ) 2.7 Plasticity Plasticity can be defined as a theory that describes permanent deformation in materials with constitutive relations. In essence plasticity is the opposite of elasticity, in that elasticity deals with reversible strains whereas plasticity deals with permanent strains. In some materials there exists a stress, called the yield stress, where any loading beyond this stress results in permanent deformation. Elastic deformation can continue to occur after the yield stress but is joined by plastic deformation as well. All materials that can be described by plasticity have a yield point which can be represented by a yield surface in six dimensional space, which is used to determine the stress under current loading conditions. Rate-independent plasticity theory deals with the relationship between stress and plastic strain. Viscoplasticity is a rate-dependent theory that deals with stress, plastic strain rate and plastic strain. Viscoplastic materials do not have to exhibit a yield surface and often times do not [Neto, 2011]. Viscoplastic material can be modeled using a theory called the viscoplasticity theory based on overstress (VBO) formulated by Krempl [Krausz, 1996]. VBO determines a 27

40 constitutive relation between stress and plastic strain depending on the long term equilibrium response,, of the material. An example of a model that uses VBO can be seen in equation (19), called the Norton s Creep Law [Neto, 2011]. Norton s Creep law determines the rate of plastic strain,, in terms of stress, σ, and the long term stress response, σ 0, at a given plastic strain, with the constants N and λ. Many other models for viscoplasticity can be found in literature, see Neto for example [Neto, 2011]. (19) ( ) The paper substrate in this research is assumed to be viscoplastic with no yield surface, which means it is yielding from onset. The reason behind this is because once hydrogen bonds are broken they cannot be reformed again without the addition of water to the material [Zauscher, 1997]. This is assuming that broken hydrogen bonds are largely responsible for the permanent deformation of the paper substrate under load. Paper is assumed to be yielding from onset because the hydrogen bonds are assumed to have varying strengths resulting in some breaking immediately. Other factors may contribute to inelastic deformation of the fiber network, such as the plastic straining of the fibers themselves and the irreversible conformational changes to the structure of the fiber network itself. 2.8 Failure In order to complete a model of the paper material in this research, there must be an idea of when the material will fail. This is done by creating a model of the stress in which the material fails, the model usually being referred to as failure criteria. According to 28

41 Theocaris, general papers strength is dependent on the strength of its fibers [Theocaris, 1989]. His research dealt with a much higher basis weight paper sheet so the determining factor for strength may be different. Hagglund concludes that for low basis weight paper sheets, like the one used in this research, the strength is dependent on the fiber network s fiber-to-fiber bonding [Hagglund, 2006]. This is most likely the failure mechanism of the paper substrate used in this research. The strength of the material, F, is also rate dependent [Daniel, 2011; Kwon, 2011]. Modeling the failure criteria usually takes the form of one of two models. The first model being the Tsai-Hill equation for plane stress, equation (20), which relates constants F11, F12, F22 and F66 to stresses σ1, σ2, and σ6 to determine the failure surface (where for this research σ1 stands for stress in MD, σ2 stands for stress in CD and σ6 stands for the shear stress between MD and CD). The other model that is typically used is the Tsai-Wu equation for plane stress, equation (21), which is the same form but adds constants F1 and F2. Rowland used the Tsai-Hill and Tsai-Wu equations to determine the failure criteria of paperboard by biaxial testing [Rowlands, 1985]. Biaxial testing is ideally how the failure criteria are determined because the shear stress term can be eliminated and fit separately but biaxial testing equipment is rare and may not be available to the user. If this is so then off-axis testing should be done and the stresses rotated into the material basis and determined that way. Tryding developed a modified version of Tsai-Wu in order to more accurately model the failure surface of paper [Tryding, 1994]. His model added a a few more constants and should really only be used after attempting Tsai-Hill and Tsai-Wu and finding that they do not provide an accurate fit. Daniel develops a strain-rate dependent failure criteria equation, equation (22), for nonlinear composites [Daniel, 2011]. Equation 29

42 (22) has the constants F( ), which is the failure strength at the reference strain rate, and mf that relate a current strain rate,, to the failure at a reference strain rate,. This way the failure criteria only needs to be fit to one strain rate and Daniel s equation used to relate to different strain rates. (20) (21) (22) ( ) ( )( ) 2.9 Literature Paper Models Many other people have attempted to model paper before. Not all of them are applicable to the research done in this model but it is worth looking at their conclusions. Makela determined an orthotropic elastic-plastic model for paper [Makela, 2003]. The problem with this model is that it is not rate dependent and it has been discussed that one of the main goals of this research is to study the rate dependencies of the paper. Since the model developed in this research was only develop for the MD and CD and Makela s work deals with stress in all three directions (MD, CD, and ZD) then this could be a good starting point for adding the ZD to this model. Castro developed an elastic-plastic model for paper as well, but this one uses a nonlinear hyperelastic fit the model [Castro, 2003]. Castro s model again does not deal with the rate dependencies that are of interest in this research but does propose a viable hyperelastic model for use in describing the nonlinear elasticplastic stress strain curve of paper. Castro s paper also uses Tsai-Hill, equation (20), to determine the failure criteria for the paper sheet [Castro, 2003]. 30

43 Nordin came up with a nonlinear viscoelastic/viscoplastic model for paper fiber composites in compression [Nordin, 2006]. Nordin s model exhibits the rate dependencies necessary for this research in the elastic and plastic regions but only for compression. Since this research is looking for a tensile model it is necessary to employ a different model. Nordin s model could be used to compare the model developed in this research to, and see how closely well this tensile model could do in compression and visa versa. 31

44 3 Experimental Methods 3.1 Tensile Specimen Preparation The sample material that was used in these experiments was conditioned at 23 C and 50% relative humidity for at least two weeks. This was done in order to make sure the sample material was acclimated to the conditions of test environment. The material came straight off of the production line in a large stack so time was needed for the material in the center of the stack to be acclimated to the conditions. Tensile specimens described in this section will be used to test uniaxial tensile tests to failure, uniaxial cyclic tensile tests, uniaxial Poisson s tests, and uniaxial stress relaxation tests. The typical sample used is 25.4 mm (1 in.) wide and mm (6 in.) long. The sample is mm long because the normal gage length used during the test is only 127 mm (5 in.) long but the sample needs the extra length to be able to be gripped by the testing equipment. A gage length of 127 mm is used in order to prevent the fiber network from having a fiber that stretches the length of the sample. If a fiber were to stretch the length of the sample then the test would just produce results for that individual fiber and not the fiber network. The mm direction of the sample is the direction in which the sample should be loaded into the test equipment. For most of the tests completed that direction is either the MD or CD. All the samples from this research are cut from the same production run of the paper substrate. This was done with the intention of minimizing scatter of the data that could be evident due to varying production conditions. The samples were cut out of the 32

45 sheet using a 25.4 mm (1 in.) double-bladed paper cutter and then cut using scissors to mm long. The double-bladed paper cutter ensures that every sample is the exact same width. The mechanical properties of the paper are dependent on the density of the paper which is known vary depending upon the location in which the sample is cut from the production material. So each sample that is prepared has its thickness, length, and width measured as well as the weight. From these measurements the density and cross sectional area can be determined. 3.2 Moisture-Sorption Isotherms The main purpose for constructing Moisture-Sorption isotherms is to understand the amount of moisture within the paper at varying temperatures and relative humidity (RH) values. The temperatures of interest were 5, 15, 23 and 28 C. These temperatures were chosen in order to understand the moisture levels in the paper over common production conditions. In order for the Time-Temperature Superposition method to be used effectively, there must be lower temperatures that shift the master curve to times that are not realistically measurable. Moisture-Sorption Isotherms for the paper substrate were constructed by using a temperature/humidity chamber. The desired temperatures were obtained by means of heating/cooling the chamber, which was heated by a heater located inside the chamber and cooled by a pair of compressed air cooling tubes connected to the chamber. The temperature and humidity were controlled by an ETS Dual Control Model A Venta Humidifier was used to add moisture to the air and an ETS dehumidifier was used to dry the air. A carriage was hung via a pass-through in the chamber from a scale, which was 33

46 located outside of the chamber, and tared so that the mass of the carriage was not taken into account. A diagram of the setup for the experiment can be seen in Figure Nine samples of the paper substrate, approximately 10 cm X 10 cm in area, were then loaded into the carriage. The temperature was set to the desired temperature and the humidity was set to 0% RH on the temperature/humidity controller used. For each test the temperature was held constant in order to produce one moisture-isotherm for that temperature. The sample was then left overnight in order to get the samples as dry as possible. The next day the RH was never below 2% RH because the chamber could not maintain a perfect 0% RH due to leaks in the chamber. The weight of the sample was taken every ten seconds for 5 C and 15 C and every minute for 23 C and 28 C for a half-hour at each RH increment from Table 3.2-1, excluding the initial dry weight. The reason for the two sample rates is because for the lower temperatures the cooling tubes were on and they added motion to the air which disturbed the samples and the added data sets were averaged to get a more accurate value. 34

47 Table Relative Humidity Increments Depending on Temperature Increment 5 C 15 C 23 C/28 C (#) 1 0% 0% 0% 2 3% 3% 10% 3 3.5% 4% 20% 4 4% 5% 35% 5 4.5% 6% 50% 6 5% 7% 65% 7 10% 8% 80% 8 20% 9% 90% 9 30% 10% % % % % - The absolute bone dry weight of the sample was found by using a different temperature/humidity chamber that allowed for higher temperatures. The same carriage was hung from a scale in order to measure the weight of the samples. The samples were then dried out and placed at 105 C for an extended period of time. After turning off the internal heater for a moment, a weight measurement was taken of the samples without any moisture in them. The reason for turning off the fan is because the hot air convection actually slightly lifts the paper samples in the carriage, causing a reported weight that is less than the actual weight of the paper samples. 35

48 Cool Tubes Scale Carriage Heater Chamber Figure Diagram of Moisture-Isotherm Test Setup It is worth noting that isotherms taken this way are adsorption isotherms. Adsorption isotherms tend to be slightly lower in MC at the same RH than a desorption isotherm [Parker, 2006; Gitte, 2003; Agrawal, 2004]. For the purposes of this experiment the desorption isotherms and adsorption isotherms for the paper substrate were assumed to be the same. 3.3 Uniaxial Tension at Varying Strain Rates The goal of running uniaxial tensile tests for varying strain rates is to determine whether or not there are rate-dependencies in tension for this paper substrate. Once it is determined if there are rate-dependencies in the tensile direction, the data can be used to develop a rate dependent model of the material. Three different strain rates were chosen over 3.5 decades. Uniaxial Tensile strain-controlled tensile tests to failure were run for varying strain rates using tensile specimens defined in the first section of this chapter. Three samples were run at each of the conditions in order to determine repeatability of the tests. The 36

49 conditions used for this test were to run tensile tests in the MD and CD directions at strain rates of 0.1, 4.5 and 45 1/s. This data was run on multiple different types of equipment in order to obtain the strain rates desired for the test. The different conditions/test setups for each of the strain rate tests can be found in Table Tensile specimens were held in place by pneumatic rubber grips at a distance of 127 mm apart. It is nearly impossible to place a specimen in between the grips perfectly taught at zero load, so the samples were placed in the grips with a preload specified to redefine the gage length at that load. The gage length is defined as the length of the sample between the two grips. The data sets for 4.5 and 45 1/s do not use a preload because the strain rate is so fast that the initial inertial load causes the load to spike. So for these strain rates the gage length is assumed to be 127 mm as the sample is loaded into the grips at 127 mm apart and then slack is induced. Data taken for the 4.5 and 45 1/s strain rates was unfiltered because the MTS 810 High Speed Test Frame used in this study did not have a load averaging data acquisition system on it. Table Uniaxial Test Information depending on Strain Rate Test Machine MTS Insight 5 kn 0.1 1/s 4.5 1/s 45 1/s MTS 810 High Speed Test Frame Frame Deformation Rate (mm/s) Preload (gf) Load Cell (N) Test Temp ( C) Test RH (%) MTS 810 High Speed Test 37

50 3.4 Off-Axis Uniaxial Tension The off-axis uniaxial tensile tests are completed exactly the same as the 0.1 1/s tests from the uniaxial tensile tests at varying strain rates. The only exception being that each sample is cut at incremental angles of 15 each, so there will be samples at angles of 0 (MD), 15, 30, 45, 60, 75 and 90 (CD). Samples are cut at an angle off the MD directions in order to construct a failure envelope for the material. MD θ 1 CD 2 Figure Diagram of Sample for Off-Axis Uniaxial Tests 3.5 Cyclic Loading at Varying Strain Rates Cyclic tests are done in order to break apart the stress-strain data into elastic and plastic strain, as well as determining if there is damage in the material. Paper is known to have permanent deformation so cyclic testing was done at multiple strain rates in order to determine if the permanent deformation was rate-dependent or not. Cyclic tests to failure 38

51 were done for strain rates of 0.01 and 0.1 1/s on the MTS Insight Electromechanical 5 kn instrument with a 50 N load cell. Three samples were tested at both directions (MD and CD) and both strain rates with the temperature at 23 C and the relative humidity at 50%, standard TAPPI conditions. Cyclic testing is when a sample is loaded to a set strain point and then unloaded to allow the sample to fully relaxes, 5 seconds was used in this study. The sample is pulled taught by an initial strain rate until a set pre-load limit is reached, for these experiments that value is 5 gf. After the pre-load is applied then the test starts and is continued to failure. Permanent strain is defined at the start of each cycle when force applied on the sample is greater than the preload value discussed above. This allows the elastic and plastic parts of the loading to be decoupled so that the elastic and plastic strain can be mapped to the total strain. Some error can be introduced to cyclic data by the load averaging algorithm that the test equipment uses. Errors in the data due to load averaging are seen at the peaks of each cycle. The averaging causes the unloading data to cross back over the load curve because the averaging is rounding out the sharp peak in the data. The load averaging algorithm was turned down on the test equipment in order to eliminate this error. By turning down the averaging algorithm, a trade-off is being made for accuracy at the peak value of each cycle but with the consequence of added mechanical noise in the data. 3.6 Stress Relaxation at Varying Temperature and Humidity From the previous chapter we have determined that paper exhibits viscoelascity. Stress relaxation tests are done in order to understand papers rate-dependencies. It has 39

52 also been determined that paper is dependent on the temperature and moisture content within the paper. In order to study how temperature and moisture content affect the paper, stress relaxation tests were run with varying temperatures and relative humidity levels in the test chamber. The end goal of these tests would be to develop a master stress relaxation curve at TAPPI standard conditions by using Time-Temperature and Time- Moisture Superposition. Stress relaxation data is taken by straining a sample to a certain strain as fast as possible and then holding the sample at that strain. Ideally the strain would be applied instantaneous to the sample but experimentally that is impossible. The stress relaxation tests were taken to 5% strain for the MD and 1.5% strain for the CD, where the sample was held for 10 minutes. These tests were run on an MTS Alliance RT 5 with a 50 N load cell and used wrench tightened rubber surfaced grips. The initial displacement rate to get to the set strain point was 63.5 mm/s (0.5 1/s). All the different conditions were run with three samples each to determine the repeatability of the relaxation tests. In order to determine the effect of temperature and moisture, the stress relaxation tests were completed at different ambient conditions that were held constant throughout the test. The temperature and relative humidity of the test were varied using the same temperature/humidity chamber used for the moisture-sorption isotherms, seen in Figure The only difference being instead of having a scale hang through the system the load train for the test frame passes through the box. The same equipment from the moisturesorption isotherms was used to control the temperature and humidity in the box for the stress relaxation tests. In order to get an understanding of how temperature effects the 40

53 stress relaxation, four different temperatures were used which were: 5 C, 15 C, 23 C and 28 C. At each of these temperatures three different moisture contents were run which were calculated using the data from the moisture-sorption isotherms. The moisture contents studied were 4%, 6% and 8%. At every change of condition, the samples were left in the chamber for 30 minutes in order to allow them to acclimate to the chamber conditions. 3.7 Uniaxial Tension to Determine Poisson s Ratio The Poisson s ratio between the MD and CD directions, v12, is important as it describes how the material deforms in the CD when loading in the MD. This value is calculated by placing a 10 mm by 10 mm plus sign on a tensile test sample with each line oriented in the MD and CD. The sample was tested in uniaxial strain-controlled tension in the MD on a MTS Insight ElectroMechanical 5 kn test frame. A strain rate of /s (1.27 mm/s with a 127 mm gage length) was used to allow for the sample to be photographed as it was strained. The sample was pre-strained to 20 gf just like previous tensile tests to get rid of slack in the sample. At the point of pre-strain an initial photograph of the sample was taken, and then photographs were taken at approximate increments of 50 gf from there to failure. There was no effort made to prevent bending along the edges of the paper because after a few test samples were run bending was negligible to the naked eye. If there was bending it would have shown values for strain in the CD to be greater than actual strain values in the paper. The Poisson s ratio is calculated by observing how the CD deforms with respect to the MD. The change in length of the plus sign drawn on the sample was measured at each 41

54 picture using ImageJ, an image processing and analysis software. This was done by taking the stock image, for example Figure 3.7-1, and cropping it down to just the region where the markings are seen as in Figure The cropped image was then subjected to color threshold analysis in order to remove anything that isn t the markings, seen as red in Figure 3.7-2, and to highlight the markings. This was done in order to obtain a consistent measure of the marking lengths for different photographs. The length in pixels was measured at each photograph and can be converted into actual length as the pixel length can be calculated from the initial photograph where the lengths are known to be 10 mm and 10 mm in the MD and CD respectively. Knowing the lengths of each direction of the markings, the Poisson s ratio can be calculated for each photograph. Figure Sample Test Set-Up for determining Poisson s Ratio 42

55 Figure Cropped Image of markings on sample pre and post color threshold analysis 43

56 3.8 Total Test Plan The table below, Table 3.8-1, details the complete test plan for this thesis project. Table List of all Tests done on the Paper Substrate Test Total Runs Variables Overview Moisture-Sorption Isotherms 4-4 temps. Sweep RH at constant Temp. over Uniaxial Tension at Varying Strain Rates Off-Axis Uniaxial Tension Cyclic Loading at Varying Strain Rates Stress Relaxation at Varying Temp. and Humidity Uniaxial Tension to Determine Poisson s Ratio 18-3 strain rates - 2 directions - 3 spec./test 21-7 angles - 3 spec./test 12-2 strain rates - 2 directions - 3 spec./test 72-2 directions - 4 temps. - 3 MC a defined range Uniaxial strain controlled tensile test to failure Uniaxial strain controlled tensile test to failure Uniaxial strain controlled cyclic tensile test to failure Stress relaxation for 10 minutes in temperature and RH controlled environ. - 3 spec./test 1 - Take pictures of sample at extremely slow strain rate 44

57 4 Modeling Experimental Data 4.1 Moisture-Sorption Isotherm Models Moisture-Sorption Isotherm Experimental Data Moisture-Sorption Isotherms were obtained by following the procedure listed in the Experimental Methods section above. Figure shows the dynamic experimental data taken at 23 C, which the same test was also run at 5, 15 and 28 C. The relative humidity (RH) was changed approximately every 30 minutes to insure that equilibrium is reached. Figure shows that the moisture content of the paper had enough time to reach equilibrium at each humidity level. This amount of time allows for the chamber to acclimate to temperature and humidity set by the user and allows for the paper acclimate once the environment reaches equilibrium. In order to determine the actual moisture content (MC) at each relative humidity equation (8) was used with a dry basis weight of g/m 2, which was obtained by cooking the samples above 100 C. The scatter seen in Figure can be attributed to the movement of the air in the chamber causing the paper samples to be raised by the air and causing the carriage to move with the airflow. This error was accounted for by averaging the weight data once the sample had come to assumed moisture content equilibrium. The moisture contents taken at varying RH values for four different constant temperatures, 278 K (5 C), 288 K (15 C), 296 K (23 C) and 301 K (28 C), were then plotted versus water activity, aw, in Figure Water activity is defined here as relative humidity divided by 100%. From Figure 4.1-2, it can be determined that as temperature 45

58 increases at constant relative humidity, the total moisture content in this paper will decrease. This is consistent with other paper materials studied by Sorenson as he found the same trend with paper molded trays [Gitte, 2003]. Although Parker studied multiple different paper products and found that not all substrates followed this trend [Parker, 2006]. Isotherms for this substrate take the shape of Type II isotherms according to Keller s study on adsorption isotherms [Keller, 2005]. An assumption can be made that the data for 5 and 15 C will continue to follow the shape of the curves for 23 and 28 C above 30% RH. Higher relative humidity data was not obtained for the lower temperatures as it was well above the MC that this study was interested in. It can be seen that there is a finer amount of data points retrieved at lower relative humidity levels for the 5 and 15 C samples in order to more accurately model the sharp increase in moisture content the substrate sees between 0% RH and 2% RH. The humidity controlling equipment wasn t accurate enough to capture this phenomenon because the chamber could not be held at humidity levels below 2.5% RH at lower temperatures. 46

59 Moisture Content (g/g) MC (g/g) Experimental Data for 296 K (23 C) % RH 20% RH 35% RH 50% RH 65% RH 80% RH 90% RH Time (s) Figure Dynamic Experimental Data for Moisture Sorption Isotherm at 296K (23 C) 0.25 Moisture Sorption Isotherm Experimental Data Exp 278 K Exp 288 K Exp 296 K Exp 301 K Water Activity, aw Figure Moisture Sorption Isotherm Experimental Data for the paper substrate 47

60 4.1.2 Moisture-Sorption Isotherm Model Description The moisture isotherm experimental data seen in Figure must be modeled so that the moisture content in the paper can be determined at any environmental conditions, varying humidity and temperature. This was accomplished by using the GAB temperature dependent model, equations (9)-(12) found on page 22, described in the literature review. The GAB temperature dependent model was used in order to account for the differences in the moisture content of the paper for varying temperatures and because the GAB is known to fit Type II adsorption isotherms well. Figure shows the moisture sorption isotherm model developed using nonlinear least squares fitting with the constants, determined by the fitting procedure, displayed in Table As it can be seen in Figure 4.1-3, the model does a great job of fitting the higher temperature curves (15, 23 and 28 C) and not as good a job of fitting the 5 C isotherm. This was done on purpose as the converting line conditions will most likely be at or above room temperature so it was important to get a better fit for that region. Even though it isn t the best fit for the 5 C isotherm, the model still provides a decent fit and accurate shape of the isotherm and because this is true the model is assumed to be valuable for all four temperatures and the inclusive temperature range (5 C to 28 C). Table GAB Fitted Constants for paper substrate M 0 ' C' 5.91E-29 K' k k k

61 Moisture Content (g/g) Moisture Sorption Isotherm Models Exp 278 K Exp 288 K Exp 296 K Exp 301 K Model 278 K Model 288 K Model 296 K Model 301 K Water Activity, aw Figure Moisture Sorption Isotherm Model at 278 K (5 C), 288 K (15 C), 296 K (23 C) and 301 K (28 C) The values obtained for M0, C and K at each temperature using equations (10), (11) and (12) can be seen in Table The value of M0 ranges from 5.8 to 7.2 g/100g of dry sample weight whereas the paper studied by Parker varies from 3.8 to 6.5 g/100g of dry sample weight [Parker, 2006]. From this it can be concluded that the monolayer weight, M0, is consistent with literature values of paper. C and K also match literature values as C is greater than zero and K is constant between 0 and 1. The constants found for this paper substrate also match the trends Sorenson found for cardboard trays [Gitte, 2003]. Parker and Sorenson studied different kinds of food industry packing paper material and not paper tissue, so the behavior should not be the same but similar. Figure displays the moisture isotherm taken at 23 C for this study alongside experimental data taken by Salmen for kraft sack paper at 23 C [Salmen, 1980]. These curves compare favorably even 49

62 Moisturer Content (g/g) though the one being researched in this paper is toilet paper and the other is not. Now that the model obtained can be considered to be accurate the GAB model can be used to obtain the moisture content in test runs for other experiments. Most experimental tests are done at TAPPI standard test conditions, 23 C and 50% RH, which when using the GAB model for this substrate an equilibrium moisture content of 8.25% is found. Table M0, C and K values at each constant temperature isotherm Temp (K) M C K Experimental Data taken at 23 C compared to Kraft Sack at 23 C Exp. Paper Substrate Kraft Sack Water Activity, aw Figure Experimental Data taken at 23 C versus Kraft Sack paper data [Salmen, 1980] 50

63 4.2 Time-Temperature-Moisture Superposition Stress Relaxation Experimental Data Now that the moisture content at any environmental conditions can be found, it is possible to determine what type of influence moisture content and temperature have on the mechanical response of the paper. Figure and Figure show the stress relaxation curves for the material at varying moisture contents, 4%, 6% and 8%, for the MD and CD respectively. For each MC there are three tests run at four different temperatures as to determine the effect temperature and moisture had on the paper and the repeatability of each tests. To model the effect of moisture and temperature on the stress relaxation, temperature and moisture shifting were used. These methods were also used to develop a stress relaxation master curve for a specific reference temperature and moisture content. Temperature and moisture shifting can be used to obtain the mechanical response for times that are not able to be measured experimentally. The master curve that was generated consists of stress relaxation curves shifted to TAPPI standard testing conditions, 23 C and 8.25% MC (only for this paper substrate). The first step was to shift each individual MC to the value of 23 C and determine the temperature dependence of the material. 51

64 Figure Experimental Stress Relaxation Curves in MD for varying Temperature and Moisture Contents Figure Experimental Stress Relaxation Curves in CD for varying Temperature and Moisture Contents 52

65 4.2.2 Temperature Shifting Model Master curves are obtained for each of the three MC s and shown in Figure and Figure at 23 C in the MD and CD respectively. The initial shifting is only done at varying temperatures and not varying moisture content so that the temperature dependence on the stress relaxation can be isolated and then modeled. Each relaxation curve was shifted horizontally along the log time axis to the relaxation data taken at 23 C for that respective moisture content. From these relaxation curves it can be seen that in both the MD and CD an increase in moisture content will decrease the stiffness of the material, which is consistent with literature discussed above in the literature review. The shifting in both the MD and CD seem to provide consistent master curves meaning that no transition temperature is observed in the region of interest. A transition temperature would be assumed to occur if the shape of the curve changed between two sample sets. This data shows what can be considered to be a continuous master curves with only some outlying data. 53

66 Figure Stress Relaxation Master Curves in MD after Temperature Shifting for 4%, 6% and 8% Moisture Contents Figure Stress Relaxation Master Curves in CD after Temperature Shifting for 4%, 6% and 8% Moisture Content 54

67 In the literature review, the WLF and Arrhenius equations were discussed as ways in which to model the temperature time shifting factors, at. The WLF equation was used as it accurately modeled the shape of the curve where the Arrhenius equation could not. The WLF model and the experimental shift factors determined by shifting to the master curves at each temperature can be seen in Figure and Figure for the MD and CD respectively. The constants used to fit the WLF function, equation (16), can be seen in Table Each of the models was fit using a nonlinear least squares fit to minimize the error. The temperature shift factors seem to be independent of moisture content as each look to fall on top of one another. There is slight scatter to the data in the MD and CD, which can largely be attributed to the density variations from sample to sample. This paper material is extremely thin so any variation in density will have a significant impact on the stiffness of the material. The WLF model can be used to obtain the shift factor, at, that will shift the data to the mechanical response that would be seen at 23 C for the respective moisture content, shown in equation (15). The temperature shift data shows that as temperature increases the stiffness of the material decreases, which falls in line with what literature suggests. Figure superimposes the WLF temperature shifting model for the MD and CD on top of each other. From this figure it can be seen that the models are almost identical, thus allowing us to assume that the temperature dependencies of the material are uniform. Even though this can be assumed, two separate models will be used for the remainder of this research in order to improve accuracy amongst the data that takes advantage of the WLF model determined here. 55

68 log(at) Table Values of Constants for WLF Temperature Shifting for MD and CD MD CD C C Temperature Shifting Model for MD % MC 6% MC 8% MC WLF Model Temperature (K) Figure WLF Temperature Shifting Model for MD 56

69 log (a t ) log(at) 14 Temperature Shifting Model for CD % MC 6% MC 8% MC WLF Model Temperature (K) Figure WLF Temperature Shifting Model for CD 14 Models for Temperature Shifting in MD and CD MD CD Temperature (K) Figure WLF Temperature Shifting Models for MD and CD 57

70 4.2.3 Moisture Shifting Model The moisture dependence of the paper substrate can be modeled in a similar manner as the temperature dependence. The three master curves for each of their respective moisture contents, 4%, 6% and 8%, obtained from temperature shifting in Figure and Figure were then shifted to the 8.25% MC. This was done in order to obtain the moisture dependency of the substrate in both directions. Figure and Figure display the master curves, obtained in the MD and CD respectively, from shifting the temperature master curves with respect to moisture content. It can be seen in both figures that the master curves for 4% and 6% moisture contents have the same shape but the master curve for 8% exhibits a slightly different shape. Usually this means that a transition temperature has been crossed and shifting is no longer a viable option but because the shape is only slightly off then shifting is still assumed to be viable. An explanation for this can come from the data being taken on the verge of a transition temperature or test operation error. In order to fully understand which it is, another test should be run to develop a master curve at a moisture content level above 8% and if the shape continues to change then a transition temperature has been reached. Literature suggest that this could be the glass transition temperature for paper substrates as Salmen found that for Kraft sack paper the glass transition temperature was approximately 45 C for an 8% MC [Salmen, 1980]. It is not hard to imagine that for this paper substrate at 8% MC that value could be around the highest temperature tested, 28 C. 58

71 Figure Stress Relaxation Master Curve in MD at 8.25% Moisture Content after Temperature and Moisture Shifting Figure Stress Relaxation Master Curve in CD at 8.25% Moisture Content after Temperature and Moisture Shifting 59

72 The time shift factors due to moisture content, am, obtained from creating these master curves are plotted in Figure , the constitutive model for the MD and CD is also plotted. The model used can be seen in equation (13) and allows for the material to account for moisture in the mechanical response by using equation (14). A nonlinear least squares curve fit was used in order to obtain the constants D1 and D2, shown in Table Like the temperature model it is observed that the moisture models for the MD and CD are remarkably similar. Again it is reasonable to assume that for this range of moisture contents the paper exhibits identical moisture dependencies. The two separate models for the MD and CD are used though in order to have improved accuracy on further data that takes advantage of this model. From the data displayed in Figure it can be determined that the moisture dependency is linear in nature and may be better suited to be modeled by a different equation. For now the WLF type moisture shifting model works well for the desired range but may be inadequate to describe a broader range of moisture contents. Table Values of Constants for WLF Moisture Shifting Model for MD and CD MD CD D D

73 log(am) 2 Moisture Shifting Model for MD and CD 1 0 0% 2% 4% 6% 8% 10% 12% -1-2 MD Shift Factors CD Shift Factors Model MD Model CD Moisture Content (%) Figure WLF Moisture Shifting Model for MD and CD Now that both the temperature and moisture constitutive relations are constructed any environmental condition in the tested range of temperatures and moisture contents can be accurately shifted to find the equivalent TAPPI standard testing conditions response. This will be important later on as some of the tests run were done in conditions that were not held at constant 23 C and 50% RH. Equation (23) is assumed to hold true in shifting the mechanical response to TAPPI conditions for this research. The time at TAPPI conditions, ttappi, is calculated by dividing the time during the test, ttest, by the multiplication of the temperature time shift factor, at, and the moisture time shift factor, am. (23) 61

74 4.3 Strain Mapping Cyclic Tensile Experimental Data Cyclic uniaxial tension tests were run in order to determine the relationship between total strain and elastic/plastic strain. The permanent strain was broken out at each cycle and equation (4) was assumed to hold for additive plastic and elastic strain. In order to determine if there were any rate-dependencies for the relationship between total strain and elastic/plastic strain, the tests were run at 0.1 1/s and /s total strain rates. Figure and Figure show the stress-strain data obtained from the cyclic tests run in the MD and CD respectively. After observing these figures they strongly indicated yielding from the onset of loading hence the initial yield stress is assumed to be zero. Figure shows that stress increases with the total strain rate indicating that viscoplasticity should we used to describe the plastic response. An equilibrium yield stress response with an initial yield stress of zero is still needed to describe the viscoplastic model. It can also be observed in the MD that the elastic strain peaks around 5% and any loading from then on only produces plastic strain. The CD does not depict this but that is probably because the material failed before it reached the maximum elastic strain level. The goal for these tests is to develop a mapping that can be used to calculated plastic strain, εp, and elastic strain, εe, from the total strain, εt, at all strain rates. 62

75 Figure Experimental Cyclic Stress-Strain Breakdown for MD Figure Experimental Cyclic Stress-Strain Breakdown for CD 63

76 4.3.2 Strain Map Model In order to develop the strain map, the total strain was plotted against the elastic and plastic strains in Figure and Figure for the MD and CD respectively. It can be seen in these figures that the relationship between total strain and elastic strain and the relationship between total strain and plastic strain is the same for both 0.1 1/s and /s strain rates. This may not be the case for all strain rates but since no other cyclic tests were able to be run at higher strain rates the relationship is assumed to be rate independent. When developing a model to fit the strain map it is important that the rate of change of the mapping function never become greater than one for the plastic strain and less than zero for elastic strain. If the rate of change of the plastic strain to total strain is greater than one that means that there is increasingly more plastic strain than total strain which is not possible under the laws of thermodynamics. For this reason a hyperbolic model should be used that approaches 0 and 1 for the rate of change of elastic and plastic strain respectively. Equations (24) and (25) were used to model the elastic and plastic strain mapping respectively. Each function takes the same form with three constants A, B and C and is dependent on the variable εt which representing total strain. 64

77 Figure Experimental Elastic/Plastic Strain versus Total Strain at 0.1 1/s and /s Strain Rates for MD Figure Experimental Elastic/Plastic Strain versus Total Strain at 0.1 1/s and /s Strain Rates for CD 65

78 Equations (24) and (25) were fitted to the total strain versus elastic and plastic strain by a nonlinear least squares method. The constants fitted to the mapping function can be found in Table The models plotted versus the experimental data can be found in Figure and Figure for the MD and CD respectively. The plastic strain fits are forced to make sure that plastic strain rate is not greater than the total strain rate before failure. The elastic strain rate fits are forced into maintaining at least an elastic strain rate of zero or greater to ensure that the energy is conserved. (24) (25) Table Strain Mapping Constants for MD and CD Elastic/Plastic Strain MD CD Elastic Plastic Elastic Plastic A B C

79 Elastic/Plastic Strain Elastic/Plastic Strain Strain Mapping MD between Elastic/Plastic Strain and Total Strain Exp. Elastic Strain Exp. Plastic Strain Model Elastic Map Model Plastic Map Total Strain Figure Elastic and Plastic Strain Mapping Model for MD Strain Mapping CD between Elastic/Plastic Strain and Total Strain Exp. Elastic Strain 0.01 Exp. Plastic Strain Model Elastic Map Model Plastic Map Total Strain Figure Elastic and Plastic Strain Mapping Model for CD 67

80 4.3.3 Uniaxial Tensile Experimental Data These strain mapping models will be used on uniaxial tension tests at varying strain rates in order to pull out the elastic and plastic strain. From the elastic and plastic strain data pulled out from the mapping function, the viscoelastic and viscoplastic models can be determined. Uniaxial tension experimental data obtained at 0.1 1/s, 4.5 1/s and 45 1/s strain rates for the MD and CD are displayed in Figure and Figure respectively. Figure shows definite rate dependencies in the material in the MD with the stiffness of the material increasing at higher strain rates. The CD response, shown in Figure 4.3-8, looks to be closer to rate independent than the MD. The 0.1 1/s data is still slightly softer than the 4.5 1/s and 45 1/s data so it is still considered rate dependent for the sake of this experiment. The scatter in the data for the 4.5 1/s and 45 1/s uniaxial tensile tests is due to the load cell data being unfiltered, which will be discussed more in the sources of error section. The other source of scatter in all three strain rates is the variations of density from sample to sample which is due to the paper manufacturing process not controlling the distribution of wood fibers accurately enough to allow for the paper to have uniform density. 68

81 True Stress (Pa) True Stress (Pa) Experimental Data in Uniaxial Tension for MD Hz Exp. Data (1) 0.1 Hz Exp. Data (2) 0.1 Hz Exp. Data (3) 4.5 Hz Exp. Data (1) 4.5 Hz Exp. Data (2) 4.5 Hz Exp. Data (3) 45 Hz Exp. Data (1) 45 Hz Exp. Data (2) 45 Hz Exp. Data (3) Total True Strain Figure Experimental Data taken in Uniaxial Tension for MD at Strain Rates of 0.1 1/s, 4.5 1/s and 45 1/s Experimental Data in Uniaxial Tension for CD Hz Exp. Data (1) 0.1 Hz Exp. Data (2) 0.1 Hz Exp. Data (3) 4.5 Hz Exp. Data (1) 4.5 Hz Exp. Data (2) 4.5 Hz Exp. Data (3) 45 Hz Exp Data (1) 45 Hz Exp. Data (2) 45 Hz Exp. Data (3) Total True Strain Figure Experimental Data taken in Uniaxial Tension for CD at Strain Rates of 0.1 1/s, 4.5 1/s and 45 1/s 69

82 4.4 Viscoelastic Model Stress Relaxation Master Curve The stress relaxation master curves for the MD and CD, obtained from the temperature and moisture shifting, includes data that could not be experimentally measured. From these master curves the generalized Maxwell model should be fit in order to determine the viscoelastic constants for both directions. It is assumed that the tests conducted to obtain the stress relaxation curves are purely viscoelastic and exhibit no inelasticity. This model/master curve is completed for TAPPI standard conditions and can be shifted to different conditions by utilizing the moisture/temperature shifting models obtained in a previous section. After the Generalized Maxwell model is fit for each direction the instantaneous,, and long term response,, of the material will need to be approximated because the viscoelastic model used in this research depends on the instantaneous elastic response and the viscoplastic model used depends on the long term plastic response. Figure and Figure show the stress relaxation master curves that will be fit to the Generalized Maxwell Model. 70

83 Figure Stress Relaxation Master Curve at 8.25% MC for MD Figure Stress Relaxation Master Curve at 8.25% MC for CD 71

84 4.4.2 Generalized Maxwell Model The Generalized Maxwell model used is based off of Figure The model used can be more clearly explained by equations (26) through (36) which describe how the generalized Maxwell model is implemented by using a Prony series of Maxwell elements (shown in Figure a)). Figure can be broken down into equation (26) which has a long term response followed by the sum of the response of the viscous dashpots found in equation (27). In equation (27) the variable α is an internal variable used to characterize the elastic strain due to the viscous dashpot. The long term stiffness of the system is defined as E whereas the instantaneous response is defined by E0 calculated from equation (33). The long term response is when all viscous dashpots are open and is represented by the all the stiffness values of the Maxwell components in parallel. (26) (27) ( ) An internal stress variable, qi, for the viscous dashpot is introduced in equation (28) in order to come up with relation for the stress in terms of the instantaneous stress response, S. The instantaneous stress response is the response of the system if the strain rate was infinite. Equation (29) can be obtained by combining equations (26), (27), (28) and (33). Equation (27) can be rewritten in terms of the internal stress variable, qi, in the form of equation (30). Solving for qi in equation (30) and plugging that into equation (29) obtains equation (31). Equation (31) allows for the stress to be determined at a certain time as long as the instantaneous stress response, S, is known. The method for determining the instantaneous stress response will be detailed later. 72

85 (28) (29) (30) (31) ( ) ( ) (32) ( ) ( ) (33) (34) (35) (36) Equation (32) can be used to fit the variables for the generalized Maxwell model. The stress relaxation master curves is normalized by an initial stress to obtain e(t). The initial stress values were calculated by fitting an exponential function to the first 50 values closest to zero for each master curve. Each of these fitted initial stress values can be found in Table Now that the master curves are normalized, equation (32) can be used to fit the constants γi and τi by using a nonlinear least squares fitting method. The constant γi is the normalized stiffness value found in equation (34). The constant τi is the time constant and is the damping value of the viscous dashpot normalized to the stiffness of that Maxwell unit; this can be found in equation (35). In this research it was found that the number of Maxwell components should be one unit for every decade of experimental data. This meant that there needs to be 8 Maxwell components in the MD and 7 in the CD for this paper substrate. 73

86 The fitted parameters for the generalized Maxwell model can be seen for the MD and CD in Table and Table respectively. The stress relaxation fit of equation (32) alongside the stress relaxation data for each master curve can be seen in Figure and Figure for the MD and CD respectively. These fits were determined to be unique when the error value of the minimizing function changed less than 1%. Although that does not guarantee a unique fit, the fits obtained are assumed unique and look to be an accurate representation of the experimentally obtained stress relaxation master curves. If this fitting technique was not assumed to be trustworthy a genetic fitting algorithm can be used to guarantee uniqueness. The Prony series fits still do not allow for the modeling of uniaxial elastic data. For this the instantaneous stress response, S, needs to be determined. Figure Maxwell Model fit to Stress Relaxation Master Curve for MD at 8.25% MC 74

87 Figure Maxwell Model fit to Stress Relaxation Master Curve for CD at 8.25% MC Table Calculated Initial Stress for Stress Relaxation Master Curves in MD and CD MD CD σ 0 (Pa) Table Prony Series Fits for MD γ τ 75

88 Table Prony Series Fits for CD γ E τ Instantaneous and Long Term Response The instantaneous stress response needs to be determined in order to complete the viscoelastic model for the paper substrate. The long term stress response is also needed in order to model the viscoplastic region of the mechanical response. Both the instantaneous and long term stress responses are determined in the same manner. Equation (31) can be used in conjunction with the uniaxial tension data and a root finding method to determine the instantaneous and long term stress responses. This can be completed by using the Secant method to determine the roots of the variable S in equation (31). Equation (31) can be numerically integrated by using a method from Simo seen in equation (37) [Simo, 1998]. For this model equation (37) uses equation (38) for the function hi because the constants γi and τi are assumed to not vary with stress. Whereas there is an alternate form of function hi that can also be found in Simo that accounts for the constants being dependent on stress. A uniaxial tension data set is used to fill in the stress values and the roots can be found from each of the experimental data sets. (37) ( ) ( ) (38) ( ) ( ) ( ) ( ) ( ( ) ( )) 76

89 The instantaneous and long term stress responses with respect to the total strain are found using the method described above for each of the three uniaxial tension strain rates, 0.1 1/s, 4.5 1/s and 45 1/s (strain rates will be referred to as 1/s or Hz for here on). The results for each of these responses in the MD and CD can be seen in Figure through Figure Each of the MD roots look to be consistent with one another but the CD roots seem to have a difference between the roots obtained from the 0.1 1/s strain rate the roots obtained from the 4.5 1/s and 45 1/s strain rates. It was discussed early that the CD conveyed only slight rate dependencies and the discrepancy in the roots is likely due to a rate dependent model not necessarily being the best model to describe the material. A nonlinear fitting function, equation (39), was then used to get a single fit for the long term and instantaneous stress response compared to the total strain. The constants used for each of those fits can be seen in Table (39) ( ) ( ( )) ( ) 77

90 True Stress (Pa) True Stress (Pa) Instantaneous Response Model for MD Hz Instant Response (1) 0.1 Hz Instant Response (2) 0.1 Hz Instant Response (3) 4.5 Hz Instant Response (1) 4.5 Hz Instant Response (2) 4.5 Hz Instant Response (3) 45 Hz Instant Response (1) 45 Hz Instant Response (2) 45 Hz Instant Response (3) Model Instant Response Total True Strain Figure Instantaneous Response in MD Long Term Response Model for MD Hz Long Response (1) 0.1 Hz Long Response (2) 0.1 Hz Long Response (3) 4.5 Hz Long Response (1) 4.5 Hz Long Response (2) 4.5 Hz Long Response (3) 45 Hz Long Response (1) 45 Hz Long Response (2) 45 Hz Long Response (3) Model Long Term Response Total True Strain Figure Long Term Response in MD 78

91 True Stress (Pa) True Stress (Pa) Instant Response Model for CD Hz Instant Response (1) 0.1 Hz Instant Response (2) 0.1 Hz Instant Response (3) 4.5 Hz Instant Response (1) 4.5 Hz Instant Response (2) 4.5 Hz Instant Response (3) 45 Hz Instant Response (1) 45 Hz Instant Response (2) 45 Hz Instant Response (3) Model Instant Response Total True Strain Figure Instantaneous Response CD Long Term Response Model for CD Hz Long Response (1) 0.1 Hz Long Response (2) 0.1 Hz Long Response (3) 4.5 Hz Long Response (1) 4.5 Hz Long Response (2) 4.5 Hz Long Response (3) 45 Hz Long Response (1) 45 Hz Long Response (2) 45 Hz Long Response (3) Long Response Model Total True Strain Figure Long Term Response CD 79

92 Table Constants for Instantaneous and Long Term Stress Responses in terms of Total Strain for MD and CD Instant Long-Term MD CD MD CD A B C D E F Viscoelastic Model Description To complete the viscoelastic model the instantaneous stress response with respect to total strain should then be mapped to obtain the instantaneous stress response with respect to elastic strain. Strain mapping constants for total strain to elastic strain can be found in Table The values obtained for the instantaneous stress response after mapping to the elastic region can be found in Table Equation (37) can now be used as the instantaneous stress response, S, for the elastic region is known for the paper substrate. This equation can then be numerically integrated to find the model viscoelastic response. The results of the models compared to the experimental data can be found in Figure The viscoelastic model fits the data taken in the MD much better than the data in the CD, which is most likely due to the reasons listed above for the discrepancies in the instantaneous/long term response. Although the CD viscoelastic model still manages to predict the elastic data at total strain rates of 4.5 1/s and 45 1/s fairly well. Figure and Figure show the complete viscoelastic model in the MD and CD, respectively, for the 0.1 1/s, 4.5 1/s and 45 1/s strain rates as well, as the instantaneous and long term responses of the material. It can be assumed in the CD that the viscoelastic model for the 80

93 material is the same as the instantaneous elastic response above a total strain rate of 4.5 1/s. At 45 1/s total strain rate the MD model is still not able to be assumed equivalent to the instantaneous elastic response. Table Constants for Elastic Instantaneous Response in MD and CD MD CD A B C D E F

94 Stress (Pa) Stress (Pa) Stress (Pa) Stress (Pa) Stress (Pa) Stress (Pa) Viscoelastic Fit for 0.1 Hz Strain Rate in MD Viscoelastic Fit for 0.1 Hz Strain Rate in CD Hz Exp. Data (1) 0.1 Hz Exp. Data (2) Hz Exp (1) 0.1 Hz Exp (2) Hz Exp. Data (3) 0.1 Hz Model Hz Exp (3) 0.1 Hz Model Elastic Strain Elastic Strain Viscoelastic Fit for 4.5 Hz Strain Rate in MD Viscoelastic Fit for 4.5 Hz Strain Rate in CD Hz Exp. Data (1) Hz Exp (1) Hz Exp. Data (2) 4.5 Hz Exp. Data (3) Hz Exp (2) 4.5 Hz Exp (3) Hz Model Hz Model Elastic Strain Elastic Strain Viscoelastic Fit for 45 Hz Strain Rate in MD Viscoelastic Fit for 45 Hz Strain Rate in CD Exp (1) 45 Exp (2) 45 Exp (3) 45 Model Hz Exp (1) 45 Hz Exp (2) 45 Hz Exp (3) 45 Hz Model ELastic Strain Elastic Strain Figure Viscoelastic Fits at Strain Rates of 0.1 1/s, 4.5 1/s and 45 1/s in the MD and CD 82

95 True Stress (Pa) Treu Stress (Pa) Viscoelastic Model for MD Long Term 0.1 Hz 4.5 Hz 45 Hz Instant Elastic Strain Figure Viscoelastic Model at Strain Rates Equal to 0 (Long Term), 0.1, 4.5, 45 and Inf (Instant) 1/s for MD Viscoelastic Model for CD Long Term 0.1 Hz Model 4.5 Hz Model 45 Hz Model Instant Elastic Strain Figure Viscoelastic Model at Strain Rates Equal to 0 (Long Term), 0.1, 4.5, 45 and Inf (instant) 1/s for CD 83

96 4.5 Plasticity Model Plastic Experimental data The other component of the material is the inelastic response, which is assumed in this research to only consist of plastic strain. Equation (25) is used to map the total strain onto the plastic region for each of the uniaxial tensile tests. Figure and Figure show the plastic strain versus stress data mapped from the total strain data found in Figure and Figure for the MD and CD respectively. Both MD and CD show rate dependencies in the plastic region with the CD showing slightly less rate dependencies than the MD. The rate dependencies in the plastic region indicate that a viscoplastic model with no initial yield stress should be used to model the plastic stress response. In order to determine the viscoplastic model, the long term plastic stress response must be obtained. By using the long term response found when determining the roots of equation (31) for total strain and mapping to plastic strain the long term stress response in terms of plastic strain is found. The constants for the nonlinear fit, equation (39), of the long term stress response in terms of plastic strain can be found in Table Discussed in the literature review, the viscous based overstress (VBO) method will be used in order to model the viscoplastic response in the MD and CD. Table Constants for Plastic Long Term Stress Response in MD and CD MD CD A B C D E F

97 True Stress (Pa) True Stress (Pa) Plastic Strain vs. Stress in the MD Hz Exp (1) 0.1 Hz Exp (2) 0.1 Hz Exp (3) 4.5 Hz Exp (1) 4.5 Hz Exp (2) 4.5 Hz Exp (3) 45 Hz Exp (1) 45 Hz Exp (2) 45 Hz Exp (3) Plastic Strain Figure Plastic Strain versus Stress Data for varying Strain Rates in the MD Plastic Strain vs. Stress in CD Hz Exp. Data (1) 0.1 Hz Exp. Data (2) 0.1 Hz Exp. Data (3) 4.5 Hz Exp. Data (1) 4.5 Hz Exp. Data (2) 4.5 Hz Exp. Data (3) 45 Hz Exp. Data (1) 45 Hz Exp. Data (2) 45 Hz Exp. Data (3) Plastic Strain Figure Plastic Strain versus Stress Data for varying Strain Rates in the CD 85

98 4.5.2 Viscoplastic Model Description The viscoplastic model for this material is a modified version of the G sell flow stress equation found in equation (40) [G sell, 1979]. G sell came up with this equation in order to model the plastic nature of solid polymers at a constant total true strain rate. This model was chosen for the paper sheet material model because the experimental data taken from the uniaxial tensile tests are done at constant total engineering strain rates and it fit the data better than Norton s creep law did. G sell developed the model to model the plastic response of PVC and HDPE. When plotting the natural log of the stress, σ, divided by the plastic strain rate,, to a constant power, m, versus the plastic strain squared,, G sell found a linear relationship equal to equation (41). When this same relationship was tested for the paper substrate, the relationship was not found to be linear. Figure and Figure show that the relationship for the paper substrate can actually better be determined by a power law. The power law relationship seen in equation (43) was used to model the data seen in Figure and Figure for the MD and CD respectively. Equation (42) now becomes the modified G sell flow stress equation used to model the viscoplasticity in this paper substrate. Equations (42) and (43) introduce the long term stress response, σ0, which is the way of incorporating the VBO method into G sell s flow stress law. G sell s flow stress law needed to be modified in order to account for this material not being a solid polymer, whereas G sell s research was done completely on solid polymers. (40) ( ) ( ) (41) ( ) ( ) (42) ( ) ( ) 86

99 Ln(Stress/StrainRate m ) Ln(Stress/StrainRate m ) (43) ( ) Fitting Modified G'sell to MD Hz Hz Model Plastic Strain Figure Fitting Modified G'Sell Flow Stress Law in MD Fitting Modified G'sell to CD Hz 4.5 Hz 45 Hz Model Plastic Strain Figure Fitting Modified G'Sell Flow Stress Law in CD 87

100 Equation (42) is fit to each of the uniaxial tensile test s plastic data for varying strain rates in order to determine the constants K, n and m. The values for these constants that make up the viscoplastic model can be seen in Table A nonlinear least squares fitting method identical to the method used to fit the viscoelastic model is used to fit the modified G Sell flow stress law. Figure displays the viscoplastic model for the MD and CD against the experimental data for each strain rate. The model does a great job of predicting the stress response due to plastic strain, especially at the lower strain rates. The complete models including the long term response for the MD and CD can be seen in Figure and Figure respectively. There is no instantaneous stress response for plastic strain because the theory of viscoplasticity assumes that plastic deformation is not limited except by failure. The long term response is found via a root finding technique but an experiment can be run at an assumed infinitely slow strain rate in order to verify this value. This study did not use that technique as there was not enough time to run tests on multiple samples for a few months each. Table Constants for Modified G'Sell Flow Stress Equation for MD and CD MD CD K (Pa s) n m

101 True Stress (Pa) True Stress (Pa) True Stress (Pa) True Stress (Pa) True Stress (Pa) True Stress (Pa) Viscoplastic Fit for 0.1 Hz Strain Rate in MD Viscoplastic Fit for 0.1 Hz Strain Rate in CD Hz Experimental Data 0.1 Hz Model Hz Experimental Data 0.1 Hz Model Plastic Strain Plastic Strain Viscoplastic Fit for 4.5 Hz Strain Rate in MD Viscoplastic Fit for 4.5 Hz Strain Rate in CD Hz Experimental Data 4.5 Hz Model Hz Experimental Data 4.5 Hz Model Plastic Strain Plastic Strain Viscoplastic Fit for 45 Hz Strain Rate in MD Hz Experimental Data 45 Hz Model Plastic Strain Viscoplastic Fit for 45 Hz Strain Rate in CD Hz Experimental Data Hz Model Plastic Strain Figure Viscoplastic Fits at Strain Rates of 0.1 1/s, 4.5 1/s and 45 1/s in the MD and CD 89

102 True Stress (Pa) True Stress (Pa) Viscoplastic Model in MD Hz Model 4.5 Hz Model 0.1 Hz Model Long Term Plastic Strain Figure Viscoplastic Model in MD at Strain Rates Equal to 0 1/s (Long Term), 0.1 1/s, 4.5 1/s and 45 1/s Viscoplastic Model in CD Hz Model 4.5 Hz Model 0.1 Hz Model Long Term Response Plastic Strain Figure Viscoplastic Model in CD at Strain Rates Equal to 0 1/s (Long Term), 0.1 1/s, 4.5 1/s and 45 1/s 90

103 4.6 Failure Envelope Model Failure Envelope Model Description The failure envelope for this paper substrate is found by observing the failure stress of the off axis uniaxial tensile data. The failure stress is rotated so it is only in terms of the stress in the MD, σ1, the stress in the CD, σ2, and the shear stress, σ6. Tsai-Hill and Tsai-Wu failure criteria found in equations (20) and (21) are then fit to the data to obtain the failure envelope. A nonlinear least squares fitting method is used again to determine the constants that will minimize the error in the Tsai-Hill and Tsai-Wu fits. Table lists the values of the constants for the fits obtained using the Tsai-Hill and Tsai-Wu failure envelopes. These models were fit to off-axis uniaxial tensile tests run at 0.1 1/s constant total strain rate. The error values, φ, for both the Tsai-Hill and Tsai-Wu fits can be seen in Figure From this figure, the Tsai-Wu failure envelope is observed to fit the data exceedingly better than the Tsai-Hill failure envelope. It can then be determined that the Tsai-Wu failure envelope determines this paper substrate s failure criterion. This Tsai-Wu failure envelope is only applicable to tests run at a constant total strain rate of 0.1 1/s. In order to accommodate other strain rates, equation (22) is fit to data taken in the MD and CD for 4.5 1/s and 45 1/s while using 0.1 1/s strain rate data as reference data. Using this relation combined with the Tsai-Hill failure envelope at 0.1 1/s, failure can be determined at any stress level/strain rate. The constants fitted to equation (22) can be found in Table Figure shows the total strain rate dependent failure model in relation to the values obtained experimentally for failure at varying strain rates. A complete model for the failure envelope of the paper substrate is determined by using 91

104 Phi both the Tsai-Wu failure criteria at 0.1 1/s total strain rate and the rate dependent failure models discussed above. Table Failure Envelope Constants for Tsai-Hill and Tsai-Wu Tsai- Hill Tsai-Wu F 11 (1/Pa 2 ) 1.71E E-12 F 22 (1/Pa 2 ) 1.14E E-11 F 12 (1/Pa 2 ) 2.62E E-12 F 66 (1/Pa 2 ) 1.39E E-11 F 1 (1/Pa) E-06 F 2 (1/Pa) E E-01 Failure Criteria Fit for Paper Substrate 2.00E E E Tsai-Hill Fit Tsai-Wu Fit -1.00E E E-01 Angle (Deg) Figure Failure Envelope Fit for Paper Substrate Using Tsai-Hill and Tsai-Wu Table Constants for Failure Envelope due to Strain Rate for MD and CD MD CD F (Pa) m f

105 Failure Stress (Pa) Rate Dependent Failure Stress Model for MD and CD MD Experimental Data CD Experimental Data Model MD Model CD Strain Rate (Hz) Figure Rate Dependent Failure Model for MD and CD with rates 0.1 1/s, 4.5 1/s and 45 1/s 4.7 Poisson s Ratio It is important to determine the Poisson s ratio in order to be able to assume if the material is volume preserving (Poisson s equal to 0.5) in the MD/CD direction. The Poisson s ratio between the MD and CD in plane, ν12, can be determined by equation (44). Strain values in the MD and CD were observed by photographing a sample pulled in the MD for a constant slow strain rate. The strain values for the MD and CD were plotted against one another in Figure with the idea of determining the Poisson s ratio for the paper substrate. It can be observed that the Poisson s ratio is initially 0.2 until 6% strain in the MD and 0.41 after this strain point. Assuming that the paper is volume preserving in the MD and CD is not completely accurate but does not input excess error as the Poisson s ratio is close enough to 0.5 to be considered volume preserving. The initial Poisson s ratio differentiating from the final Poisson s ratio could be due to the sample not being 93

106 Strain CD completely taught in the grips before loading begins or could signify that yielding starts after 6% total strain. If the yielding starts after 6% than the assumption that the material is yielding from onset is incorrect. For this research we are assuming that this initial Poisson s ratio is due to the sample not being taught in the grips when loading starts. (44) Poisson's Ratio as Strain Increases in the MD y = x y = x Initial Secondary Linear (Initial) Linear (Secondary) Strain MD Figure Strain in MD versus Strain in CD as strain increases in the MD 94

107 4.8 Flow Chart used to obtain Viscoelastic and Viscoplastic Equations Figure Flow chart of work done with teal, white, blue and black boxes representing experiments, inbetween calculations, models and final models respectively 95