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THREEDIMENSIONAL VISCOELASTIC WINDING MODELS FOR THE ANALYSIS OF BAGGY LANES FORMATION By HAOWEN YU Bachelor of Science in Naval Architecture Second Bachelor of Science in Technological Economy Shanghai Jiaotong University Shanghai, China 1998 Master of Science in Naval Architecture Shanghai Jiaotong University Shanghai, China 2001 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY December, 2007 i i THREEDIMENSIONAL VISCOELASTIC WINDING MODELS FOR THE ANALYSIS OF BAGGY LANES FORMATION Dissertation Approved: Dr. Hongbing Lu Dissertation Adviser Dr. J. Keith Good Dr. Demir Coker Dr. G. Steven Gipson Dr. A. Gordon Emslie Dean of the Graduate College ii i ACKNOWLEDGEMENT I would like to express my sincere appreciation to my advisor Dr. Hongbing Lu for his guidance and support. He offered his academic expertise and insightful knowledge all the way through completion of this dissertation. I appreciate his understanding, patience, and affable manner of communication, which made the tense research process less stressful. I also thank the other three committee members, Dr. James K. Good, Dr. Demir Coker, and Dr. G. Steven Gipson, for their thoughtful comments, suggestions, and the valuable time they spent on reviewing my documents. I have to give my special thanks to Dr. Gipson, who actively participated in every stage of this process despite his illness. I sincerely wish him defeat the disease and recover at the earliest date. As always, my husband Huawen Xu has provided the much needed understanding and encouragement. His love presents me the courage to face the difficulties in daily life. I also thank my parents for their love and the selfless support. I am also grateful for the help I got from all of the research group members and my friends. I wish them all successful in their future endeavor. Thanks also to Department of Mechanical and Aerospace Engineering and Web Handling Research Center for their direction and financial support during this research. i v TABLE OF CONTENTS Chapter Page 1 INTRODUCTION........................................................................................................... 1 2 LITERATURE REVIEW................................................................................................ 7 2.1 Elastic Analysis ....................................................................................................... 7 2.1.1 Anisotropic Linear Elastic Winding Model ....................................................... 8 2.1.2 Anisotropic Nonlinear Elastic Winding Model ............................................... 11 2.2 Viscoelastic Analysis ............................................................................................. 12 2.2.1 Isotropic Linear Viscoelastic Winding Model ................................................. 13 2.2.2 Orthotropic Nonlinear Viscoelastic Winding Model ....................................... 15 2.3 Pseudo 3D Model and 3D Finite Element Model ............................................... 17 2.3.1 Pseudo 3D Winding Model.............................................................................. 18 2.3.2 3D FEM Elastic Winding Models.................................................................... 20 3 PSEUDO 3D VISCOELASTIC WINDING/UNWINDING MODEL ........................ 22 3.1 Pseudo 3D Winding Model................................................................................... 24 3.2 Viscoelastic Effects during Winding ................................................................... 28 3.3 Viscoelastic Effects after Winidng....................................................................... 32 3.4 Viscoelastic Effects during Unwinding ............................................................... 33 3.5 Finite Element Winding Simulations for Checking Outer Boundary Condition and Effects of Asymmetric Structure................................................. 35 3.5.1 Evaluation of Boundary Conditions using Dynamic Analysis via ABAQUS / Explicit .................................................................................................. 36 3.5.2 2D Viscoelastic FEM Winding Model ............................................................ 37 v Chapter Page 3.5.3 Results and Discussion .................................................................................... 41 4 RESULTS FROM 3D VISCOELASTIC WINDING/UNWINDING MODEL.......... 48 4.1 Comparison of Results from Pseudo 3D Viscoelastic Model and Lin & Westmann’s 2D Viscoelastic Model ..................................................................... 50 4.2 Comparison of Results from Pseudo 3D Viscoelastic Model and the 3D Analytical Solution................................................................................................. 51 4.3 Comparison of Results from Qualls and Good’s 2D Viscoelastic Model......... 53 4.4 Comparison of Results from Pseudo 3D Viscoelastic Model and Experimental Data................................................................................................. 55 4.5 Special Case One: Formation of a Cambered Web........................................... 58 4.6 Discussion about Hakiel’s Mode l [2] and Cole and Hakiel’s Model [46] ....... 61 4.7 Special Case Two: Formation of Wavy Edge..................................................... 63 4.8 Special Case Three: Formation of Baggy Web .................................................. 66 4.9 Consideration of Thickness Variation in Both CMD and MD......................... 68 4.10 Consideration of Winding and Unwinding....................................................... 71 5 CONCLUSIONS ........................................................................................................... 73 6 FUTURE WORK........................................................................................................... 76 6.1 Hygrothermal Effects on Viscoelastic Material Properties .............................. 76 6.2 Dimensionless Study ............................................................................................. 79 6.3 Optimization of Tension History......................................................................... 80 6.4 Experimental Validations..................................................................................... 80 REFERENCES................................................................................................................ 81 APPENDIX A: A PSEUDO 3D TRANSIENT VISCOELASTIC WINDING / UNWINDING CODE ...................................................................................................... 86 v i LIST OF TABLES Table Page Table 31: Viscoelastic material parameters ..................................................................... 40 Table 41: Material properties used in pseudo 3D winding model ................................... 50 Table 42: Deviation of deformed length.......................................................................... 71 v ii LIST OF FIGURES Figure Page Figure 11: A schematic diagram of a laboratory winder ................................................... 1 Figure 12: A wound roll in threedimensional situation.................................................... 2 Figure 13: Baggy lanes examples. Left: cambered web; Right: baggy edge in circled area............................................................................................................... 3 Figure 21: Exaggerated view of a wound roll with thickness variation .......................... 18 Figure 31: Pseudo 3D models.......................................................................................... 24 Figure 32: Exaggerated view of a wound roll with thickness variation, Hakiel [2]........ 25 Figure 33: Outer lap geometry, Cole [46]........................................................................ 27 Figure 34: Viscoelastic effects during winding ............................................................... 31 Figure 35: An arbitrary external force P(t) ...................................................................... 34 Figure 36: Flow chart of algorithm for viscoelastic unwinding ...................................... 35 Figure 37: Schematic of the 2D FEM winding model..................................................... 38 Figure 38: Schematic of 2D FEM storage model ............................................................ 38 Figure 39: Definition of the contact in 2D FEM winding model .................................... 39 Figure 310: Definition of the contact in 2D FEM storage model.................................... 39 Figure 311: Three sets of viscoelastic material properties .............................................. 41 Figure 312: The outer layer timedependent circumferential strain for Set 1 ................. 42 Figure 313: The outer layer timedependent circumferential stress for Set 1 ................. 42 Figure 314: The outer layer timedependent circumferential strain for Set 2 ................. 43 Figure 315: The outer layer timedependent circumferential stress for Set 2 ................. 43 vi ii Figure Page Figure 316: The outer layer timedependent circumferential stain for Set 3................... 44 Figure 317: The outer layer timedependent circumferential stress for Set 3 ................. 44 Figure 318: Circumferential stresses in layer 1 to layer 10............................................. 47 Figure 41: Comparison with the results of Lin and Westmann [10]................................ 51 Figure 42: Comparison with the results of 3D analytical solution.................................. 53 Figure 43: Comparison with the results of Qualls and Good’s model, v=1000 in/s ....... 54 Figure 44: Comparison with the results of Qualls and Good’s model, v=30 fpm........... 55 Figure 45: Schematic diagram for inserting a triangular web for every four plies for use in the investigation of formation of a cambered web .................................... 55 Figure 46: The experimental data for both edges ............................................................ 56 Figure 47: Comparison with the experimental data......................................................... 58 Figure 48: The length of deformed web at ten locations................................................. 59 Figure 49: Numerical transformation for cambered web ................................................ 60 Figure 410: Maximum bow length .................................................................................. 60 Figure 411: Deformed shape after storage....................................................................... 61 Figure 412: Comparison of the pressure distributions..................................................... 62 Figure 413: Comparison of deformed shapes after storage............................................. 63 Figure 414: Formation of wavy edge .............................................................................. 64 Figure 415: The length of deformed web at ten locations............................................... 64 Figure 416: Numerical transformation for wavy edge .................................................... 65 Figure 417: Formation of baggy web .............................................................................. 66 Figure 418: The lengths of deformed web at forty locations .......................................... 67 Figure 419: Numerical transformation for baggy web. ................................................... 68 i x Figure Page Figure 420: The thickness profiles with 50% deviation of average thickness for layer #1 and Location #1...................................................................................... 70 Figure 421: The whole thickness profile ......................................................................... 70 Figure 422: Deformed length for 10%, 20%, 50% and 80% cases ................................. 71 Figure 423: Comparison of stresses during winding and unwinding .............................. 72 Figure 61: Comparison of master creep curves for PVAc [45] ....................................... 76 Figure 62: (a) Creep curves for dry PVAc at several temperature; (b) Creep curves for PVAc at c o 24 at several absorbed moisture levels.................................... 77 x ABSTRACT A pseudo 3D winding model, with the consideration of viscoelastic effects during winding, storage and unwinding, has been developed and implemented in a Fortran code. The pseudo 3D model is based on the 2D viscoelastic winding model and the associated boundary conditions developed by Qualls and Good, and the tension partition method proposed by Cole and Hakiel. In the model, the thickness variation along the cross machine direction (CMD) has been considered. A web in the CMD direction is discretized into smaller segments (or lanes) of various lengths, each having a constant web thickness. The number and size of segments can be changed to represent the actual thickness profile. Tension is assigned to each segment using the Cole and Hakiel approach, and is updated after winding of each lap based on the deformed radius of the segment relative to the profile of the relaxed radii of that lap. In each segment, a 2D viscoelastic winding/storage/unwinding model is applied. The results from the pseudo 3D viscoelastic winding model have been compared with some winding models, analytical solutions and experimental data. They include (1) the comparison with Lin & Westmann’s 2D isotropic viscoelastic windng model; (2) the comparison with the 3D analytical solution; (3) the comparison with Qualls and Good’s 2D orthotropic viscoelastic winding model; and (4) the comparison with experimental data. x i The pseudo 3D viscoelastic winding model has also been used to analyze three problems involving formation of baggy lanes. They are (1) formation of cambered web (inplane imperfection) due to webnonuniformity; (2) formation of localized baggy lanes due to edge burr following slitting; and (3) formation of baggy web (inplane imperfection) due to webnonuniformity. Simulation results are compared with experimental data. FEM simulations using ABAQUS/Explicit code have been conducted to determine the actual boundary condition at the outer layer. The winding and storage processes were simulated to determine the stress and strain in a wound roll and at the outer layer. There are three sets of viscoelastic material parameters used in this FEM simulation to investigate the viscoelastic effects on the outside boundary condition. 1 CHAPTER 1 1 INTRODUCTION Web handling is described as the engineering science underlying the transport of webs through processes successfully without incurring material defects and losses [3]. Figure 11 shows a schematic diagram of a laboratory winder used in this investigation. 1, 2 7 6 5 4 3 8 Figure 11: A schematic diagram of a laboratory winder In industry, paper towels, toilet tissue and photographic film are all considered as webs in the manufacturing process. A web is a thin continuous sheet or strip, such as plastic films, paper, textiles, metal sheets, composite prepregs, foils, wovens, and nonwoven materials. Webs are often stored in the form of wound rolls, convenient for ransporting and storing and widely used in most industries. A roller is a revolving cylinder over which a web or substrate is moved for the purpose of transporting, pressing, shaping, or shaping, or smoothing a web or substrate. The process of wrapping a continuous flexible web onto a roller (core) to form a wound roll is called winding. 1. Unwinding roller 2. Brake 3. Web 4. Idle roller 5. Tension measuring roller 6. Idle roller 7. Idle roller 8. Winding roller 2 Figure 12 shows schematic diagram of a 3D wound roll. Core Web Figure 12: A wound roll in threedimensional situation Generally, it is difficult to obtain the analytical solution for the stress, strain and displacement fields because of the complexity of 3D geometry. Numerous efforts have been made to determine the stress distribution in a 3D wound roll. Altmann [5] and Hakiel [6] et al. simplified this real 3D winding situation into a 2D winding model that can be solved using solutions for an axisymmetric hollow cylinder. For the investigation of baggy lanes, however, the realistic winding structures have to be considered to take into account of the effects of varying web thickness in the CMD on the stress / strain distribution. Consequently, it is necessary to develop a 3D winding model to deal with realistic 3D winding issues. Usually, wound roll defects are formed in the processes of manufacture, transportation and storage. It is important to develop methods to reduce product defects and losses through fundamental understanding of the mechanisms of defect formation. There are various kinds of defects in wound rolls; these usually include wrinkles, tears, baggy lanes etc. All these defects could lead to damage in web, causing poor appearance and consequently poor quality of the wound roll. The research in this report will focus on 3 developing (pseudo) 3D viscoelastic winding models and 3D FEM winding models for purpose of understanding the baggy lanes formation in viscoelastic webs. A baggy lane is a defect commonly observed in a wound roll and has many aliases such as camber, layflat and puckers. It can be found in most webs as diverse as tissue, printing paper, carpet, nonwovens, plastic film, and sheet metals [4]. An example of baggy lane formation in the web is shown in Figure 13. Figure 13: Baggy lanes examples. Left: cambered web; Right: baggy edge in circled area When baggy lanes occur, the visual appearance of a web is no longer flat or straight, causing some poor effects. For instance, the baggy portion may float over rollers causing processing difficulties and may not wind in the same way as the tight portions. Deformation in webs may remain, evolve and deteriorate and the losses may be unpredictable. Thus, it is very important to investigate the mechanisms of baggy lane formation and to develop methods to reduce or eliminate it during winding, unwinding or storage processes. Baggy lanes are expected to be associated with the existence of compression or shear zones, and with variations of stress and strain. Generally, baggy 4 lanes could occur at locations where there is high gradient of stresses/strains in the cross machine direction (CMD). Therefore, in this report, winding models will be developed to analyze the stress distribution in wound rolls. Later efforts will include the development of a criterion for predicting baggy lanes formation. Theoretical, numerical and experimental investigations will be carried out in the development of 3D viscoelastic winding models. Theoretical studies will be focused on the analysis of stress and strain distributions in wound rolls. Through analyzing stress and strain fields, the mechanism of defect formation in the web will be revealed. So far, some winding models have been developed to determine stress and stain distributions; these will be introduced in detail in the following chapters. It may be noted that theoretical models for determining stress and stain distributions need to be validated by experimental results. In experimental studies, the winding process needs to be conducted until baggy lanes occur in the web. To investigate the mechanical behavior of the web in winding, it is essential to determine the material properties of the web, such as the creep compliance for a viscoelastic material. Another essential aspect in experiments is to find appropriate methodology and equipment to measure the stress and strain distributions in a wound roll, especially in the CMD direction. Some theoretical winding models have been developed to determine stress and strain distributions in a wound roll. In general, the existing winding models can be categorized into 2D models and 3D models. Each category can further be classified into elastic models and viscoelstic models. The first rigorous analysis of a 2D elastic wound roll was given by Altmann [5]. The web material was assumed to be homogeneous, anisotropic, and linear elastic in this model. Formulas were derived for the calculation of the inroll 5 stresses and can be easily implemented for numerical computation. Hakiel [6] extended Altmann’s model and presented a numerical method employing a nonlinear radial stiffness. In Hakiel’s model, the web was assumed to be an orthotropic elastic material with linear elastic properties in the circumferential direction and nonlinear elastic properties in the radial direction. On 2D viscoelastic winding models, Qualls and Good [1] developed a realistic and adaptive viscoelastic model for the prediction of transient stress distributions in a wound roll. In this model, the generalized Kelvin model was used to represent viscoelastic behavior of materials. Numerical results have a very good agreement with analytical solutions for an isotropic linear viscoelastic 2D winding problem. Several 3D winding models have been proposed, these include winding models developed by Hakiel [2], Cole and Hakiel [46], Kedl [7], Lee and Wickert [8], and Hoffecker [9]. In order to consider the effects of thickness variations along the width direction on wound rolls, Hakiel, Cole & Hakiel and Kedl proposed pseudo 3D models for a wound roll. In their models, it was assumed that the wound roll can be split into multiple independent widthwise segments and in each segment wound roll stresses can be computed separately through incorporating the winding tensions into the existing 2D model. Lee and Wickert, and Hoffecker found their ways to determine the stress distributions in a wound roll using 3D FEM. It was assumed that a wound roll has an axisymmetric geometry and stress distributions are only functions of radius and are independent of angle. Winding is an accretion problem, in which the total stress in a wound roll is equal to the summation of all the stress increments, as the radius increases from inner layer to outer layer. 6 Although some theoretical winding models have been developed in the past several decades, there are still unresolved problems associated with the winding models. For example, the viscoelastic model for a wound roll has not been fully developed, especially during winding and unwinding process. In the model developed by Qualls and Good, nonlinear elastic behavior is considered during winding and viscoelastic behavior is considered after winding. Lin and Westmann [10] considered viscoelastic winding during winding, storage and unwinding. However, this model was limited to isotropic linear viscoelastic materials. The FEM has become a useful tool to compute the stress distributions. However, it is not very easy to deal with winding problems using the commercial FEM codes. Since a web usually has thickness variations in both the MD and CMD, there might be gaps between two layers. If the gap is considered and the finite elements are assigned for each layer, the number of finite elements will be too large to handle when the number of layers becomes very large. Moreover, it is difficult to implement the pressure dependent radial modulus of the wound roll into commercial FEM code. This difficulty will impose restrictions on the application of FEM in winding simulations; thus it is necessary to explore some other methods to determine the stress distribution in a wound roll. It may be noted that most existing pseudo 3D models and 3D FEM models are for elastic materials only. It is necessary to extend previous elastic pseudo 3D and 3D FEM models to consider viscoelastic behavior. In conclusion from the above review, 3D elastic and viscoelastic winding models need to be developed for the identification of baggy lanes. A criterion needs to be established to predict baggy lanes formation. 7 CHAPTER 2 2 LITERATURE REVIEW In this chapter, several major winding models will be reviewed and discussed. These models will be divided into several sections and presented separately. In the section of elastic analysis and viscoelastic analysis, we will introduce only 2D elastic and viscoelastic winding models. Pseudo 3D models and 3D FEM models for elastic material will be introduced separately in Section 2.3. 2.1 Elastic Analysis Elastic winding model may be the most fullygrown model of a wound roll. Detailed analysis of stresses in an elastic wound roll was traced to the work of Gutterman [11] in 1959. Altman [5] followed his work and presented an analytical solution to the elastic winding problems including complete derivation, assumption and detailed formula. Then the elastic model was extended by Pfeiffer [12, 13] and Yogada [14] by considering nonlinear radial stiffness. In 1987, Hakiel [6] developed a practical numerical method incorporating advantages in the previous work. Moreover, in Hakiel’s model the nonlinear radial stiffness, orthotropic material properties have been considered. In this section, two elastic models developed by Atmann and Hakiel will be reviewed. 8 2.1.1 Anisotropic Linear Elastic Winding Model In the anisotropic linear elastic winding model [5], Altmann presented detailed stress analysis for the wound roll of anisotropic linear elastic material. Some assumptions were made in the derivation and formulas were presented for the computation of stress distribution in a wound roll. These formulas can be evaluated easily on a computer. However, in some cases, when these assumptions are violated, the results will not be correct. The following assumptions were made in the work of Altmann: 1. The core is geometrically cylindrical and remains a cylinder during and after winding. 2. The web is uniform in thickness and its thickness is small compared to its width, and therefore offers no resistance to bending. 3. During and following the winding, the roll may be considered to be a homogeneous cylinder with anisotropic properties r E , Eθ , r μ , μθ all considered to be constant throughout the roll. 4. The stresses in the roll are only functions of roll radius, not of roll angle, because of its axisymmetric structure. 5. No nonelastic displacements and betweenlayer motions occur during winding. Based on the above assumptions, the stresses, strains, and displacements are only the functions of roll radius. Equilibrium equation can be expressed as + − = 0 ∂ ∂ θ σ σ σ r r r r (2.1) where r is the radius ratio ( r = r / c ), r is the radius to a point in the roll, c is the radius of the core, r σ is the radial stress, σθ is the circumferential stress. 9 Straindisplacement relations are r u r u r ε = ∂ ∂ ε = , θ (2.2) where r ε is radial strain, εθ is circumferential strain, u is dimensionless radial deformation ( u = u / c ),u is the radial deformation. The stressstrain relations under the plane stress condition are θ μθ ⋅σθ − σ ε = E E r r r r r r E E μ ⋅σ − σ ε = θ θ θ (2.3) where r E is radial modulus, Eθ is circumferential modulus, r μ and μθ are Poisson’s ratio. Solving Equations (2.2, 2.3) leads to + μ ∂ ∂ − μ ⋅μ = ε μ + ε − μ ⋅μ σ = θ θ θ θ θ r u r E E u r r r r r r 1 ( ) 1 + ∂ ∂ μ − μ ⋅μ = ε + ε ⋅ μ − μ ⋅μ σ = θ θ θ θ θ θ r u r E E u r r r r r 1 ( ) 1 (2.4) Consider that u = Arα + Br −β (2.5) 1. Boundary conditions at the core. when r = 1, A B E u c = r = + (1) (1) σ (2.6) where [ A B] E r r r ( ) ( ) 1 ) 1 ( β − μ + α + μ − μ ⋅μ σ = θ θ θ 2. Boundary conditions at outer radius of the wound roll. when r = s , [ ] ds s T s As Bs E s w r r r − = μ + α + μ − β − μ ⋅μ σ = −β θ α θ θ 1 ( ) ( ) 1 ( ) (2.7) 1 0 Then A, B and θ σ ,σ r can be determined from Equations (2.4~2.7). As the number of layer increases, the stress change will be ds s T as s r ar dP w b r b )( ) 1 ) ( 1 ( 2 2 − γ − γ + × + = −σ = ds s T as s r a r dT w b b )( ) 1 ( ) ( 2 2 γ γ θ α β σ − − + × − ⋅ = = − (2.8) where: c c e e a γ + μ + γ − μ − = r γ = δ2 + e r r e = Eθ / E c c e = Eθ / E b = 1−α α =γ −δ β = γ + δ The isotropic component of Poisson’s ratio is ( ) 2 1 r r μ = μθ + e μ The anisotropic component of Poisson’s ratio can be written as ( ) 2 1 r r δ = μθ − e μ The final stresses P and T are therefore given by ds s T as s r ar P dP w R r b b )( ) 1 ) ( 1 ( 2 2 ∫ ∫ − − + × + = = γ γ (2.9) ds s T as s r a r T T dT T w R r b w w b )( ) 1 ( ) ( 2 2 ∫ ∫ − − + × − ⋅ = + = − γ α β γ (2.10) 1 1 where R is dimensionless outside radius of the roll ( R = R / c ), R is the outside radius of roll. These formulas can be evaluated on a computer. The computational results have shown to agree well with some experimental data. These formulas are limited to linear elastic materials. Nonlinear response has been found to exist in a wound roll. For improved accuracy, it is necessary to investigate nonlinear effects in winding. 2.1.2 Anisotropic Nonlinear Elastic Winding Model Hakiel developed a nonlinear elastic winding model [6].The most important attribute of Hakiel’s model is to consider nonlinear material behavior in radial direction. Hakiel’s work extended the previous analysis by considering radial modulus as a function of pressure, instead of a constant as used in the previous models. A numerical solution was developed to solve the boundary value problem. For a roll with N laps, a set of N+ 1 linear algebraic equation will be solved to obtain the stresses in the web. This numerical solution can also be evaluated conveniently on a computer. The equations used in the development of Hakiel’s model are summarized below. Hakiel defined t r g E / E 2 ≡ (or r t E g E ≡ 2 ) (2.11) 1. Nonlinear model The radial modulus was assumed to depend on the radial stress. Then the radial modulus is a function of radial stress. E E (P) r r = (2.12) 2. A numerical solution to the boundary value problem For a roll with N laps, the boundary condition can be presented by 1 2 0 1 1 + + = i+ i i i i− i δP A δPB δP C (i=2,3…N) (2.13) where i δP is the pressure at ri caused by the winding on of lap N+1. Eq (2.13) can be rewritten as 0 1 1 + + = N+ N N N N− N δP A δP B δP C 0 1 1 1 2 1 + + = N N− N− N− N − N− δP A δP B δP C … … 0 3 2 2 2 1 2 δP A +δP B +δPC = (2.14) The N1 equations above contain N+1 undetermined coefficients. These constants can be determined using boundary conditions on the inner and outer surfaces. On the outer surface, h r T P N W N N 1 , 1 1 + + + δ = , (2.15) where W,N+1 T is the winding tension stress for Lap (N+1). On the inner surface, 2 1 1 ( P P ) / h [(E / E ) 1 ] P t c δ −δ = − +ν δ (2.16) Hakiel’s model has been validated by experimental data and verified by comparing results obtained from some of the previous models. Nevertheless, Hakiel’s model does not consider viscoelastic effects. Thus, viscoelastic analysis needs to be carried out for materials with pronounced viscoelastic effects. 2.2 Viscoelastic Analysis Transient analysis of wound roll was first introduced by Tramposch in 1965 [15,16]. The web material was considered to be an isotropic viscoelastic material in his work. Lin and Westmann [10] extended Tramposch’s work in viscoelastic winding mechanics by taking into account of histories for winding, windingpause, and windingpauseunwinding. Qualls and Good [1, 17] developed a solution for viscoelastic analysis that 1 3 has considered viscoelastic effects after winding and the thermal influence on viscoelastic winding. The material in the model by Qualls and Good is orthotropic with a nonlinear elastic radial stiffness depending upon interlayer pressure. In this section, the two viscoelastic models developed by Lin and Westmann, and Qualls and Good will be reviewed. 2.2.1 Isotropic Linear Viscoelastic Winding Model A viscoelastic winding model for isotropic linear material was presented by Lin and Westmann [10]. In that paper, Lin and Westmann considered viscoelastic response during winding, i.e. winding velocity can be considered in viscoelastic analysis. Moreover, a new method was introduced to solve viscoelastic unwinding problems. It was assumed that viscoelastic unwinding could be considered by the superposition of two states, an initial state and a correction state. The initial state is determined from the initial condition when unwinding begins and the correction state is determined after the initial state so that the superposition of the initial and correction states satisfies the traction free boundary condition at the current unwinding radius. However, the nonlinear radial stiffness was not considered in this model. Moreover, this model is limited to isotropic material. Actual web properties are orthotropic and the radial modulus is nonlinear. Thus considering such features will provide more realistic predictions for stresses in a wound roll. In this section, an outline of Lin and Westmann’s work will be presented. At first, in the same way as used in other elastic isotropic models, all the stresses, strains and displacements distributions are determined in cylindrical coordinates through equilibrium, stressstrain and straindisplacement equations. The solution procedure for 1 4 the viscoelastic case parallels the method used for the elastic winding problem. The solution to the viscoelasic winding problem was obtained by employing elasticviscoelastic corresponding principle by way of Laplace transform. In order to consider the viscoelastic effects during winding, the current roll radius will depend on winding speed and time passed. Two special cases were considered: R v th 0 2 π ( −1) = at constant linear velocity; (2.17) R th 0 2π ( −1) =ω at constant angular velocity, (2.18) And numerical examples were presented for a viscoelastic web represented by the generalized Kelvin model. The creep compliance of materials function was: n t N n n e E E J t /τ 0 1 1 1 ( ) − = Σ = − (2.19) For viscoelastic unwinding, the system can be treated by superposing two timedependent states, an initial state and a correction state. At current unwinding radius, the superposition of the initial and correction states should satisfy the traction free boundary condition, which can be expressed as: ) 0 ~ ( , , ) ( , , ) ( , , 1 0 1 0 1 0 R R t = R R t + R R t = c r i r r σ σ σ (2.20) where, R1 is the current unwinding radius, R0 is the fixed outer radius, which is assumed to be constant during unwinding, t is the time starting from the initial winding, t ~ is the time starting from the unwinding. This traction free boundary condition is used to determine the arbitrary external pressure history P (t ~ ) at the correction state. Finally, stresses and displacement at any layer can be obtained as follows c r i r r σ =σ +σ , i c θ θ θ σ =σ +σ , i c u = u + u (2.21) 1 5 The analysis presented by Lin and Westmann provides a methodology for analyzing isotropic linear viscoelastic winding problems. Especially, the unwinding case has been discussed and the stress distribution has been determined. Numerical results for several examples have been given in that paper. However, this model is for isotropic linear viscoelastic material. Many actual web materials are orthotropic with a nonlinear radial stiffness. Incorporation of these features will provide more realistic predictions of the stress distribution in the wound roll. 2.2.2 Orthotropic Nonlinear Viscoelastic Winding Model Qualls and Good [1] developed a viscoelastic winding model for orthotropic nonlinear viscoelastic webs. The modeling results were found to be in a good agreement with some other existing winding models. The generalized Kelvin model was used to represent the viscoelastic function in this orthotropic nonlinear viscoelastic winding model. A numerical solution has been developed to predict transient stress profile in wound rolls composed of orthotropic viscoelastic web with nonlinear radial modulus. In this section, the derivation of the solution will be reviewed briefly. Equilibrium equation is expressed as + − = 0 ∂ ∂ θ σ σ σ r r r r (2.22) Strain compatibility is written as + − = 0 ∂ ∂ r r r ε ε ε θ θ (2.23) Constitutive equations for an orthotropic viscoelastic material are expressed by dt t J t t t J t t t r r r r ′ ∂ ′ ∂ + − ′ ∂ ′ ∂ = ∫ − ′ 0 ( ) ( ) θ θ σ σ ε 1 6 dt t J t t t J t t t r r ′ ∂ ′ ∂ + − ′ ∂ ′ ∂ = ∫ − ′ 0 ( ) ( ) σ σ ε θ θ θ θ (2.24) The above equations lead to [ ( ) ( ) {3 ( ) ( ) ( ) 2 2 2 0 J t t J t t J t t r r t J t t r r r t + − ′ + − ′ − − ′ ∂ ∂ ∂ ′ ′ ∂ ∫ θ − θ θ θ σ ( )} ( ) { (J (t t ) J (t t )) J (t t ) r r r r t J t t r r r r − ′ + − ′ + − ′ ∂ ∂ + ∂ ∂ ∂ ′ ′ ∂ − ∂ ∂ + θ θ θ θ σ ( ) ( ) ( )} ] ′ = 0 ∂ ′ ′ ∂ + − ′ − − ′ − − dt t J t t J t t J t t r r r r σ θ θ (2.25) The generalized Kelvin model is used to represent the viscoelastic behavior. i t t N i iJ t t J J e ( ) /τ 1 0 ( ) − − ′ = Σ − ′ = + (2.26) Equation (2.25) can be discretized to the following form ) ( ) 0 2 ) ( ( ) 2 ( ) ) ( ( ) 2 ( ) 2 ( ( ) ( ) 2 ( 1) 4 2 2 2 ( ) 1 2 2 2 ( 1) 3 1 2 1 + % + + − % + − % − + F r = h r F r h r F r h r F r F r h r F r h r F r r i i i r i i r i i i σ σ σ (2.27) There are N+1 unknown parameters in N1 equations. After two boundary conditions are applied, all parameters can be determined by solving Equation (2.27) with two boundary conditions. The inner boundary condition is given as: j c r j E ( ) ( ) θ ε σ = (2.28) where Ec is the core stiffness, defined as the radial pressure divided by hoop strain at the outer diameter of the core. When t=tj, the inner boundary condition can be expressed as 1 7 dt t J t t t J t t r r t J t t E r r r r t c r j r j ′ ∂ ′ ∂ + − ′ ∂ ′ ∂ + − ′ ∂ ∂ ∂ ′ ′ ∂ = − + % ∫ − [ ( ) ( ) ( ) ( ) ] ( ) ( ) 0 1 σ σ σ σ σ θ θ θ (2.29) The boundary condition on the outer surface of the roll is dt t J t t t T J J t t r t w ′ ∂ ′ ∂ + − ′ ∂ ′ ∂ (0) = ∫ [ ( − ′) ( ) ] 0 θ θ θ θ θ σ σ (2.30) With the change in circumferential stress, the corresponding change in radial stress beneath the outer layer can be determined by employing the following condition. 0 ( ) ( ) r h j r j θ σ σ % % = (2.31) This viscoelastic winding model has considered viscoelastic effects after winding. Results from this model agree well with those from some other models and experimental data. The model, however, did not consider viscoelastic effects during winding. Also unwinding was not considered in the model. This model is a 2D winding model and cannot be used to predict the stress profile along CMD. A 3D winding model is needed to consider the variations in CMD direction. 2.3 Pseudo 3D Model and 3D Finite Element Model Some pseudo 3D models and 3D FEM models have been developed by Hakiel [2], Cole and Hakiel [46], Kedl [7], Hoffecker [9], and Lee and Wickert [8] to predict the stresses in both radial and crossweb (transverse) direction. Hakiel assumed that the roll could be partitioned across its width into small strips or segments and stresses or displacements in each segment can be calculated separately through a twodimensional analysis. Cole and Hakiel refined the Hakiel’s model and considered the effect of radial displacement on tension distribution. Kedl made a similar assumption as Hakiel and Cole 1 8 & Hakiel’s, dividing the roll into an arbitrary number of crossweb segments and using a twodimensional winding model to predict wound roll stresses in each segments. Hoffecker, and Lee and Wickert have developed axisymmetric FEM models to predict the widthwise variation of stresses in wound rolls and have compared the results with those obtained from previous models. 2.3.1 Pseudo 3D Winding Model The pseudo 3D winding model developed by Hakiel [2] will be introduced in this section. This model can be used to predict the widthwise variability in the wound roll. In the model the effect of thickness variations along the width direction is considered. However this model is not suitable for highly compressible material. And there are still some problems remaining in dealing with the deformation compatibility between neighboring segments. Figure 21: Exaggerated view of a wound roll with thickness variation ρ (i, j) : widthwise distribution of radius to the outside surface of lap i. r(i, j) : widthwise distribution of radius to the inside surface of lap i. i: the lap number within the roll j: widthwise position 1 9 ( ) 0 R i is relaxation radius for lap i. It is assumed that the lap would be stress free at relaxation radius. Firstly, the initial value of ( ) 0 R i will be estimated. The circumferential stress can be written as − − ν σω = ( ) ( , ) ( ) 1 ( , ) 0 0 2 R i E r i j R i i j (2.32) Then, we have the circumferential stress, as given by } ( ) ( , ) ( ) { (1 ) ( ) { ( , ) } 0 0 1 2 1 R i r i j R i M Ebh M bh T i i j M j M j − − =Σ = Σ − − ν σω ω (2.33) In order to find the relaxation radius, let the predicted winding tension be equal to the actual tension force in the outer lap. It is an iteration process to find the relaxation radius. T (i) T (i) a = ω (2.34) Widthwise distribution of radius to the outside surface of lap is ρ(i, j) = r(i, j) + h( j); j = 1,...M (2.35) The inner radius of the lap being wound on is r(i, j) Max{ (i 1, j),R (i); j 1,...M} 0 = ρ − = (2.36) Based on Equations (2.33~2.36), the predicted widthwise distribution of radius and tension can be obtained. After that, the roll could be discretized into independent widthwise segments and the wound roll stresses could be determined in each segments by using any of existing models to compute the wound roll stresses. However, problems exist in dealing with the deformation compatibility between neighboring segments. 3D winding model is needed to solve this problem to provide more accurate computation of stresses in a wound roll. 2 0 2.3.2 3D FEM Elastic Winding Models Two 3D FEM elastic winding models have been presented by Lee and Wickert [8], and Hoffecker [9] separately. These two FEM models are somewhat similar and their results agree. The FEM winding model developed by Lee and Wickert focuses on various core shape, while the FEM model developed by Hoffecker pays attention on the thickness variation in CMD. The basic formulation in 3D FEM model is similar to other winding models. In all these winding models, the stress state within a wound roll with N laps is the superposition of stress states resulting from the addition of each additional layer from 1 to N. The wound roll is considered to be composed of N sub rolls or substructures. The stress state of each subroll consists of the result from the addition of that subroll’s outermost layer. The total stress of a single layer in the actual wound roll is found by superimposing the stress state corresponding to that layer from all subrolls. In these two axisymmetric FEM models, fournode rectangular elements were used. These finite elements can be obtained through discretizing the wound roll along both the radius and cross machine directions. Since the stress state in the wound roll is the result of superposition of stress states of substructures, it is very difficult for commercial FEM software to determine the stress distributions in webs. In these two 3D FEM models mentioned previously, special FEM code has been developed to solve this problem and the numerical results have been obtained. In LinWickert model, the widthwise variation of stresses in a wound roll is investigated by using a 3D axisymmetric finite element model. The results obtained for hollow core and cupshaped core show the effects of different core shapes. In the two 2 1 examples given for hollow core and cupshaped core, the widthwise variation of stresses happens only near the interface of the core and the web. In the 3D FEM code developed by Hoffecker [9], the core and web in the wound roll are represented by axisymmetric finite elements and FEM code was developed. The web can have up to 30 different thickness regions (referred to as segments) across its width. The output files give stresses (including radial pressure, and circumferential tension) and displacement or radius profiles. This model can capture the effect of varying tensions and thickness profiles. There are still remaining issues in using FEM to find the stress distribution in such a special structure like a wound roll, when the air entrainment and gaps between two layers are considered. Moreover, as the lap number increases, the number of elements that can be simulated is restricted by computing capability. The primary objective of this proposal is to develop methodologies for winding viscoelastic webs to produce rolls that have dimension stability to minimize the formation of baggy lanes. Research will focus on the understanding of baggy lane formation mechanism in viscoelastic webs with a varying thickness profile along the width direction. The thesis is organized as follows: (1) a pseudo 3D viscoelastic winding model will be developed based on the work by Qualls and Good [1], and Cole and Hakiel [46]; (2) commercial FEM codes ABAQUS, will be used to generate the FEM model to find the appropriate boundary condition on the outer layer; (3) experimental investigation will be conducted to examine the baggy lane formation and experimental data will be used to examine results from the models developed in this research. 2 2 CHAPTER 3 3 PSEUDO 3D VISCOELASTIC WINDING/UNWINDING MODEL In this investigation, a pseudo 3D viscoelastic winding/unwinding model was developed and implemented in a code revised from the 2D viscoelastic winding code developed by Qualls and Good [1]. This pseudo 3D viscoelastic winding/unwinding model was based on the pseudo 3D winding model by Hakiel [2], Cole and Hakiel [46] and the 2D linear viscoelastic winding/unwinding model by Lin and Westmann [10]. The web material is assumed to be orthotropic with a nonlinear radial stiffness depending on the interlayer pressure. This model allows for the consideration of a viscoelastic response during winding, storage and unwinding. In this model, the varying thickness profile along CMD has been considered in winding. In this pseudo 3D model, the wound roll is discretized into smaller segments of varying length in width direction and each segment has a constant web thickness. Tension is applied to each segment using the Hakiel approach [2]; the tension is updated after the winding of each lap based on the deformed radius of the segment relative to the relaxed radius profile of that lap. In each segment, a 2D model winding/unwinding model is applied to determine stress distribution in the web. A pseudo 3D transient viscoelastic winding model that has considered viscoelastic effects during winding has been developed for an orthotropic, nonlinear viscoelastic web based on the work by Qualls and Good. The pseudo 3D model can consider (1) varying 2 3 thickness profile in both CMD(cross machine direction) and MD(machine direction); (2) winding tension variation as a function of winding laps; (3) varying core stiffness in the width direction; and (4) hygrothermal influences. The most important feature is that the model allows the consideration of viscoelastic effects in a wound roll during winding at a tension, so that this model is especially suitable for viscoealstic materials with short characteristic relaxation times, such as nonwovens and plastic webs with glass transition temperature close to room temperature. Tapered tension profiles were employed to investigate winding at varying tension histories. The tension histories were used as input to the viscoelastic winding model to determine the stress distribution in a wound roll. Winding tension was changed through the change of the tension taper factor based on the work by Shelton [31] and Feiertag [32]. Stress distribution was determined for ten different tension profiles (ranging from constant tension to constant torque) for a roll of 1000 laps. The number of laps can be varied based on the capacity of the computer RAM. The pseudo 3D viscoelastic winding model has been used to analyze three baggy lanes problems for validation. They are (1) formation of cambered web (inplane imperfection) due to webnonuniformity; (2) formation of localized baggy lanes due to edge burr following slitting; and (3) formation of baggy web (inplane imperfection) due to webnonuniformity. Simulation results are compared with experimental data. The pseudo 3D model has considered the unwinding of a wound roll as well. During unwinding process, the viscoelastic memory effects are also considered. The viscoelastic effects built in this model depend on both viscoelastic properties and winding conditions, such as unwinding speed and tension. In this model, the stress distribution in 2 4 a roll comes from two sources, one from the initial winding, and the other from a correction state that starts from unwinding. Figure 31: Pseudo 3D models As shown in Figure 31, the wound roll can be divided into several small segments. In each segment, a 2D viscoelastic model is applied to determine stress distributions in the web. Numerical solution for the stress distribution in a wound roll during winding, storage and unwinding parts will be discussed in following sections separately. 3.1 Pseudo 3D Winding Model In this work, we use the approach by Hakiel [2] and Cole and Hakiel [46] to partition a 3D wound roll into a number of 2D segments. Some modifications will be made to the their model and the varying segment width will be considered. 2 5 Figure 32: Exaggerated view of a wound roll with thickness variation, Hakiel [2] As shown in Figure 32, before the lap i is wound on the roll, the widthwise radius distribution of the wound roll is ρ(i −1, j) . Consider that the surface of wound roll is no longer cylindrical due to thickness variation and the winding tension is not high enough nor is the radial modulus low enough to result in large deformations. The lap i may or may not be in full contact with the wound roll surface. In the areas where the contact is made between the lap i and the wound roll, the radius of inside surface of lap i r(i, j) is equal to the radius of the previous wound roll surface ρ(i −1, j) . In the areas where there is no contact, it will be assumed that the radius of inside surface of lap i r(i, j) is constant and is equal to the relaxation radius ( ) 0 R i which is assumed as the inner radius of a nonstretched part of the web. In order to determine the relaxation radius, the summation of tension applied on all segments at the current outer lap is set to be equal to the web tension. Tθ (i) = T(i) (3.1) The predicted winding tensile force is equal to the summation of tensile force in all the widthwise positions. 2 6 } ( ) ( , ) ( ) ( , ) ( ) { (1 ) ( ) { ( , ) ( ) ( , )} 0 0 1 2 1 w j h i j R i E r i j R i T i i j w j h i j M j M j − − ν =Σ σ = Σ − θ θ − θ (3.2) where w( j) is the width of segment j; h(i, j) is the thickness of segment j at lap i. The width of segment could be different in this modified model. Usually, Equation (3.1) cannot be satisfied the first time. However, in an iterating process, the relaxation radius ( ) 0 R i can be determined by extrapolating over previous estimates until the calculated winding tensile force is equal to the applied tensile force at the current radius and the tension in each widthwise position can be determined. The inner radius of the lap being wound on is ( , ) { ( 1, ), ( ); 1,... } 0 r i j = Max ρ i − j R i j = M (3.3) The widthwise radius distribution of wound roll is ρ(i, j) = r(i, j) + h( j); j =1,...M (3.4) Based on Equations (3.1) ~ (3.4) the widthwise distribution of radius and tension can be determined at a designated wound roll radius. After that, the wound roll can be divided into several small segments. In each segment, a 2D viscoelastic winding model is applied to determine the timedependent stress distributions in the segment. It is noted that in the previous equations, the radial deformation of web during winding is not considered. Radial deformation can change significantly the tension distribution. The effect of radial deformation on tension distribution can be considered using Cole and Hakiel’s model [46]. As mentioned previously, while each layer is wound, the layer may or may not make full contact with the wound roll surface. In the areas where the contact is made, the lap being wound will induce radial displacement. A higher winding tension will result in higher radial displacement, which will reduce the 2 7 effect of thickness variation and prevent from sudden changes in thickness profile in the wound roll. Figure 33: Outer lap geometry, Cole [46] As shown in Figure 33, the r (i, j ) d is deflected roll outer radius; U ′(i, j ) is the radial displacement of the segment due to winding of the ith. In the segments where there is no gapping, r (i, j ) d is defined as r (i 1, j ) r (i, j ) U (i, j ) h(i, j) d d + = + ′ + (3.5) In the segments where there is gap, r (i, j ) d is defined as ( 1, ) ( ) ( , ) 0 r i j R i h i j d + = + (3.6) where ( ) 0 R i is relaxation radius as defined before. Then the predicted winding tensile stress is equal to + − − − ν σ = θ θ ( ) ( ) 2 ( ) ( 1, ) (1 ) ( ) 0 0 2 R i R i h j r i j E i d (3.7) 2 8 The predicted winding tensile force is equal to the summation of tensile forces in all the widthwise positions. ( ) { ( , ) ( ) ( , )} 1 T i i j w j h i j M j θ − θ =Σ σ (3.8) Similarly, in order to determine the relaxation radius, the summation of tension applied on all segments at the current outer lap is set to be equal to the web tension. Tθ (i) = T(i) (3.9) An iteration is conducted to satisfy the above Equation (3.9) and determine the relaxation and. After the relaxation radius ( ) 0 R i is determined, the tension could be distributed into each segment. In each segment, a 2D viscoelastic winding model is applied to determine the timedependent stress distributions in the segment. The models for the winding and storage will be discussed in the following sections separately. 3.2 Viscoelastic Effects during Winding In this section, viscoelastic effects during winding will be considered. For viscoelastic materials with relatively short characteristic relaxation times, compared with the time it takes to wind a roll, it is necessary to consider the viscoelastic effects during winding. This part of work is based on the 2D viscoelastic winding model developed by Qualls and Good [1]. In the QuallsGood model viscoelastic effects are considered after winding is finished. Their model has been extended in this work to allow the consideration of viscoelastic effects from the beginning of the winding. With the consideration of viscoelastic effects during winding, some winding conditions, such as the winding velocity and winding tension as a function of radius or time can be considered. 2 9 Consider an orthotropic viscoelastic material with radial modulus depending on radial stress in 2D case in polar coordinates. The equilibrium equation is + σ − σ = 0 ∂ ∂σ r θ r r r (3.10) The strain compatibility in the rθ plane is given by: + ε − ε = 0 ∂ ∂ε θ θ r r r (3.11) It should be noted that in the current model there is no strain compatibility enforced between segments. The constitutive equations for an orthotropic viscoelastic material are dt t J t t t J t t t r r r r ′ ∂ ′ ∂ + − ′ ∂ ′ ∂ = ∫ − ′ 0 ( ) ( ) θ θ σ σ ε dt t J t t t J t t t r r ′ ∂ ′ ∂ + − ′ ∂ ′ ∂ = ∫ − ′ 0 ( ) ( ) σ σ ε θ θ θ θ (3.12) By solving Equation (3.10), σθ can be expressed in terms of r σ . r r r r + σ ∂ ∂σ σθ = (3.13) We next insert Equation (3.13) into Equation (3.12) to eliminate σθ , and then substitute εθ , r ε in Equation (3.11) to obtain [ ( ) ( ) {3 ( ) ( ) ( ) 2 2 2 0 J t t J t t J t t r r t J t t r r r t + − ′ + − ′ − − ′ ∂ ∂ σ ∂ ′ ′ ∂ ∫ θ − θ θ θ ( )} ( ) { (J (t t ) J (t t )) J (t t ) r r r r t J t t r r r r − ′ + − ′ + − ′ ∂ ∂ + ∂ ∂σ ∂ ′ ′ ∂ − ∂ ∂ + θ θ θ θ 3 0 ( ) ( ) ( )} ] ′ = 0 ∂ ′ ′ ∂σ + θ − ′ − − ′ − θ − dt t J t t J t t J t t r r r r (3.14) Although the radial stiffness r E is related to roll radius, the ratio, r r νθ / E and ν θ Eθ r / , are extremely small and could be assumed to be radially independent, as the radial Poisson’s ratio νθr and ν θ r are comparatively small. Therefore, Qualls and Good [1] suggested that Jθ , rθ J , r Jθ are radially independent. Consider that rθ J , r Jθ are small and r Jθ is equal to rθ J at some special time, we could assume that θ = θ r r J J . Therefore the Equation (3.14) can be simplified as ( ) ( ) ( ) ( ) ( ) 0 0 2 2 3 2 2 1 = ′ ⋅ ∂ ′ ′ ∂σ + − ∂ ∂σ ∂ ′ ′ ∂ + − ∂ ∂ σ ∂ ′ ′ ∂ ∫ − dt t F t t r r t F t t r r t F t t t r r r (3.5) where ( ) ( ) 1 F t −t′ = Jθ t −t′ ; ( ) 3 ( ) 2 F t −t′ = Jθ t −t′ ; ( ) ( ) ( ) 3 F t t J t t J t t r − ′ = θ − ′ − − ′ . r J and Jθ follow the generalized Kelvin model. It should be noted that r J is also a function of the radial pressure. The generalized Kelvin model for creep compliance is expressed as ( ) (1 ) / 1 0 i t N i i J t J J e − ξ = Σ = + − (3.6) In order to solve Equation (3.15), a general form is assumed. dt t f I F t t t ⋅ ′ ∂ ′ ∂ = ∫ − ′ 0 ( ) (3.7) 3 1 As shown in Figure 34, for a wound roll with n laps we divide the entire winding time into n time steps. We have k t = t after winding lap k (radius k r = r , 1≤ k < j ≤ n ). Therefore, at the current outer layer, lap j, j t = t . Figure 34: Viscoelastic effects during winding For the case of constant velocity 0 v(t) = v , the time increment from tj to tk can be derived as follows. Since R dR h v dt r t r j k π∫ ⋅ = ∫ ⋅ % 0 0 2 , we have ( ) v h r r t t t j k j k 0 π⋅ 2 − 2 % = − = (3.8) And the integral I can be written as j j j j j j j j k k I = F(t − t − ) ⋅%f + F(t −t − ) ⋅%f − + + F(t −t − ) ⋅%f 1 2 1 1 L (3.9) where % = = − = −1 j t j t j f f f . Thus j j k %f − ,%f − , ,%f 1 2 K are already known when j t = t . And ( ) − −1 j j F t t , ( ) − −2 j j F t t ,K, ( ) − −1 j k F t t can be calculated at any radius using the formula for the creep compliance. Thus the only unknown is j %f . Lap n, t=tn Lap j, t=tj Lap k, t=tk 3 2 Comparing Equation (3.15) and (3.17), we can find that the function f could be either 2 2 2 r r r ∂ ∂ σ , or r r r ∂ ∂σ , or r σ . Therefore j %f (the increment of f from = −1 j t t to j t = t ) can be expressed in terms of 2 2 2 r r r ∂ ∂ %σ , r r r ∂ ∂%σ or r %σ , where % refers to the increment from = −1 j t t to j t = t . After using Equation (3.20), j %f can be rewritten in terms of ( 1) ( ) ( 1) + , , − %σ %σ %σ r k r k r k . 2 ( 1) ( ) ( 1) 2 2 2 r h r r k + r k r k− %σ − %σ + %σ ≈ ∂ ∂ %σ r h r r k r k 2 ( +1) ( −1) %σ − %σ ≈ ∂ ∂%σ (3.20) Then the general form I in Equation (3.17) can be represented by ( 1) ( ) ( 1) + , , − %σ %σ %σ r k r k r k . At radius k r = r , Equation (3.15) will be represented by a general form. ( , , ) 0 ( 1) ( ) ( 1) %σ %σ %σ = k r k+ r k r k− G (k=1,2,…j1) (3.21) Equation (3.21) can be written for each radial location within the wound roll. When j t = t there are j1 simultaneous algebraic equations with j+1 unknowns. After both the outer and inner boundary conditions at j t = t are applied, this system of equations can be solved. 3.3 Viscoelastic Effects after Winidng After winding process is finished, the total number of laps is n. For radius k r = r , at time m t = t (n < m), the integral I can be written as 3 3 m m m m m m m m k k I = F(t −t − ) ⋅%f + F(t − t − ) ⋅%f − + + F(t −t − ) ⋅%f 1 2 1 1 L (3.22) where % = = − = −1 m t m t m f f f . When m t = t , m m k %f − ,%f − , ,%f 1 2 K are already known. ( ), ( ), , ( ) − −1 − −2 − −1 m m m m m k F t t F t t K F t t can be calculated at any radius using the formula for the creep compliance. Thus the unknown is m %f only when m t = t . Substitution of the finite difference approximation gives 2 ( 1) ( ) ( 1) 2 2 2 r h r r k + r k r k− %σ − %σ + %σ ≈ ∂ ∂ %σ r h r r k r k 2 ( +1) ( −1) %σ − %σ ≈ ∂ ∂%σ (3.23) Similarly, at radius k r = r , the relation among ( 1) ( ) ( 1) + , , − %σ %σ %σ r k r k r k can be written in terms of a general form as follows: ( , , ) 0 ( 1) ( ) ( 1) %σ %σ %σ = k r k + r k r k − H (k=1,2,…n1) (3.24) Equation (3.24) can be written for each radial location within the wound roll. At m t = t , there are n1 simultaneous algebraic equations with n+1 unknowns. When both the outer and inner boundary conditions at m t = t are applied, this system of equations can be solved. 3.4 Viscoelastic Effects during Unwinding In this model, the stress distribution in a roll comes from two sources, one from the initial winding, and the other from a correction state that starts from unwinding. The summation of the initial and correction states satisfies the traction free boundary 3 4 condition for the current unwinding radius. The initial state includes both winding and after winding period. The stress in initial state is the basis of later calculation. It starts from the beginning of winding until unwinding begins. σinitial = σduringwinding + σafterwinding (3.10) Correction state starts from the beginning of unwinding. It is assumed that the final stress at any layer is equal to the sum of both initial state and correction state. The correction state is determined by solving the problem of a wound roll with fixed radius subjected to an arbitrary external force P(t). P(t) Figure 35: An arbitrary external force P(t) The unknown external force P(t) can be determined by enforcing the traction free condition at the current winding radius. In unwinding case, the external force direction is outward. After the external force is found, stress distributions can be determined at any time and at any layer. σ final =σ initial +σ correction only ifσ final = 0 at current unwinding radius (3.11) The flow chart of algorithm for viscoelastic unwinding is shown in Figure 36. 3 5 Figure 36: Flow chart of algorithm for viscoelastic unwinding 3.5 Finite Element Winding Simulations for Checking Outer Boundary Condition and Effects of Asymmetric Structure In this section, 2D FEM simulation of winding is conducted to check for the outer boundary condition after winding process is finished. Since the winding tension has been removed at that time, the outside layer boundary condition is different from that used in during winding process. The winding and storage are simulated using ABAQUS/Explicit code to find the outer boundary conditions. Moreover, the influence due to asymmetric structure will be investigated using the present FEM approach. The “spiral” structure of wound roll was considered and simulated in ABAQUS. Calculate the stress change i=1 Unwind first layer Update initial stress at remaining layers Assume one external Start Pressure, P(t). End at current unwinding radius N N Y Y σ initial + %σ correction = 0 ? %σ correction σ initial =σ initial + %σ correction i>N? 3 6 3.5.1 Evaluation of Boundary Conditions using Dynamic Analysis via ABAQUS / Explicit The dynamic analysis algorithm in ABAQUS/Explicit code is used. The equation of motion is (i ) (i ) (i ) M ⋅ ü = F − I (3.12) where ü is acceleration, M is the diagonal lumped mass matrix, F is the applied load vector, and I is the internal force vector. The explicit dynamics analysis in ABAQUS/Explicit is based on an explicit integration rule with the use of diagonal or “lumped” element mass matrices. The acceleration ü is computed from Equation (3.27), (i) 1 ( (i) (i) ) ü = M ⋅ F − I − (3.13) The equations of motion for the body are integrated using the explicit central difference integration rule, ( ) ( ) (i ) i i i i u t t u& u& && 2 1 2 1 2 1 + = + + − + (3.14) ( ) ( ) ( ) + + = + % + 2 1 i 1 i 1 i i u u t u& (3.30) where u& is velocity, and u is the state vector (displacements or rotation angles). The superscript (i) refers to the increment number and 2 1 i − and 2 1 i + refer to midincrement values. The central difference integration operator is explicit in that the kinematic state can be advanced using known values of − 2 1 i u& and (i ) u&& from the previous increment. However, the central difference operator is not selfstarting. The initial values (at time t = 0) of velocity and acceleration need to be given by user before the procedure starts. In 3 7 ABAQUS/Explicit, if the initial values are not given by the user, the default initial values of velocity and acceleration are set to zero. After the initial values of velocity and acceleration are defined, the mean velocity + 2 1 u& and − 2 1 u& can be determined by the following Equations (3.31) and (3.32). Then the displacement, velocity and acceleration at any time can be obtained using Equation (3.28) ~ Equation (3.30). ( ) ( ) (0) 1 2 0 1 2 u t u& = u& + && + (3.31) ( ) ( ) (0) 0 2 0 1 2 u t u& = u& − && − (3.32) The central difference operator is conditionally stable. ABAQUS/Explicit adjusts the time increment automatically and requires no user intervention. 3.5.2 2D Viscoelastic FEM Winding Model Geometry and Boundary Conditions The 2D viscoelastic FEM winding model contains two parts. One is the winding part and the other one is the storage part. Schematic of the 2D FEM winding model is shown in Figure 37. The winding model can be divided into two substructures which are web and core. In this model, the core is considered as a rigid cylinder to simplify the model. In the winding part, the core is restricted in both horizontal and vertical directions and can only rotate with respect to the axis of cylinder at a constant angular velocity. One end of the continuous web was tied to the core while a horizontal tension was applied on the other end of web as shown in the Figure 37. Therefore the web can be wound onto the core layer by layer after the core starts to rotate. 3 8 Figure 37: Schematic of the 2D FEM winding model After winding was complete, the free end of web was attached to the second layer of wound roll. The geometry and boundary condition after winding are shown in Figure 38. In the storage part, the core is fixed at the reference point and does not allow for any movement in all directions. Although, there are no additional loadings added on wound roll or core, the stress and strain fields still change with time due to the viscoelastic effects of the web material. Figure 38: Schematic of 2D FEM storage model Contact Problems During the winding part, contacts are defined as follows: 1) contact between the web and the core surface; 2) self contact of the web. The core is designed as a rigid cylinder and the surface of the core is defined as rigid surface. The definition of the contact is shown in Figure 39. 3 9 Figure 39: Definition of the contact in 2D FEM winding model Similarly, in the storage part the definition of the contact surfaces is shown in Figure 310. Figure 310: Definition of the contact in 2D FEM storage model Viscoelstic Material Properties In the numerical examples, three sets of viscoelastic material parameters, representing different levels of viscoelastic behavior, were used to investigate the viscoelastic effects on stress or strain of the outer layer. The relaxation modulus is given by a Prony series. 2 1 3 4 • Contact between surface (1) and surface (2) • Contact between surface (3) and surface (5) • Tie surface (4) to surface (5) Web • Surface (5) is rigid surface 5 Core 6 7 Web 4 Do not allow normal and tangential movements between surface (6) and surface (7) after contact. 5 Core 4 0 (1 (1 e ))  1 0 i t/τ N i P R i  g  E (t) E Σ= = (3.33) where 0 E is the instantaneous Young's modulus; P i g is the modulus ratio in the i th term in the Prony series expansion of the shear relaxation modulus; i τ is the relaxation time for the i th term in the Prony series expansion. There are three sets of viscoelastic material parameters used in this 2D viscoelastic winding simulation to investigate the viscoelastic effects. The three sets of viscoelastic material parameters are listed in Table 31. Table 31: Viscoelastic material parameters Set 1 Set 2 Set 3 P i g i τ P i g i τ P i g i τ 1th 0.128 20 0.3 40 0.4 40 2nd 0.072 10 0.2 20 0.25 20 The relaxation modulustime curve for three sets of viscoelastic material plotted in Figure 311. Set 3 has the highest viscoelastic properties while Set 1 has the lowest viscoelastic properties. 4 1 0 20,000 40,000 60,000 80,000 100,000 0 20 40 60 80 Time(s) E ( t ) ( p s i ) Set 1 Set 2 set 3 Figure 311: Three sets of viscoelastic material properties Dimension and Loading Three 2D FEM viscoelastic winding models are established using three sets of viscoelastic properties correspondingly. Dimension and loading condition are same for the three models. As shown in Figure 37, the total length of the web is 300 in, the thickness of the web is 0.05 in and the Poisson’s ratio is 0.3. It is assumed that the core is rigid cylinder. The radius of core is 5 in. The tensile stress applied at one end of web is 1000 psi. In the winding part, the angular velocity of core is 2 rad/s. 3.5.3 Results and Discussion Outer Layer Boundary Condition after Winding Was Finished The outer layer timedependent circumferential stresses and strains are plotted for Set 1~3. For viscoelastic material parameters in Set 1, the outer layer timedependent circumferential stress and strain are shown in Figure 312 and Figure 313. 4 2 Figure 312: The outer layer timedependent circumferential strain for Set 1 Figure 313: The outer layer timedependent circumferential stress for Set 1 0 20000 40000 60000 80000 100000 0 10 20 30 40 50 60 Time(s) E(t) (psi) Set 1 Set 2 Set 3 0 20000 40000 60000 80000 100000 0 10 20 30 40 50 60 Time(s) E(t) (psi) Set 1 Set 2 Set 3 4 3 For viscoelastic material parameters Set 2, the outer layer timedependent circumferential stress and strain are shown in Figure 314 and Figure 315. Figure 314: The outer layer timedependent circumferential strain for Set 2 Figure 315: The outer layer timedependent circumferential stress for Set 2 0 20000 40000 60000 80000 100000 0 10 20 30 40 50 60 Time(s) E(t) (psi) Set 1 Set 2 Set 3 0 20000 40000 60000 80000 100000 0 10 20 30 40 50 60 Time(s) E(t) (psi) Set 1 Set 2 Set 3 4 4 Similarly, for viscoelastic material parameters in Set 3, the outer layer timedependent circumferential stress and strain are shown in Figure 316 and Figure 317. Figure 316: The outer layer timedependent circumferential stain for Set 3 Figure 317: The outer layer timedependent circumferential stress for Set 3 0 20000 40000 60000 80000 100000 0 10 20 30 40 50 60 Time(s) E (t) (p s i) Set 1 Set 2 Set 3 0 20000 40000 60000 80000 100000 0 10 20 30 40 50 60 Time(s) E (t) (p s i) Set 1 Set 2 Set 3 4 5 Based on the Figure 312~Figure 317, it is seen that, after the winding process is finished, the percentage of drop in circumferential stress is much larger than circumferential strain at outermost layer. A highly viscoelastic material has less percentage of drop in both circumferential stress and strain. As shown in Figure 316, the circumferential strain is nearly constant. Therefore, for the highly viscoelastic materials, a constant strain can be used as the boundary condition at the outer layer of a wound roll. Moreover, the boundary and loading condition used in this simulation will be discussed further. In this AQAQUS model, during the winding process, the winding tension was added at the end of web. When the winding process finishes, the tension is removed from the web immediately. When the storage process starts, a restriction which does not allow normal and tangential movement between the end of web and the second layer of wound roll will be added. However, this restriction might not be effective immediately although we define so. Since, in ABAQUS model, we only can restrict the relation between the nodes. Therefore the web will slide a little bit until the nodes on the end of web contact with the nearest nodes on the second layer. During this short period, the winding tension has already been removed and the stress and strain might drop. The percentage of drop will depend on the level of viscoelastic effects. High viscoelastic material will have fewer drops. This phenomenon could be found the Figure 312~Figure 317. Effect of Asymmetric Structure In pseudo 3D winding model and most other models, it is assumed that the wound roll is an axisymmetric structure. However, the real wound roll is a “spiral’ structure. In order 4 6 to find the influence due to asymmetric structure, in the present FEM approach, the “spiral” structure of wound roll was considered and simulated in ABAQUS. The circumferential stresses vs. angle curves for layer 1 to layer 10 are plotted in Figure 318, which show that the asymmetric structure has effects on stress distribution in the wound roll. The curves for Layer 1 and Layer 10 have most wave amplitude changes. Asymmetric structure also has effects on the layers close to the outermost and innermost layers. Layer 2 and Layer 7, 8, 9 also show high stress variation. Those layers far away from the asymmetric structure show less fluctuation in stress. Consequently, those layers can be considered as axisymmetric structure to simplify the calculation in the development of winding models. Layer 1 0 200 400 600 800 1000 1200 1400 0 1 2 3 4 5 6 Angle (radians) Circumferential Stress (psi) Layer 2 0 200 400 600 800 1000 1200 0 1 2 3 4 5 6 Angle (radians) Circumferential Stress (psi) Layer 3 0 200 400 600 800 1000 1200 0 1 2 3 4 5 6 Angle (radians) Circumferential Stress (psi) Layer 4 0 100 200 300 400 500 600 700 800 900 1000 0 1 2 3 4 5 6 Angle (radians) Circumferential Stress (psi) 4 7 Figure 318: Circumferential stresses in layer 1 to layer 10 Layer 5 0 100 200 300 400 500 600 700 800 900 1000 0 1 2 3 4 5 6 Angle (radians) Circumferential Stress (psi) Layer 6 0 200 400 600 800 1000 0 1 2 3 4 5 6 Angle (radians) Circumferential Stress (psi) Layer 8 0 100 200 300 400 500 600 700 800 900 0 1 2 3 4 5 6 Angle (radians) Circumferential Stress (psi) Layer 7 0 100 200 300 400 500 600 700 800 900 1000 0 1 2 3 4 5 6 Angle (radians) Circumferential Stress (psi) Layer 9 0 100 200 300 400 500 600 700 800 900 1000 0 1 2 3 4 5 6 Angle (radians) Circumferential Stress (psi) Layer 10 200 0 200 400 600 800 1000 1200 1400 1600 1800 0 1 2 3 4 5 6 4 8 CHAPTER 4 4 RESULTS FROM 3D VISCOELASTIC WINDING/UNWINDING MODEL As mentioned in previous sections, the pseudo 3D viscoelastic winding model is capable of dealing with the thickness variation in both CMD and MD directions, as well as winding tension variation with the winding laps. In this section, both verification and validation results will be presented for the pseudo 3D viscoelastic winding model. In verifications, we compare the results from the pseudo 3D viscoelastic winding model with analytical or numerical results. In these examples, the results from the pseudo 3D viscoelastic winding model have been compared with some other winding models and analytical solutions. They include (1) the comparison with Lin & Westmann’s 2D Viscoelastic windng model; (2) the comparison with the 3D analytical solution; and (3) the comparison with Qualls and Good’s 2D viscoelastic model. In validations, we created CMD thickness variations in the winding experiment by adding a stack of web strips of the same web material being wound at every nth lap [44]. In addition, in the numerical calculation all the extra thickness and related volume are assumed to be allocated evenly in these n laps. Therefore, in these several laps, the thickness varies only in the CMD direction and the thickness profile is persistent in the MD direction. Three special cases have been considered for validation: (1) The formation of cambered web due to linearly varying thickness; (2) The formation of localized baggy lanes due to edge burr following slitting; and (3) The formation of baggy web due to 4 9 thickness increment in the middle of the web. The simulation results using the pseudo 3D viscoelastic winding model will be compared with experimental data and the deformation results from pseudo 3D viscoelastic winding model. The dimensional changes estimated by numerical simulations will be compared with measured data. After that, the pseudo 3D winding model also has been used to investigate the effects of some winding parameters and winding conditions. They are (1) consideration of thickness variation in both CMD and MD; (2) consideration of varying tension histories; (3) consideration of during winding and after winding; and (4) consideration of winding and unwinding. The material used in all the experiments is polyethylene web of 2.5 mil thick and 6 in wide. The inplane and outofplane creep compliance data are taken from Qualls [47]. The outofplane creep compliance follows 3 5 5 / 2.78 10 5 / 2.613 10 5 2 3 2.034 10 1.024 10 3.058 10 167.24 0.09855 0.000422 1 ( ) − − × − − × − − × ⋅ − × + × σ − σ − σ = t t r r r r e e J t (4.1) The inplane creep compliance follows 4 7 4 5 /1 10 4 /1 10 1.62 10 1.285 10 1.491 10 24000 1 ( ) − − − × − − × θ = + × − × ⋅ − × t t J t e e (4.2) where r σ is in psi, t is in seconds and J (t) r is in 1/psi in these inplane and outofplane creep compliance data. 5 0 4.1 Comparison of Results from Pseudo 3D Viscoelastic Model and Lin & Westmann’s 2D Viscoelastic Model Lin and Westmann’s model is linear isotropic viscoelastic winding model. In order to compare with their model, isotropic material properties are used and shown in Table 41. Table 41: Material properties used in pseudo 3D winding model to compare with Lin and Westmann’s results Web Width 6 in Effective Modulus of Core (psi) 5 3.0851×10 J0 (1/psi) 6 1.631 10× − J (1/psi) τ (sec) 1st 9 0.90823 10× − 0.1 2nd 7 0.16903 10× − 1 0.1×10 3rd 6 0.10489 10× − 2 0.1×10 4th 6 0.17270 10× − 3 0.1×10 5th 6 0.39090 10× − 4 0.1×10 6th 6 0.62461 10× − 5 0.1×10 7th 6 0.64518 10× − 6 0.1×10 The dimensionless radial stress has been obtained under uniform tension to compare with the results from Lin and Westmann’s 2D viscoelastic winding model. The results were plotted in Figure 41. Based on the comparison, we can find that the results are matched. 5 1 CMD (in) 2 0 2 D emensio nless Radius 1 1.2 1.4 1.6 1.8 2 Dimesionless Radial Stress 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Pseudo 3D Viscoelastic Winding Model Lin and Westmann [10] Figure 41: Comparison with the results of Lin and Westmann [10] 4.2 Comparison of Results from Pseudo 3D Viscoelastic Model and the 3D Analytical Solution In this section, before the comparison, the solution for 3D isotropic viscoelastic winding models will be introduced briefly. The incremental radial stress ( ( ) * r r σ ) for 3D isotropic elastic winding model is ( ) p a ad bc a aE b c p aE b cE d r r ⋅ − ⋅ + ⋅ = + + + σ = * 1 (4.3) where: o w r T h p = − ( ) ( ) ( ) ( ) 2 i 1 o 1 i 2 o a = F r F r − F r F r [ ( ) ( ) ( )] ( ) [ ( ) ( ) ( )] ( ) c 2 i c 6 i c 8 i 1 o c 1 i c 5 i c 7 i 2 o b = νE F r − E F r + νE F r F r − νE F r − E F r + νE F r F r cos [ ( ) ( ) ( ) ( )] 2 1 1 2 c kz F r F r F r F r i i = ⋅ − cos { [ ( ) ( ) ( )] ( ) [ ( ) ( ) ( )] ( )} 1 5 7 2 2 6 8 1 d kz E F r E F r E F r F r E F r E F r E F r F r c i c i c i c i c i c i = ⋅ − ν − + ν + ν − + ν ( ) ( ) ( ) 1 2 0 3 1 B ikr r ik F r = −k B ikr − ; ( ) 2 ( ) ( ) ( ) 1 4 0 3 0 3 2 F r = − k νB ikr + k B ikr − ik rB ikr ; 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 1.2 1.4 1.6 1.8 2 r, in Radial stress/Initial tension r/rin 5 2 ( ) ( ) 1 3 3 F r = −ik B ikr ; ( ) 2 (1 ) ( ) ( ) 0 4 1 3 4 F r = ik − ν B ikr − k rB ikr ; ( ) ( ) 1 2 5 B ikr r ik F r = ; ( ) 2 ( ) ( ) 0 3 0 3 6 F r = − k νB ikr + k B ikr ; ( ) ( ) 0 3 7 F r = k B ikr ; ( ) 2 (2 ) ( ) ( ) ( ) 1 4 0 3 8 F r = − k − ν B ikr + ik r B ikr . Applying elasticviscoelastic corresponding principle, we have ( ) ( ) ( ) ( ) − ⋅ + = ⋅ + − ⋅ + σ = + a ad bc as E s sb p a c s p a ad bc a asE s b c r s p s r 2 * ) 1 , ( (4.4) where: ( ) ( ) ( ) ( ) a sbJ (s) J s sb sJ s as E s sb sa s + = + = + φ = 1 1 1 2 . E(t) is the Young’s relaxation modulus; J(t) is the (uniaxial) creep compliance; and s is the Laplace transform variable. Inverting the L.T. leads to the function of φ(t ) ( ) ( ) ( ) ( ) + ξ − ξ − ξ φ = − ∫ φ ξ (0) ( ) 1 0 a bJ d d t dJ t t J t b t (4.5) Then we have cos [ ( ) ( )] 0 1 1 2 kz a F r a F r o R r r σ = ∫ ⋅ + . Similarly, we can find other stress components. Results from the pseudo 3D viscoelastic winding model are used to compare with the results form 3D analytical solution. The linear isotropic viscoelastic material as shown in Table 41 was used in this comparison. The winding velocity is 0.1 in/sec. The results are plotted in Figure 42. Based on the comparison, we can find that the results are matched. The maximum difference is around 5%. 5 3 Figure 42: Comparison with the results of 3D analytical solution 4.3 Comparison of Results from Qualls and Good’s 2D Viscoelastic Model The pseudo 3D viscoelastic model was developed and implemented in a code revised from the 2D viscoelastic winding code developed by Qualls and Good [1]. The difference between these two models is that in Qualls and Good’s model viscoelastic effect is considered after the winding process is finished while in the pseudo 3D model viscoelastic effect has been considered from the beginning of winding process. The material used in this comparison is polyethylene, with its inplane and outofplane creep compliance data taken from Qualls [47] as shown in Equations (4.1) and (4.2). The total number of layers is 100 and the storage time is 5 6×10 seconds (~7 days). The winding tensile stress is 100 psi. In the first example, we choose a very high winding speed, 1000 in/s, to reduce the winding time. The radial pressures were obtained from both code and were plotted in Figure 43 for winding process and storage process CMD (in) 3 2 1 0 1 2 3 Radius (in) 1.8 1.85 1.9 1.95 2 2.05 Radial Stress (psi) 0 10 20 30 40 Pressure (psi) 3D Analytical Solution Pseudo 3D Model 5 4 correspondingly. As shown in Figure 43, results from both models agree very well since there is not much viscoelastic effect in such a short winding time. 0 2 4 6 8 10 12 14 1.8 1.85 1.9 1.95 2 2.05 2.1 Radius (in) Pressure (psi) Pseudo 3D: Winding Qualls and Good's, Winding Pseudo 3D: Storage, t=6e+5 s Qualls and Good's: Storage, t=6e+5 s Winding Storage Figure 43: Comparison with the results of Qualls and Good’s model, v=1000 in/s In the second example, we compare with the results from Qualls [47]. In this example, the inside radius is 1.75 in and outside radius is 5.25 inch. Winding tension is 100 psi and winding speed is 30 fpm. The material used in this comparison is also polyethylene, with its inplane and outofplane creep compliance data taken from Qualls [47] as shown in Equations (4.1) and (4.2). The radial pressure was obtained from pseudo 3D model and was plotted with Qualls’ data in Figure 44 for winding process and storage process for 10 days, 33 days and 64 days correspondingly. As shown in Figure 4 4, the results from pseudo 3D model are close to the results from Qualls and Good’s model. It seems that viscoelastic effects during winding for a polyethylene web are not significant in this case. 5 5 0 25 50 75 100 1.75 2.25 2.75 3.25 3.75 4.25 4.75 5.25 Radius (in) Pressure (psi) Qualls, T=0 Pseudo 3D, T=0 Qualls, T=10 days Pseudo 3D, T=10 days Qualls, T=33 days Pseudo 3D, T=33 days Qualls, T=64 days Pseudo 3D, T=64 days Measured, T=0 Figure 44: Comparison with the results of Qualls and Good’s model, v=30 fpm 4.4 Comparison of Results from Pseudo 3D Viscoelastic Model and Experimental Data The pseudo 3D winding code was used to calculate the stress in a roll under conditions used in experiments; results are compared with experimental data. A CMD thickness variation was created during winding by adding a stack of 3 inserts of equilateral triangular webs at every fourth lap after winding 170 layers. The schematic of this insertion is shown in Figure 45. Figure 45: Schematic diagram for inserting a triangular web for every four plies for use in the investigation of formation of a cambered web On average, a thinner edge On average, a thicker edge Location #12 Equilateral triangular web 7 in 6 in Location #1 Increasing Time 5 6 The total number of layers is 200. The first pull tab was inserted on the 10th layer. After that, pull tabs were inserted on every 20th layer. After the 8th pull tab was inserted, 20 more layers were wound on the roll. Then a stack of 3 trianglar web inserts were inserted every 4th layer. The winding speed was 4.3 ft/min and winding tensile stress was 320 psi. The material used was polyethylene. The calibrated pull tabs were inserted at both left and right hand sides of the roll prior to the insertion of triangular inserts. Only one inch of the pull tabs was inserted into the winding roll, in order to measure the end roll pressure. The experimental data were obtained for both ends of the wound roll and plotted in Figure 46. 0 10 20 30 40 50 60 70 80 90 0 20 40 60 80 100 120 140 160 180 200 Layer # Pressure (psi) Figure 46: The experimental data for both edges Pseudo 3D winding model was used to simulate this process. Since the pseudo 3D winding model is an axisymmetric model, it is assumed that for each 2D segment the extra thickness and associated volume will be allocated evenly in that layer. In the simulation, 12 segments, each 0.5 inch wide, were used. The average values of the stresses in segment 1 & 2, segment 11 & 12 have been taken to compare pressure as Thinner Edge Thicker Edge 5 7 measured by pull tabs. The inplane and outofplane creep compliance data are taken from Qualls [47] and listed as shown in Equation (4.1) and (4.2). It should be noted that Qualls’ material data were obtained more than 10 years ago. In 2006, Poh [48] conducted experiment to measure the material properties of these polyethylene webs after 10 years storage. Poh compared the inplane creep compliance of polyethylene webs with the data by Qualls and found that the results are close. For the outofplane creep compliance, he did not make the comparison. We might need to conduct additional creep compliance experiment to measure the radial compliance. Before we could obtain the further material properties data from experiment, we will use Qualls’ data. Although there might be some error, the effect of 10 years storage is not as big as we thought considering that a good comparison has been indicated for inplane creep compliance. The radial stress distribution from pseudo 3D winding model was plotted with experimental data in Figure 47. 5 8 0 10 20 30 40 50 60 70 80 90 0 20 40 60 80 100 120 140 160 180 200 Layer # Pressure (psi) Figure 47: Comparison with the experimental data From Figure 47, it is seen that stresses at the thicker edge are higher than that at the thinner edge. The results from numerical model and experiments agree reasonably well. It should be noted that in the pull tab tests, a pull tab was inserted only one inch into the wound roll. The quality of the web edge and the pull tab is very important in this experiment. The web with wavy edge could not be used, since in those areas where wavy edge occurs, the layertolayer contact could not be achieved easily, which will lead to the error in pull tab test. The polyethylene web used in this pull tab test was never been used, so that there is no wavy edge at both edges. 4.5 Special Case One: Formation of a Cambered Web A CMD thickness variation was created during winding by adding a stack of 3 inserts of equilateral triangular webs at every fourth lap. The schematic of this insertion is same Thinner Edge Thicker Edge 5 9 as that shown in Figure 45. While using the pseudo 3D winding model, the web was divided into ten small segments. Since the pseudo 3D winding model is an axisymmetric model, it is assumed that for each 2D segment the extra thickness and associated volume will be allocated evenly in these four laps respectively. From location #1 to #10, the thickness increases linearly. The pseudo 3D viscoelastic winding model has been used to simulate this viscoelastic winding problem. The winding speed is 4.3 ft/min and winding tensile stress is 433.3 psi. The total wound roll length is around 72 ft 10 in and the number of laps is 110 layers. The storage time is 86400 seconds (24 hours). The length of deformed web for last 13 layers at the ten locations is obtained using the pseudo 3D winding model and is plotted in Figures 48. Location Number inWidth Direction Length of Deformed Web at Different Locations (in) 0 1 2 3 4 5 6 7 8 9 10 168.5 169 169.5 170 170.5 171 171.5 Figure 48: The length of deformed web at ten locations From the data shown in Figure 48, we can find that the length of deformed web changes linearly along the CMD direction as shown in Figure 49I. Since the web edges 6 0 are straight in a cambered web, results in Figure 49I can be plotted as shown in Figure 49II, with the use of an appropriate radius and central angle as shown in Figure 49III. Figure 49: Numerical transformation for cambered web Figure 410: Maximum bow length In Figure 49, b1, b2, ..., b10 represent the lengths of segments. a1, a2, …, a10 stand for the distances from inner edge to the upper edge of these segments. They satisfy the following equations + ⋅ θ = + ⋅θ = + ⋅θ = 10 10 2 2 1 1 ( ) ( ) ( ) r a b r a b r a b M M (4.6) I I III Maximum Bow Length I II III 6 1 In this case, the numerical simulation gives the results r =464.58 in and θ = 20.83 degrees (0.3635 radians). Therefore, we can obtain the maximum bow length as shown in Figure 410 is around 7.65 in. The value of maximum bow length from experiment is 7.5 in. The deformed shapes of web at ten locations are simulated and shown in Figure 411. The numerical results and experimental data are in same order of magnitude though there are still some errors. As mentioned in the previous three examples, it is assumed that the extra thickness and volume will be allocated evenly in several related laps at each location respectively. In some situations, this assumption might have resulted in errors. Machine Direction (in) Bow Length (in) 0 25 50 75 100 125 150 175 0 5 10 15 Figure 411: Deformed shape after storage 4.6 Discussion about Hakiel’s Mode l [2] and Cole and Hakiel’s Model [46] Current modified pseudo 3D winding model has used a refined version of Hakiel’s model which is Cole and Hakiel’s model. The original code has been revised to consider the effects of radial displacement on the tension distribution. As mentioned in chapter 3, Location # 10 Location # 1 6 2 the radial displacement will decrease the effect of thickness variation and abrupt changes in thickness profile will not be exaggerated dramatically in the wound roll shape. In this section, the special case one, formation of cambered web, will be recalculated using Cole and Hakiel’s model to discuss the difference between the original model and the refined model. The problem is exactly same as described in section 4.5 except we used these two models respectively. The comparisons were shown in Figure 412 and Figure 413. Location # in CMD 0 2 4 6 8 10 Radius (in) 1.8 1.9 2 2.1 Pressure (psi) 0 20 40 60 Y X Z Figure 412: Comparison of the pressure distributions Cole and Hakiel’s Model Hakiel’s Model 6 3 Length of Deformed Web at Different Locations (in) Length of Deformed Web at Different Locations (in) 0 1 2 3 4 5 6 7 8 9 10 169 170 171 172 Figure 413: Comparison of deformed shapes after storage According to Figure 413, we find the tangential angles of these two curves are different. The tangential angle of curve from Hakiel’s model is larger than that from Cole and Hakiel’s model. Based on the length of deformed web, the maximum bow length from Hakiel’s model is 8.89 in and the maximum bow length from Cole and Hakiel’s model is 7.65 in. The experimental result is 7.5 in. The results from Cole and Hakiel’s model are closer to the experimental results. In Cole and Hakiel’s [46] paper, they also tried several examples and showed that the refined the model is better than the previous Hakiel’s model [2] in all these cases. As a result, we updated the all three special cases we provided before and get better results. 4.7 Special Case Two: Formation of Wavy Edge One of the factors contributing to formation of a wavy edge is winding a web with an edge burr following slitting. In this validation case the effect of slit edge burr on the formation of baggy lanes is investigated. A web with edge burr is shown in Figure 414. Hakiel’s model Cole and Hakiel’s model 6 4 The segment close to one edge is thicker than the other segments. The thickness profile is assumed to be persistent along MD. Figure 414: Formation of wavy edge The pseudo 3D viscoelastic winding model was used to simulate this wavy edge problem. As mentioned before, the material used is polyethylene web of 2.5 mil thick and 6 in wide. The height of edge burr is 0.803 mil and the winding tensile stress is 300 psi. The number of laps is 80. The storage time is 86400 seconds (24 hours). The entire length of deformed web at ten locations is plotted in Figures 415. Location Number in Width Direction Length of Deformed Web at Different Locations (in) 1 2 3 4 5 6 7 8 9 10 11 12 13 958 960 962 964 966 968 970 972 974 976 978 Figure 415: The length of deformed web at ten locations edge burr 6 5 As shown in Figure 415, the deformation in the edge burr area is much larger than the rest of the web. The deformations in the rest of the web are constant. In the experiment, we observed the wavy edge and found that the wavy edge follows approximately a sinusoidal oscillation. Therefore, we assume that the deformed shape follows Equation (4.7) at the edge burr, and the deformation of web can be converted from shape I into shape II as defined in Figure 416. ) 2 sin( λ π = ⋅ x y A (4.7) where λ is the wave length, A is the amplitude of wavy edge. These two parameters can be determined in numerical simulation and measured in experiment. Then the shape of wavy edge can be simulated and examined. Figure 416: Numerical transformation for wavy edge In the experiment, the wave length and amplitude have been obtained through image analysis. λ is 0.923 in and the A is 31.26 mil. In the numerical simulation, the wave length (λ) and amplitude (A) were extracted from Equation (4.8). dx A x Length Length rest burr ∫λ λ π λ π = + λ 0 2 2 2 2 2 cos 4 1 / (4.8) Lengthburr Lengthrest I II 6 6 It is assumed that the wavy edge has the same wavy length as that in the experiment to determine the amplitude. The amplitude of wavy edge can be determined by inserting the average length of deformed web at burr area into Equation (4.8). In this numerical simulation, the amplitude is determined as 30.57 mil based on the deformation results from the pseudo 3D winding model and wavy length data from experiment. The numerical results and experimental data have a very good agreement. 4.8 Special Case Three: Formation of Baggy Web In this case, CMD thickness variation was generated by adding a stack of 6 inserts of strip webs at every sixth lap. The example setup is shown in Figure 417. In the pseudo 3D winding model, the web was divided into forty segments. Since the pseudo 3D winding model is an axisymmetric model, it is assumed that the extra thickness and volume will be allocated evenly in these six laps at each location respectively. The segments at the middle of web are thicker that the rest. Figure 417: Formation of baggy web The pseudo 3D viscoelastic winding model has been used to simulate this baggy web problem. The winding speed is 4.3 ft/min and the winding tensile stress is 333.3 psi. The number of laps is around 50. The storage time is 108000 seconds (30 hours). The entire length of deformed web at forty locations determined from simulations is plotted in Figures 418. a stack of 6 inserts; width is 1 in; length is equal to the circumference at location of insertion 6 in 6 7 Location Number in Width Direction Length of Deformed Web (in) 0 4 8 12 16 20 24 28 32 36 40 545 547.5 550 552.5 555 557.5 560 Figure 418: The lengths of deformed web at forty locations From results shown in Figure 418, the lengths of deformed web reach the maximum value at the middle segments. Results are quantified in this example in terms of h as shown in Figure 419, a measure of the outofplane deformation of the web that resulted from the thickness variation. The amplitude of h is expressed by Equation (4.9). Numerical solution for h, based on the model results was 30.11 mil. The experimental result for h is 27.56 mil. The numerical results and experimental have a reasonably agreement. However, from the comparison it can be seen the numerical result is higher than the experimental result. In current pseudo 3D approach, the tension is distributed to each segment based on the thickness profile. In the areas where the thickness is high, the winding tension will be high in those areas as well. If there are only a few areas in the widthwise thickness profile which are significantly higher than the rest of the web, severe tension will be distributed to those areas. Nevertheless, in the experiment, sometimes we 6 8 could find that the web material is somewhat loose or damaged in those areas. Therefore the web material might not bear such severe tension in those areas as we predicted in numerical model. The actual tension distributed to the baggy lane areas should be less than the predicted value. This might lead to the error between numerical result and experimental result this baggy lane formation case. Figure 419: Numerical transformation for baggy web. ⋅ π − ⋅ π ≈ − ≈ 2 2 2 1 2 1 n L n L h r r ( ) ⋅ π = − 2 2 1 n L L (4.9) The dimensional changes estimated by numerical simulation are compared favorably with measurement data. However, there are still some issues in the pseudo 3D viscoelastic winding model. Similar to most of existing winding models, the pseudo 3D viscoelastic winding model is an axisymmetric model. Asymmetric local deformation can not be considered using this winding model. As mentioned in the previous three examples, it is assumed that the extra thickness and volume will be allocated evenly in several related laps at each location respectively. In some situations, this assumption might have resulted in errors. 4.9 Consideration of Thickness Variation in Both CMD and MD In ideal case, the web thickness is uniform in both CMD and MD. Restricted by the manufacturing technology, the web produced will inevitably have thickness variation in CMD. For example, in order to avoid the stress and strain to accumulate too much at r1 r2 h L1 L2 6 9 certain area which may lead to web imperfections, rotary die is used in blownfilm machine. The rotary die turns around, leading to nearly periodic thickness variation. As a result, the roll radius will distribute more evenly in stead of increasing too much at certain area. In this section, we will make an attempt to simulation this problem to find the influence of thickness variation on final deformations. In this section, we assume that the thickness variation in CMD and MD follows the sinusoidal wave as shown in Equation (4.10). B l n y l m x z A x y + ⋅ π + ⋅ π = ⋅ ) 2 2 sin( (4.10) where m, n are the cycle numbers in MD and CMD; and lx , ly are the total length in MD and CMD; A is the amplitude; B is the average thickness. The cycle number is set by the code and could be different in CMD and MD. In this example, we used 2 cycles in CMD and 3 cycles in MD. The amplitude in this example is the average thickness 0.0025 in. The thickness profiles with 50% deviation of average thickness for layer #1 and Location #1 was plotted separately in Figure 420. LayerNumber in MD Thickness (in) 0 20 40 60 80 100 0 0.0025 0.005 Location Number in CMD Thickness (in) 2 4 6 8 10 12 14 16 18 20 0 0.0025 0.005 7 0 Figure 420: The thickness profiles with 50% deviation of average thickness for layer #1 and Location #1 The deviation of average thickness used here is defined as ×100% B A . The whole thickness profile with 50% deviation of average thickness was shown in Figure 421. Location # in CMD 4 8 12 16 20 Layer # in MD 0 20 40 60 80 100 Thickness (in) 0 0.001 0.002 0.003 0.004 0.005 X Y Z 50% Figure 421: The whole thickness profile The pseudo 3D viscoelastic winding model has been used to simulate this problem. The material used in this simulation is polyethylene web of 2.5 mil average thickness and 6 in wide. The deviation of average thickness is 10%, 20%, 50% and 80% respectively. The winding tensile stress is 300 psi. The number of laps is 100. The lengths of deformed web at twenty locations determined from simulations are plotted in Figures 422. The deviation of deformed length for each case is listed in Table 42. 7 1 Location # in CMD Deformed Length (in) 2 4 6 8 10 12 14 16 18 20 1210 1212 1214 1216 1218 1220 1222 1224 1226 10% 20% 50% 80% Figure 422: Deformed length for 10%, 20%, 50% and 80% cases Table 42: Deviation of deformed length 10% 20% 50% 80% Avg Deformed Length (in) 1218.146 1218.188 1218.322 1218.472 Deviation (%) 0.066311 0.132571 0.330985 0.528435 From the results in Table 42, we find that the maximum deviation of deformed length is 0.528%. It is pretty low. Therefore, it can be concluded that the periodic thickness variation in both MD and CMD direction can mitigate the influence of the thickness variation on the formation of baggy lanes. 4.10 Consideration of Winding and Unwinding Using the 3D viscoelastic winding code, the stress distributions in the web can be calculated during winding/unwinding and after winding. In order to determine the stress difference between winding and unwinding, the comparison of stresses during winding and unwinding is shown in Figure 423. The difference between stresses during winding 7 2 and unwinding decreases as the number of layer decreases. Near the core, the stresses in these two stages are almost identical. For each sub roll, we could find the pressure near the core drops much more than other areas. For the smallest sub roll, there is no obvious drop since the pressure in this sub roll is low comparing with other bigger sub rolls. The material used in this case is polyethylene. The web has constant thickness and is under constant winding tension. The total number of layers is 50 and the winding tension is 20 psi. The winding speed is 0.086 in/s and storage time is 10 days. The pressure distribution is plotted for one location in CMD. 0 0.5 1 1.5 2 2.5 1.7 1.75 1.8 1.85 1.9 1.95 2 2.05 2.1 Radius (in) Pressure (psi) Layer=5 (winding) Layer=5 (unwinding) Layer=10 (winding) Layer=10 (unwinding) Layer=20 (winding) Layer=20 (unwinding) Layer=30 (winding) Layer=30 (unwinding) Layer=40 (winding) Layer=40 (unwinding) Layer=50 (winding) Layer=50 (unwinding) Figure 423: Comparison of stresses during winding and unwinding 7 3 CHAPTER 5 5 CONCLUSIONS A pseudo 3D winding model with the consideration of viscoelastic effects during winding and storage has been developed and implemented in a code. The following conclusions can be drawn. 1) The pseudo 3D transient viscoelastic winding model is capable of dealing with (1) a varying thickness profile in both MD and CMD direction; (2) a winding tension variation with the winding laps; (3) varying core stiffness in the width direction. Moreover, the viscoelastic effects built in this model allow the consideration of winding conditions, such as winding speed and tension. The model is especially suitable for viscoelastic materials with relatively short characteristic relaxation times, such as plastic webs with glass transition temperature close to room temperature. 2) The pseudo 3D transient viscoelastic winding model has been compared with some viscoelastic winding models, analytical solutions and experimental data. They include (1) the comparison with Lin & Westmann’s 2D Viscoelastic windng model; (2) the comparison with the 3D analytical solution; (3) the comparison with Qualls and Good’s 2D Viscoelastic Model; and (4) the comparison with experimental data. The results are reasonably matched. 7 4 3) For the polyethylene web in the validation of this model it has been shown that the effects of winding velocity are negligible in comparison to the storage effect on the viscoelastic decay of pressure in a roll. 4) The pseudo 3D transient viscoelastic winding model has been used to analyze three baggy lane problems for validation. They are (1) the formation of cambered web (inplane imperfection) due to linearly varying thickness; and (2) the formation of localized baggy lanes due to an edge burr that resulted from slitting; and (3) the formation of a baggy web (outofplane imperfection) due to thickness increment in the middle of the web. Simulation results are compared with the experimental data, and a reasonably agreement was reached. 5) There are still some errors when we compare the numerical results with experimental results. For special case one (the formation of cambered web), the error is 2%. For special case two (the formation of localized baggy lanes due to an edge burr that resulted from slitting), the error is 2.2%. For special case three (formation of baggy web), the error is 9.2%, which is highest in these three cases. It shows that the shape of thickness profile is important in the estimation accuracy. 2D viscoelastic winding models based on finite element method (FEM) have been developed to determine boundary condition at the outer layer. In the numerical examples, three sets of viscoelastic material parameters, representing different levels of viscoelastic effects were used to investigate the viscoelastic effects on the timedependent stress or strain at the outer layer. After the winding process was finished and the tension was removed, the drop of stress at the outer layer is larger than that of outer layer strain. For highly viscoelastic material, the circumferential strain at the outer layer is nearly constant 7 5 after winding is complete. Moreover, the percentage of drop in circumferential stress is much larger than circumferential strain at outermost layer and the circumferential strain at outermost layer almost keeps constant for all three sets of viscoelastic material parameters. This has proven the boundary condition of constant circumferential strain in the outer lap chosen and experimentally verified by Qualls [47] is valid. 7 6 CHAPTER 6 6 FUTURE WORK 6.1 Hygrothermal Effects on Viscoelastic Material Properties In this section the hygrothermal effects induced by temperature change and moisture absorption will be considered. The work by Knauss and Kenner [45] indicated that the polymer response to increased temperature corresponds closely to that resulting from the absorption of moisture if the volume increased by temperature is the same as that increased by moisture. One comparison of master curves for polyvinyl acetate (PVAc) obtained from varying temperature and varying moisture is shown in Figure 61. According to the master curves shown in Figure 61, it seems hygroviscoelasticity and themoviscoleasticity are similar subjects that can be treated with similar models. Figure 61: Comparison of master creep curves for PVAc [45] 7 7 Defined the moisture as the ratio of weight gain to original weight, %m/m. As one example, thermomechanical behavior and hygromechanical behavior of PVAc were shown in Figure 62. Figure 62: (a) Creep curves for dry PVAc at several temperature; (b) Creep curves for PVAc at c o 24 at several absorbed moisture levels As shown in Figure 62, the moisture has similar effect on creep compliance as temperature. For example, during certain period ( 4 10 s), the effect on creep compliance caused by moisture increase from 0% to 0.9% is almost same as the effect caused by temperature increase from c o 15.4 to c o 33.7 . However, there are still some differences between these two factors. Usually material can only absorb limited changes in moisture before it saturates. Therefore, even though the moisture content of the environment is increased, it may still have not much impact on the material properties. Therefore, we may focus on thermomechanical behavior only for most web materials. For those materials which are easy to absorb moisture under the room temperature such as paper, it is necessary to consider the effect of hygrothermal change on material properties. 7 8 Moreover, it is assumed that the wound roll is subjected to a homogeneous hygrothermal change. The hygrothermal change will have influences on the following two aspects: (1) Hygrothermal effects on (temperature/moisture) shift factor; (2) Hydrothermal effects on stressstrain relationships. 1. Hygrothermal effects on (temperature/moisture) shift factor Using the timetemperature/moisture superposition principle, master curves for relaxation modulus can be described by ( , , ) = (ς, , ) ref ref E t T M E T M , with a (T M) t TM , ς = (6.1) The subscripts ref refers to the reference temperature or reference moisture; T is temperature; M is moisture; ς is reduced time; aTM is (temperature/moisture) shift factor. ) 1  1 log ( ref TM f f a = B (6.2) f is the fractional free volume; fref is the fractional free volume at Tref and Mref; B is the Doolittle constant. The fractional free volume includes the freevolume fraction due to both temperature and moisture. f f f f T M = T + m = ref + 3α% + 3β% (6.3) where fT and fm represent the free volume fraction due to temperature and moisture, respectively; α is the coefficient of thermal expansion; β is the coefficient of hygroscopic expansion; ΔT is the change in temperature; ΔM is the change in moisture. 2. Hydrothermal effects on stressstrain relationships T M E E r r r r r r + α % + β % σ ν σ ε = θ θ θ  (6.4) 7 9 T M E Er r r + α + β ν σ  σ ε = θ θ θ θ θ θ (6.5) The subscripts r and θ refer to the radial and circumferential directions; ν is Poisson’s ratio. The viscoelasic constitutive equations are given as dt T M t J t t t J t t r r t r r r r % β + % α + ′ ∂ ′ ∂σ + − ′ ∂ ′ ∂σ ε = ∫ − ′ θ θ 0 ( ) ( ) (6.6) dt T M t J t t t J t t t r r % β + % α + ′ ∂ ′ ∂σ + − ′ ∂ ′ ∂σ ε = − ′ θ θ θ θ θ ∫ θ 0 ( ) ( ) (6.7) These equations were discretized and implemented in a Fortran code to solve for stress distribution. It may be noted that the material properties in this model depends not only on time but also on temperature/moisture. The work in this section has been implemented into pseudo 3D winding code. However, the validation will be needed in the future. 6.2 Dimensionless Study 7 Dimensionless parameters, such as the ratio of maximum thickness to average thickness, storage time to retardation time, will be varied and used in simulations to determine a phase map showing the relationship between the web conditions (thickness variation, relaxation times) and the extent of web bagginess. With the development of the phase map, we hope that it can be used directly without recourse to the numerical code in the investigation of baggy lanes in some situations. 8 0 6.3 Optimization of Tension History 8 Optimization of tension history to reach predetermined timedependent tangential stress distribution in a roll. The tension history will be simulated by a polynomial with some control parameters. Through changing these parameters to achieve the minimum tangential stress in the wound roll. 6.4 Experimental Validations More experimental validations and verifications are needed to investigate stress variations in CMD and hygrothermal influences on would roll subjected to changing temperature and humidity environments. New experimental methodology is needed to measure the stress profile in CMD and find the temperature/moisture material coefficients. 8 1 REFERENCES [1] Qualls, W. R. and Good, J. K. (1997) “An Orthotropic Viscoelastic Winding Model Including a Nonlinear Radial Stiffness,” Journal of Applied Mechanics, 64, 201 208. [2] Hakiel, Z. (1991) “On the Effect of Width Direction Thickness Variations in Wound Rolls,” Web Handling Proceeding, 7998. [3] Good, J. K. (1999) “Preface,” Proceeding of the Fifth International Conference on Web Handling, Oklahoma State University, Stillwater, Oklahoma, 1999. [4] Roisum, D. R. (2001) “Baggy Webs: Making, Measurement & Mitigation thereof,” http://roisum.com/documents/Baggy.pdf [5] Altmann, H. C. (1968, April) “Formulas for Computing the Stresses in Centerwound rolls,” TAPPI, 51(4), 176179. [6] Hakiel, Z. (1987, May) “Nonlinear Model for Wound Roll Stress,” TAPPI, 70(5), 113117. [7] Kedl, D. M. (1991) “Using a Two Dimensional Winding Model to Predict Wound Roll Stresses That Occur due to Circumferential Steps in Core Diameter or to Crossweb Caliper Variation,” Web Handling Proceeding, 99112. [8] Lee, Y. M. and Wickert, J. A. (2002), “Stress Field in Finite Width Axisymmetric Wound Rolls,” Vol. 69, 130138. 8 2 [9] Hoffecker, P. (2006, May) “The Analysis of A Nip Impinged, Three Dimensional Wound Roll,” PhD thesis. [10] Lin, J. Y. And Westmann, R. A. (1989) “Viscoelastic Winding Mechanics,” ASME Journal of Applied Mechanics, 56, 821827. [11] Gutterman, R. P. (1959) “Theoretical and Practical Studies of Magnetic Tape Winding Tensions and of Environmental Roll Stability,” General Kinetics, contract no. DA18119SC42, Arlington, VA. [12] Pfeiffer, J. D. (1966, august) “Internal Pressures in a Wound Roll,” TAPPI, 49(8), 342347. [13] Pfeiffer, J. D. (1979) “Prediction of Roll Defects from Roll Structure Formula,” TAPPI, 62, 8388. [14] Yagoda, H. P. (1980) “Resolution of the Core Problem in Wound Rolls”. ASME Journal of Applied Mechanics, 47, 847854. [15] Tramposch, H. (1965) “Relaxation of Internal Forces in a Wound Reel of Magnetic Tape,” ASME Journal of Applied Mechanics, 32, 865873. [16] Tramposch, H. (1967) “Anisotropic Relaxation of Internal Forces in a Wound Reel of Magnetic Tape”. ASME Journal of Applied Mechanics, 34, 888894. [17] Qualls, W. R. and Good, J. K. (1999) “Thermal Analysis of a Wound Roll,” Journal of Applied Mechanics, 64, 871876. [18] Kotousov, A. and Wang, C. H. (2002) “Fundamental Solution for the Generalized Plane Strain Theory,” International Journal of Engineering Science, 40, 1775 1790. 8 3 [19] Timoshenko, S. P. And Goddier, J. N. (1970), Theory of Elasticity, 3rd ed., McGrawhall, new York, 422425. [20] Flügge, W. (1973) Stresses in Shells, 2nd ed, New York. [21] Minahen, T. M. And Knauss, W. G. (1993) “Creep Buckling of Viscoelastic Structures,” International Journal of Solids and Structures, 30(8), 10751092. [22] Keshavan, M. B. and Wickert, J. A. (1997) “Air Entrainment During Steady State Web Winding,” ASME Journal of Applied Mechanics, 64, 916922. [23] Kotousov, A. and Wang, C. H. (2002) “Threedimensional Stress Constrain in an Elastic Plate With a Notch,” International Journal of Solids and Structures, 39, 43114326. [24] Kotousov, A. and Wang, C. H. (2002) “Threedimensional Solution for Transversally Isotropic Composite Plates,” Composite Structure, 57, 445452. [25] Lai, J. and Bakker, A.(1995) “An Integral Constitutive Equation for Nonlinear PlastoViscoelastic Behavior of HighDensity Polyethylene,” Vol. 35, No. 17, p 1339. [26] Lekhnitskii, S. G. (1963) Theory of Elasticity of an Anisotropic Elastic Body, HoldenDay, Inc. San Francisco. [27] Olsen, J. E. (1999) “Modeling of Edge Rolls Defects,” Web Handling Proceeding. [28] Shelton, J. J. (1993) “Buckling of Webs from Lateral Compressive Forces,” Web Handling Proceeding, 303321. [29] Yuan, F.G., Yang, W. and Kim, H. (2000) “Analysis of Axisymmetricallyloaded Filament Wound Composite Cylindrical Shells,” Composite Structures, 50, 115 130. 8 4 [30] Zabaras, N., Liu, S., Koppuzha, J., And Donaldson, E. (1994) “A Hypoelastic Model for Computing the Stress in CenterWound rolls of Magnetic Tape,” ASME Journal of Applied Mechanics, 61, 290295. [31] Shelton, J. J. (2003) Private Communication on Tension Profiles. [32] Feiertag, B. (2001, March) Web Handling Seminar, Tab 4, Page 46, Oklahoma State University, Stillwater, Oklahoma. [33] Fischer, F. D., Rammerstorfer, F. G., Friedl, N. and Wieser, W. (2000) “Buckling phenomena related to rolling and levelling of sheet metal,” International Journal of Mechanical Sciences, 42, 18871910. [34] Timoshenko, S. P. and Gere, J. M. (1961) Theory of Elastic Stability, 2nd ed., McGrawHall, New York, 348439. [35] Shimizu, S. and Yoshida, S. (1991) “Buckling of Plates with a Hole under Tension,” ThinWalled Structures, 12, 3549. [36] Gilabert, A., Sibillot, P., Sornette, D., Vanneste, C., Maugis, D. and Muttin, F. (1992) “Buckling Instability and Pattern around Holes or Cracks in Thin Plates under a Tension Load,” European Journal of Mechanics, A/Solids, 11(1), 6589. [37] Shaw, D. and Huang, Y. H. (1990) “Buckling Behavior of a Central Cracked Thin Plate under Tension,” Engineering Fracture Mechanics, 35(6), 10191027. [38] Friedl, N., Rammerstorfer, F. G. and Fischer, F. D. (2000) “Buckling of Stretched Strips,” Computer and Structures, 78, 185190. [39] Cerda, E., RaviChandar, K. and Mahadevan, L. (2002) “Wrinkling of an Elastic Sheet under Tension,” NATURE, Vol.419, 579580. 8 5 [40] Tomita, Y. and Shindo, A. (1988) “Onset and Growth of Wrinkles in Thin Square Plates Subjected to Diagonal Tension,” International Journal of Mechanical Sciences, 30(12), 921931. [41] Lu, H., Good, J. K., Yu, H. and Poh, E. (2005, November) “Viscoelastic Effects on the Formation of Baggy Lanes in Webs (02003),” WHRC Project Report, 4 1~442. [42] Kandadai, B. (2006) “The Development of WoundOnTension in Webs Wound into Rolls,” PhD thesis. [43] Ferry, J. D. (1970) Viscoelasitc Properties of Ploymers, 2nd ed., J. Wiley, NY. [44] Swanson, R. P. (1999) “Mechanics of Nonuniform Webs,” Proceeding of the Fifth International Conference on Web Handling, Oklahoma State University, Stillwater, Oklahoma, pp. 443459. [45] Knauss, W.G. and Kenner, V. H. (1980) “On the Hygrothermomechanical Characterization of Polyvinyl Acetate,” Journal of Applied Physics, Vol. 51, No. 10, 51315136. [46] Cole, K. A. and Hakiel, Z. (1992) “A Nonlinear Wound Roll Stress Model Accounting for Widthwise Web Thickness Nonuniformities,” Proceedings of the 1992 Winter Annual meeting of the American Society of Mechanical Engineering, Symposium on Web Handling, AMDVolume 149, pp. 1324. [47] Qualls, W. R. (1995, May) “Hygrothermomechaical Characterization of Viscoelastic Centerwound Rolls,” PhD thesis. [48] Poh, E. (2007, May) “Cambered and Nonflat Polyethylene Webs Induced by Thickness Variation,” Master thesis. 8 6 APPENDIX A: A PSEUDO 3D TRANSIENT VISCOELASTIC WINDING / UNWINDING CODE The pseudo 3D transient viscoelastic winding / unwinding code has been validated by available experimental data in the 3D case. In the case where uniform thickness is used in the 3D code, the model is reduced to the viscoelastic code by Qualls and Good. The manual for the program is provided for users. Purpose: This Pseudo 3D Viscoelastic Winding Code is designed to model winding of a center wound roll with thickness variations on a personal computer. The rolls may have up to 100 different thicknesses across their width (limits are easily changed in the source code). The result files include pressure and radius profiles. This code includes several parts. One part is viscoelastic winding model that has considered viscoelastic effects after winding. This part is revised from the 2D viscoelastic winding code developed by Qualls and Good [1]. It is based on the work of Qualls and Good [1] and Cole and Hakiel [46]. The model considers winding of webs with varying widthdirection thickness. The model discretizes the width into smaller segments of varying length, each having a constant web thickness. Tension is assigned to each segment using the approach by Cole and Hakiel [46]; the tension is updated after the winding of each lap based on the deformed radius of the segment relative to the relaxed radius profile of that lap. In each segment, a 2D model developed by Qualls and Good is applied. The pseudo 3D model is capable of dealing with (1) a varying thickness profile; 8 7 (2) winding tension variation with the winding laps; and (3) varying core stiffness in the width direction. The second part is a pseudo 3D viscoelastic winding model that has considered viscoelastic effects during winding. This part has been developed for an orthotropic, nonlinear viscoelastic web based on the work by Qualls and Good [2]. The pseudo 3D model can consider (1) varying thickness profile; (2) winding tension variation as a function of winding laps; and (3) varying core stiffness in the width direction. The most important feature is that the model allows the consideration of viscoelastic effects in a wound roll during winding at a tension, so that the model is especially suitable for viscoealstic materials with short characteristic relaxation times, such as nonwovens and plastic webs with glass transition temperature close to room temperature. The third part is a pseudo 3D model has considered unwinding of a wound roll. During unwinding process, the viscoelastic memory effects are also considered in this unwinding model. The viscoelastic effects built in this model depend on both viscoelastic properties and winding conditions, such as unwinding speed and tension. In this model, the stress distribution in a roll comes from two sources, one from the initial winding, and the other from a correction state that starts from unwinding. The pseudo 3D viscoelastic winding code has been refined to shorten the computational time and allow for up to 1500 laps to investigate the effect of winding tension variations. The number of laps can be increased based on the capacity of the computer RAM. The improved pseudo 3D model allows an easy handling of tension history in winding and unwinding. If tapered tension option is chosen, winding tension can be changed through the change of the tension taper factor based on the work by 8 8 Shelton [31] and Feiertag [32]. Stress distribution can be determined for ten different tension profiles (ranging from constant tension to constant torque) for a roll with up to 1500 laps. Moreover the enhanced pseudo 3D model has been revised to allow nonpersistent thickness variation. The thickness profile can be different not only in CMD (Cross Machine Direction) but also in MD (Machine Direction). This manual includes a brief description of the underlying theory behind the code, a description of the needed input variables and their units, an example program run, an overview of the output files, and some notes about the program’s performance. Theory: The methodology behind the calculations used in this code is mostly based on the theory of pseudo 3D and viscoelastic winding mechanics. In the pseudo 3D theory the effects of width direction thickness variations in wound rolls are considered. Briefly, at first, we need to find the winding tensions at designated wound roll radii. We next determine the widthwise distribution of tension for all the laps in the wound roll. 8 9 Then the roll is split into independent widthwise segments. The winding tensions is then incorporated into any of the several existing models to compute the wound roll stresses (Qualls and Good model [1]). Qualls and Good model used the general Kelvin model to represent the viscoelastic behavior. A realistic numerical solution was developed to get reasonable results. Using numerical solution in the discretized form, we have ) ( ) 0 2 ) ( ( ) 2 ( ) ) ( ( ) 2 ( ) 2 ( ( ) ( ) 2 2 ( 1) 4 2 2 ( ) 1 2 2 2 ( 1) 3 1 2 1 + % + − % + − % + = + − F r h r F r h r F r h r F r F r h r F r h r F r r i i i r i i r i i i σ σ σ where F1,F2,F3,F4 contain all history effects of the viscoelastic material. Now there are N+1 unknown parameters, and have N1 equations. Plus 2 boundary conditions, we can get the results. When we consider the viscoelastic effects during winding, the current roll radius is a function of time. The relationship follows d(Vol) = 2π
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Title  Threedimensional Viscoelastic Winding Models for the Analysis of Baggy Lanes Formation 
Date  20071201 
Author  Yu, Haowen 
Keywords  Mechanical engineering 
Department  Mechanical Engineering 
Document Type  
Full Text Type  Open Access 
Abstract  A pseudo 3D winding model, with the consideration of viscoelastic effects during winding, storage and unwinding, has been developed and implemented in a Fortran code. The pseudo 3D model is based on the 2D viscoelastic winding model and the associated boundary conditions developed by Qualls and Good, and the tension partition method proposed by Cole and Hakiel. In the model, the thickness variation along the cross machine direction (CMD) has been considered. A web in the CMD direction is discretized into smaller segments (or lanes) of various lengths, each having a constant web thickness. The number and size of segments can be changed to represent the actual thickness profile. Tension is assigned to each segment using the Cole and Hakiel approach, and is updated after winding of each lap based on the deformed radius of the segment relative to the profile of the relaxed radii of that lap. In each segment, a 2D viscoelastic winding/storage/unwinding model is applied.\nThe results from the pseudo 3D viscoelastic winding model have been compared with some winding models, analytical solutions and experimental data. They include (1) the comparison with Lin & Westmann's 2D isotropic viscoelastic windng model; (2) the comparison with the 3D analytical solution; (3) the comparison with Qualls and Good's 2D orthotropic viscoelastic winding model; and (4) the comparison with experimental data. The pseudo 3D viscoelastic winding model has also been used to analyze three problems involving formation of baggy lanes. They are (1) formation of cambered web (inplane imperfection) due to webnonuniformity; (2) formation of localized baggy lanes due to edge burr following slitting; and (3) formation of baggy web (inplane imperfection) due to webnonuniformity. Simulation results are compared with experimental data. FEM simulations using ABAQUS/Explicit code have been conducted to determine the actual boundary condition at the outer layer. The winding and storage processes were simulated to determine the stress and strain in a wound roll and at the outer layer. There are three sets of viscoelastic material parameters used in this FEM simulation to investigate the viscoelastic effects on the outside boundary condition. 
Note  Dissertation 
Rights  © Oklahoma Agricultural and Mechanical Board of Regents 
Transcript  THREEDIMENSIONAL VISCOELASTIC WINDING MODELS FOR THE ANALYSIS OF BAGGY LANES FORMATION By HAOWEN YU Bachelor of Science in Naval Architecture Second Bachelor of Science in Technological Economy Shanghai Jiaotong University Shanghai, China 1998 Master of Science in Naval Architecture Shanghai Jiaotong University Shanghai, China 2001 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY December, 2007 i i THREEDIMENSIONAL VISCOELASTIC WINDING MODELS FOR THE ANALYSIS OF BAGGY LANES FORMATION Dissertation Approved: Dr. Hongbing Lu Dissertation Adviser Dr. J. Keith Good Dr. Demir Coker Dr. G. Steven Gipson Dr. A. Gordon Emslie Dean of the Graduate College ii i ACKNOWLEDGEMENT I would like to express my sincere appreciation to my advisor Dr. Hongbing Lu for his guidance and support. He offered his academic expertise and insightful knowledge all the way through completion of this dissertation. I appreciate his understanding, patience, and affable manner of communication, which made the tense research process less stressful. I also thank the other three committee members, Dr. James K. Good, Dr. Demir Coker, and Dr. G. Steven Gipson, for their thoughtful comments, suggestions, and the valuable time they spent on reviewing my documents. I have to give my special thanks to Dr. Gipson, who actively participated in every stage of this process despite his illness. I sincerely wish him defeat the disease and recover at the earliest date. As always, my husband Huawen Xu has provided the much needed understanding and encouragement. His love presents me the courage to face the difficulties in daily life. I also thank my parents for their love and the selfless support. I am also grateful for the help I got from all of the research group members and my friends. I wish them all successful in their future endeavor. Thanks also to Department of Mechanical and Aerospace Engineering and Web Handling Research Center for their direction and financial support during this research. i v TABLE OF CONTENTS Chapter Page 1 INTRODUCTION........................................................................................................... 1 2 LITERATURE REVIEW................................................................................................ 7 2.1 Elastic Analysis ....................................................................................................... 7 2.1.1 Anisotropic Linear Elastic Winding Model ....................................................... 8 2.1.2 Anisotropic Nonlinear Elastic Winding Model ............................................... 11 2.2 Viscoelastic Analysis ............................................................................................. 12 2.2.1 Isotropic Linear Viscoelastic Winding Model ................................................. 13 2.2.2 Orthotropic Nonlinear Viscoelastic Winding Model ....................................... 15 2.3 Pseudo 3D Model and 3D Finite Element Model ............................................... 17 2.3.1 Pseudo 3D Winding Model.............................................................................. 18 2.3.2 3D FEM Elastic Winding Models.................................................................... 20 3 PSEUDO 3D VISCOELASTIC WINDING/UNWINDING MODEL ........................ 22 3.1 Pseudo 3D Winding Model................................................................................... 24 3.2 Viscoelastic Effects during Winding ................................................................... 28 3.3 Viscoelastic Effects after Winidng....................................................................... 32 3.4 Viscoelastic Effects during Unwinding ............................................................... 33 3.5 Finite Element Winding Simulations for Checking Outer Boundary Condition and Effects of Asymmetric Structure................................................. 35 3.5.1 Evaluation of Boundary Conditions using Dynamic Analysis via ABAQUS / Explicit .................................................................................................. 36 3.5.2 2D Viscoelastic FEM Winding Model ............................................................ 37 v Chapter Page 3.5.3 Results and Discussion .................................................................................... 41 4 RESULTS FROM 3D VISCOELASTIC WINDING/UNWINDING MODEL.......... 48 4.1 Comparison of Results from Pseudo 3D Viscoelastic Model and Lin & Westmann’s 2D Viscoelastic Model ..................................................................... 50 4.2 Comparison of Results from Pseudo 3D Viscoelastic Model and the 3D Analytical Solution................................................................................................. 51 4.3 Comparison of Results from Qualls and Good’s 2D Viscoelastic Model......... 53 4.4 Comparison of Results from Pseudo 3D Viscoelastic Model and Experimental Data................................................................................................. 55 4.5 Special Case One: Formation of a Cambered Web........................................... 58 4.6 Discussion about Hakiel’s Mode l [2] and Cole and Hakiel’s Model [46] ....... 61 4.7 Special Case Two: Formation of Wavy Edge..................................................... 63 4.8 Special Case Three: Formation of Baggy Web .................................................. 66 4.9 Consideration of Thickness Variation in Both CMD and MD......................... 68 4.10 Consideration of Winding and Unwinding....................................................... 71 5 CONCLUSIONS ........................................................................................................... 73 6 FUTURE WORK........................................................................................................... 76 6.1 Hygrothermal Effects on Viscoelastic Material Properties .............................. 76 6.2 Dimensionless Study ............................................................................................. 79 6.3 Optimization of Tension History......................................................................... 80 6.4 Experimental Validations..................................................................................... 80 REFERENCES................................................................................................................ 81 APPENDIX A: A PSEUDO 3D TRANSIENT VISCOELASTIC WINDING / UNWINDING CODE ...................................................................................................... 86 v i LIST OF TABLES Table Page Table 31: Viscoelastic material parameters ..................................................................... 40 Table 41: Material properties used in pseudo 3D winding model ................................... 50 Table 42: Deviation of deformed length.......................................................................... 71 v ii LIST OF FIGURES Figure Page Figure 11: A schematic diagram of a laboratory winder ................................................... 1 Figure 12: A wound roll in threedimensional situation.................................................... 2 Figure 13: Baggy lanes examples. Left: cambered web; Right: baggy edge in circled area............................................................................................................... 3 Figure 21: Exaggerated view of a wound roll with thickness variation .......................... 18 Figure 31: Pseudo 3D models.......................................................................................... 24 Figure 32: Exaggerated view of a wound roll with thickness variation, Hakiel [2]........ 25 Figure 33: Outer lap geometry, Cole [46]........................................................................ 27 Figure 34: Viscoelastic effects during winding ............................................................... 31 Figure 35: An arbitrary external force P(t) ...................................................................... 34 Figure 36: Flow chart of algorithm for viscoelastic unwinding ...................................... 35 Figure 37: Schematic of the 2D FEM winding model..................................................... 38 Figure 38: Schematic of 2D FEM storage model ............................................................ 38 Figure 39: Definition of the contact in 2D FEM winding model .................................... 39 Figure 310: Definition of the contact in 2D FEM storage model.................................... 39 Figure 311: Three sets of viscoelastic material properties .............................................. 41 Figure 312: The outer layer timedependent circumferential strain for Set 1 ................. 42 Figure 313: The outer layer timedependent circumferential stress for Set 1 ................. 42 Figure 314: The outer layer timedependent circumferential strain for Set 2 ................. 43 Figure 315: The outer layer timedependent circumferential stress for Set 2 ................. 43 vi ii Figure Page Figure 316: The outer layer timedependent circumferential stain for Set 3................... 44 Figure 317: The outer layer timedependent circumferential stress for Set 3 ................. 44 Figure 318: Circumferential stresses in layer 1 to layer 10............................................. 47 Figure 41: Comparison with the results of Lin and Westmann [10]................................ 51 Figure 42: Comparison with the results of 3D analytical solution.................................. 53 Figure 43: Comparison with the results of Qualls and Good’s model, v=1000 in/s ....... 54 Figure 44: Comparison with the results of Qualls and Good’s model, v=30 fpm........... 55 Figure 45: Schematic diagram for inserting a triangular web for every four plies for use in the investigation of formation of a cambered web .................................... 55 Figure 46: The experimental data for both edges ............................................................ 56 Figure 47: Comparison with the experimental data......................................................... 58 Figure 48: The length of deformed web at ten locations................................................. 59 Figure 49: Numerical transformation for cambered web ................................................ 60 Figure 410: Maximum bow length .................................................................................. 60 Figure 411: Deformed shape after storage....................................................................... 61 Figure 412: Comparison of the pressure distributions..................................................... 62 Figure 413: Comparison of deformed shapes after storage............................................. 63 Figure 414: Formation of wavy edge .............................................................................. 64 Figure 415: The length of deformed web at ten locations............................................... 64 Figure 416: Numerical transformation for wavy edge .................................................... 65 Figure 417: Formation of baggy web .............................................................................. 66 Figure 418: The lengths of deformed web at forty locations .......................................... 67 Figure 419: Numerical transformation for baggy web. ................................................... 68 i x Figure Page Figure 420: The thickness profiles with 50% deviation of average thickness for layer #1 and Location #1...................................................................................... 70 Figure 421: The whole thickness profile ......................................................................... 70 Figure 422: Deformed length for 10%, 20%, 50% and 80% cases ................................. 71 Figure 423: Comparison of stresses during winding and unwinding .............................. 72 Figure 61: Comparison of master creep curves for PVAc [45] ....................................... 76 Figure 62: (a) Creep curves for dry PVAc at several temperature; (b) Creep curves for PVAc at c o 24 at several absorbed moisture levels.................................... 77 x ABSTRACT A pseudo 3D winding model, with the consideration of viscoelastic effects during winding, storage and unwinding, has been developed and implemented in a Fortran code. The pseudo 3D model is based on the 2D viscoelastic winding model and the associated boundary conditions developed by Qualls and Good, and the tension partition method proposed by Cole and Hakiel. In the model, the thickness variation along the cross machine direction (CMD) has been considered. A web in the CMD direction is discretized into smaller segments (or lanes) of various lengths, each having a constant web thickness. The number and size of segments can be changed to represent the actual thickness profile. Tension is assigned to each segment using the Cole and Hakiel approach, and is updated after winding of each lap based on the deformed radius of the segment relative to the profile of the relaxed radii of that lap. In each segment, a 2D viscoelastic winding/storage/unwinding model is applied. The results from the pseudo 3D viscoelastic winding model have been compared with some winding models, analytical solutions and experimental data. They include (1) the comparison with Lin & Westmann’s 2D isotropic viscoelastic windng model; (2) the comparison with the 3D analytical solution; (3) the comparison with Qualls and Good’s 2D orthotropic viscoelastic winding model; and (4) the comparison with experimental data. x i The pseudo 3D viscoelastic winding model has also been used to analyze three problems involving formation of baggy lanes. They are (1) formation of cambered web (inplane imperfection) due to webnonuniformity; (2) formation of localized baggy lanes due to edge burr following slitting; and (3) formation of baggy web (inplane imperfection) due to webnonuniformity. Simulation results are compared with experimental data. FEM simulations using ABAQUS/Explicit code have been conducted to determine the actual boundary condition at the outer layer. The winding and storage processes were simulated to determine the stress and strain in a wound roll and at the outer layer. There are three sets of viscoelastic material parameters used in this FEM simulation to investigate the viscoelastic effects on the outside boundary condition. 1 CHAPTER 1 1 INTRODUCTION Web handling is described as the engineering science underlying the transport of webs through processes successfully without incurring material defects and losses [3]. Figure 11 shows a schematic diagram of a laboratory winder used in this investigation. 1, 2 7 6 5 4 3 8 Figure 11: A schematic diagram of a laboratory winder In industry, paper towels, toilet tissue and photographic film are all considered as webs in the manufacturing process. A web is a thin continuous sheet or strip, such as plastic films, paper, textiles, metal sheets, composite prepregs, foils, wovens, and nonwoven materials. Webs are often stored in the form of wound rolls, convenient for ransporting and storing and widely used in most industries. A roller is a revolving cylinder over which a web or substrate is moved for the purpose of transporting, pressing, shaping, or shaping, or smoothing a web or substrate. The process of wrapping a continuous flexible web onto a roller (core) to form a wound roll is called winding. 1. Unwinding roller 2. Brake 3. Web 4. Idle roller 5. Tension measuring roller 6. Idle roller 7. Idle roller 8. Winding roller 2 Figure 12 shows schematic diagram of a 3D wound roll. Core Web Figure 12: A wound roll in threedimensional situation Generally, it is difficult to obtain the analytical solution for the stress, strain and displacement fields because of the complexity of 3D geometry. Numerous efforts have been made to determine the stress distribution in a 3D wound roll. Altmann [5] and Hakiel [6] et al. simplified this real 3D winding situation into a 2D winding model that can be solved using solutions for an axisymmetric hollow cylinder. For the investigation of baggy lanes, however, the realistic winding structures have to be considered to take into account of the effects of varying web thickness in the CMD on the stress / strain distribution. Consequently, it is necessary to develop a 3D winding model to deal with realistic 3D winding issues. Usually, wound roll defects are formed in the processes of manufacture, transportation and storage. It is important to develop methods to reduce product defects and losses through fundamental understanding of the mechanisms of defect formation. There are various kinds of defects in wound rolls; these usually include wrinkles, tears, baggy lanes etc. All these defects could lead to damage in web, causing poor appearance and consequently poor quality of the wound roll. The research in this report will focus on 3 developing (pseudo) 3D viscoelastic winding models and 3D FEM winding models for purpose of understanding the baggy lanes formation in viscoelastic webs. A baggy lane is a defect commonly observed in a wound roll and has many aliases such as camber, layflat and puckers. It can be found in most webs as diverse as tissue, printing paper, carpet, nonwovens, plastic film, and sheet metals [4]. An example of baggy lane formation in the web is shown in Figure 13. Figure 13: Baggy lanes examples. Left: cambered web; Right: baggy edge in circled area When baggy lanes occur, the visual appearance of a web is no longer flat or straight, causing some poor effects. For instance, the baggy portion may float over rollers causing processing difficulties and may not wind in the same way as the tight portions. Deformation in webs may remain, evolve and deteriorate and the losses may be unpredictable. Thus, it is very important to investigate the mechanisms of baggy lane formation and to develop methods to reduce or eliminate it during winding, unwinding or storage processes. Baggy lanes are expected to be associated with the existence of compression or shear zones, and with variations of stress and strain. Generally, baggy 4 lanes could occur at locations where there is high gradient of stresses/strains in the cross machine direction (CMD). Therefore, in this report, winding models will be developed to analyze the stress distribution in wound rolls. Later efforts will include the development of a criterion for predicting baggy lanes formation. Theoretical, numerical and experimental investigations will be carried out in the development of 3D viscoelastic winding models. Theoretical studies will be focused on the analysis of stress and strain distributions in wound rolls. Through analyzing stress and strain fields, the mechanism of defect formation in the web will be revealed. So far, some winding models have been developed to determine stress and stain distributions; these will be introduced in detail in the following chapters. It may be noted that theoretical models for determining stress and stain distributions need to be validated by experimental results. In experimental studies, the winding process needs to be conducted until baggy lanes occur in the web. To investigate the mechanical behavior of the web in winding, it is essential to determine the material properties of the web, such as the creep compliance for a viscoelastic material. Another essential aspect in experiments is to find appropriate methodology and equipment to measure the stress and strain distributions in a wound roll, especially in the CMD direction. Some theoretical winding models have been developed to determine stress and strain distributions in a wound roll. In general, the existing winding models can be categorized into 2D models and 3D models. Each category can further be classified into elastic models and viscoelstic models. The first rigorous analysis of a 2D elastic wound roll was given by Altmann [5]. The web material was assumed to be homogeneous, anisotropic, and linear elastic in this model. Formulas were derived for the calculation of the inroll 5 stresses and can be easily implemented for numerical computation. Hakiel [6] extended Altmann’s model and presented a numerical method employing a nonlinear radial stiffness. In Hakiel’s model, the web was assumed to be an orthotropic elastic material with linear elastic properties in the circumferential direction and nonlinear elastic properties in the radial direction. On 2D viscoelastic winding models, Qualls and Good [1] developed a realistic and adaptive viscoelastic model for the prediction of transient stress distributions in a wound roll. In this model, the generalized Kelvin model was used to represent viscoelastic behavior of materials. Numerical results have a very good agreement with analytical solutions for an isotropic linear viscoelastic 2D winding problem. Several 3D winding models have been proposed, these include winding models developed by Hakiel [2], Cole and Hakiel [46], Kedl [7], Lee and Wickert [8], and Hoffecker [9]. In order to consider the effects of thickness variations along the width direction on wound rolls, Hakiel, Cole & Hakiel and Kedl proposed pseudo 3D models for a wound roll. In their models, it was assumed that the wound roll can be split into multiple independent widthwise segments and in each segment wound roll stresses can be computed separately through incorporating the winding tensions into the existing 2D model. Lee and Wickert, and Hoffecker found their ways to determine the stress distributions in a wound roll using 3D FEM. It was assumed that a wound roll has an axisymmetric geometry and stress distributions are only functions of radius and are independent of angle. Winding is an accretion problem, in which the total stress in a wound roll is equal to the summation of all the stress increments, as the radius increases from inner layer to outer layer. 6 Although some theoretical winding models have been developed in the past several decades, there are still unresolved problems associated with the winding models. For example, the viscoelastic model for a wound roll has not been fully developed, especially during winding and unwinding process. In the model developed by Qualls and Good, nonlinear elastic behavior is considered during winding and viscoelastic behavior is considered after winding. Lin and Westmann [10] considered viscoelastic winding during winding, storage and unwinding. However, this model was limited to isotropic linear viscoelastic materials. The FEM has become a useful tool to compute the stress distributions. However, it is not very easy to deal with winding problems using the commercial FEM codes. Since a web usually has thickness variations in both the MD and CMD, there might be gaps between two layers. If the gap is considered and the finite elements are assigned for each layer, the number of finite elements will be too large to handle when the number of layers becomes very large. Moreover, it is difficult to implement the pressure dependent radial modulus of the wound roll into commercial FEM code. This difficulty will impose restrictions on the application of FEM in winding simulations; thus it is necessary to explore some other methods to determine the stress distribution in a wound roll. It may be noted that most existing pseudo 3D models and 3D FEM models are for elastic materials only. It is necessary to extend previous elastic pseudo 3D and 3D FEM models to consider viscoelastic behavior. In conclusion from the above review, 3D elastic and viscoelastic winding models need to be developed for the identification of baggy lanes. A criterion needs to be established to predict baggy lanes formation. 7 CHAPTER 2 2 LITERATURE REVIEW In this chapter, several major winding models will be reviewed and discussed. These models will be divided into several sections and presented separately. In the section of elastic analysis and viscoelastic analysis, we will introduce only 2D elastic and viscoelastic winding models. Pseudo 3D models and 3D FEM models for elastic material will be introduced separately in Section 2.3. 2.1 Elastic Analysis Elastic winding model may be the most fullygrown model of a wound roll. Detailed analysis of stresses in an elastic wound roll was traced to the work of Gutterman [11] in 1959. Altman [5] followed his work and presented an analytical solution to the elastic winding problems including complete derivation, assumption and detailed formula. Then the elastic model was extended by Pfeiffer [12, 13] and Yogada [14] by considering nonlinear radial stiffness. In 1987, Hakiel [6] developed a practical numerical method incorporating advantages in the previous work. Moreover, in Hakiel’s model the nonlinear radial stiffness, orthotropic material properties have been considered. In this section, two elastic models developed by Atmann and Hakiel will be reviewed. 8 2.1.1 Anisotropic Linear Elastic Winding Model In the anisotropic linear elastic winding model [5], Altmann presented detailed stress analysis for the wound roll of anisotropic linear elastic material. Some assumptions were made in the derivation and formulas were presented for the computation of stress distribution in a wound roll. These formulas can be evaluated easily on a computer. However, in some cases, when these assumptions are violated, the results will not be correct. The following assumptions were made in the work of Altmann: 1. The core is geometrically cylindrical and remains a cylinder during and after winding. 2. The web is uniform in thickness and its thickness is small compared to its width, and therefore offers no resistance to bending. 3. During and following the winding, the roll may be considered to be a homogeneous cylinder with anisotropic properties r E , Eθ , r μ , μθ all considered to be constant throughout the roll. 4. The stresses in the roll are only functions of roll radius, not of roll angle, because of its axisymmetric structure. 5. No nonelastic displacements and betweenlayer motions occur during winding. Based on the above assumptions, the stresses, strains, and displacements are only the functions of roll radius. Equilibrium equation can be expressed as + − = 0 ∂ ∂ θ σ σ σ r r r r (2.1) where r is the radius ratio ( r = r / c ), r is the radius to a point in the roll, c is the radius of the core, r σ is the radial stress, σθ is the circumferential stress. 9 Straindisplacement relations are r u r u r ε = ∂ ∂ ε = , θ (2.2) where r ε is radial strain, εθ is circumferential strain, u is dimensionless radial deformation ( u = u / c ),u is the radial deformation. The stressstrain relations under the plane stress condition are θ μθ ⋅σθ − σ ε = E E r r r r r r E E μ ⋅σ − σ ε = θ θ θ (2.3) where r E is radial modulus, Eθ is circumferential modulus, r μ and μθ are Poisson’s ratio. Solving Equations (2.2, 2.3) leads to + μ ∂ ∂ − μ ⋅μ = ε μ + ε − μ ⋅μ σ = θ θ θ θ θ r u r E E u r r r r r r 1 ( ) 1 + ∂ ∂ μ − μ ⋅μ = ε + ε ⋅ μ − μ ⋅μ σ = θ θ θ θ θ θ r u r E E u r r r r r 1 ( ) 1 (2.4) Consider that u = Arα + Br −β (2.5) 1. Boundary conditions at the core. when r = 1, A B E u c = r = + (1) (1) σ (2.6) where [ A B] E r r r ( ) ( ) 1 ) 1 ( β − μ + α + μ − μ ⋅μ σ = θ θ θ 2. Boundary conditions at outer radius of the wound roll. when r = s , [ ] ds s T s As Bs E s w r r r − = μ + α + μ − β − μ ⋅μ σ = −β θ α θ θ 1 ( ) ( ) 1 ( ) (2.7) 1 0 Then A, B and θ σ ,σ r can be determined from Equations (2.4~2.7). As the number of layer increases, the stress change will be ds s T as s r ar dP w b r b )( ) 1 ) ( 1 ( 2 2 − γ − γ + × + = −σ = ds s T as s r a r dT w b b )( ) 1 ( ) ( 2 2 γ γ θ α β σ − − + × − ⋅ = = − (2.8) where: c c e e a γ + μ + γ − μ − = r γ = δ2 + e r r e = Eθ / E c c e = Eθ / E b = 1−α α =γ −δ β = γ + δ The isotropic component of Poisson’s ratio is ( ) 2 1 r r μ = μθ + e μ The anisotropic component of Poisson’s ratio can be written as ( ) 2 1 r r δ = μθ − e μ The final stresses P and T are therefore given by ds s T as s r ar P dP w R r b b )( ) 1 ) ( 1 ( 2 2 ∫ ∫ − − + × + = = γ γ (2.9) ds s T as s r a r T T dT T w R r b w w b )( ) 1 ( ) ( 2 2 ∫ ∫ − − + × − ⋅ = + = − γ α β γ (2.10) 1 1 where R is dimensionless outside radius of the roll ( R = R / c ), R is the outside radius of roll. These formulas can be evaluated on a computer. The computational results have shown to agree well with some experimental data. These formulas are limited to linear elastic materials. Nonlinear response has been found to exist in a wound roll. For improved accuracy, it is necessary to investigate nonlinear effects in winding. 2.1.2 Anisotropic Nonlinear Elastic Winding Model Hakiel developed a nonlinear elastic winding model [6].The most important attribute of Hakiel’s model is to consider nonlinear material behavior in radial direction. Hakiel’s work extended the previous analysis by considering radial modulus as a function of pressure, instead of a constant as used in the previous models. A numerical solution was developed to solve the boundary value problem. For a roll with N laps, a set of N+ 1 linear algebraic equation will be solved to obtain the stresses in the web. This numerical solution can also be evaluated conveniently on a computer. The equations used in the development of Hakiel’s model are summarized below. Hakiel defined t r g E / E 2 ≡ (or r t E g E ≡ 2 ) (2.11) 1. Nonlinear model The radial modulus was assumed to depend on the radial stress. Then the radial modulus is a function of radial stress. E E (P) r r = (2.12) 2. A numerical solution to the boundary value problem For a roll with N laps, the boundary condition can be presented by 1 2 0 1 1 + + = i+ i i i i− i δP A δPB δP C (i=2,3…N) (2.13) where i δP is the pressure at ri caused by the winding on of lap N+1. Eq (2.13) can be rewritten as 0 1 1 + + = N+ N N N N− N δP A δP B δP C 0 1 1 1 2 1 + + = N N− N− N− N − N− δP A δP B δP C … … 0 3 2 2 2 1 2 δP A +δP B +δPC = (2.14) The N1 equations above contain N+1 undetermined coefficients. These constants can be determined using boundary conditions on the inner and outer surfaces. On the outer surface, h r T P N W N N 1 , 1 1 + + + δ = , (2.15) where W,N+1 T is the winding tension stress for Lap (N+1). On the inner surface, 2 1 1 ( P P ) / h [(E / E ) 1 ] P t c δ −δ = − +ν δ (2.16) Hakiel’s model has been validated by experimental data and verified by comparing results obtained from some of the previous models. Nevertheless, Hakiel’s model does not consider viscoelastic effects. Thus, viscoelastic analysis needs to be carried out for materials with pronounced viscoelastic effects. 2.2 Viscoelastic Analysis Transient analysis of wound roll was first introduced by Tramposch in 1965 [15,16]. The web material was considered to be an isotropic viscoelastic material in his work. Lin and Westmann [10] extended Tramposch’s work in viscoelastic winding mechanics by taking into account of histories for winding, windingpause, and windingpauseunwinding. Qualls and Good [1, 17] developed a solution for viscoelastic analysis that 1 3 has considered viscoelastic effects after winding and the thermal influence on viscoelastic winding. The material in the model by Qualls and Good is orthotropic with a nonlinear elastic radial stiffness depending upon interlayer pressure. In this section, the two viscoelastic models developed by Lin and Westmann, and Qualls and Good will be reviewed. 2.2.1 Isotropic Linear Viscoelastic Winding Model A viscoelastic winding model for isotropic linear material was presented by Lin and Westmann [10]. In that paper, Lin and Westmann considered viscoelastic response during winding, i.e. winding velocity can be considered in viscoelastic analysis. Moreover, a new method was introduced to solve viscoelastic unwinding problems. It was assumed that viscoelastic unwinding could be considered by the superposition of two states, an initial state and a correction state. The initial state is determined from the initial condition when unwinding begins and the correction state is determined after the initial state so that the superposition of the initial and correction states satisfies the traction free boundary condition at the current unwinding radius. However, the nonlinear radial stiffness was not considered in this model. Moreover, this model is limited to isotropic material. Actual web properties are orthotropic and the radial modulus is nonlinear. Thus considering such features will provide more realistic predictions for stresses in a wound roll. In this section, an outline of Lin and Westmann’s work will be presented. At first, in the same way as used in other elastic isotropic models, all the stresses, strains and displacements distributions are determined in cylindrical coordinates through equilibrium, stressstrain and straindisplacement equations. The solution procedure for 1 4 the viscoelastic case parallels the method used for the elastic winding problem. The solution to the viscoelasic winding problem was obtained by employing elasticviscoelastic corresponding principle by way of Laplace transform. In order to consider the viscoelastic effects during winding, the current roll radius will depend on winding speed and time passed. Two special cases were considered: R v th 0 2 π ( −1) = at constant linear velocity; (2.17) R th 0 2π ( −1) =ω at constant angular velocity, (2.18) And numerical examples were presented for a viscoelastic web represented by the generalized Kelvin model. The creep compliance of materials function was: n t N n n e E E J t /τ 0 1 1 1 ( ) − = Σ = − (2.19) For viscoelastic unwinding, the system can be treated by superposing two timedependent states, an initial state and a correction state. At current unwinding radius, the superposition of the initial and correction states should satisfy the traction free boundary condition, which can be expressed as: ) 0 ~ ( , , ) ( , , ) ( , , 1 0 1 0 1 0 R R t = R R t + R R t = c r i r r σ σ σ (2.20) where, R1 is the current unwinding radius, R0 is the fixed outer radius, which is assumed to be constant during unwinding, t is the time starting from the initial winding, t ~ is the time starting from the unwinding. This traction free boundary condition is used to determine the arbitrary external pressure history P (t ~ ) at the correction state. Finally, stresses and displacement at any layer can be obtained as follows c r i r r σ =σ +σ , i c θ θ θ σ =σ +σ , i c u = u + u (2.21) 1 5 The analysis presented by Lin and Westmann provides a methodology for analyzing isotropic linear viscoelastic winding problems. Especially, the unwinding case has been discussed and the stress distribution has been determined. Numerical results for several examples have been given in that paper. However, this model is for isotropic linear viscoelastic material. Many actual web materials are orthotropic with a nonlinear radial stiffness. Incorporation of these features will provide more realistic predictions of the stress distribution in the wound roll. 2.2.2 Orthotropic Nonlinear Viscoelastic Winding Model Qualls and Good [1] developed a viscoelastic winding model for orthotropic nonlinear viscoelastic webs. The modeling results were found to be in a good agreement with some other existing winding models. The generalized Kelvin model was used to represent the viscoelastic function in this orthotropic nonlinear viscoelastic winding model. A numerical solution has been developed to predict transient stress profile in wound rolls composed of orthotropic viscoelastic web with nonlinear radial modulus. In this section, the derivation of the solution will be reviewed briefly. Equilibrium equation is expressed as + − = 0 ∂ ∂ θ σ σ σ r r r r (2.22) Strain compatibility is written as + − = 0 ∂ ∂ r r r ε ε ε θ θ (2.23) Constitutive equations for an orthotropic viscoelastic material are expressed by dt t J t t t J t t t r r r r ′ ∂ ′ ∂ + − ′ ∂ ′ ∂ = ∫ − ′ 0 ( ) ( ) θ θ σ σ ε 1 6 dt t J t t t J t t t r r ′ ∂ ′ ∂ + − ′ ∂ ′ ∂ = ∫ − ′ 0 ( ) ( ) σ σ ε θ θ θ θ (2.24) The above equations lead to [ ( ) ( ) {3 ( ) ( ) ( ) 2 2 2 0 J t t J t t J t t r r t J t t r r r t + − ′ + − ′ − − ′ ∂ ∂ ∂ ′ ′ ∂ ∫ θ − θ θ θ σ ( )} ( ) { (J (t t ) J (t t )) J (t t ) r r r r t J t t r r r r − ′ + − ′ + − ′ ∂ ∂ + ∂ ∂ ∂ ′ ′ ∂ − ∂ ∂ + θ θ θ θ σ ( ) ( ) ( )} ] ′ = 0 ∂ ′ ′ ∂ + − ′ − − ′ − − dt t J t t J t t J t t r r r r σ θ θ (2.25) The generalized Kelvin model is used to represent the viscoelastic behavior. i t t N i iJ t t J J e ( ) /τ 1 0 ( ) − − ′ = Σ − ′ = + (2.26) Equation (2.25) can be discretized to the following form ) ( ) 0 2 ) ( ( ) 2 ( ) ) ( ( ) 2 ( ) 2 ( ( ) ( ) 2 ( 1) 4 2 2 2 ( ) 1 2 2 2 ( 1) 3 1 2 1 + % + + − % + − % − + F r = h r F r h r F r h r F r F r h r F r h r F r r i i i r i i r i i i σ σ σ (2.27) There are N+1 unknown parameters in N1 equations. After two boundary conditions are applied, all parameters can be determined by solving Equation (2.27) with two boundary conditions. The inner boundary condition is given as: j c r j E ( ) ( ) θ ε σ = (2.28) where Ec is the core stiffness, defined as the radial pressure divided by hoop strain at the outer diameter of the core. When t=tj, the inner boundary condition can be expressed as 1 7 dt t J t t t J t t r r t J t t E r r r r t c r j r j ′ ∂ ′ ∂ + − ′ ∂ ′ ∂ + − ′ ∂ ∂ ∂ ′ ′ ∂ = − + % ∫ − [ ( ) ( ) ( ) ( ) ] ( ) ( ) 0 1 σ σ σ σ σ θ θ θ (2.29) The boundary condition on the outer surface of the roll is dt t J t t t T J J t t r t w ′ ∂ ′ ∂ + − ′ ∂ ′ ∂ (0) = ∫ [ ( − ′) ( ) ] 0 θ θ θ θ θ σ σ (2.30) With the change in circumferential stress, the corresponding change in radial stress beneath the outer layer can be determined by employing the following condition. 0 ( ) ( ) r h j r j θ σ σ % % = (2.31) This viscoelastic winding model has considered viscoelastic effects after winding. Results from this model agree well with those from some other models and experimental data. The model, however, did not consider viscoelastic effects during winding. Also unwinding was not considered in the model. This model is a 2D winding model and cannot be used to predict the stress profile along CMD. A 3D winding model is needed to consider the variations in CMD direction. 2.3 Pseudo 3D Model and 3D Finite Element Model Some pseudo 3D models and 3D FEM models have been developed by Hakiel [2], Cole and Hakiel [46], Kedl [7], Hoffecker [9], and Lee and Wickert [8] to predict the stresses in both radial and crossweb (transverse) direction. Hakiel assumed that the roll could be partitioned across its width into small strips or segments and stresses or displacements in each segment can be calculated separately through a twodimensional analysis. Cole and Hakiel refined the Hakiel’s model and considered the effect of radial displacement on tension distribution. Kedl made a similar assumption as Hakiel and Cole 1 8 & Hakiel’s, dividing the roll into an arbitrary number of crossweb segments and using a twodimensional winding model to predict wound roll stresses in each segments. Hoffecker, and Lee and Wickert have developed axisymmetric FEM models to predict the widthwise variation of stresses in wound rolls and have compared the results with those obtained from previous models. 2.3.1 Pseudo 3D Winding Model The pseudo 3D winding model developed by Hakiel [2] will be introduced in this section. This model can be used to predict the widthwise variability in the wound roll. In the model the effect of thickness variations along the width direction is considered. However this model is not suitable for highly compressible material. And there are still some problems remaining in dealing with the deformation compatibility between neighboring segments. Figure 21: Exaggerated view of a wound roll with thickness variation ρ (i, j) : widthwise distribution of radius to the outside surface of lap i. r(i, j) : widthwise distribution of radius to the inside surface of lap i. i: the lap number within the roll j: widthwise position 1 9 ( ) 0 R i is relaxation radius for lap i. It is assumed that the lap would be stress free at relaxation radius. Firstly, the initial value of ( ) 0 R i will be estimated. The circumferential stress can be written as − − ν σω = ( ) ( , ) ( ) 1 ( , ) 0 0 2 R i E r i j R i i j (2.32) Then, we have the circumferential stress, as given by } ( ) ( , ) ( ) { (1 ) ( ) { ( , ) } 0 0 1 2 1 R i r i j R i M Ebh M bh T i i j M j M j − − =Σ = Σ − − ν σω ω (2.33) In order to find the relaxation radius, let the predicted winding tension be equal to the actual tension force in the outer lap. It is an iteration process to find the relaxation radius. T (i) T (i) a = ω (2.34) Widthwise distribution of radius to the outside surface of lap is ρ(i, j) = r(i, j) + h( j); j = 1,...M (2.35) The inner radius of the lap being wound on is r(i, j) Max{ (i 1, j),R (i); j 1,...M} 0 = ρ − = (2.36) Based on Equations (2.33~2.36), the predicted widthwise distribution of radius and tension can be obtained. After that, the roll could be discretized into independent widthwise segments and the wound roll stresses could be determined in each segments by using any of existing models to compute the wound roll stresses. However, problems exist in dealing with the deformation compatibility between neighboring segments. 3D winding model is needed to solve this problem to provide more accurate computation of stresses in a wound roll. 2 0 2.3.2 3D FEM Elastic Winding Models Two 3D FEM elastic winding models have been presented by Lee and Wickert [8], and Hoffecker [9] separately. These two FEM models are somewhat similar and their results agree. The FEM winding model developed by Lee and Wickert focuses on various core shape, while the FEM model developed by Hoffecker pays attention on the thickness variation in CMD. The basic formulation in 3D FEM model is similar to other winding models. In all these winding models, the stress state within a wound roll with N laps is the superposition of stress states resulting from the addition of each additional layer from 1 to N. The wound roll is considered to be composed of N sub rolls or substructures. The stress state of each subroll consists of the result from the addition of that subroll’s outermost layer. The total stress of a single layer in the actual wound roll is found by superimposing the stress state corresponding to that layer from all subrolls. In these two axisymmetric FEM models, fournode rectangular elements were used. These finite elements can be obtained through discretizing the wound roll along both the radius and cross machine directions. Since the stress state in the wound roll is the result of superposition of stress states of substructures, it is very difficult for commercial FEM software to determine the stress distributions in webs. In these two 3D FEM models mentioned previously, special FEM code has been developed to solve this problem and the numerical results have been obtained. In LinWickert model, the widthwise variation of stresses in a wound roll is investigated by using a 3D axisymmetric finite element model. The results obtained for hollow core and cupshaped core show the effects of different core shapes. In the two 2 1 examples given for hollow core and cupshaped core, the widthwise variation of stresses happens only near the interface of the core and the web. In the 3D FEM code developed by Hoffecker [9], the core and web in the wound roll are represented by axisymmetric finite elements and FEM code was developed. The web can have up to 30 different thickness regions (referred to as segments) across its width. The output files give stresses (including radial pressure, and circumferential tension) and displacement or radius profiles. This model can capture the effect of varying tensions and thickness profiles. There are still remaining issues in using FEM to find the stress distribution in such a special structure like a wound roll, when the air entrainment and gaps between two layers are considered. Moreover, as the lap number increases, the number of elements that can be simulated is restricted by computing capability. The primary objective of this proposal is to develop methodologies for winding viscoelastic webs to produce rolls that have dimension stability to minimize the formation of baggy lanes. Research will focus on the understanding of baggy lane formation mechanism in viscoelastic webs with a varying thickness profile along the width direction. The thesis is organized as follows: (1) a pseudo 3D viscoelastic winding model will be developed based on the work by Qualls and Good [1], and Cole and Hakiel [46]; (2) commercial FEM codes ABAQUS, will be used to generate the FEM model to find the appropriate boundary condition on the outer layer; (3) experimental investigation will be conducted to examine the baggy lane formation and experimental data will be used to examine results from the models developed in this research. 2 2 CHAPTER 3 3 PSEUDO 3D VISCOELASTIC WINDING/UNWINDING MODEL In this investigation, a pseudo 3D viscoelastic winding/unwinding model was developed and implemented in a code revised from the 2D viscoelastic winding code developed by Qualls and Good [1]. This pseudo 3D viscoelastic winding/unwinding model was based on the pseudo 3D winding model by Hakiel [2], Cole and Hakiel [46] and the 2D linear viscoelastic winding/unwinding model by Lin and Westmann [10]. The web material is assumed to be orthotropic with a nonlinear radial stiffness depending on the interlayer pressure. This model allows for the consideration of a viscoelastic response during winding, storage and unwinding. In this model, the varying thickness profile along CMD has been considered in winding. In this pseudo 3D model, the wound roll is discretized into smaller segments of varying length in width direction and each segment has a constant web thickness. Tension is applied to each segment using the Hakiel approach [2]; the tension is updated after the winding of each lap based on the deformed radius of the segment relative to the relaxed radius profile of that lap. In each segment, a 2D model winding/unwinding model is applied to determine stress distribution in the web. A pseudo 3D transient viscoelastic winding model that has considered viscoelastic effects during winding has been developed for an orthotropic, nonlinear viscoelastic web based on the work by Qualls and Good. The pseudo 3D model can consider (1) varying 2 3 thickness profile in both CMD(cross machine direction) and MD(machine direction); (2) winding tension variation as a function of winding laps; (3) varying core stiffness in the width direction; and (4) hygrothermal influences. The most important feature is that the model allows the consideration of viscoelastic effects in a wound roll during winding at a tension, so that this model is especially suitable for viscoealstic materials with short characteristic relaxation times, such as nonwovens and plastic webs with glass transition temperature close to room temperature. Tapered tension profiles were employed to investigate winding at varying tension histories. The tension histories were used as input to the viscoelastic winding model to determine the stress distribution in a wound roll. Winding tension was changed through the change of the tension taper factor based on the work by Shelton [31] and Feiertag [32]. Stress distribution was determined for ten different tension profiles (ranging from constant tension to constant torque) for a roll of 1000 laps. The number of laps can be varied based on the capacity of the computer RAM. The pseudo 3D viscoelastic winding model has been used to analyze three baggy lanes problems for validation. They are (1) formation of cambered web (inplane imperfection) due to webnonuniformity; (2) formation of localized baggy lanes due to edge burr following slitting; and (3) formation of baggy web (inplane imperfection) due to webnonuniformity. Simulation results are compared with experimental data. The pseudo 3D model has considered the unwinding of a wound roll as well. During unwinding process, the viscoelastic memory effects are also considered. The viscoelastic effects built in this model depend on both viscoelastic properties and winding conditions, such as unwinding speed and tension. In this model, the stress distribution in 2 4 a roll comes from two sources, one from the initial winding, and the other from a correction state that starts from unwinding. Figure 31: Pseudo 3D models As shown in Figure 31, the wound roll can be divided into several small segments. In each segment, a 2D viscoelastic model is applied to determine stress distributions in the web. Numerical solution for the stress distribution in a wound roll during winding, storage and unwinding parts will be discussed in following sections separately. 3.1 Pseudo 3D Winding Model In this work, we use the approach by Hakiel [2] and Cole and Hakiel [46] to partition a 3D wound roll into a number of 2D segments. Some modifications will be made to the their model and the varying segment width will be considered. 2 5 Figure 32: Exaggerated view of a wound roll with thickness variation, Hakiel [2] As shown in Figure 32, before the lap i is wound on the roll, the widthwise radius distribution of the wound roll is ρ(i −1, j) . Consider that the surface of wound roll is no longer cylindrical due to thickness variation and the winding tension is not high enough nor is the radial modulus low enough to result in large deformations. The lap i may or may not be in full contact with the wound roll surface. In the areas where the contact is made between the lap i and the wound roll, the radius of inside surface of lap i r(i, j) is equal to the radius of the previous wound roll surface ρ(i −1, j) . In the areas where there is no contact, it will be assumed that the radius of inside surface of lap i r(i, j) is constant and is equal to the relaxation radius ( ) 0 R i which is assumed as the inner radius of a nonstretched part of the web. In order to determine the relaxation radius, the summation of tension applied on all segments at the current outer lap is set to be equal to the web tension. Tθ (i) = T(i) (3.1) The predicted winding tensile force is equal to the summation of tensile force in all the widthwise positions. 2 6 } ( ) ( , ) ( ) ( , ) ( ) { (1 ) ( ) { ( , ) ( ) ( , )} 0 0 1 2 1 w j h i j R i E r i j R i T i i j w j h i j M j M j − − ν =Σ σ = Σ − θ θ − θ (3.2) where w( j) is the width of segment j; h(i, j) is the thickness of segment j at lap i. The width of segment could be different in this modified model. Usually, Equation (3.1) cannot be satisfied the first time. However, in an iterating process, the relaxation radius ( ) 0 R i can be determined by extrapolating over previous estimates until the calculated winding tensile force is equal to the applied tensile force at the current radius and the tension in each widthwise position can be determined. The inner radius of the lap being wound on is ( , ) { ( 1, ), ( ); 1,... } 0 r i j = Max ρ i − j R i j = M (3.3) The widthwise radius distribution of wound roll is ρ(i, j) = r(i, j) + h( j); j =1,...M (3.4) Based on Equations (3.1) ~ (3.4) the widthwise distribution of radius and tension can be determined at a designated wound roll radius. After that, the wound roll can be divided into several small segments. In each segment, a 2D viscoelastic winding model is applied to determine the timedependent stress distributions in the segment. It is noted that in the previous equations, the radial deformation of web during winding is not considered. Radial deformation can change significantly the tension distribution. The effect of radial deformation on tension distribution can be considered using Cole and Hakiel’s model [46]. As mentioned previously, while each layer is wound, the layer may or may not make full contact with the wound roll surface. In the areas where the contact is made, the lap being wound will induce radial displacement. A higher winding tension will result in higher radial displacement, which will reduce the 2 7 effect of thickness variation and prevent from sudden changes in thickness profile in the wound roll. Figure 33: Outer lap geometry, Cole [46] As shown in Figure 33, the r (i, j ) d is deflected roll outer radius; U ′(i, j ) is the radial displacement of the segment due to winding of the ith. In the segments where there is no gapping, r (i, j ) d is defined as r (i 1, j ) r (i, j ) U (i, j ) h(i, j) d d + = + ′ + (3.5) In the segments where there is gap, r (i, j ) d is defined as ( 1, ) ( ) ( , ) 0 r i j R i h i j d + = + (3.6) where ( ) 0 R i is relaxation radius as defined before. Then the predicted winding tensile stress is equal to + − − − ν σ = θ θ ( ) ( ) 2 ( ) ( 1, ) (1 ) ( ) 0 0 2 R i R i h j r i j E i d (3.7) 2 8 The predicted winding tensile force is equal to the summation of tensile forces in all the widthwise positions. ( ) { ( , ) ( ) ( , )} 1 T i i j w j h i j M j θ − θ =Σ σ (3.8) Similarly, in order to determine the relaxation radius, the summation of tension applied on all segments at the current outer lap is set to be equal to the web tension. Tθ (i) = T(i) (3.9) An iteration is conducted to satisfy the above Equation (3.9) and determine the relaxation and. After the relaxation radius ( ) 0 R i is determined, the tension could be distributed into each segment. In each segment, a 2D viscoelastic winding model is applied to determine the timedependent stress distributions in the segment. The models for the winding and storage will be discussed in the following sections separately. 3.2 Viscoelastic Effects during Winding In this section, viscoelastic effects during winding will be considered. For viscoelastic materials with relatively short characteristic relaxation times, compared with the time it takes to wind a roll, it is necessary to consider the viscoelastic effects during winding. This part of work is based on the 2D viscoelastic winding model developed by Qualls and Good [1]. In the QuallsGood model viscoelastic effects are considered after winding is finished. Their model has been extended in this work to allow the consideration of viscoelastic effects from the beginning of the winding. With the consideration of viscoelastic effects during winding, some winding conditions, such as the winding velocity and winding tension as a function of radius or time can be considered. 2 9 Consider an orthotropic viscoelastic material with radial modulus depending on radial stress in 2D case in polar coordinates. The equilibrium equation is + σ − σ = 0 ∂ ∂σ r θ r r r (3.10) The strain compatibility in the rθ plane is given by: + ε − ε = 0 ∂ ∂ε θ θ r r r (3.11) It should be noted that in the current model there is no strain compatibility enforced between segments. The constitutive equations for an orthotropic viscoelastic material are dt t J t t t J t t t r r r r ′ ∂ ′ ∂ + − ′ ∂ ′ ∂ = ∫ − ′ 0 ( ) ( ) θ θ σ σ ε dt t J t t t J t t t r r ′ ∂ ′ ∂ + − ′ ∂ ′ ∂ = ∫ − ′ 0 ( ) ( ) σ σ ε θ θ θ θ (3.12) By solving Equation (3.10), σθ can be expressed in terms of r σ . r r r r + σ ∂ ∂σ σθ = (3.13) We next insert Equation (3.13) into Equation (3.12) to eliminate σθ , and then substitute εθ , r ε in Equation (3.11) to obtain [ ( ) ( ) {3 ( ) ( ) ( ) 2 2 2 0 J t t J t t J t t r r t J t t r r r t + − ′ + − ′ − − ′ ∂ ∂ σ ∂ ′ ′ ∂ ∫ θ − θ θ θ ( )} ( ) { (J (t t ) J (t t )) J (t t ) r r r r t J t t r r r r − ′ + − ′ + − ′ ∂ ∂ + ∂ ∂σ ∂ ′ ′ ∂ − ∂ ∂ + θ θ θ θ 3 0 ( ) ( ) ( )} ] ′ = 0 ∂ ′ ′ ∂σ + θ − ′ − − ′ − θ − dt t J t t J t t J t t r r r r (3.14) Although the radial stiffness r E is related to roll radius, the ratio, r r νθ / E and ν θ Eθ r / , are extremely small and could be assumed to be radially independent, as the radial Poisson’s ratio νθr and ν θ r are comparatively small. Therefore, Qualls and Good [1] suggested that Jθ , rθ J , r Jθ are radially independent. Consider that rθ J , r Jθ are small and r Jθ is equal to rθ J at some special time, we could assume that θ = θ r r J J . Therefore the Equation (3.14) can be simplified as ( ) ( ) ( ) ( ) ( ) 0 0 2 2 3 2 2 1 = ′ ⋅ ∂ ′ ′ ∂σ + − ∂ ∂σ ∂ ′ ′ ∂ + − ∂ ∂ σ ∂ ′ ′ ∂ ∫ − dt t F t t r r t F t t r r t F t t t r r r (3.5) where ( ) ( ) 1 F t −t′ = Jθ t −t′ ; ( ) 3 ( ) 2 F t −t′ = Jθ t −t′ ; ( ) ( ) ( ) 3 F t t J t t J t t r − ′ = θ − ′ − − ′ . r J and Jθ follow the generalized Kelvin model. It should be noted that r J is also a function of the radial pressure. The generalized Kelvin model for creep compliance is expressed as ( ) (1 ) / 1 0 i t N i i J t J J e − ξ = Σ = + − (3.6) In order to solve Equation (3.15), a general form is assumed. dt t f I F t t t ⋅ ′ ∂ ′ ∂ = ∫ − ′ 0 ( ) (3.7) 3 1 As shown in Figure 34, for a wound roll with n laps we divide the entire winding time into n time steps. We have k t = t after winding lap k (radius k r = r , 1≤ k < j ≤ n ). Therefore, at the current outer layer, lap j, j t = t . Figure 34: Viscoelastic effects during winding For the case of constant velocity 0 v(t) = v , the time increment from tj to tk can be derived as follows. Since R dR h v dt r t r j k π∫ ⋅ = ∫ ⋅ % 0 0 2 , we have ( ) v h r r t t t j k j k 0 π⋅ 2 − 2 % = − = (3.8) And the integral I can be written as j j j j j j j j k k I = F(t − t − ) ⋅%f + F(t −t − ) ⋅%f − + + F(t −t − ) ⋅%f 1 2 1 1 L (3.9) where % = = − = −1 j t j t j f f f . Thus j j k %f − ,%f − , ,%f 1 2 K are already known when j t = t . And ( ) − −1 j j F t t , ( ) − −2 j j F t t ,K, ( ) − −1 j k F t t can be calculated at any radius using the formula for the creep compliance. Thus the only unknown is j %f . Lap n, t=tn Lap j, t=tj Lap k, t=tk 3 2 Comparing Equation (3.15) and (3.17), we can find that the function f could be either 2 2 2 r r r ∂ ∂ σ , or r r r ∂ ∂σ , or r σ . Therefore j %f (the increment of f from = −1 j t t to j t = t ) can be expressed in terms of 2 2 2 r r r ∂ ∂ %σ , r r r ∂ ∂%σ or r %σ , where % refers to the increment from = −1 j t t to j t = t . After using Equation (3.20), j %f can be rewritten in terms of ( 1) ( ) ( 1) + , , − %σ %σ %σ r k r k r k . 2 ( 1) ( ) ( 1) 2 2 2 r h r r k + r k r k− %σ − %σ + %σ ≈ ∂ ∂ %σ r h r r k r k 2 ( +1) ( −1) %σ − %σ ≈ ∂ ∂%σ (3.20) Then the general form I in Equation (3.17) can be represented by ( 1) ( ) ( 1) + , , − %σ %σ %σ r k r k r k . At radius k r = r , Equation (3.15) will be represented by a general form. ( , , ) 0 ( 1) ( ) ( 1) %σ %σ %σ = k r k+ r k r k− G (k=1,2,…j1) (3.21) Equation (3.21) can be written for each radial location within the wound roll. When j t = t there are j1 simultaneous algebraic equations with j+1 unknowns. After both the outer and inner boundary conditions at j t = t are applied, this system of equations can be solved. 3.3 Viscoelastic Effects after Winidng After winding process is finished, the total number of laps is n. For radius k r = r , at time m t = t (n < m), the integral I can be written as 3 3 m m m m m m m m k k I = F(t −t − ) ⋅%f + F(t − t − ) ⋅%f − + + F(t −t − ) ⋅%f 1 2 1 1 L (3.22) where % = = − = −1 m t m t m f f f . When m t = t , m m k %f − ,%f − , ,%f 1 2 K are already known. ( ), ( ), , ( ) − −1 − −2 − −1 m m m m m k F t t F t t K F t t can be calculated at any radius using the formula for the creep compliance. Thus the unknown is m %f only when m t = t . Substitution of the finite difference approximation gives 2 ( 1) ( ) ( 1) 2 2 2 r h r r k + r k r k− %σ − %σ + %σ ≈ ∂ ∂ %σ r h r r k r k 2 ( +1) ( −1) %σ − %σ ≈ ∂ ∂%σ (3.23) Similarly, at radius k r = r , the relation among ( 1) ( ) ( 1) + , , − %σ %σ %σ r k r k r k can be written in terms of a general form as follows: ( , , ) 0 ( 1) ( ) ( 1) %σ %σ %σ = k r k + r k r k − H (k=1,2,…n1) (3.24) Equation (3.24) can be written for each radial location within the wound roll. At m t = t , there are n1 simultaneous algebraic equations with n+1 unknowns. When both the outer and inner boundary conditions at m t = t are applied, this system of equations can be solved. 3.4 Viscoelastic Effects during Unwinding In this model, the stress distribution in a roll comes from two sources, one from the initial winding, and the other from a correction state that starts from unwinding. The summation of the initial and correction states satisfies the traction free boundary 3 4 condition for the current unwinding radius. The initial state includes both winding and after winding period. The stress in initial state is the basis of later calculation. It starts from the beginning of winding until unwinding begins. σinitial = σduringwinding + σafterwinding (3.10) Correction state starts from the beginning of unwinding. It is assumed that the final stress at any layer is equal to the sum of both initial state and correction state. The correction state is determined by solving the problem of a wound roll with fixed radius subjected to an arbitrary external force P(t). P(t) Figure 35: An arbitrary external force P(t) The unknown external force P(t) can be determined by enforcing the traction free condition at the current winding radius. In unwinding case, the external force direction is outward. After the external force is found, stress distributions can be determined at any time and at any layer. σ final =σ initial +σ correction only ifσ final = 0 at current unwinding radius (3.11) The flow chart of algorithm for viscoelastic unwinding is shown in Figure 36. 3 5 Figure 36: Flow chart of algorithm for viscoelastic unwinding 3.5 Finite Element Winding Simulations for Checking Outer Boundary Condition and Effects of Asymmetric Structure In this section, 2D FEM simulation of winding is conducted to check for the outer boundary condition after winding process is finished. Since the winding tension has been removed at that time, the outside layer boundary condition is different from that used in during winding process. The winding and storage are simulated using ABAQUS/Explicit code to find the outer boundary conditions. Moreover, the influence due to asymmetric structure will be investigated using the present FEM approach. The “spiral” structure of wound roll was considered and simulated in ABAQUS. Calculate the stress change i=1 Unwind first layer Update initial stress at remaining layers Assume one external Start Pressure, P(t). End at current unwinding radius N N Y Y σ initial + %σ correction = 0 ? %σ correction σ initial =σ initial + %σ correction i>N? 3 6 3.5.1 Evaluation of Boundary Conditions using Dynamic Analysis via ABAQUS / Explicit The dynamic analysis algorithm in ABAQUS/Explicit code is used. The equation of motion is (i ) (i ) (i ) M ⋅ ü = F − I (3.12) where ü is acceleration, M is the diagonal lumped mass matrix, F is the applied load vector, and I is the internal force vector. The explicit dynamics analysis in ABAQUS/Explicit is based on an explicit integration rule with the use of diagonal or “lumped” element mass matrices. The acceleration ü is computed from Equation (3.27), (i) 1 ( (i) (i) ) ü = M ⋅ F − I − (3.13) The equations of motion for the body are integrated using the explicit central difference integration rule, ( ) ( ) (i ) i i i i u t t u& u& && 2 1 2 1 2 1 + = + + − + (3.14) ( ) ( ) ( ) + + = + % + 2 1 i 1 i 1 i i u u t u& (3.30) where u& is velocity, and u is the state vector (displacements or rotation angles). The superscript (i) refers to the increment number and 2 1 i − and 2 1 i + refer to midincrement values. The central difference integration operator is explicit in that the kinematic state can be advanced using known values of − 2 1 i u& and (i ) u&& from the previous increment. However, the central difference operator is not selfstarting. The initial values (at time t = 0) of velocity and acceleration need to be given by user before the procedure starts. In 3 7 ABAQUS/Explicit, if the initial values are not given by the user, the default initial values of velocity and acceleration are set to zero. After the initial values of velocity and acceleration are defined, the mean velocity + 2 1 u& and − 2 1 u& can be determined by the following Equations (3.31) and (3.32). Then the displacement, velocity and acceleration at any time can be obtained using Equation (3.28) ~ Equation (3.30). ( ) ( ) (0) 1 2 0 1 2 u t u& = u& + && + (3.31) ( ) ( ) (0) 0 2 0 1 2 u t u& = u& − && − (3.32) The central difference operator is conditionally stable. ABAQUS/Explicit adjusts the time increment automatically and requires no user intervention. 3.5.2 2D Viscoelastic FEM Winding Model Geometry and Boundary Conditions The 2D viscoelastic FEM winding model contains two parts. One is the winding part and the other one is the storage part. Schematic of the 2D FEM winding model is shown in Figure 37. The winding model can be divided into two substructures which are web and core. In this model, the core is considered as a rigid cylinder to simplify the model. In the winding part, the core is restricted in both horizontal and vertical directions and can only rotate with respect to the axis of cylinder at a constant angular velocity. One end of the continuous web was tied to the core while a horizontal tension was applied on the other end of web as shown in the Figure 37. Therefore the web can be wound onto the core layer by layer after the core starts to rotate. 3 8 Figure 37: Schematic of the 2D FEM winding model After winding was complete, the free end of web was attached to the second layer of wound roll. The geometry and boundary condition after winding are shown in Figure 38. In the storage part, the core is fixed at the reference point and does not allow for any movement in all directions. Although, there are no additional loadings added on wound roll or core, the stress and strain fields still change with time due to the viscoelastic effects of the web material. Figure 38: Schematic of 2D FEM storage model Contact Problems During the winding part, contacts are defined as follows: 1) contact between the web and the core surface; 2) self contact of the web. The core is designed as a rigid cylinder and the surface of the core is defined as rigid surface. The definition of the contact is shown in Figure 39. 3 9 Figure 39: Definition of the contact in 2D FEM winding model Similarly, in the storage part the definition of the contact surfaces is shown in Figure 310. Figure 310: Definition of the contact in 2D FEM storage model Viscoelstic Material Properties In the numerical examples, three sets of viscoelastic material parameters, representing different levels of viscoelastic behavior, were used to investigate the viscoelastic effects on stress or strain of the outer layer. The relaxation modulus is given by a Prony series. 2 1 3 4 • Contact between surface (1) and surface (2) • Contact between surface (3) and surface (5) • Tie surface (4) to surface (5) Web • Surface (5) is rigid surface 5 Core 6 7 Web 4 Do not allow normal and tangential movements between surface (6) and surface (7) after contact. 5 Core 4 0 (1 (1 e ))  1 0 i t/τ N i P R i  g  E (t) E Σ= = (3.33) where 0 E is the instantaneous Young's modulus; P i g is the modulus ratio in the i th term in the Prony series expansion of the shear relaxation modulus; i τ is the relaxation time for the i th term in the Prony series expansion. There are three sets of viscoelastic material parameters used in this 2D viscoelastic winding simulation to investigate the viscoelastic effects. The three sets of viscoelastic material parameters are listed in Table 31. Table 31: Viscoelastic material parameters Set 1 Set 2 Set 3 P i g i τ P i g i τ P i g i τ 1th 0.128 20 0.3 40 0.4 40 2nd 0.072 10 0.2 20 0.25 20 The relaxation modulustime curve for three sets of viscoelastic material plotted in Figure 311. Set 3 has the highest viscoelastic properties while Set 1 has the lowest viscoelastic properties. 4 1 0 20,000 40,000 60,000 80,000 100,000 0 20 40 60 80 Time(s) E ( t ) ( p s i ) Set 1 Set 2 set 3 Figure 311: Three sets of viscoelastic material properties Dimension and Loading Three 2D FEM viscoelastic winding models are established using three sets of viscoelastic properties correspondingly. Dimension and loading condition are same for the three models. As shown in Figure 37, the total length of the web is 300 in, the thickness of the web is 0.05 in and the Poisson’s ratio is 0.3. It is assumed that the core is rigid cylinder. The radius of core is 5 in. The tensile stress applied at one end of web is 1000 psi. In the winding part, the angular velocity of core is 2 rad/s. 3.5.3 Results and Discussion Outer Layer Boundary Condition after Winding Was Finished The outer layer timedependent circumferential stresses and strains are plotted for Set 1~3. For viscoelastic material parameters in Set 1, the outer layer timedependent circumferential stress and strain are shown in Figure 312 and Figure 313. 4 2 Figure 312: The outer layer timedependent circumferential strain for Set 1 Figure 313: The outer layer timedependent circumferential stress for Set 1 0 20000 40000 60000 80000 100000 0 10 20 30 40 50 60 Time(s) E(t) (psi) Set 1 Set 2 Set 3 0 20000 40000 60000 80000 100000 0 10 20 30 40 50 60 Time(s) E(t) (psi) Set 1 Set 2 Set 3 4 3 For viscoelastic material parameters Set 2, the outer layer timedependent circumferential stress and strain are shown in Figure 314 and Figure 315. Figure 314: The outer layer timedependent circumferential strain for Set 2 Figure 315: The outer layer timedependent circumferential stress for Set 2 0 20000 40000 60000 80000 100000 0 10 20 30 40 50 60 Time(s) E(t) (psi) Set 1 Set 2 Set 3 0 20000 40000 60000 80000 100000 0 10 20 30 40 50 60 Time(s) E(t) (psi) Set 1 Set 2 Set 3 4 4 Similarly, for viscoelastic material parameters in Set 3, the outer layer timedependent circumferential stress and strain are shown in Figure 316 and Figure 317. Figure 316: The outer layer timedependent circumferential stain for Set 3 Figure 317: The outer layer timedependent circumferential stress for Set 3 0 20000 40000 60000 80000 100000 0 10 20 30 40 50 60 Time(s) E (t) (p s i) Set 1 Set 2 Set 3 0 20000 40000 60000 80000 100000 0 10 20 30 40 50 60 Time(s) E (t) (p s i) Set 1 Set 2 Set 3 4 5 Based on the Figure 312~Figure 317, it is seen that, after the winding process is finished, the percentage of drop in circumferential stress is much larger than circumferential strain at outermost layer. A highly viscoelastic material has less percentage of drop in both circumferential stress and strain. As shown in Figure 316, the circumferential strain is nearly constant. Therefore, for the highly viscoelastic materials, a constant strain can be used as the boundary condition at the outer layer of a wound roll. Moreover, the boundary and loading condition used in this simulation will be discussed further. In this AQAQUS model, during the winding process, the winding tension was added at the end of web. When the winding process finishes, the tension is removed from the web immediately. When the storage process starts, a restriction which does not allow normal and tangential movement between the end of web and the second layer of wound roll will be added. However, this restriction might not be effective immediately although we define so. Since, in ABAQUS model, we only can restrict the relation between the nodes. Therefore the web will slide a little bit until the nodes on the end of web contact with the nearest nodes on the second layer. During this short period, the winding tension has already been removed and the stress and strain might drop. The percentage of drop will depend on the level of viscoelastic effects. High viscoelastic material will have fewer drops. This phenomenon could be found the Figure 312~Figure 317. Effect of Asymmetric Structure In pseudo 3D winding model and most other models, it is assumed that the wound roll is an axisymmetric structure. However, the real wound roll is a “spiral’ structure. In order 4 6 to find the influence due to asymmetric structure, in the present FEM approach, the “spiral” structure of wound roll was considered and simulated in ABAQUS. The circumferential stresses vs. angle curves for layer 1 to layer 10 are plotted in Figure 318, which show that the asymmetric structure has effects on stress distribution in the wound roll. The curves for Layer 1 and Layer 10 have most wave amplitude changes. Asymmetric structure also has effects on the layers close to the outermost and innermost layers. Layer 2 and Layer 7, 8, 9 also show high stress variation. Those layers far away from the asymmetric structure show less fluctuation in stress. Consequently, those layers can be considered as axisymmetric structure to simplify the calculation in the development of winding models. Layer 1 0 200 400 600 800 1000 1200 1400 0 1 2 3 4 5 6 Angle (radians) Circumferential Stress (psi) Layer 2 0 200 400 600 800 1000 1200 0 1 2 3 4 5 6 Angle (radians) Circumferential Stress (psi) Layer 3 0 200 400 600 800 1000 1200 0 1 2 3 4 5 6 Angle (radians) Circumferential Stress (psi) Layer 4 0 100 200 300 400 500 600 700 800 900 1000 0 1 2 3 4 5 6 Angle (radians) Circumferential Stress (psi) 4 7 Figure 318: Circumferential stresses in layer 1 to layer 10 Layer 5 0 100 200 300 400 500 600 700 800 900 1000 0 1 2 3 4 5 6 Angle (radians) Circumferential Stress (psi) Layer 6 0 200 400 600 800 1000 0 1 2 3 4 5 6 Angle (radians) Circumferential Stress (psi) Layer 8 0 100 200 300 400 500 600 700 800 900 0 1 2 3 4 5 6 Angle (radians) Circumferential Stress (psi) Layer 7 0 100 200 300 400 500 600 700 800 900 1000 0 1 2 3 4 5 6 Angle (radians) Circumferential Stress (psi) Layer 9 0 100 200 300 400 500 600 700 800 900 1000 0 1 2 3 4 5 6 Angle (radians) Circumferential Stress (psi) Layer 10 200 0 200 400 600 800 1000 1200 1400 1600 1800 0 1 2 3 4 5 6 4 8 CHAPTER 4 4 RESULTS FROM 3D VISCOELASTIC WINDING/UNWINDING MODEL As mentioned in previous sections, the pseudo 3D viscoelastic winding model is capable of dealing with the thickness variation in both CMD and MD directions, as well as winding tension variation with the winding laps. In this section, both verification and validation results will be presented for the pseudo 3D viscoelastic winding model. In verifications, we compare the results from the pseudo 3D viscoelastic winding model with analytical or numerical results. In these examples, the results from the pseudo 3D viscoelastic winding model have been compared with some other winding models and analytical solutions. They include (1) the comparison with Lin & Westmann’s 2D Viscoelastic windng model; (2) the comparison with the 3D analytical solution; and (3) the comparison with Qualls and Good’s 2D viscoelastic model. In validations, we created CMD thickness variations in the winding experiment by adding a stack of web strips of the same web material being wound at every nth lap [44]. In addition, in the numerical calculation all the extra thickness and related volume are assumed to be allocated evenly in these n laps. Therefore, in these several laps, the thickness varies only in the CMD direction and the thickness profile is persistent in the MD direction. Three special cases have been considered for validation: (1) The formation of cambered web due to linearly varying thickness; (2) The formation of localized baggy lanes due to edge burr following slitting; and (3) The formation of baggy web due to 4 9 thickness increment in the middle of the web. The simulation results using the pseudo 3D viscoelastic winding model will be compared with experimental data and the deformation results from pseudo 3D viscoelastic winding model. The dimensional changes estimated by numerical simulations will be compared with measured data. After that, the pseudo 3D winding model also has been used to investigate the effects of some winding parameters and winding conditions. They are (1) consideration of thickness variation in both CMD and MD; (2) consideration of varying tension histories; (3) consideration of during winding and after winding; and (4) consideration of winding and unwinding. The material used in all the experiments is polyethylene web of 2.5 mil thick and 6 in wide. The inplane and outofplane creep compliance data are taken from Qualls [47]. The outofplane creep compliance follows 3 5 5 / 2.78 10 5 / 2.613 10 5 2 3 2.034 10 1.024 10 3.058 10 167.24 0.09855 0.000422 1 ( ) − − × − − × − − × ⋅ − × + × σ − σ − σ = t t r r r r e e J t (4.1) The inplane creep compliance follows 4 7 4 5 /1 10 4 /1 10 1.62 10 1.285 10 1.491 10 24000 1 ( ) − − − × − − × θ = + × − × ⋅ − × t t J t e e (4.2) where r σ is in psi, t is in seconds and J (t) r is in 1/psi in these inplane and outofplane creep compliance data. 5 0 4.1 Comparison of Results from Pseudo 3D Viscoelastic Model and Lin & Westmann’s 2D Viscoelastic Model Lin and Westmann’s model is linear isotropic viscoelastic winding model. In order to compare with their model, isotropic material properties are used and shown in Table 41. Table 41: Material properties used in pseudo 3D winding model to compare with Lin and Westmann’s results Web Width 6 in Effective Modulus of Core (psi) 5 3.0851×10 J0 (1/psi) 6 1.631 10× − J (1/psi) τ (sec) 1st 9 0.90823 10× − 0.1 2nd 7 0.16903 10× − 1 0.1×10 3rd 6 0.10489 10× − 2 0.1×10 4th 6 0.17270 10× − 3 0.1×10 5th 6 0.39090 10× − 4 0.1×10 6th 6 0.62461 10× − 5 0.1×10 7th 6 0.64518 10× − 6 0.1×10 The dimensionless radial stress has been obtained under uniform tension to compare with the results from Lin and Westmann’s 2D viscoelastic winding model. The results were plotted in Figure 41. Based on the comparison, we can find that the results are matched. 5 1 CMD (in) 2 0 2 D emensio nless Radius 1 1.2 1.4 1.6 1.8 2 Dimesionless Radial Stress 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Pseudo 3D Viscoelastic Winding Model Lin and Westmann [10] Figure 41: Comparison with the results of Lin and Westmann [10] 4.2 Comparison of Results from Pseudo 3D Viscoelastic Model and the 3D Analytical Solution In this section, before the comparison, the solution for 3D isotropic viscoelastic winding models will be introduced briefly. The incremental radial stress ( ( ) * r r σ ) for 3D isotropic elastic winding model is ( ) p a ad bc a aE b c p aE b cE d r r ⋅ − ⋅ + ⋅ = + + + σ = * 1 (4.3) where: o w r T h p = − ( ) ( ) ( ) ( ) 2 i 1 o 1 i 2 o a = F r F r − F r F r [ ( ) ( ) ( )] ( ) [ ( ) ( ) ( )] ( ) c 2 i c 6 i c 8 i 1 o c 1 i c 5 i c 7 i 2 o b = νE F r − E F r + νE F r F r − νE F r − E F r + νE F r F r cos [ ( ) ( ) ( ) ( )] 2 1 1 2 c kz F r F r F r F r i i = ⋅ − cos { [ ( ) ( ) ( )] ( ) [ ( ) ( ) ( )] ( )} 1 5 7 2 2 6 8 1 d kz E F r E F r E F r F r E F r E F r E F r F r c i c i c i c i c i c i = ⋅ − ν − + ν + ν − + ν ( ) ( ) ( ) 1 2 0 3 1 B ikr r ik F r = −k B ikr − ; ( ) 2 ( ) ( ) ( ) 1 4 0 3 0 3 2 F r = − k νB ikr + k B ikr − ik rB ikr ; 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 1.2 1.4 1.6 1.8 2 r, in Radial stress/Initial tension r/rin 5 2 ( ) ( ) 1 3 3 F r = −ik B ikr ; ( ) 2 (1 ) ( ) ( ) 0 4 1 3 4 F r = ik − ν B ikr − k rB ikr ; ( ) ( ) 1 2 5 B ikr r ik F r = ; ( ) 2 ( ) ( ) 0 3 0 3 6 F r = − k νB ikr + k B ikr ; ( ) ( ) 0 3 7 F r = k B ikr ; ( ) 2 (2 ) ( ) ( ) ( ) 1 4 0 3 8 F r = − k − ν B ikr + ik r B ikr . Applying elasticviscoelastic corresponding principle, we have ( ) ( ) ( ) ( ) − ⋅ + = ⋅ + − ⋅ + σ = + a ad bc as E s sb p a c s p a ad bc a asE s b c r s p s r 2 * ) 1 , ( (4.4) where: ( ) ( ) ( ) ( ) a sbJ (s) J s sb sJ s as E s sb sa s + = + = + φ = 1 1 1 2 . E(t) is the Young’s relaxation modulus; J(t) is the (uniaxial) creep compliance; and s is the Laplace transform variable. Inverting the L.T. leads to the function of φ(t ) ( ) ( ) ( ) ( ) + ξ − ξ − ξ φ = − ∫ φ ξ (0) ( ) 1 0 a bJ d d t dJ t t J t b t (4.5) Then we have cos [ ( ) ( )] 0 1 1 2 kz a F r a F r o R r r σ = ∫ ⋅ + . Similarly, we can find other stress components. Results from the pseudo 3D viscoelastic winding model are used to compare with the results form 3D analytical solution. The linear isotropic viscoelastic material as shown in Table 41 was used in this comparison. The winding velocity is 0.1 in/sec. The results are plotted in Figure 42. Based on the comparison, we can find that the results are matched. The maximum difference is around 5%. 5 3 Figure 42: Comparison with the results of 3D analytical solution 4.3 Comparison of Results from Qualls and Good’s 2D Viscoelastic Model The pseudo 3D viscoelastic model was developed and implemented in a code revised from the 2D viscoelastic winding code developed by Qualls and Good [1]. The difference between these two models is that in Qualls and Good’s model viscoelastic effect is considered after the winding process is finished while in the pseudo 3D model viscoelastic effect has been considered from the beginning of winding process. The material used in this comparison is polyethylene, with its inplane and outofplane creep compliance data taken from Qualls [47] as shown in Equations (4.1) and (4.2). The total number of layers is 100 and the storage time is 5 6×10 seconds (~7 days). The winding tensile stress is 100 psi. In the first example, we choose a very high winding speed, 1000 in/s, to reduce the winding time. The radial pressures were obtained from both code and were plotted in Figure 43 for winding process and storage process CMD (in) 3 2 1 0 1 2 3 Radius (in) 1.8 1.85 1.9 1.95 2 2.05 Radial Stress (psi) 0 10 20 30 40 Pressure (psi) 3D Analytical Solution Pseudo 3D Model 5 4 correspondingly. As shown in Figure 43, results from both models agree very well since there is not much viscoelastic effect in such a short winding time. 0 2 4 6 8 10 12 14 1.8 1.85 1.9 1.95 2 2.05 2.1 Radius (in) Pressure (psi) Pseudo 3D: Winding Qualls and Good's, Winding Pseudo 3D: Storage, t=6e+5 s Qualls and Good's: Storage, t=6e+5 s Winding Storage Figure 43: Comparison with the results of Qualls and Good’s model, v=1000 in/s In the second example, we compare with the results from Qualls [47]. In this example, the inside radius is 1.75 in and outside radius is 5.25 inch. Winding tension is 100 psi and winding speed is 30 fpm. The material used in this comparison is also polyethylene, with its inplane and outofplane creep compliance data taken from Qualls [47] as shown in Equations (4.1) and (4.2). The radial pressure was obtained from pseudo 3D model and was plotted with Qualls’ data in Figure 44 for winding process and storage process for 10 days, 33 days and 64 days correspondingly. As shown in Figure 4 4, the results from pseudo 3D model are close to the results from Qualls and Good’s model. It seems that viscoelastic effects during winding for a polyethylene web are not significant in this case. 5 5 0 25 50 75 100 1.75 2.25 2.75 3.25 3.75 4.25 4.75 5.25 Radius (in) Pressure (psi) Qualls, T=0 Pseudo 3D, T=0 Qualls, T=10 days Pseudo 3D, T=10 days Qualls, T=33 days Pseudo 3D, T=33 days Qualls, T=64 days Pseudo 3D, T=64 days Measured, T=0 Figure 44: Comparison with the results of Qualls and Good’s model, v=30 fpm 4.4 Comparison of Results from Pseudo 3D Viscoelastic Model and Experimental Data The pseudo 3D winding code was used to calculate the stress in a roll under conditions used in experiments; results are compared with experimental data. A CMD thickness variation was created during winding by adding a stack of 3 inserts of equilateral triangular webs at every fourth lap after winding 170 layers. The schematic of this insertion is shown in Figure 45. Figure 45: Schematic diagram for inserting a triangular web for every four plies for use in the investigation of formation of a cambered web On average, a thinner edge On average, a thicker edge Location #12 Equilateral triangular web 7 in 6 in Location #1 Increasing Time 5 6 The total number of layers is 200. The first pull tab was inserted on the 10th layer. After that, pull tabs were inserted on every 20th layer. After the 8th pull tab was inserted, 20 more layers were wound on the roll. Then a stack of 3 trianglar web inserts were inserted every 4th layer. The winding speed was 4.3 ft/min and winding tensile stress was 320 psi. The material used was polyethylene. The calibrated pull tabs were inserted at both left and right hand sides of the roll prior to the insertion of triangular inserts. Only one inch of the pull tabs was inserted into the winding roll, in order to measure the end roll pressure. The experimental data were obtained for both ends of the wound roll and plotted in Figure 46. 0 10 20 30 40 50 60 70 80 90 0 20 40 60 80 100 120 140 160 180 200 Layer # Pressure (psi) Figure 46: The experimental data for both edges Pseudo 3D winding model was used to simulate this process. Since the pseudo 3D winding model is an axisymmetric model, it is assumed that for each 2D segment the extra thickness and associated volume will be allocated evenly in that layer. In the simulation, 12 segments, each 0.5 inch wide, were used. The average values of the stresses in segment 1 & 2, segment 11 & 12 have been taken to compare pressure as Thinner Edge Thicker Edge 5 7 measured by pull tabs. The inplane and outofplane creep compliance data are taken from Qualls [47] and listed as shown in Equation (4.1) and (4.2). It should be noted that Qualls’ material data were obtained more than 10 years ago. In 2006, Poh [48] conducted experiment to measure the material properties of these polyethylene webs after 10 years storage. Poh compared the inplane creep compliance of polyethylene webs with the data by Qualls and found that the results are close. For the outofplane creep compliance, he did not make the comparison. We might need to conduct additional creep compliance experiment to measure the radial compliance. Before we could obtain the further material properties data from experiment, we will use Qualls’ data. Although there might be some error, the effect of 10 years storage is not as big as we thought considering that a good comparison has been indicated for inplane creep compliance. The radial stress distribution from pseudo 3D winding model was plotted with experimental data in Figure 47. 5 8 0 10 20 30 40 50 60 70 80 90 0 20 40 60 80 100 120 140 160 180 200 Layer # Pressure (psi) Figure 47: Comparison with the experimental data From Figure 47, it is seen that stresses at the thicker edge are higher than that at the thinner edge. The results from numerical model and experiments agree reasonably well. It should be noted that in the pull tab tests, a pull tab was inserted only one inch into the wound roll. The quality of the web edge and the pull tab is very important in this experiment. The web with wavy edge could not be used, since in those areas where wavy edge occurs, the layertolayer contact could not be achieved easily, which will lead to the error in pull tab test. The polyethylene web used in this pull tab test was never been used, so that there is no wavy edge at both edges. 4.5 Special Case One: Formation of a Cambered Web A CMD thickness variation was created during winding by adding a stack of 3 inserts of equilateral triangular webs at every fourth lap. The schematic of this insertion is same Thinner Edge Thicker Edge 5 9 as that shown in Figure 45. While using the pseudo 3D winding model, the web was divided into ten small segments. Since the pseudo 3D winding model is an axisymmetric model, it is assumed that for each 2D segment the extra thickness and associated volume will be allocated evenly in these four laps respectively. From location #1 to #10, the thickness increases linearly. The pseudo 3D viscoelastic winding model has been used to simulate this viscoelastic winding problem. The winding speed is 4.3 ft/min and winding tensile stress is 433.3 psi. The total wound roll length is around 72 ft 10 in and the number of laps is 110 layers. The storage time is 86400 seconds (24 hours). The length of deformed web for last 13 layers at the ten locations is obtained using the pseudo 3D winding model and is plotted in Figures 48. Location Number inWidth Direction Length of Deformed Web at Different Locations (in) 0 1 2 3 4 5 6 7 8 9 10 168.5 169 169.5 170 170.5 171 171.5 Figure 48: The length of deformed web at ten locations From the data shown in Figure 48, we can find that the length of deformed web changes linearly along the CMD direction as shown in Figure 49I. Since the web edges 6 0 are straight in a cambered web, results in Figure 49I can be plotted as shown in Figure 49II, with the use of an appropriate radius and central angle as shown in Figure 49III. Figure 49: Numerical transformation for cambered web Figure 410: Maximum bow length In Figure 49, b1, b2, ..., b10 represent the lengths of segments. a1, a2, …, a10 stand for the distances from inner edge to the upper edge of these segments. They satisfy the following equations + ⋅ θ = + ⋅θ = + ⋅θ = 10 10 2 2 1 1 ( ) ( ) ( ) r a b r a b r a b M M (4.6) I I III Maximum Bow Length I II III 6 1 In this case, the numerical simulation gives the results r =464.58 in and θ = 20.83 degrees (0.3635 radians). Therefore, we can obtain the maximum bow length as shown in Figure 410 is around 7.65 in. The value of maximum bow length from experiment is 7.5 in. The deformed shapes of web at ten locations are simulated and shown in Figure 411. The numerical results and experimental data are in same order of magnitude though there are still some errors. As mentioned in the previous three examples, it is assumed that the extra thickness and volume will be allocated evenly in several related laps at each location respectively. In some situations, this assumption might have resulted in errors. Machine Direction (in) Bow Length (in) 0 25 50 75 100 125 150 175 0 5 10 15 Figure 411: Deformed shape after storage 4.6 Discussion about Hakiel’s Mode l [2] and Cole and Hakiel’s Model [46] Current modified pseudo 3D winding model has used a refined version of Hakiel’s model which is Cole and Hakiel’s model. The original code has been revised to consider the effects of radial displacement on the tension distribution. As mentioned in chapter 3, Location # 10 Location # 1 6 2 the radial displacement will decrease the effect of thickness variation and abrupt changes in thickness profile will not be exaggerated dramatically in the wound roll shape. In this section, the special case one, formation of cambered web, will be recalculated using Cole and Hakiel’s model to discuss the difference between the original model and the refined model. The problem is exactly same as described in section 4.5 except we used these two models respectively. The comparisons were shown in Figure 412 and Figure 413. Location # in CMD 0 2 4 6 8 10 Radius (in) 1.8 1.9 2 2.1 Pressure (psi) 0 20 40 60 Y X Z Figure 412: Comparison of the pressure distributions Cole and Hakiel’s Model Hakiel’s Model 6 3 Length of Deformed Web at Different Locations (in) Length of Deformed Web at Different Locations (in) 0 1 2 3 4 5 6 7 8 9 10 169 170 171 172 Figure 413: Comparison of deformed shapes after storage According to Figure 413, we find the tangential angles of these two curves are different. The tangential angle of curve from Hakiel’s model is larger than that from Cole and Hakiel’s model. Based on the length of deformed web, the maximum bow length from Hakiel’s model is 8.89 in and the maximum bow length from Cole and Hakiel’s model is 7.65 in. The experimental result is 7.5 in. The results from Cole and Hakiel’s model are closer to the experimental results. In Cole and Hakiel’s [46] paper, they also tried several examples and showed that the refined the model is better than the previous Hakiel’s model [2] in all these cases. As a result, we updated the all three special cases we provided before and get better results. 4.7 Special Case Two: Formation of Wavy Edge One of the factors contributing to formation of a wavy edge is winding a web with an edge burr following slitting. In this validation case the effect of slit edge burr on the formation of baggy lanes is investigated. A web with edge burr is shown in Figure 414. Hakiel’s model Cole and Hakiel’s model 6 4 The segment close to one edge is thicker than the other segments. The thickness profile is assumed to be persistent along MD. Figure 414: Formation of wavy edge The pseudo 3D viscoelastic winding model was used to simulate this wavy edge problem. As mentioned before, the material used is polyethylene web of 2.5 mil thick and 6 in wide. The height of edge burr is 0.803 mil and the winding tensile stress is 300 psi. The number of laps is 80. The storage time is 86400 seconds (24 hours). The entire length of deformed web at ten locations is plotted in Figures 415. Location Number in Width Direction Length of Deformed Web at Different Locations (in) 1 2 3 4 5 6 7 8 9 10 11 12 13 958 960 962 964 966 968 970 972 974 976 978 Figure 415: The length of deformed web at ten locations edge burr 6 5 As shown in Figure 415, the deformation in the edge burr area is much larger than the rest of the web. The deformations in the rest of the web are constant. In the experiment, we observed the wavy edge and found that the wavy edge follows approximately a sinusoidal oscillation. Therefore, we assume that the deformed shape follows Equation (4.7) at the edge burr, and the deformation of web can be converted from shape I into shape II as defined in Figure 416. ) 2 sin( λ π = ⋅ x y A (4.7) where λ is the wave length, A is the amplitude of wavy edge. These two parameters can be determined in numerical simulation and measured in experiment. Then the shape of wavy edge can be simulated and examined. Figure 416: Numerical transformation for wavy edge In the experiment, the wave length and amplitude have been obtained through image analysis. λ is 0.923 in and the A is 31.26 mil. In the numerical simulation, the wave length (λ) and amplitude (A) were extracted from Equation (4.8). dx A x Length Length rest burr ∫λ λ π λ π = + λ 0 2 2 2 2 2 cos 4 1 / (4.8) Lengthburr Lengthrest I II 6 6 It is assumed that the wavy edge has the same wavy length as that in the experiment to determine the amplitude. The amplitude of wavy edge can be determined by inserting the average length of deformed web at burr area into Equation (4.8). In this numerical simulation, the amplitude is determined as 30.57 mil based on the deformation results from the pseudo 3D winding model and wavy length data from experiment. The numerical results and experimental data have a very good agreement. 4.8 Special Case Three: Formation of Baggy Web In this case, CMD thickness variation was generated by adding a stack of 6 inserts of strip webs at every sixth lap. The example setup is shown in Figure 417. In the pseudo 3D winding model, the web was divided into forty segments. Since the pseudo 3D winding model is an axisymmetric model, it is assumed that the extra thickness and volume will be allocated evenly in these six laps at each location respectively. The segments at the middle of web are thicker that the rest. Figure 417: Formation of baggy web The pseudo 3D viscoelastic winding model has been used to simulate this baggy web problem. The winding speed is 4.3 ft/min and the winding tensile stress is 333.3 psi. The number of laps is around 50. The storage time is 108000 seconds (30 hours). The entire length of deformed web at forty locations determined from simulations is plotted in Figures 418. a stack of 6 inserts; width is 1 in; length is equal to the circumference at location of insertion 6 in 6 7 Location Number in Width Direction Length of Deformed Web (in) 0 4 8 12 16 20 24 28 32 36 40 545 547.5 550 552.5 555 557.5 560 Figure 418: The lengths of deformed web at forty locations From results shown in Figure 418, the lengths of deformed web reach the maximum value at the middle segments. Results are quantified in this example in terms of h as shown in Figure 419, a measure of the outofplane deformation of the web that resulted from the thickness variation. The amplitude of h is expressed by Equation (4.9). Numerical solution for h, based on the model results was 30.11 mil. The experimental result for h is 27.56 mil. The numerical results and experimental have a reasonably agreement. However, from the comparison it can be seen the numerical result is higher than the experimental result. In current pseudo 3D approach, the tension is distributed to each segment based on the thickness profile. In the areas where the thickness is high, the winding tension will be high in those areas as well. If there are only a few areas in the widthwise thickness profile which are significantly higher than the rest of the web, severe tension will be distributed to those areas. Nevertheless, in the experiment, sometimes we 6 8 could find that the web material is somewhat loose or damaged in those areas. Therefore the web material might not bear such severe tension in those areas as we predicted in numerical model. The actual tension distributed to the baggy lane areas should be less than the predicted value. This might lead to the error between numerical result and experimental result this baggy lane formation case. Figure 419: Numerical transformation for baggy web. ⋅ π − ⋅ π ≈ − ≈ 2 2 2 1 2 1 n L n L h r r ( ) ⋅ π = − 2 2 1 n L L (4.9) The dimensional changes estimated by numerical simulation are compared favorably with measurement data. However, there are still some issues in the pseudo 3D viscoelastic winding model. Similar to most of existing winding models, the pseudo 3D viscoelastic winding model is an axisymmetric model. Asymmetric local deformation can not be considered using this winding model. As mentioned in the previous three examples, it is assumed that the extra thickness and volume will be allocated evenly in several related laps at each location respectively. In some situations, this assumption might have resulted in errors. 4.9 Consideration of Thickness Variation in Both CMD and MD In ideal case, the web thickness is uniform in both CMD and MD. Restricted by the manufacturing technology, the web produced will inevitably have thickness variation in CMD. For example, in order to avoid the stress and strain to accumulate too much at r1 r2 h L1 L2 6 9 certain area which may lead to web imperfections, rotary die is used in blownfilm machine. The rotary die turns around, leading to nearly periodic thickness variation. As a result, the roll radius will distribute more evenly in stead of increasing too much at certain area. In this section, we will make an attempt to simulation this problem to find the influence of thickness variation on final deformations. In this section, we assume that the thickness variation in CMD and MD follows the sinusoidal wave as shown in Equation (4.10). B l n y l m x z A x y + ⋅ π + ⋅ π = ⋅ ) 2 2 sin( (4.10) where m, n are the cycle numbers in MD and CMD; and lx , ly are the total length in MD and CMD; A is the amplitude; B is the average thickness. The cycle number is set by the code and could be different in CMD and MD. In this example, we used 2 cycles in CMD and 3 cycles in MD. The amplitude in this example is the average thickness 0.0025 in. The thickness profiles with 50% deviation of average thickness for layer #1 and Location #1 was plotted separately in Figure 420. LayerNumber in MD Thickness (in) 0 20 40 60 80 100 0 0.0025 0.005 Location Number in CMD Thickness (in) 2 4 6 8 10 12 14 16 18 20 0 0.0025 0.005 7 0 Figure 420: The thickness profiles with 50% deviation of average thickness for layer #1 and Location #1 The deviation of average thickness used here is defined as ×100% B A . The whole thickness profile with 50% deviation of average thickness was shown in Figure 421. Location # in CMD 4 8 12 16 20 Layer # in MD 0 20 40 60 80 100 Thickness (in) 0 0.001 0.002 0.003 0.004 0.005 X Y Z 50% Figure 421: The whole thickness profile The pseudo 3D viscoelastic winding model has been used to simulate this problem. The material used in this simulation is polyethylene web of 2.5 mil average thickness and 6 in wide. The deviation of average thickness is 10%, 20%, 50% and 80% respectively. The winding tensile stress is 300 psi. The number of laps is 100. The lengths of deformed web at twenty locations determined from simulations are plotted in Figures 422. The deviation of deformed length for each case is listed in Table 42. 7 1 Location # in CMD Deformed Length (in) 2 4 6 8 10 12 14 16 18 20 1210 1212 1214 1216 1218 1220 1222 1224 1226 10% 20% 50% 80% Figure 422: Deformed length for 10%, 20%, 50% and 80% cases Table 42: Deviation of deformed length 10% 20% 50% 80% Avg Deformed Length (in) 1218.146 1218.188 1218.322 1218.472 Deviation (%) 0.066311 0.132571 0.330985 0.528435 From the results in Table 42, we find that the maximum deviation of deformed length is 0.528%. It is pretty low. Therefore, it can be concluded that the periodic thickness variation in both MD and CMD direction can mitigate the influence of the thickness variation on the formation of baggy lanes. 4.10 Consideration of Winding and Unwinding Using the 3D viscoelastic winding code, the stress distributions in the web can be calculated during winding/unwinding and after winding. In order to determine the stress difference between winding and unwinding, the comparison of stresses during winding and unwinding is shown in Figure 423. The difference between stresses during winding 7 2 and unwinding decreases as the number of layer decreases. Near the core, the stresses in these two stages are almost identical. For each sub roll, we could find the pressure near the core drops much more than other areas. For the smallest sub roll, there is no obvious drop since the pressure in this sub roll is low comparing with other bigger sub rolls. The material used in this case is polyethylene. The web has constant thickness and is under constant winding tension. The total number of layers is 50 and the winding tension is 20 psi. The winding speed is 0.086 in/s and storage time is 10 days. The pressure distribution is plotted for one location in CMD. 0 0.5 1 1.5 2 2.5 1.7 1.75 1.8 1.85 1.9 1.95 2 2.05 2.1 Radius (in) Pressure (psi) Layer=5 (winding) Layer=5 (unwinding) Layer=10 (winding) Layer=10 (unwinding) Layer=20 (winding) Layer=20 (unwinding) Layer=30 (winding) Layer=30 (unwinding) Layer=40 (winding) Layer=40 (unwinding) Layer=50 (winding) Layer=50 (unwinding) Figure 423: Comparison of stresses during winding and unwinding 7 3 CHAPTER 5 5 CONCLUSIONS A pseudo 3D winding model with the consideration of viscoelastic effects during winding and storage has been developed and implemented in a code. The following conclusions can be drawn. 1) The pseudo 3D transient viscoelastic winding model is capable of dealing with (1) a varying thickness profile in both MD and CMD direction; (2) a winding tension variation with the winding laps; (3) varying core stiffness in the width direction. Moreover, the viscoelastic effects built in this model allow the consideration of winding conditions, such as winding speed and tension. The model is especially suitable for viscoelastic materials with relatively short characteristic relaxation times, such as plastic webs with glass transition temperature close to room temperature. 2) The pseudo 3D transient viscoelastic winding model has been compared with some viscoelastic winding models, analytical solutions and experimental data. They include (1) the comparison with Lin & Westmann’s 2D Viscoelastic windng model; (2) the comparison with the 3D analytical solution; (3) the comparison with Qualls and Good’s 2D Viscoelastic Model; and (4) the comparison with experimental data. The results are reasonably matched. 7 4 3) For the polyethylene web in the validation of this model it has been shown that the effects of winding velocity are negligible in comparison to the storage effect on the viscoelastic decay of pressure in a roll. 4) The pseudo 3D transient viscoelastic winding model has been used to analyze three baggy lane problems for validation. They are (1) the formation of cambered web (inplane imperfection) due to linearly varying thickness; and (2) the formation of localized baggy lanes due to an edge burr that resulted from slitting; and (3) the formation of a baggy web (outofplane imperfection) due to thickness increment in the middle of the web. Simulation results are compared with the experimental data, and a reasonably agreement was reached. 5) There are still some errors when we compare the numerical results with experimental results. For special case one (the formation of cambered web), the error is 2%. For special case two (the formation of localized baggy lanes due to an edge burr that resulted from slitting), the error is 2.2%. For special case three (formation of baggy web), the error is 9.2%, which is highest in these three cases. It shows that the shape of thickness profile is important in the estimation accuracy. 2D viscoelastic winding models based on finite element method (FEM) have been developed to determine boundary condition at the outer layer. In the numerical examples, three sets of viscoelastic material parameters, representing different levels of viscoelastic effects were used to investigate the viscoelastic effects on the timedependent stress or strain at the outer layer. After the winding process was finished and the tension was removed, the drop of stress at the outer layer is larger than that of outer layer strain. For highly viscoelastic material, the circumferential strain at the outer layer is nearly constant 7 5 after winding is complete. Moreover, the percentage of drop in circumferential stress is much larger than circumferential strain at outermost layer and the circumferential strain at outermost layer almost keeps constant for all three sets of viscoelastic material parameters. This has proven the boundary condition of constant circumferential strain in the outer lap chosen and experimentally verified by Qualls [47] is valid. 7 6 CHAPTER 6 6 FUTURE WORK 6.1 Hygrothermal Effects on Viscoelastic Material Properties In this section the hygrothermal effects induced by temperature change and moisture absorption will be considered. The work by Knauss and Kenner [45] indicated that the polymer response to increased temperature corresponds closely to that resulting from the absorption of moisture if the volume increased by temperature is the same as that increased by moisture. One comparison of master curves for polyvinyl acetate (PVAc) obtained from varying temperature and varying moisture is shown in Figure 61. According to the master curves shown in Figure 61, it seems hygroviscoelasticity and themoviscoleasticity are similar subjects that can be treated with similar models. Figure 61: Comparison of master creep curves for PVAc [45] 7 7 Defined the moisture as the ratio of weight gain to original weight, %m/m. As one example, thermomechanical behavior and hygromechanical behavior of PVAc were shown in Figure 62. Figure 62: (a) Creep curves for dry PVAc at several temperature; (b) Creep curves for PVAc at c o 24 at several absorbed moisture levels As shown in Figure 62, the moisture has similar effect on creep compliance as temperature. For example, during certain period ( 4 10 s), the effect on creep compliance caused by moisture increase from 0% to 0.9% is almost same as the effect caused by temperature increase from c o 15.4 to c o 33.7 . However, there are still some differences between these two factors. Usually material can only absorb limited changes in moisture before it saturates. Therefore, even though the moisture content of the environment is increased, it may still have not much impact on the material properties. Therefore, we may focus on thermomechanical behavior only for most web materials. For those materials which are easy to absorb moisture under the room temperature such as paper, it is necessary to consider the effect of hygrothermal change on material properties. 7 8 Moreover, it is assumed that the wound roll is subjected to a homogeneous hygrothermal change. The hygrothermal change will have influences on the following two aspects: (1) Hygrothermal effects on (temperature/moisture) shift factor; (2) Hydrothermal effects on stressstrain relationships. 1. Hygrothermal effects on (temperature/moisture) shift factor Using the timetemperature/moisture superposition principle, master curves for relaxation modulus can be described by ( , , ) = (ς, , ) ref ref E t T M E T M , with a (T M) t TM , ς = (6.1) The subscripts ref refers to the reference temperature or reference moisture; T is temperature; M is moisture; ς is reduced time; aTM is (temperature/moisture) shift factor. ) 1  1 log ( ref TM f f a = B (6.2) f is the fractional free volume; fref is the fractional free volume at Tref and Mref; B is the Doolittle constant. The fractional free volume includes the freevolume fraction due to both temperature and moisture. f f f f T M = T + m = ref + 3α% + 3β% (6.3) where fT and fm represent the free volume fraction due to temperature and moisture, respectively; α is the coefficient of thermal expansion; β is the coefficient of hygroscopic expansion; ΔT is the change in temperature; ΔM is the change in moisture. 2. Hydrothermal effects on stressstrain relationships T M E E r r r r r r + α % + β % σ ν σ ε = θ θ θ  (6.4) 7 9 T M E Er r r + α + β ν σ  σ ε = θ θ θ θ θ θ (6.5) The subscripts r and θ refer to the radial and circumferential directions; ν is Poisson’s ratio. The viscoelasic constitutive equations are given as dt T M t J t t t J t t r r t r r r r % β + % α + ′ ∂ ′ ∂σ + − ′ ∂ ′ ∂σ ε = ∫ − ′ θ θ 0 ( ) ( ) (6.6) dt T M t J t t t J t t t r r % β + % α + ′ ∂ ′ ∂σ + − ′ ∂ ′ ∂σ ε = − ′ θ θ θ θ θ ∫ θ 0 ( ) ( ) (6.7) These equations were discretized and implemented in a Fortran code to solve for stress distribution. It may be noted that the material properties in this model depends not only on time but also on temperature/moisture. The work in this section has been implemented into pseudo 3D winding code. However, the validation will be needed in the future. 6.2 Dimensionless Study 7 Dimensionless parameters, such as the ratio of maximum thickness to average thickness, storage time to retardation time, will be varied and used in simulations to determine a phase map showing the relationship between the web conditions (thickness variation, relaxation times) and the extent of web bagginess. With the development of the phase map, we hope that it can be used directly without recourse to the numerical code in the investigation of baggy lanes in some situations. 8 0 6.3 Optimization of Tension History 8 Optimization of tension history to reach predetermined timedependent tangential stress distribution in a roll. The tension history will be simulated by a polynomial with some control parameters. Through changing these parameters to achieve the minimum tangential stress in the wound roll. 6.4 Experimental Validations More experimental validations and verifications are needed to investigate stress variations in CMD and hygrothermal influences on would roll subjected to changing temperature and humidity environments. New experimental methodology is needed to measure the stress profile in CMD and find the temperature/moisture material coefficients. 8 1 REFERENCES [1] Qualls, W. R. and Good, J. K. (1997) “An Orthotropic Viscoelastic Winding Model Including a Nonlinear Radial Stiffness,” Journal of Applied Mechanics, 64, 201 208. [2] Hakiel, Z. (1991) “On the Effect of Width Direction Thickness Variations in Wound Rolls,” Web Handling Proceeding, 7998. [3] Good, J. K. (1999) “Preface,” Proceeding of the Fifth International Conference on Web Handling, Oklahoma State University, Stillwater, Oklahoma, 1999. [4] Roisum, D. R. (2001) “Baggy Webs: Making, Measurement & Mitigation thereof,” http://roisum.com/documents/Baggy.pdf [5] Altmann, H. C. (1968, April) “Formulas for Computing the Stresses in Centerwound rolls,” TAPPI, 51(4), 176179. [6] Hakiel, Z. (1987, May) “Nonlinear Model for Wound Roll Stress,” TAPPI, 70(5), 113117. [7] Kedl, D. M. (1991) “Using a Two Dimensional Winding Model to Predict Wound Roll Stresses That Occur due to Circumferential Steps in Core Diameter or to Crossweb Caliper Variation,” Web Handling Proceeding, 99112. [8] Lee, Y. M. and Wickert, J. A. (2002), “Stress Field in Finite Width Axisymmetric Wound Rolls,” Vol. 69, 130138. 8 2 [9] Hoffecker, P. (2006, May) “The Analysis of A Nip Impinged, Three Dimensional Wound Roll,” PhD thesis. [10] Lin, J. Y. And Westmann, R. A. (1989) “Viscoelastic Winding Mechanics,” ASME Journal of Applied Mechanics, 56, 821827. [11] Gutterman, R. P. (1959) “Theoretical and Practical Studies of Magnetic Tape Winding Tensions and of Environmental Roll Stability,” General Kinetics, contract no. DA18119SC42, Arlington, VA. [12] Pfeiffer, J. D. (1966, august) “Internal Pressures in a Wound Roll,” TAPPI, 49(8), 342347. [13] Pfeiffer, J. D. (1979) “Prediction of Roll Defects from Roll Structure Formula,” TAPPI, 62, 8388. [14] Yagoda, H. P. (1980) “Resolution of the Core Problem in Wound Rolls”. ASME Journal of Applied Mechanics, 47, 847854. [15] Tramposch, H. (1965) “Relaxation of Internal Forces in a Wound Reel of Magnetic Tape,” ASME Journal of Applied Mechanics, 32, 865873. [16] Tramposch, H. (1967) “Anisotropic Relaxation of Internal Forces in a Wound Reel of Magnetic Tape”. ASME Journal of Applied Mechanics, 34, 888894. [17] Qualls, W. R. and Good, J. K. (1999) “Thermal Analysis of a Wound Roll,” Journal of Applied Mechanics, 64, 871876. [18] Kotousov, A. and Wang, C. H. (2002) “Fundamental Solution for the Generalized Plane Strain Theory,” International Journal of Engineering Science, 40, 1775 1790. 8 3 [19] Timoshenko, S. P. And Goddier, J. N. (1970), Theory of Elasticity, 3rd ed., McGrawhall, new York, 422425. [20] Flügge, W. (1973) Stresses in Shells, 2nd ed, New York. [21] Minahen, T. M. And Knauss, W. G. (1993) “Creep Buckling of Viscoelastic Structures,” International Journal of Solids and Structures, 30(8), 10751092. [22] Keshavan, M. B. and Wickert, J. A. (1997) “Air Entrainment During Steady State Web Winding,” ASME Journal of Applied Mechanics, 64, 916922. [23] Kotousov, A. and Wang, C. H. (2002) “Threedimensional Stress Constrain in an Elastic Plate With a Notch,” International Journal of Solids and Structures, 39, 43114326. [24] Kotousov, A. and Wang, C. H. (2002) “Threedimensional Solution for Transversally Isotropic Composite Plates,” Composite Structure, 57, 445452. [25] Lai, J. and Bakker, A.(1995) “An Integral Constitutive Equation for Nonlinear PlastoViscoelastic Behavior of HighDensity Polyethylene,” Vol. 35, No. 17, p 1339. [26] Lekhnitskii, S. G. (1963) Theory of Elasticity of an Anisotropic Elastic Body, HoldenDay, Inc. San Francisco. [27] Olsen, J. E. (1999) “Modeling of Edge Rolls Defects,” Web Handling Proceeding. [28] Shelton, J. J. (1993) “Buckling of Webs from Lateral Compressive Forces,” Web Handling Proceeding, 303321. [29] Yuan, F.G., Yang, W. and Kim, H. (2000) “Analysis of Axisymmetricallyloaded Filament Wound Composite Cylindrical Shells,” Composite Structures, 50, 115 130. 8 4 [30] Zabaras, N., Liu, S., Koppuzha, J., And Donaldson, E. (1994) “A Hypoelastic Model for Computing the Stress in CenterWound rolls of Magnetic Tape,” ASME Journal of Applied Mechanics, 61, 290295. [31] Shelton, J. J. (2003) Private Communication on Tension Profiles. [32] Feiertag, B. (2001, March) Web Handling Seminar, Tab 4, Page 46, Oklahoma State University, Stillwater, Oklahoma. [33] Fischer, F. D., Rammerstorfer, F. G., Friedl, N. and Wieser, W. (2000) “Buckling phenomena related to rolling and levelling of sheet metal,” International Journal of Mechanical Sciences, 42, 18871910. [34] Timoshenko, S. P. and Gere, J. M. (1961) Theory of Elastic Stability, 2nd ed., McGrawHall, New York, 348439. [35] Shimizu, S. and Yoshida, S. (1991) “Buckling of Plates with a Hole under Tension,” ThinWalled Structures, 12, 3549. [36] Gilabert, A., Sibillot, P., Sornette, D., Vanneste, C., Maugis, D. and Muttin, F. (1992) “Buckling Instability and Pattern around Holes or Cracks in Thin Plates under a Tension Load,” European Journal of Mechanics, A/Solids, 11(1), 6589. [37] Shaw, D. and Huang, Y. H. (1990) “Buckling Behavior of a Central Cracked Thin Plate under Tension,” Engineering Fracture Mechanics, 35(6), 10191027. [38] Friedl, N., Rammerstorfer, F. G. and Fischer, F. D. (2000) “Buckling of Stretched Strips,” Computer and Structures, 78, 185190. [39] Cerda, E., RaviChandar, K. and Mahadevan, L. (2002) “Wrinkling of an Elastic Sheet under Tension,” NATURE, Vol.419, 579580. 8 5 [40] Tomita, Y. and Shindo, A. (1988) “Onset and Growth of Wrinkles in Thin Square Plates Subjected to Diagonal Tension,” International Journal of Mechanical Sciences, 30(12), 921931. [41] Lu, H., Good, J. K., Yu, H. and Poh, E. (2005, November) “Viscoelastic Effects on the Formation of Baggy Lanes in Webs (02003),” WHRC Project Report, 4 1~442. [42] Kandadai, B. (2006) “The Development of WoundOnTension in Webs Wound into Rolls,” PhD thesis. [43] Ferry, J. D. (1970) Viscoelasitc Properties of Ploymers, 2nd ed., J. Wiley, NY. [44] Swanson, R. P. (1999) “Mechanics of Nonuniform Webs,” Proceeding of the Fifth International Conference on Web Handling, Oklahoma State University, Stillwater, Oklahoma, pp. 443459. [45] Knauss, W.G. and Kenner, V. H. (1980) “On the Hygrothermomechanical Characterization of Polyvinyl Acetate,” Journal of Applied Physics, Vol. 51, No. 10, 51315136. [46] Cole, K. A. and Hakiel, Z. (1992) “A Nonlinear Wound Roll Stress Model Accounting for Widthwise Web Thickness Nonuniformities,” Proceedings of the 1992 Winter Annual meeting of the American Society of Mechanical Engineering, Symposium on Web Handling, AMDVolume 149, pp. 1324. [47] Qualls, W. R. (1995, May) “Hygrothermomechaical Characterization of Viscoelastic Centerwound Rolls,” PhD thesis. [48] Poh, E. (2007, May) “Cambered and Nonflat Polyethylene Webs Induced by Thickness Variation,” Master thesis. 8 6 APPENDIX A: A PSEUDO 3D TRANSIENT VISCOELASTIC WINDING / UNWINDING CODE The pseudo 3D transient viscoelastic winding / unwinding code has been validated by available experimental data in the 3D case. In the case where uniform thickness is used in the 3D code, the model is reduced to the viscoelastic code by Qualls and Good. The manual for the program is provided for users. Purpose: This Pseudo 3D Viscoelastic Winding Code is designed to model winding of a center wound roll with thickness variations on a personal computer. The rolls may have up to 100 different thicknesses across their width (limits are easily changed in the source code). The result files include pressure and radius profiles. This code includes several parts. One part is viscoelastic winding model that has considered viscoelastic effects after winding. This part is revised from the 2D viscoelastic winding code developed by Qualls and Good [1]. It is based on the work of Qualls and Good [1] and Cole and Hakiel [46]. The model considers winding of webs with varying widthdirection thickness. The model discretizes the width into smaller segments of varying length, each having a constant web thickness. Tension is assigned to each segment using the approach by Cole and Hakiel [46]; the tension is updated after the winding of each lap based on the deformed radius of the segment relative to the relaxed radius profile of that lap. In each segment, a 2D model developed by Qualls and Good is applied. The pseudo 3D model is capable of dealing with (1) a varying thickness profile; 8 7 (2) winding tension variation with the winding laps; and (3) varying core stiffness in the width direction. The second part is a pseudo 3D viscoelastic winding model that has considered viscoelastic effects during winding. This part has been developed for an orthotropic, nonlinear viscoelastic web based on the work by Qualls and Good [2]. The pseudo 3D model can consider (1) varying thickness profile; (2) winding tension variation as a function of winding laps; and (3) varying core stiffness in the width direction. The most important feature is that the model allows the consideration of viscoelastic effects in a wound roll during winding at a tension, so that the model is especially suitable for viscoealstic materials with short characteristic relaxation times, such as nonwovens and plastic webs with glass transition temperature close to room temperature. The third part is a pseudo 3D model has considered unwinding of a wound roll. During unwinding process, the viscoelastic memory effects are also considered in this unwinding model. The viscoelastic effects built in this model depend on both viscoelastic properties and winding conditions, such as unwinding speed and tension. In this model, the stress distribution in a roll comes from two sources, one from the initial winding, and the other from a correction state that starts from unwinding. The pseudo 3D viscoelastic winding code has been refined to shorten the computational time and allow for up to 1500 laps to investigate the effect of winding tension variations. The number of laps can be increased based on the capacity of the computer RAM. The improved pseudo 3D model allows an easy handling of tension history in winding and unwinding. If tapered tension option is chosen, winding tension can be changed through the change of the tension taper factor based on the work by 8 8 Shelton [31] and Feiertag [32]. Stress distribution can be determined for ten different tension profiles (ranging from constant tension to constant torque) for a roll with up to 1500 laps. Moreover the enhanced pseudo 3D model has been revised to allow nonpersistent thickness variation. The thickness profile can be different not only in CMD (Cross Machine Direction) but also in MD (Machine Direction). This manual includes a brief description of the underlying theory behind the code, a description of the needed input variables and their units, an example program run, an overview of the output files, and some notes about the program’s performance. Theory: The methodology behind the calculations used in this code is mostly based on the theory of pseudo 3D and viscoelastic winding mechanics. In the pseudo 3D theory the effects of width direction thickness variations in wound rolls are considered. Briefly, at first, we need to find the winding tensions at designated wound roll radii. We next determine the widthwise distribution of tension for all the laps in the wound roll. 8 9 Then the roll is split into independent widthwise segments. The winding tensions is then incorporated into any of the several existing models to compute the wound roll stresses (Qualls and Good model [1]). Qualls and Good model used the general Kelvin model to represent the viscoelastic behavior. A realistic numerical solution was developed to get reasonable results. Using numerical solution in the discretized form, we have ) ( ) 0 2 ) ( ( ) 2 ( ) ) ( ( ) 2 ( ) 2 ( ( ) ( ) 2 2 ( 1) 4 2 2 ( ) 1 2 2 2 ( 1) 3 1 2 1 + % + − % + − % + = + − F r h r F r h r F r h r F r F r h r F r h r F r r i i i r i i r i i i σ σ σ where F1,F2,F3,F4 contain all history effects of the viscoelastic material. Now there are N+1 unknown parameters, and have N1 equations. Plus 2 boundary conditions, we can get the results. When we consider the viscoelastic effects during winding, the current roll radius is a function of time. The relationship follows d(Vol) = 2π 



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