ORGANIZATION OF WOOD ELEMENTS IN PARTIALLY ORIENTED FLAKEBOARD MATS By Congjin L u M.Sc. Chinese Academy of Forestry, 1986 B. Sc. Nanjing Forestry University, 1983 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF DOCTOR OF PHILOSOPHY in T H E F A C U L T Y OF G R A D U A T E STUDIES Department of Forestry We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH C O L U M B I A October, 1999 © Congjin Lu, 1999 In presenting degree freely at this the thesis in partial fulfilment University of British Columbia, available for copying of department publication this or of reference thesis by this for his and scholarly or thesis for her of f~V /^JT7>"' The University of British Vancouver, Canada Date DE-6 (2/88) Columbia (ycj-. '4- , I further purposes the requirements I agree gain shall that agree may representatives. financial permission. Department study. of It not be is that the permission granted allowed an advanced Library shall by understood be for for the that without make it extensive head of my copying or my written ABSTRACT Partially oriented flakeboard mats play a significant role in commercial flake-based products, such as three-layered oriented strand board (OSB). The study presented in this thesis mathematically investigates the structure of partially oriented flakeboard mats. To better understand the nature of the structures of flakeboard mats, a simulation program Winmat®, based on the Monte Carlo technique, has been written to compute the horizontal distribution of overlap and density, free flake length and its distribution, number of flake crossings, the location and distribution of void sizes, the autocorrelation function, variance function and the degree of orientation of flakes in both simulated mats and experimental mats. This program can also determine the effect of sampling zone sizes on the density/overlap distribution. In the model development, flake position was considered to be random. The orientation angle of the flake was assumed to be random following either the Von Mises distribution or the uniform distribution. A mathematical model based on these distributions was developed. The autocorrelation function and variance function of the horizontal density distribution were investigated at different k values and 6 angles. The characteristic area concept from random field theory was first introduced to evaluate the degree of orientation of the flakes in a mat. In the process of estimating the degree of orientation of a flakeboard, the horizontal density distribution is needed to compute the autocorrelation function and the characteristic area. A non-destructive method, X-ray scanning technique, was used to determine the density profiles from experimental flakeboard mats. A model that maps X-ray voltage levels to ii overlaps and/or density was presented and discussed. The density and overlap were found to be a logarithm function of the X-ray intensity ratio (/(//: the intensity of the incident radiation to the intensity of radiation at location (JC, y) in a mat). A study of the relationships between thickness swelling and mat structure in robot-formed flakeboard mats made without wax was conducted under 95% and 90% relative humidity conditions and 24-hour water soaking tests. A model describing such relationships was established for two relative humidity conditions. With this model the thickness swelling of flakeboard mats (without wax) can be predicted, provided that the amount of moisture absorbed and the density distribution of the mat are known. Finally, a case study was presented to demonstrate the application of the models developed in the thesis. Two kinds of mats, partially oriented flakeboard mats and OSB mats, of size 2440 mm x 1220 mm were simulated and characterized. Their density/overlap profiles and degree of orientations were then compared with a commercial OSB panel whose density profile was obtained by X-ray scanning technique. The thickness swelling values of these simulated mats were predicted and the degree of orientation of the commercial OSB panel was presented. iii T A B L E O F CONTENTS ABSTRACT ii TABLE OFCONTENTS iv LIST O F TABLES viii LIST O FFIGURES ix ACKNOWLEDGEMENTS XV CHAPTER! L I T E R A T U R E R E V I E W A N D SCOPE O FTHESIS 1 1.1. Introduction 1 1.2. Literature Review 2 1.2.1. Density variation 3 1.2.2. Veneer strip model in a flakeboard mat 4 1.2.3. Uniform flake randomly-formed mat 5 Scope of Thesis 12 1.3.1. Definition of flake in mat network and computer simulation 13 1.3.2. Partially oriented flakeboard mat network 15 1.3.3. X-ray scanning technique 16 1.3.4. Thickness swelling 17 1.3.5. Application of the model 17 References 18 1.3. 1.4. CHAPTER II COMPUTER SIMULATION O FW O O D - F L A K E COMPOSITE M A T STRUCTURES 21 2.1. Introduction 22 2.2. Monte Carlo Simulation 23 2.3. Program Input and Output 25 2.4. Robot Control System 26 iv 2.5. Results and Discussions 27 2.5.1. Density and overlap distributions 27 2.5.2. Free flake length and its distribution 30 2.5.3. Void size and its distribution 31 2.5.4. Effect of sampling zone size 33 2.5.5. Comparison between different flake types 35 2.6. Conclusions 36 2.7. References 37 CHAPTER III R A N D O M F I E L D REPRESENTATION OF H O R I Z O N T A L DENSITY D I S T R I B U T I O N I N PARTIALLY ORIENTED FLAKEBOARD M A T S 39 3.1. Introduction 40 3.2. Theoretical Model 42 3.2.1. Probability density function for flake orientation 42 3.2.2. Point-to-point variance of the density 44 3.2.3. Variance function of the density 45 3.2.4. Correlation coefficient for two points in a rectangle 45 3.2.5. Probability density function for two points in a rectangle 51 3.2.6. Characteristic area - a measure of correlation 53 3.2.7. Degree of orientation 54 3.3. Evaluation of Density Image Autocorrelation 57 3.4. Results and Discussions 58 3.4.1. Correlation coefficient 58 3.4.2. Variance function of density 65 3.4.3. Characteristic area 69 3.4.4. Degree of orientation 71 3.5. Conclusions 71 3.6. References 73 v CHAPTER IV STUDY O N T H E X - R A Y C A L I B R A T I O N A N D O V E R L A P MEASUREMENTS I N ROBOT-FORMED FLAKEBOARD M A T S 76 4.1. Introduction 76 4.2. X-ray Theory and Overlap Model 78 4.2.1. X-ray theory and calibration 78 4.2.2. Mass absorption coefficient 80 4.2.3. Relationship between overlaps and intensity ratio 82 4.2.4. Image filter 83 4.3. Materials and Methods 84 4.4. Results and Discussions 86 4.4.1. X-ray calibration 86 4.4.2. Relationship between X-ray intensity ratio and flake overlaps 88 4.4.3. Flake overlaps from X-ray scanning images 89 4.5. Conclusions 95 4.6. References 95 CHAPTER V RELATIONSHIP BETWEEN THICKNESS S W E L L I N G A N D M A T STRUCTURES I N ROBOT-FORMED FLAKEBOARD M A T S 98 5.1. Introduction 98 5.2. Model 102 5.2.1. Strain and stress relationship 102 5.2.2. Effect of mat structures 104 5.3. Materials and Methods 105 5.4. Results and Discussions 110 5.4.1. Relationship between moisture content and time 110 5.4.2. Relationship between thickness swelling and moisture absorbed 113 5.4.3. Relationship between density and absorption coefficient 119 vi 5.4.4. Verification of the model 119 5.5. Conclusions 122 5.6. References 122 CHAPTER VI M O D E L A P P L I C A T I O N : A CASE STUDY 125 6.1. Simulation Parameters 125 6.2. Densities and Overlaps 127 6.3. Void Size and Distribution 129 6.4. Autocorrelation Functions and Variance Functions 131 6.5. Prediction of Thickness Swelling 134 6.6. Degree of Orientation of Commercial OSB 139 6.7. Future Research Work 141 6.8. Limitations 142 6.9. References 142 CHAPTER VII CONCLUSIONS 143 APPENDIX A PSEUDO-CODE FOR V A R I A N C E C A L C U L A T I O N P R O G R A M 146 A. 1. PDF for Two Points in a Square with Side Length of A 146 A.2. Correlation Coefficient for Two Points in a Rectangle 146 A.3. Variance Function 147 A.4. Variance Calculation for Von Mises Distributed Flake Orientation 147 A.5. Variance Calculation for Uniform Distributed Flake Orientation 150 vii LIST OF TABLES Table 2.1 Void sizes in the area of each layer 32 Table 2.2 Effect of sampling zone size on the density distribution 35 Table 4.1 Calculated mass absorption coefficients 82 Table 4.2 Scaling factors for X-ray intensity under different number of calibration plates 87 Table 4.3 Calibration coefficients for one pixel 87 Table 4.4 The statistical results for the X-ray analysis of three images of each mat structure as compared to computer simulation 93 Table 6.1 Simulation parameters 126 Table 6.2 Basic properties of the mats 128 Table 6.3 Void areas in the simulated OSB mat and the simulated partially oriented mat 130 Table 6.4 Predicted thickness swelling under 95% and 90% relative humidity conditions 135 Table 6.5 Distribution of thickness swelling of simulated OSB in horizontal plane 138 viii LIST OF FIGURES Figure 1.1 Scope of the thesis research 12 Figure 1.2 Flake position and orientation defined in three-dimensional space 13 Figure 1.3 Horizontal density variation in a flake mat 16 Figure 2.1 Position and orientation of the flake in a mat 24 Figure 2.2 Robot control system 26 Figure 2.3 Flake overlap and density distributions in the composite mat 28 Figure 2.4a Graphical representation of horizontal density distribution by surface map 29 Figure 2.4b Graphical representation of horizontal density distribution by contour map 30 Figure 2.5 The free flake length distributions with the data predicted by Dai's mathematical model and the current simulation study 31 Figure 2.6 The effect of sampling zone sizes on the standard deviation of density of flakeboard mat Figure 2.7 Comparison of area ratio obtained from three different flake types 34 36 Figure 3.1a Schematic diagram for the range of angles within Range 1 for a flake with length X and width co during integration 48 Figure 3.1b Schematic diagram for the range of angles within Range 2 for a flake with length X and width co during integration 49 Figure 3.1c Schematic diagram for the range of angles within Range 3 for a flake with length A and width co during integration Figure 3.2 Probability density for the distance between two points in square zones 50 52 Figure 3.3 Correlation coefficients for different values of concentration parameter k in Von Mises distribution 55 ix Figure 3.4 Correlation coefficients for different ranges of angles 0\ in uniform distribution 56 Figure 3.5 Correlation coefficient between two points in the mat with various mean direction (#<?) in Von Mises distribution 60 Figure 3.6a Correlation coefficient between two points in a mat (3D graphical representation) in completely randomized distribution of flake location and orientation 60 Figure 3.6b Correlation coefficient between two points in a mat (contour map) in completely randomized distribution of flake location and orientation 61 Figure 3.6c Correlation coefficient between two points in a mat (3D graphical representation) with partial orientation of flakes 61 Figure 3.6d Correlation coefficient between two points in a mat (contour map) with partial orientation of flakes 62 Figure 3.6e Correlation coefficient between two points in a mat (3D graphical representation) in perfectly aligned flake orientation (0°) and random location 62 Figure 3.6f Correlation coefficient between two points in a mat (contour map) in perfectly aligned flake orientation (0°) and random location 63 Figure 3.6g Correlation coefficient between two points in a mat (3D graphical representation) in perfectly aligned flake orientation (45°) and random location 63 Figure 3.6h Correlation coefficient between two points in a mat (contour map) in perfectly aligned flake orientation (0°) and random location 64 Figure 3.7 Comparison of correlation coefficients between model prediction and computer simulation 64 x Figure 3.8 The variance reduction with respect to different k values in Von Mises distribution and different side length of square zones 66 Figure 3.9 The variance reduction with respect to different ranges of angles in uniform distribution and different side length of square zones 66 Figure 3.10 The variance reduction rate with respect to k values in Von Mises distribution 67 Figure 3.11 The variance reduction rate with respect to ranges of angles in uniform distribution 67 Figure 3.12 The comparison of variance function for the perfect aligned flake by Von Mises distribution (k - 700) and uniform distribution (9\ =0) 68 Figure 3.13 The comparison of variance function for the randomly aligned (+90) and perfectly aligned flakes by model prediction and simulation 68 Figure 3.14 Characteristic area in relation to the concentration parameter k in Von Mises distribution of flakes 69 Figure 3.15 Characteristic area in relation to the ranges of angles in uniform distribution of flakes 70 Figure 3.16 Characteristic area in relation to the average variance reduction from random orientation to perfect alignment 70 Figure 3.17 Degree of orientation of flakes with respect to concentration parameter k in Von Mises distribution 72 Figure 3.18 Degree of orientation of flakes with respect to the ranges of angles in uniform distribution 72 Figure 4.1 Schematic representation of the mat structures 85 Figure 4.2 Schematic diagram of the X-ray scanning system 85 Figure 4.3 The relationship between X-ray intensity ratio and voltage levels 88 xi Figure 4.4 Flake overlaps in relation to X-ray intensity ratio 89 Figure 4.5a Horizontal density distribution images from X-ray measurements in structure I 91 Figure 4.5b Horizontal density distribution images from X-ray measurements in structure II 92 Figure 4.6 Flake overlaps in a particular scanning line in the mat area for the simulation and X-ray measurements in structure I 94 Figure 4.7 Comparing the normalized standard deviation of density for the simulation and X-ray measurements in structure I 94 Figure 5.1 Schematic diagram of mat structures 107 Figure 5.2 Contour map of horizontal density distribution 107 Figure 5.3. Specimen cutting pattern corresponding to each square in Figure 5.2 108 Figure 5.4a Local density averages of robot-formed flakeboard mats (TS1, TS2, TS3) as compared to the simulated mat 108 Figure 5.4b Local density averages of robot-formed flakeboard mats (TS4, TS5, TS6) as compared to the simulated mat 109 Figure 5.5a Absorbed moisture and thickness swelling in relation to test time under 95% relative humidity test conditions 111 Figure 5.5b Absorbed moisture and thickness swelling in relation to test time under 90% relative humidity test conditions 112 Figure 5.6 Absorbed moisture of flakeboard in relation to square root of time under 95% and 90% relative humidity test conditions 112 Figure 5.7 Water absorption and thickness swelling in relation to soaking time during 24-hour water soaking test 113 xii Figure 5.8a The correlation between thickness swelling and absorbed moisture under 95% relative humidity condition 114 Figure 5.8b The correlation between thickness swelling and absorbed moisture under 90% relative humidity condition 115 Figure 5.8c The correlation between thickness swelling and absorbed moisture under 24-hour water soaking test 115 Figure 5.9a The correlation between the relative thickness swelling and relative moisture absorbed at each time interval under 95% relative humidity condition 116 Figure 5.9b The correlation between the relative thickness swelling and relative moisture absorbed at each time interval under 90% relative humidity condition 116 Figure 5.9c The correlation between the relative thickness swelling and relative moisture absorbed at each time interval under 24-hour water soaking test 117 Figure 5.10a The correlation between the rate of thickness swelling and the rate of moisture changes under 95% relative humidity condition 117 Figure 5.10b The correlation between the rate of thickness swelling and the rate of moisture changes under 90% relative humidity condition 118 Figure 5.10c The correlation between the rate of thickness swelling and the rate of moisture changes under 24-hour water soaking test 118 Figure 5.11 The relationship between absorption coefficient and density of flakeboard under 95% and 90% relative humidity test conditions 120 Figure 5.12a The predicted and measured thickness swelling in relation to absorbed moisture for three density levels (0.66, 0.62 and 0.56 g/cm3) under 95% relative humidity test condition 120 Figure 5.12b The predicted and measured thickness swelling in relation to absorbed moisture for three density levels (0.66, 0.62 and 0.56 g/cm3) under 90% xiii relative humidity test condition 121 Figure 5.13 The predicted and measured absorption coefficients in relation to density under 95% and 90% relative humidity test conditions 121 Figure 6.1 Density variation of the simulated OSB and the commercial OSB 128 Figure 6.2 Void measurement and distribution in a part of a randomly formed layer 129 Figure 6.3 Autocorrelation function for the simulated partially oriented flakeboard 132 Figure 6.4 Autocorrelation function for the simulated OSB 133 Figure 6.5 Autocorrelation functions of the simulated partially oriented flakeboard, the simulated OSB and the commercial OSB 133 Figure 6.6 Variance functions of the simulated partially oriented flakeboard, the simulated OSB and the commercial OSB 134 Figure 6.7a 3D representation of the predicted thickness swelling of the simulated partially oriented flakeboard under 95% relative humidity condition 136 Figure 6.7b 2D representation of the predicted thickness swelling of the simulated partially oriented flakeboard under 95% relative humidity condition 136 Figure 6.8a 3D representation of the predicted thickness swelling of the simulated OSB under 95% relative humidity condition 137 Figure 6.8b 2D representation of the predicted thickness swelling of the simulated OSB under 95% relative humidity condition 137 Figure 6.9 Degree of orientation of commercial OSB in corresponding to a constant characteristic area Figure 6.10 Degree of orientation of commercial OSB in corresponding to a constant characteristic area 140 140 xiv ACKNOWLEDGMENTS I would like to thank my supervisor Dr. Frank Lam, for his invaluable advice, tremendous suggestions, and help throughout the thesis research. My special thanks should go to my previous supervisor Dr. Paul R. Steiner (deceased) for his initial discussion and the suggestions for the development of the thesis topic. Thanks also go to my supervisory committee members: Dr. David Barrett, Dr. Simon Ellis, Faculty of Forestry, U B C , and Dr. David Plackett, formerly Forintek Canada Corp. Acknowledgement also goes to Bob Myronuk and Avtar Sidhu for their laboratory assistance, Ms. Kaiyuan Wang for providing some test panels for X-ray scanning. Other help from Dr. Liping Cai, Dr. Jack Biernacki, Dr. Chunping Dai, Sunguo Wang, Bingye Hao, Bingning Zhou and Pablo Carcia is also readily acknowledged. I also would like to thank Ainsworth Lumber Co. for contributing testing materials, C A E Industries and U B C Carpenter shop for providing flaking equipment. Fellowships received for the thesis study from Weyerhaeuser in Wood Design, University of British Columbia, and Donald McPhee are gratefully acknowledged. Finally, my greatest gratitude goes to my wife - Hua and daughter - Dina for their patience, understanding and support for my educational studies. xv Chapter I: Literature Review and Scope of Thesis CHAPTER I LITERATURE REVIEW AND SCOPE OF THESIS 1.1. Introduction The engineering of modern composite materials has had a significant impact on their technology of design and construction. By combining two or more materials together, it is possible to make advanced materials which may be lighter, stiffer and stronger, and better materials to fulfill end-use requirements than any single material ever used before. The structure of a wood flake composite mat may be defined as the geometric arrangement of the constituent flakes or strands in the mat. If only the mat structure is concerned, the mechanical properties of flakeboards depend primarily on the number of flake-to-flake contacts or flake-crossings and the physical properties depend primarily on the compression behavior of the mat. A better understanding of the internal mat structure will help us to characterize properties of such wood composites and to utilize them more efficiently. Wood composites require consolidation during the manufacturing process in order to reach a certain strength. This process often leads to dimensional stability problems and higher than desirable panel densities which have a negative effect on production cost and weight. A critical, but not yet well understood factor in performance is the packing arrangement (i.e., orientation and position) of the wood elements in a mat. This factor is believed to affect the horizontal density variation (Suchsland and Xu 1989). More attention has been paid to this research area recently and a two-dimensional multi-layer flake mat model has been developed (Dai 1993, Dai and Steiner 1994a, b and c). Within this model, the formation of a 1 Chapter I: Literature Review and Scope of Thesis short fiber composite mat was thought of as a random process due to the random nature of the constituent deposition. The structural properties of the flake network were random variables which could be characterized by Poisson and exponential distributions (Dai 1993). However, most of the earlier studies in this area assumed uniform flake geometry and completely randomized orientation of flakes in the mat, which is not realistic. These results cannot be directly applied to commercial products, such as a three-layer oriented strand board (OSB) where the flakes are partially oriented. In this thesis, emphasis was placed on improving the understanding of the relationship between horizontal density variation and properties of wood composite mats. To achieve this aim, mathematical models along with the computer simulation, robot mat formation and Xray scanning techniques were used. Knowledge gained from the combination of the mathematical models, simulation and predefined mat structures made by a robot, and quantitative data on density distribution scanned by an X-ray system, will help to guide improvements in present mat forming technology and also provide some guidance of future wood composites development. 1.2. Literature Review Wood products can be broadly classified into two categories: solid sawn products and reconstituted wood products such as flakeboards and fiberboards. The second category is the topic of the current study. The properties of different wood composites vary greatly depending on different raw materials, geometry and arrangement of their constituents and manufacturing parameters. 2 Chapter I: Literature Review and Scope of Thesis 1.2.1. Density variation Short element wood composites consist of wood elements inter-dispersed with voids in a manner that results in a distribution of density along the horizontal plane. In a mat, the density distribution can be further subdivided into a vertical component and a horizontal component. The vertical density distribution, which is the combined results of temperature, moisture, and compressive stress of wood perpendicular to the panel during hot pressing, has been extensively studied (Heebink 1972, Plath and Schnitzler 1974, Suchsland 1962, etc.) and well documented (Kelly 1977). The horizontal density distribution as well as physical and mechanical properties in the plane, such as thickness swelling, bonding strength, modulus of rupture and modulus of elasticity are determined by the geometry and arrangement of constituent elements (Dai and Steiner 1994a, 1994b and 1994c, Suchsland 1959, Xu 1994). The two most important factors controlling the mean final density of a mat are the density of raw materials and the compaction of the mat in a hot press. The density of raw material can be assumed to be constant for a given wood species. However, the pressing operation eliminates many void volumes in a mat and consolidates the mat to a desired thickness. The compaction of the mat to an average density higher than the density of the raw material will allow better surface contacts among individual flakes in the mat (Kelly 1977). Suchsland (1959 and 1962) developed a statistical model for a flakeboard mat relating the degree of densification to flake geometry, relative void volume and the density of wood components. He concluded that a higher pressure is required to reach a desired density for narrower and thicker flakes as opposed to wider and thinner flakes and that the relative compression area, 3 Chapter I: Literature Review and Scope of Thesis which is a portion of total mat area under compression, is a significant factor in developing bending strength in a flakeboard. The distribution of total flake coverage varies randomly in a mat so that the regions of higher flake coverage are obviously compressed to a greater extent in comparison with corresponding regions of lower flake coverage when the entire mat is compressed to a given uniform thickness (Suchsland 1959). In randomly formed mats, the larger relative void volume corresponds to a larger flake length to width ratio (aspect ratio) and the smaller relative void volume corresponds to a smaller aspect ratio (Suchsland 1959). 1.2.2. Veneer strip model in a flakeboard mat Suchsland and Xu (1989) demonstrated a model using parallel veneer strips to study the horizontal density variation in a flakeboard. This model is a layered mat with each layer consisting of a number of uniform size veneer strips organized in parallel with air space interspersed. The structure of a multiple-layer mat is the same as that of plywood, i.e. 90° in any two adjacent layers. An increase in the interspersed air gap in the mat layers results in more severe variation of the horizontal density in the densified panel. If we build a veneer strip mat in such a way that there is no air gap involved, the number of layers, N, will be achieved by the following equation N = JL {av) M =0 (1.1) where T = total number of veneer strips, M = number of flakes in one layer, and 4 Chapter I: Literature Review and Scope of Thesis 0(av) = average number of overlaps. In order to characterize the variation of the number of veneer overlaps in a mat, some air spaces will be allowed in each layer. Therefore the total number of layers will be larger than that of the perfectly packed mat if the number of veneer strips remains the same, resulting in some points with more overlaps than others. The distribution of the number of veneer strips over any small area follows a binomial distribution (Suchsland 1959): P(i) = •p'q " n (1.2) where P(i) = fraction of total area over which the number of solid veneer elements equal i, n = total number of flake layers, p = relative wood volume of each veneer layer, and q = 1- p = relative air volume of each veneer layer. This model allows the systematic modification of the horizontal density variation and the evaluation of its effects on the quality of the densified and consolidated mat. Although this model has many limitations in the practical sense, such as flake position and orientation, it does reveal some important facts, such as overlaps and voids, in a randomly formed flakeboard mat and provides the basic concept for further model development. 1.2.3. Uniform flake randomly-formed mat A mathematical model was developed to describe the structural characteristics of flakeboard, 5 Chapter I: Literature Review and Scope of Thesis assuming a uniform flake size and a random formation process (Dai, 1993). Since wood flake mats or flakeboards can be considered as a number of vertically stacked layers in two dimensions, this model deals with the most basic structural units of an idealized wood composite - randomly formed flake layers. The random process here refers to a random flake deposition and a random flake orientation. In the spatial structure of the wood composite mat in relation to processing and performance characteristics, flake geometry will affect relative void volume in a mat. Orientation of elements plays an important role in optimizing directional strength properties and improving packing behavior. By developing a model which incorporates flake overlap probability together with linear and non-linear compression stress perpendicular to the grain for a column of randomly packed flakes, predictions can be made of the stress-strain behavior in a mat (Steiner and Dai, 1993). In the two-dimensional mathematical model, the structure related concepts in each layer are described as: distribution of free flake length and distribution of void size (Dai and Steiner 1994a). The multi-layered spatial model allows calculation of the distribution of flake centers, distribution of flake area coverage, the distribution of local density averages, relative flake to flake contact area, internal stress and relative void volume (Steiner and Dai 1993). Most of these concepts are common to the structure of paper because of the similar structures in both flakeboard and paper (Kallmes and Corte 1960, 1961, 1963). Distribution of flake centroids: A randomly formed flake layer is defined as a horizontal plane with an area A where a limited number of flakes, Nf, with length X, width co and thickness t are independently deposited by a random process (Dai and Steiner 1994a). It is assumed that all the flakes are 6 Chapter I: Literature Review and Scope of Thesis horizontally positioned and the total flake coverage area A = N A.co. The flake centers in N f L multi-layers can be described by a Poisson process when the number of flakes (N xN ) L f is very large (Dai 1993). If a large network is divided into a number of small squares with area S, the probability of finding i centers in a square, P(i), is defined by (Hall 1988): n .e (1.3) P(i) = —— v. where n = (N N )S I A, the average value of flake centers in a square S. L f Distribution of flake area coverage: Since the mat formation follows a random process, some areas will inevitably have more than one flake, resulting in overlaps (Dai and Steiner 1994a). The distribution of these overlaps in any arbitrarily chosen point is also given by the Poisson distribution (Hall 1988), which is an approximation of the binomial distribution of Equation 1.2: ni.e~ Sf (1-4) P,(i) = - -^— tl J where the mean number of flakes covering any point equals to n f - ^ L regardless of how the flakes are distributed over the area. Distribution of free flake length: The free flake length is defined here as the distance between any two adjacent flake crossings over one flake, which is analogous to free fiber length in paper structure. The distribution of 7 Chapter I: Literature Review and Scope of Thesis free fiber length is considered to be related to the deformation behavior of paper and the void distribution (Kallmes and Bernier 1963). The number of crossings per fiber can be considered as the number of "anchors" which can influence the mechanical strength of paper (Corte 1982). The probability of free flake length (the distance between two adjacent intersects between m and m + dm) , Putim), follows an exponential distribution (Dai and Steiner, 1994a, Hall 1988, Kallmes et al, 1960, 1963), m P (m) M = m ^ (1.5) m where ra = the mean distance. Distribution of void size: The basis for the void size distribution is the distribution of polygon areas generated by random lines (Miles 1964). The number of crossings is a statistical parameter, which provides the link to the mechanical properties of the network. For random points on a line, the distribution of gaps is of a negative exponential type (Miles 1964). For random lines in a plane, the distribution of inter-crossing distances is therefore also negative exponential (Deng and Dodson 1994). These crossing distances (free flake length) form the sides of polygons and the distribution of the polygon areas for random lines in a plane is given by (Kallmes 1960, Dai and Steiner 1994a) (1.6) where 8 Chapter I: Literature Review and Scope of Thesis a = the area of individual polygon (void) which is assumed to be proportional to v the square of the free flake length m, i.e. a = am 2 v where a - constant coefficient, obtained through mathematical derivation (Dai and Steiner 1994a) Ae ~ > (P xrn 2(N -N )m a 2 C f where A = the area of plane Nf = number of flakes N = total number of crossings c B — —J_ — is the function of average length, A, and width, a>. \ + col A The polygons, into which the area of the network is divided, provide the link to its porous structure and porosity properties. If in a large network each line is changed into a constant width of plane, some small polygons will disappear and larger ones will be reduced in size (Corte 1982). Variation of local mass density averages: A flake mat can be considered as a two-dimensional fibrous network with a structure 9 Chapter I: Literature Review and Scope of Thesis exhibiting local variations of the area mass density (MD) in the horizontal plane. The mean mass density, E(MD), over a mat area was obtained by Dai and Steiner (1994b): E(MD) = NfAcorD f (1.7) A The variance of regional mass density average, Var(MD ), is expressed as (Vanmarcke, a 1983): Var(MD ) = y(s , s )Var{MD) a x y (1.8) where y(s ,s ) x is defined as the variance function of local density averages, which measures the reduction of the point variance Var(MD) under regional averaging, Var(MD) = rp E(MD), f the variance of local mass density (Dai and Steiner, 1994b), Pf is the flake density, and ris the flake thickness. Relative bonded area: Since flake overlaps have a distribution of the form of Equation 1.4, areas with more flakes will be compressed during hot pressing more than the others. The relative bonded area, which depends mainly on the compaction ratio (the board density to flake density ratio) and flake thickness, is defined by Dai and Steiner (1993) as: RBA = (1.9) where 10 Chapter I: Literature Review and Scope of Thesis T = board thickness, r= average flake thickness. Relative void volume: The void volume in a mat consists of two parts: voids between flakes and pores inside flakes. The total relative void volume, RV , in a board can be obtained by: t (1.10) where V = void volume v V, - overall board volume po = wood cell-wall density (relatively constant at 1.5-1.55 g/cm ) 3 p = overall board density The relative void volume between flakes, RVbf, is determined as (Dai and Steiner 1993): Tit (1.11) therefore the relative void volume inside flakes, i?v,j, is calculated by subtracting the relative void volume between flakes from the total relative void volume, i.e., RV =RV,-RV if b (1.12) The total relative void volume decreases linearly with the increasing compaction ratio while 11 Chapter I: Literature Review and Scope of Thesis the voids between flakes decrease dramatically with increasing compaction ratio. 1.3. Scope of Thesis The scope of this research thesis consists of the computer simulation of wood flake composite mats, development of mathematical models for partially oriented mat structures and thickness swelling prediction, use of a robot mat forming system to make experimental mats, and application of X-ray scanning and gravimetric techniques to measure horizontal density distribution of the composite mats (Figure 1.1). Computer Simulation Winmat® Autocorrelation Flake orientation Variano Model Mat Structures Partially Oriented Flake positionN. lake orientatiorjX Degree of orientation^ 7 Experimental Mat Robot ® mat Forming System Gravimetric measurement Horizontal Density and/or Overlap Distribution Figure 1.1 Scope of the thesis research 12 Chapter I: Literature Review and Scope of Thesis 1.3.1. Definition of flake in mat network and computer simulation Consider a thin flake with length A and width co arbitrarily located and oriented in space. The position and orientation of the flake can be defined by 1) the centroid coordinate of the flake, P (x, y, z), and 2) the orientation of the flake plane with respect to the X Y Z Cartesian coordinate system. Further consider the flake plane on which the flake is resided, one can project a vector from the origin to the flake plane such that is normal to the plane. The vector will make an angle #(0<c2<90 ) with respect to the z-axis. The projection of this vector onto 0 the X - Y plane will make another angle cp ( - 9 0 ° <<p<90°) with respect to the x-axis (Figure 1.2). Therefore the flake is well defined in the mat network by the five parameters (x, y, z, 0, <p). x Figure 1.2 Flake position and orientation defined in three-dimensional space. However, a flake can be oriented at any angle on this plane if no further restrictions are specified. Therefore, the direction of flake length is selected as a reference to the flake orientation. The center line in flake length direction can be expressed by Oi ( 0 < # / < 9 0 ° ) , the 13 Chapter I: Literature Review and Scope of Thesis angle between the z axis and the line itself, and cpi (-90°<^/<90°), the angle between the x axis and the normal projection of the center line onto the X-Y plane (Figure 1.2). The position and orientation of a flake can therefore be explicitly described by seven parameters (x, y, z, 0, q>, di, (pi). However, there are only six independent variables because the normal axis of a flake is perpendicular to any lines on the flake plane. Their product of direction cosines equals to zero. It can be shown that the following relationship (Equation 1.13) holds for four orientation angles (0, cp, Qi, (pi). sin#-sin#, • cos(<2>-#>,) +cos cos =0 (1-13) From the formation point of view, the evolution of a mat is really a random array of flakes in three dimensions since the flakes are not always deposited in a plane normal to the direction of the applied pressure and some of them might stand on the edge or on the end (Suchsland 1959). However, once it is formed by a forming belt, the mat is essentially a layered structure (Suchsland 1967, Dai and Steiner 1994b). This can be seen crudely by delaminating a mat and the observation that almost all the flakes lie parallel to the horizontal plane of the mat. Here a "layer" is defined as the average coverage over an area A with the thickness of one flake, and a mat could be considered as a multi-layered random flake mat as a summation of a series of two-dimensional randomly formed flake layers. Therefore, assuming all the flakes are deposited parallel to the horizontal plane (# = 0) in a predefined area, L x W mm plane, each flake has a definite position in the mat. The parameters needed to define the position and orientation of a flake in space are reduced to three variables, which can be described by x and y coordinates (0<x<L, 0<y<W) for flake centroids and orientation angles cz>(-90°<cz><90 ). The data (x.y.tp) can be numerically 0 14 Chapter I: Literature Review and Scope of Thesis simulated by computer or the results from experimental measurements. Based on this definition, a simulation program for mat formation process (Winmat®) program for controlling a robot mat former (Robot®) and a have been developed. This simulation work was carried out to study the nature of a random or partially random formation process of a mat. The variation of horizontal density depends mainly on the flake position and orientation in a mat, while the standard deviation of the horizontal density variation depends on the sampling zone sizes, that is, the smaller the sampling zone size, the higher the horizontal density variation. This part of the thesis work will be presented in Chapter n . 1.3.2. Partially oriented flakeboard mat network The nature of the flake mat formation is a random process unless the flake orientations or locations are controlled or predefined. The concept of flake distribution is designed to facilitate description of the arrangement of flakes in a mat and such an arrangement is based on two dimensional or multi-layered models. A common feature of these models is the more or less implicit assumption that three-dimensional assemblies of flakes can be represented by superimposing two dimensional flake structures. The mechanical properties of the network are essentially dependent on the properties, geometry and compaction behaviors of the constituent flakes in a complex manner. Two distribution functions are used for describing the flake orientations. One is the uniform distribution, which has an equal probability of choosing any angle within the range of -9\ to +6\, and the other is the Von Mises distribution characterized by a shape factor, called the concentration parameter k. Both distributions have extreme conditions, i.e., completely 15 Chapter I: Literature Review and Scope of Thesis randomized orientation when 9 is between - 9 0 ° and +90° or the concentration parameter k -» 0, and the perfectly aligned orientation when 9 is 0 or k —» oo. However, the partially oriented flake mat network is used to refer to the conditions with 0<c9i<90° in the uniform distribution and with k > 0 but finite in the Von Mises distribution. Details will be presented in Chapter III. 1.3.3. X-ray scanning technique Figure 1.3 Horizontal density variation in a flake mat. The horizontal density variation of experimental mats (Figure 1.3) was determined by the Xray scanning technique. With the X-ray device used, a single line of data with 128 pixels (1 pixel = 1 byte) in 150 mm wide strip of material can be measured at a time. One 16 Chapter I: Literature Review and Scope of Thesis measurement takes 1725 lines of data in a few seconds and stores it in one file. The noise can be reduced by averaging 1725 data at the same point. However, what we get from the scanned data is the voltage level representing the attenuated X-ray radiation. The voltage data is then calibrated and converted to X-ray intensity, which is used to evaluate the horizontal density distribution or overlap distribution. Chapter IV presents the detailed procedures and results from this study. 1.3.4. Thickness swelling Thickness swelling is a physical property of a material subject to the absorption of moisture. It is a measure of the relative thickness change based on the original thickness. The thickness swelling model is derived from the strain and stress rate model proposed by Martenson (1994). Use is made of the fact that the definition of strain is the same as the definition of thickness swelling. The thickness swelling is then related to both relative moisture content and the density (or flake overlap) of flakeboard. More details will be presented in Chapter V. 1.3.5. Application of the model Chapter VI presents a case study on how the models developed in the thesis can be used to analyze the characteristics of the simulated partially oriented flakeboard mats, simulated OSB mats and a commercial OSB panel. The specifications of these mats are given first. The partially oriented flakeboard model developed in Chapter III is then applied to predict the horizontal density distribution, void size distribution, autocorrelation function, variance function, degree of orientation and thickness swelling. Finally some future research work on 17 Chapter I: Literature Review and Scope of Thesis the current model is discussed. In the last chapter, Chapter VII, a brief summary on the general approach and the nature of the proposed model is given and some major conclusions are drawn based on the analysis presented. 1.4. References Corte, H. 1982. The structure of paper. Handbook of Paper Science, Volume 2. The structure and physical properties of paper, edited by H. F. Ranee. Elsevier Scientific Publishing Co., Amsterdam. Corte, H. and O. J. Kallmes. 1962. The interpretation of paper properties in terms of structure. In The Formation and Structure of Paper. Vol. 1 (F. Bolam Ed.) British Paper and Board makers Association, London. pp351-368 Dai, C. 1993. Modeling structure and processing characteristics of a randomly formed woodflake composite mat. Ph.D. Thesis. Department of Wood Science, U B C , October. Dai, C. and P. R. Steiner. 1993. Compression behavior of randomly formed wood flake mats. Wood and Fiber Science, 25(4): 349-358 Dai, C. and P. R. Steiner. 1994a. Spatial structure of wood composites in relation to simulation of a randomly formed flake layer network. Part 2. Modeling and simulation of a randomly formed flake layer network. Wood Science and Technology, 28:135-146 Dai, C. and P. R. Steiner. 1994b. Spatial structure of wood composites in relation to 18 Chapter I: Literature Review and Scope of Thesis processing and performance characteristics. Part 3. Modeling the formation of multi-layered random flake mats. Wood Science and Technology, 28:229-239 Dai, C. and P. R. Steiner. 1994c. Analysis and implication of structure in short fiber wood composites. Second Pacific Rim Bio-Based Composites Symposium, November 6-9, Vancouver, Canada, pi7-24 Deng, M. and C.T.J., Dodson. 1994. Paper: an engineered stochastic structure. Tappi Press, Atlanta, Ga. 308pp Hall, P. 1988. Introduction to the theory of coverage processes. New York: John Wiley & Sons. Heebink, B. G. 1972. Irreversible dimensional changes in panel materials. Forest Products Journal, 22(5): 44-48 Kallmes, O., and H., Corte. 1960. The structure of paper: I. The statistical geometry of an ideal two-dimensional network. Tappi 43(9): 737-752 Kallmes, O.; H., Corte. and G. Bernier. 1961. The structure of paper: II. The statistical geometry of an multiplanar fiber network. Tappi 44(7): 519-528 Kallmes, O., and G. Bernier. 1963. The structure of paper: IV. The free flake length of a multiplanar sheet. Tappi 46(2): 108-114 Kelly, M. W., 1977. Critical literature review of relationships between processing parameters and physical properties of particleboard. General Technical Report FPL-10. Forest Prod. Lab., Forest Service, USDA. 65pp 19 Chapter I: Literature Review and Scope of Thesis Miles, R. E. 1964. Random polygons determined by random lines in a plane. Proceedings of the National Academy of Sciences of the United States of America. 52:901-907, 1157-1160 Plath, L. and E. Schnitzler. 1974. The density profile, a criterion for evaluating particleboard. Holz Roh-Werkst, 32(11): 443-449 Steiner, P. R. and C. Dai. 1993. Spatial structure of wood composites in relation to processing and performance characteristics. Part 1. Rationale for model development. Wood Science and Technology, 28: 45-51 Suchsland, O. 1959. An analysis of the particle board process. Quarterly Bulletin. Michigan Agricultural Experimental Station, Michigan State University, East Lansing, Michigan. Vol. 42(2): 350-372 Suchsland, O. 1962. The density distribution in flake boards. Michigan Quarterly Bulletin 45(1): 104-121 Suchsland, O. 1967. Behavior of a particleboard mat during the pressing cycle. Forest Products Journal, 17(2): 51-57 Suchsland, O. and H., Xu. 1989. A simulation of the horizontal density distribution in a flakeboard. Forest Products Journal, 39(5): 29-33 Xu, W. 1994. Horizontal density distribution of particleboard: origin and implications. Ph.D. thesis, Department of Wood Science, U B C . 20 Chapter II: Computer Simulation of Wood-Flake Composite Mat Structures CHAPTER II COMPUTER SIMULATION OF WOOD-FLAKE COMPOSITE MAT STRUCTURES * Abstract Based on the mathematical model developed by Dai and Steiner (1993, 1994a and 1994b) and structural characteristics defined in this thesis, a Monte Carlo simulation program was written to run in a Microsoft Windows environment. Several kinds of simulations and calculations can be performed using the program. In this study, simulated mats that were randomly generated by a computer were analyzed in terms of the following structural characteristics: flake deposition, flake area coverage, free flake length and its distribution, flake crossings, location and distribution of void size, and density and overlap distributions. The simulation program can also determine the effect of sampling zone size on the maximum, minimum and mean density, standard deviation and mean deviation of density distribution. A similar analysis could also be performed on experimental designed mats. This chapter demonstrates the use of the simulation program as a tool to improve the understanding of internal mat structures and the related properties. * A paper prepared from this chapter has been published in the Forest Products Journal, 1998, 48(5): 89-93. 21 Chapter II: Computer Simulation of Wood-Flake Composite Mat Structures 2.1. Introduction The physical and mechanical properties of flake-based wood composites depend on the properties of the constituents and the manufacturing process. A better understanding of its internal structure is needed to improve the characterization of material properties and efficiency in the utilization of wood-based composites. A two-dimensional mathematical model was developed to describe the structural characteristics of flakeboards, assuming uniform flake size and random process (Dai 1993, Dai and Steiner 1993, 1994a and 1994b). In relating to wood-flake composite mat formation, the most noticeable structural feature of a random mat is the non-uniform wood element coverage, i.e., the varying number of elements overlapping at different positions within the mat. This non-uniform wood element coverage affects heat and mass transfer in the mat during hot pressing and horizontal density distribution in the final panel. Steiner and Dai (1993) discussed some other important features in the spatial structure of wood-based composites 1) the strong influence of flake geometry on the relative void volume in a mat; 2) the important role of element orientation on optimizing directional strength properties; and 3) the reduction of variation of horizontal density distribution by improving packing behavior and board alignment. Although the mathematical model (Dai 1993) shows that the flake deposition and area coverage follow Poisson process, a closed form solution to describe the real mat formation process was not developed. Furthermore, a purely empirical approach to this problem can be extremely expensive and time-consuming because of the large number of interacting variables. 22 Chapter II: Computer Simulation of WoodrFlake Composite Mat Structures In this study, an MS-Window-based simulation program was written. In addition to the simulation and calculation of the structural characteristics of the mat, a robotic control system was also developed, where flake placements can be exactly controlled for experimental study. The simulation program linked with the robot allows verification of the theory and random generation of new databases of mats with various input parameters and mat structures. The objective of the current study is to demonstrate the use of the simulation program as a tool to improve the understanding of internal mat structures and the related properties. 2.2. M o n t e C a r l o S i m u l a t i o n A two dimensional model of a wood flake composite mat is considered as multi-layers of flakes deposited in a predefined area (length (L) (mm) x width (W) (mm)) with each flake having a definite position in the horizontal plane of the mat. It is termed as a "twodimensional" model because the model outputs are referenced only to the horizontal domain. The flake centroid location can be described by x and y coordinates ( 0 < x<L, 0 < y<W) and flake orientation can be described by angle 9 (-90°<t9<90°), which is defined as the angle between the x axis and the longitudinal axis of the flake (Figure 2 . 1 ) . If P is the location in the mat area where the flake is placed, then the coordinates (x, y) for P and flake orientation angle 9 can be numerically simulated by computer or the results from experimental measurements. As we define the flake centers in a mat area (L x W), inevitably some flakes will protrude beyond the defined mat area despite having their centers restricted to lie within the mat area, causing "side effects" (Dai 1993). To solve this problem, the "torus conversion" concept 23 Chapter II: Computer Simulation of Wood-Flake Composite Mat Structures (Hall 1988) was employed in the program. The part of a flake that protrudes from one side of the mat area was considered to re-enter again from the opposite side. This concept becomes more logical and meaningful when the present mat is part of a much larger network in which flakes protruding out of one unit inevitably enter another or vice-versa. In cases where an experimental designed mat is analyzed, side effects do exist and the torus conversion is no longer suitable for this situation. Therefore a maximum of half of the flake length from every edge of the mat must be removed before any test specimens are made unless side effects were explicitly considered during the experimental design by using more flakes around the edges than in the middle. 24 Chapter II: Computer Simulation of Wood-Flake Composite Mat Structures 2.3. Program Input and Output The composite mat simulation program can randomly generate data to represent the structural characteristics of flakeboards. A simulated mat can be analyzed by the program to obtain information, such as flake coverage, free flake length and void size. The horizontal density distribution of these mats can also be graphically presented. A robotic control system, which is a link between the computer simulation and a robot controller, was also developed. Such a robotic mat formation system can be used to build experimental mats with predefined structures. Such experimental mats can also be analyzed using the program to obtain similar types of information. The inputs to the simulation program include: mat size (length, width and thickness in mm), flake geometry (length, width and thickness in mm), flake density (g/cm ), flake orientation 3 angles (-90°<6k90°), flake centroid locations, and final panel density (g/cm ). The flake 3 geometry (length and width), orientation, and centroid location can be considered as random parameters defined by various probability distribution functions. A complete randomized flakeboard mat would have: 1) the flake orientation angle specified within a range of -90° to +90° under a uniform probability distribution; and 2) the flake centroid location specified within the range of the horizontal domain under a Poisson probability distribution (Dai and Steiner 1994a and 1994b). In cases of partially randomized flakeboard composite mats, a reduced range of flake orientation angles can be specified (for example, -5 to +5 degrees) under a uniform distribution and/or the centroid location of each flake can be directly specified as desired. The length and width of each "flake" can be simulated independently from normal distribution (specified mean and standard deviation of the mean) with upper and 25 Chapter II: Computer Simulation of Wood-Flake Composite Mat Structures lower bounds within the mean ± three times the standard deviation. Alternatively, a mixed flake geometry with a predefined proportion of flake sizes can be used. 2.4. Robot Control System The developed robotic controlling system links the computer simulation and experimental mat. Data such as the orientation and centroid location of each flake can be generated from the simulation program and stored in a file. The robotic mat-forming system can access the file and the information can be transferred into the robot controller in the format of robot commands (Figure 2.2). Experimental mats can be made flake by flake using the robot to verify the simulation procedures and build a test database on mats with high repeatability. This type of work is beyond the scope of this thesis and can be found in a publication by Wang and Lam (1998). PC Computer Robot Controller Figure 2.2 Robot Robot control system 26 Chapter II: Computer Simulation of Wood-Flake Composite Mat Structures 2.5. Results and Discussions 2.5.1. Density and overlap distributions By using the constant mass in a given volume, we are able to obtain the relationship between the overlaps (O) which is the number of flake layers measured at any local area (a) and the local densities (/?), as follows: (2.1) Or =T fPf P O = —-?= RR f Pf T p (2.2) T where r= panel thickness, p = panel local density, T/= flake thickness, Pf = flake density, R = thickness ratio between panel and flake, and T R = density ratio between panel and flake. p The assumptions are that the flake thickness, flake density and panel thickness are fixed for a given mat. Under these assumptions, the overall overlap distribution is linearly proportional to the panel local density distribution. In this study, a flake thickness of 1 mm, a flake density of 0.4 g/cm , a panel thickness of 11 mm, and a target panel density of 0.6 g/cm were chosen for simulation. Generally, the flake overlap is a function of thickness ratio and density ratio between panels and flakes as shown in Equation 2.2. 27 Chapter H: Computer Simulation of Wood-Flake Comvosite Mat Structures In this study, three classes of flake sizes are considered, uniform, mixed and distributed. Uniform refers to a mat generated from flakes of uniform size (80mm x 20mm x 1mm). Mixed means a mat made of five different sizes of flakes with the same ratio of weight (100mm x 22mm x 1mm, 80mm x 20mm x 1mm, 70mm x 18mm x 1mm, 60mm x 16mm x 1mm, and 50mm x 14mm x 1mm). Distributed indicates a mat made of the average flake size of 80mm x 20mm x 1mm where the flake length and width follow a normal distribution and the standard deviations of length and width are 8mm and 2mm, respectively. 12 Figure 2.3 Flake overlap and density distributions in the composite mat. (Simulated mat geometry: 250mm x 250mm x 11mm, target board density: 0.6g/cm , flake geometry: 80mm x 20mm x 1mm, flake density: 0.4 g/cm , 3 3 and resolution: 1mm x 1mm.) The typical density and overlap distributions obtained using uniform flake size are plotted in Figure 2.3. These distributions are obtained by first digitizing the 250mm x 250mm domain 28 Chapter II: Computer Simulation of Wood-Flake Composite Mat Structures into squares with 1mm x 1mm resolution. Within each 1mm x 1mm square, the number of flakes through the depth of the "simulated" panel is obtained. Based on this information the overlaps and density with a 1mm x 1mm zone can be estimated. The average point density, maximum density, minimum density, and standard deviation of density from the simulated mat are 0.600, 1.108, 0.188, and 0.143 g/cm , respectively. As shown in Figure 2.3, the 3 computer-simulated data are in good agreement with mathematical model predictions defined by Poisson distribution (Dai and Steiner 1994a). The horizontal density variation for the simulated mat is plotted as shown in Figures 2.4a and 2.4b. Figure 2.4a Graphical representation of horizontal density distribution by surface map (conditions the same as Figure 2.3). 29 Chapter II: Computer Simulation of Wood-Flake Composite Mat Structures X Axis (mm) Figure 2.4b Graphical representation of horizontal density distribution by contour map (conditions the same as Figure 2.3). 2.5.2. Free flake length and its distribution Free flake length is defined as the distance between two adjacent crossings of flakes. So if there is only one crossing or all the flakes are parallel to each other, no free flake length can be calculated. The distribution of free flake length (average of 5 layers) is illustrated in Figure 2.5 for a flake length of 100 mm, and a flake width of 10mm and 20mm. Flake width is an important factor that influences the free flake length distribution. Thus the original 30 Chapter II: Computer Simulation of Wood-Flake Composite Mat Structures model (Dai and Steiner 1994a) needs to be modified to incorporate this factor. From Figure 2.5, it is clear that model prediction and program simulation results fit better at higher aspect ratios (the ratio of flake length to width). At higher aspect ratios, the result agrees well with the free fiber length distribution studies in paper structure (Kallmes and Bernier 1963). 12 0 20 40 60 80 100 120 Free flake length (mm) Figure 2.5 The free flake length distributions with the data predicted by Dai's mathematical model and the current simulation study, (flake length is 100 mm, width 10 mm and 20 mm, other conditions are the same as Figure 2.3) 2.5.3. Void size and its distribution By definition, a layer is one flake deep. If a given number of flakes (Nf) are perfectly aligned one by one in the plane of one layer with NfAa = A (mat area), there is no void at all. All the 31 Chapter II: Computer Simulation of Wood-Flake Comvosite Mat Structures area in that layer, whether it is big or small, will be covered by only one flake thickness. However, if the centers of flakes and their orientation angles are randomly distributed or partially randomly predefined in the experimental design, some areas will not be covered and some areas will be covered by more than one flake. Therefore, voids inevitably exist in the layer, causing the horizontal density distribution and the variation of the panel properties. Table 2.1 lists the void areas in the first 10 layers of the simulated mat. The data indicate that in a randomly formed mat the void area in any layer in a mat is between 33 to 40 percent of the total layer area. Table 2.1 Void sizes in the area of each layer Layer No. of voids No. Total void Average void area Proportion of area (mm ) (mm ) voids (%) 2 2 1 40 23401 585 37 2 29 23514 811 38 3 35 20905 597 33 4 30 21778 726 35 5 32 21540 673 34 6 29 24729 853 40 7 23 24000 1043 38 8 27 24069 891 39 9 26 20545 790 33 10 42 20836 496 33 32 Chapter II: Computer Simulation of Wood-Flake Composite Mat Structures 2.5.4. Effect of sampling zone size A sampling zone is defined here as a rectangle of certain length and width, where the local board density is evaluated. The effects of various parameters of the sampling zones on the standard deviation of density of the simulated mats are illustrated in Figures 2.6a through 2.6d, where Figure 2.6a represents the mat made of flakes with uniform size and Figures 2.6b, 2.6c, and 2.6d represent mat made of three different types of flakes (uniform, mixed and distributed). The average density remains the same in all combinations of different lengths and widths of sampling zones (Table 2.2). The maximum density, mean deviation and standard deviation of density decrease as the area of the sampling zone increases. The minimum density increases as the area of sampling zone increases. Figures 2.6a through 2.6d also show that the standard deviation of density for mats of uniform flakes is higher than that of the mixed or distributed flakes in any combination of sampling zone length, width or area; this is because more small size flakes are deposited on mats with the selection of mixed or distributed flakes. The standard deviations of density for the mats of the mixed and distributed flakes are quite similar to each other. 33 Chapter II: Computer Simulation of Wood-Flake Composite Mat Structures Figure 2.6 The effect of sampling zone sizes on the standard deviation of density of flakeboard mat. a) the effect of sampling zone length and width on the standard deviation of density for a uniform flake mat; b) the effect of sampling zone area on the standard deviation of density for three different types of mats; the effect of sampling zone width on the standard deviation of density for three different types of mats in the case of sampling zone length is 10mm (c) and 50mm (d). 34 Chapter II: Computer Simulation of Wood-Flake Composite Mat Structures Table 2.2 Effect of sampling zone size on the density distribution (g/cm ) 3 Zone Size Maximum Minimum Average Std. dev. Cov (%) lxl 1.11 0.19 0.6 0.14 23.95 2x2 1.07 0.21 0.6 0.14 23.43 5x5 0.98 0.27 0.6 0.13 .21.67 10x10 0.88 0.32 0.6 0.11 18.78 25x25 0.76 0.41 0.6 0.08 13.51 50x50 0.69 0.52 0.6 0.05 8.00 125x125 0.67 0.55 0.6 0.04 7.23 (xxy mm ) 2 2.5.5. Comparison between different flake types The uniform flake type is an ideal model available only for research purposes. Different flake sizes are used in commercial products. This simulation, therefore, includes the more useful mixed or distributed flake sizes. The results indicate that there is little difference of density distribution in the panels between the three types of flakes (Figure 2.7). Although different numbers of flakes were used (2652 for uniform flakes, 3689 for mixed flakes and 2703 for distributed flakes), the total flake coverage remained the same because there was the same volume of flakes in each mat. However, Figures 2.6b through 2.6d show that the standard deviation of density for the uniform flake size is quite different from those of the mixed and 35 Chapter II: Computer Simulation of Wood-Flake Composite Mat Structures distributed flake types. This phenomenon is because the sampling zones used in Figure 2.6 are much larger than the 1 x 1 mm resolution used to obtain the information shown in Figure 2.7. 10 Figure 2.7 Comparison of area ratio of density obtained from three different flake types (Mat size: 500 x 500mm x 20mm). 2.6. Conclusions With the developed simulation program, one can easily edit the input parameters, generate flake deposition information, calculate flake area coverage in any layer of a mat or the entire 36 Chapter II: Computer Simulation of Wood-Flake Composite Mat Structures mat, analyze distribution of flake overlaps and densities, and provide statistical information concerning density distribution under different sampling zones. Free flake length and void size, and their distributions can also be calculated. The flake coverage distribution matches the mathematical model developed by Dai and Steiner (1994a and 1994b) quite well. However, their model, which describes the free flake length distribution, should be modified to consider flake width (or the aspect ratio of flake length to width). The mixed and distributed flake sizes show a lower standard deviation of density than that of uniform flake size. The total void area in any layer in a mat takes more than 1/3 of the total mat area for random formation. The standard deviation of density variation decreases dramatically as sampling zone size increases (zone length, width or area). 2.7. References Dai, C. 1993. Modeling structure and processing characteristics of a randomly formed woodflake composite mat. Ph.D. Thesis. Department of Wood Science, U B C , October Dai, C. and P. R. Steiner. 1993. Compression behavior of randomly formed wood flake mats. Wood and Fiber Science, 25(4): 349-358 Dai, C. and P. R. Steiner. 1994a. Spatial structure of wood composites in relation to processing and performance characteristics. Part 2. Modeling and simulation of a randomly formed flake layer network. Wood Science and Technology, 28: 135-146 Dai, C. and P. R. Steiner. 1994b. Spatial structure of wood composites in relation to processing and performance characteristics. Part 3. Modeling the formation of multi-layered 37 Chapter II: Computer Simulation of Wood-Flake Composite Mat Structures random flake mats. Wood Science and Technology, 28: 229-239 Hall, P. 1988. Introduction to the theory of coverage processes. New York: John Wiley & Sons Kallmes, O., and G. Bernier. 1963. The structure of paper: IV. The free flake length of a multiplanar sheet. Tappi 46(2): 108-114 Steiner, P. R. and C. Dai. 1993. Spatial structure of wood composites in relation to processing and performance characteristics. Part 1. Rationale for model development. Wood Science and Technology, 28: 45-51 Wang, K. and F. Lam. 1998. Robot-based research on three-layer oriented flakeboard. Wood and Fiber Science, 30(4): 339-347 38 Chapter III: Random Field Representation of Horizontal Density Distribution CHAPTER III RANDOM FIELD REPRESENTATION OF HORIZONTAL DENSITY DISTRIBUTION IN PARTIALLY ORIENTED FLAKEBOARD M A T Abstract A random field representation of the horizontal density distribution in partially oriented flakeboard mats was investigated. The orientation of flakes can be characterized by both the Von Mises distribution and the uniform distribution within a range of angles. Theoretical models of the correlation coefficients of any two points simultaneously covered by one flake, variance functions of local density, characteristic area of the correlation, and the degree of orientation were developed. Results indicate that the concentration factor k = 700 is sufficiently large to represent a perfectly aligned flake arrangement in oriented flakeboards based on the Von Mises distribution. The correlation coefficients of two points in a mat have a lower bound (random case) and an upper bound (perfectly aligned) in both Von Mises and uniform distributions. The variance is considerably reduced when the sampling zone size is within the length of flake. Based on the concept of characteristic area, where the minimum characteristic area is the area of a flake and the maximum characteristic area is approximate to the square of flake length, the degree of orientation in a panel can be represented as a function of characteristic area. This value is found to be very close to the percent alignment definition. 39 Chapter III: Random Field Representation of Horizontal Density Distribution 3.1. Introduction Oriented strand boards (OSB) are made by processing relatively low value and under-utilized wood materials into strands or flakes, which are approximately 100 mm in length, 15-20 mm in width and roughly 0.6-0.8 mm in thickness. After the flakes are dried, adhesives and wax are applied, and the flakes are typically formed into a three-layer mat. The top and bottom (face) layers are partially oriented along the long axis of the panel to provide added bending strength and stiffness in the direction of orientation while flakes in the middle (core) layer of the mat are randomly oriented. The mat is bonded together under heat and pressure to produce panels of various dimensions and thicknesses. OSB panels are very versatile as their mechanical and physical properties can be controlled by adjustment of production parameters to meet end users' demands for such characteristics as size, density and strength. The various production parameters that influence OSB performance include species, flake geometry, flake lay up / orientation process (flakeboard structure), adhesive type and distribution mechanism, and pressing process. Many researchers have shown great interest in flakeboard structures and properties (Dai and Steiner 1994, Lang and Wolcott 1996, Suchsland and X u 1989, Triche and Hunt 1993, Xu and Steiner 1995). In particular, Dai and Steiner (1994) studied the spatial correlation of flake coverage by making use of the theories in random fibrous network (Dodson 1971) to define the horizontal density distribution in panels with completely random flake orientation and position. The statistics of the horizontal density distribution in OSB are of particular interest in the assessment of product quality and performance. While most of the research has focused on completely random orientation of flakes and its influences on physical and mechanical strength properties of the panels, the 40 Chapter III: Random Field Representation of Horizontal Density Distribution flake orientation in both face layers of OSB is in reality, partially random. There has been no theoretical consideration of the relationship between partially random orientation of flakeboards with physical properties such as the variation of board density and degree of orientation in the horizontal direction in the OSB products. Since two orientation angles 6 and 6 + nn (n = 1,2,3,...) cannot be distinguished for a rectangular flake placed in a mat, the range of angles from -f to +f radian can completely describe all possible choices for flake orientation. The random distribution of flake orientation within a mat is an excellent example of axial data, which can be best described by the Von Mises distribution (Mardia 1972). Harris and Johnson (1982) pointed out that the concentration parameter of the Von Mises distribution (k) can be used to characterize the distribution of flake orientations in flakeboards. Shaler (1991) compared two measures of flake alignment based on the Von Mises distribution of flake alignment and the classical definition of percent alignment given by Geimer (1976), which in fact is a linear transformation of the first moment (arithmetic mean) of the absolute value of angles in the range of ±f radian. Computer simulations yielded good agreement with theoretical calculations (Shaler 1991). Lau (1981) linked the standard deviation of normal distribution with the range of orientation angles by a factor of Jf . This section presents a theoretical model of the horizontal density distribution of partially oriented flakeboard mat. The objectives of this study are: 1) to develop a random field representation of horizontal density distribution for partially oriented flakeboards following the Von Mises distribution and the uniform distribution, 41 Chapter III: Random Field Representation of Horizontal Density Distribution 2) to introduce the concept of characteristic area as a measure of within member correlation of horizontal density distribution, and 3) to characterize the degree of orientation as compared with the traditional definition of percent alignment (Geimer 1976). 3.2. Theoretical Model 3.2.1. Probability density function for flake orientation Due to the forming process of OSB, the orientation of the principal direction of eachflakeis partially random. Mardia (1972) provided a comprehensive description of the statistics of directional data. One of the most important properties of such a random variable 6 (in radian) is that its probability density function/(c?) has a circular form: f(9±nn) = f(0) (-|<0<|) (II = 1,2,3,...) (3.1) since a flake oriented at 0 to the x direction is the same as a flake oriented at 6± tin. A function for characterizing directional axial data is the Von Mises probability density function (PDF) whose axial functional form is given: (Harris and Johnson 1982): 1 f(6,e„k) = {29 I (k) x 0 (-0, <0<6 ,and0<6 x x <-) 0 (3.2) otherwise where k is the concentration parameter to specify the shape of the distribution, 42 Chapter III: Random Field Representation of Horizontal Density Distribution 0o is the mean value of the orientation angles, 6\ equals to f-when k > 0, which means that the range of flake orientation angles in the Von Mises distribution must be between -f and f, and Io(k) is the modified Bessel function of the first kind, order zero. I- W = t, \ , v (3-3) Another property of this PDF is that the integration of the PDF over the range of - f <0 <f equals to one, i.e., f' f(0,0 ,k)d0 = { f ™w- °»d0 l k d e =\ (3.4) Based on Equation 3.4, a PDF without the Bessel function can be obtained from Equation 3.2 as: *cos[2(0-0„)] f(0,0„k) =f e - " d(p kM e (3.5) )] g When k -> 0, f{0,0 O) = u which represents a uniform distribution of orientation angles. When k —> oo, the distribution reduces to a point distribution, which can be described by the Dirac delta function (the perfectly aligned case). 43 Chapter III: Random Field Representation of Horizontal Density Distribution 3.2.2. Point-to-point variance of the density Let D(x, y) represents the point-to-point horizontal density profile in the x-y plane in a mat. The variation of the point-to-point density in the two-dimensional random flake network was obtained from Dai and Steiner (1994): var(Z>) = ^ ^ (3.6) where Xf is the flake thickness, pf is the flake density, r is the thickness of the mat, and p is the mean density of the mat. Consider dividing the two-dimensional domain into a large number of small, nonoverlapping, congruent, rectangular, sampling zones, A(x, y), with side length S and width x S , where x, y represents the coordinates of the centroid of a sampling zone. The local y average of density is given by integrating the density profile D(x, y) over the rectangular area of A (Vanmarcke 1983): DAx,y) = ~ ^ rlriD(x,y)dxdy (3.7) The information about point-to-point variation is contained in the variance function, as the variance between values of the random field at two locations. The variance of density between the regions, var(DA), can be expressed as (Vanmarcke 1983): var(D ) = y(x,y)var(D) A (3.8) where y(x, y) is defined as the variance function of the density. It can be noted that y(x, y) has 44 Chapter III: Random Field Representation of Horizontal Density Distribution the limiting value of one as the sampling zone reduces to a point and zero as the zone size approaches infinity. 3.2.3. Variance function of the density The variance function, y(x,y), in a two-dimensional domain is defined as the ratio of the variance of the local average, D (x, y), over a rectangular area A (= S x S ) and the "point" A x y variance of the original process, D(x, y). The variance function is related to the correlation function (a) and the probability density function (fi) for two points within a rectangle as (Dodson 1971), y(S ,S ,A,co,0 ,k)= x where r max y l % ' P(S ,S ,r)-a(X,o)AMdr m x x y (3.9) is a random variable depending on the flake geometry (X, co) and the sampling zone size (S , S ); 9\ is another random variable in the uniform distribution (k = 0), and x y 0, = f in the Von Mises distribution (k * 0). 3.2.4. Correlation coefficient for two points in a rectangle Consider the distribution of two-dimensional flakes with length X and width co (co < X), it is assumed that the long axes of these flakes are oriented at an angle of 6 + dO from a given reference direction, usually x direction. The random location of the flake centroids are assumed to follow a Poisson distribution (Dai and Steiner 1994) of overlap intensity dp(0) per unit area. Therefore, the superposition of such processes, for -y<#<y, produces a distribution of overlaps with intensity p per unit area for the total number of flake centres as: 45 Chapter III: Random Field Representation of Horizontal Density Distribution *cos[2(0-0 )] O dp(0,9 ,k) - pf(9,9,, k)d9 = -f^e (3.10) d9 x ^ d<p kcos[2 )] Correlation coefficient, a, between the number of flake overlaps at two points, p and q separated by a distance of r is given as (Dodson 1971) a(A,co,0 ,k,r)= l where N pq var(7V ) Dfl (3.11) var(A^) is the number of flakes that cover both p and q, and A^ is the total number of flakes that cover only p. It therefore follows that both N and N are Poisson distributed with pq p the variances equal to varfW ) = f Xcodp(9) = hop var(tf„)= ^ ( f f l - r s i n ^ X A - r c o s ^ y / / ^ ) (3.12) (3.13) where (&>-,»"sin|#|)(A-rcoscV)is the total area containing the centres of flakes that cover both p and q. Applying dp(9) from Equation 3.10 into Equation 3.13 and combining Equations 3.11 to 3.13, it can be shown that the correlation coefficient, a, is: a(^0 I f var(^) l ^ — ^ - ^ f f ^ ' ^ d e *,r) =— 1 - ^ =-5 2—. * (3.14) where r equals to the distance separated by two points p and q. 46 Chapter III: Random Field Representation of Horizontal Density Distribution For all possible values of 9 within the rectangular domain Ax co, using the symmetry properties of (co-rsm\9\)(A-rcos9), three ranges of angles are necessary to be taken into consideration where Q represents the integration intervals for 0 at 9 = f : X 1) 0<r<a>, corresponding to Q = {0 < 9 < \ n]; 2) co<r<A, corresponding to Q = {O < 9 < arcsin(f-)}; 3) + co , corresponding to Q = (arccos^) < 9 < arcsin(f-)}. A<r< 2 The correlation coefficient a(A,ca,9],k,r) can be evaluated at arbitrary r and k (k # 0). O f special interest is the limiting cases of k = 0 and k = <x> which will be discussed in more detail as follows. In cases where k = 0 and 9 < f , the Von Mises distribution becomes the uniform l distribution with PDF of the following form: f(9,9 ,Q>) = ^ 26/] x (-9, <9<9 ,and0<9, 1 <^) 2 (3.15) Here ±9\ are the upper and lower bounds for orientation angle 9. It can take arbitrary values from 0 to -f-. If 9\ approaches zero, all flakes are oriented parallel to each other; if 9\ equals to j, it is completely randomized orientation. In the general case of partially random orientation, 0 < 9 < f , the exact form of the correlation coefficient is given for three ranges X of 9\ (Figures 3.1a to 3.1c). 47 Chapter III: Random Field Representation of Horizontal Density Distribution Range 1: arcsin(—) <£?,< — X 2 — sine?, (cos#, -1) sin X co 2Xco . ,co. co r -Jr -co arcsin(—) + r 2X co co 2 a(X,co, <9, ,0,r) = co ) (0<r<-— sin#, 2 2 co (——<r<X) sin 9. . ,co X^ X +co +r arcsrn(—) - arccos(—) r r 2Xco 2 s r 2 2 Vr -co + co 2 2 •yjr -X + X 2 (3.16) ' K 2 (X<r< 4X +co ) 2 2 Figure 3.1a Schematic diagram for the range of angles within Range 1 for a flake with length X and width co during integration. 48 Chapter III: Random Field Representation of Horizontal Density Distribution Range 2: arcsin(-7==t==) < 0 < arcsin(—) 2 2 + C0 i-L a(X,co,0„O, r) = r — sine?, X r co (cost?, -1) ,X. r . . arccos(—) + — suit?, r X X ,co - arccos(—) arcsin(—) r r s s r 2 X 2Xco r r . X +r sin 0 H 2Xco 2Xco 2 a cost?, co 2 2Xco 2 l Jr -X 2 2 Q X X +co +r 2 (0 < r < X) sin 0 2 V^ 2 co' " CO 2 co X 4r -X 2 [ 2 2 (-^—<r<ylX +co ) sind. 2 2 Figure 3.1b Schematic diagram for the range of angles within Range 2 for a flake with length X and width co during integration. 49 Chapter III: Random Field Representation of Horizontal Density Distribution Figure 3.1c Schematic diagram for the range of angles within Range 3 for a flake with length X and width co during integration. When 9 =•§-, Equation 3.16 represents the completely random case in the following form, X which agrees with the finding of Dodson (1971): 50 Chapter III: Random Field Representation of Horizontal Density Distribution 1— r r r X co 2Xco . ,co. a(A,<y,f,0,r) = co r yfr ~-co~ ~ 2X co arcsrn(—) r (0<r<co) 2 2 (3.19) (co<r<X) v ..a. ,X r r co X +co +r 2 S arcsin(—) - arccos(—) 2 2 ylr -co 2 h 2Xco 2 y/r -X 2 h co 2 (X<r< Jx 2 +co ) 2 X If all flakes are perfectly aligned (d\ = 0), the correlation coefficient is a function of flake length (X) and the distance (r) between two arbitrarily chosen points on a line. When r = 0, the two points are common and the correlation coefficient equals unity. When the distance r between two points is greater than X, the correlation coefficient is zero. It can be summarized as: a(X,o),0,0,r) = 1— (0<r<X) 0 (r>X) (3.20) Equation 3.20 can also be obtained from Equation 3.18 assuming 0\ -> 0. 3.2.5. The probability density function for two points in a rectangle The probability density function for the distance r between two points chosen independently and at random within a rectangle, of side length S and width 5^ (S >S ), was given by Ghosh x x y (1951) as: P(S ,S ,r)=$ x y Ar r -^-(1-—sin0)(ls s, x -cos 6) d9 (3.21) s y Particularly, the integration results are as follows: 51 Chapter III: Random Field Representation of Horizontal Density Distribution 4r ?-S S -r(S S ) - 2 x 4r P(S ,S ,r)-. x y (0<r<S ) r ss 2 y x+ y + y arcsin(-^)-^- ^ r - S r 2 2 + sisl 4r s:s 2 -Sr 2 y (S <r<S ) y V S S [arcsin(^)-arccos(^)]r x y r LllzJl 2 + $ l * -S * 2 r +S x Jr -S 2 y (3.22) 2 X (S <r<p:+S ) 2 x For square zones (S = S ), the central range (S < r < S ) for r is not required. Some typical x y y x distribution curves for the probability density in square zones are shown in Figure 3.2. It is noted that the distribution curves tend to flatten as the sampling zone size increases. Side length of square zones 0 20 40 60 80 100 Distance between two points (mm) Figure 3.2 Probability density for the distance between two points in square zones. 52 Chapter III: Random Field Representation of Horizontal Density Distribution 3.2.6. Characteristic area - a measure of correlation In random field theory, the "characteristic area" is an important statistical parameter, which defines a measure of correlation in terms of the asymptotic form of the variance function in two dimensions. It can be expressed in terms of the variance and correlation functions. When the sampling zone (S , S ) becomes relatively large, the variance function (f) x y converges to an asymptotic expression. This property of variance function is defined by Vanmarcke (1983) as: y(S ,S ,Z,co,0 ,k) x y i = (S ,S ^<v) x y (3.23) where A is called a "characteristic area" or "correlation area". It is a proportional constant which also equals to the integration of the correlation function (a) (Vanmarcke 1983): (3.24) The validity of the asymptotic expression for the variance function (Equation 3.23) is subject to certain conditions on the "moments" of the correlation function. Consider Equation 3.21, when & and S tend toward infinity, the probability density function has only y one term left which is equal to jf-, provided the correlation function (or) decays sufficiently rapidly. In the case of perfectly oriented situation, the characteristic area is given by: (3.25) 53 Chapter III: Random Field Representation of Horizontal Density Distribution For completely randomized flake orientation, the characteristic area is evaluated by: A(X,co,—,0) = \ 2 J o 2wa(X,(o—,0,r)dr 2 = Xco (3.26) which is equal to the area of a flake. 3.2.7. Degree of orientation A conventional definition of percent alignment (Equation 3.27) was introduced by Geimer (1976), which measures the mean angle deviation from 45° to the geometric axes of the sample. Align% = 4 5 ~ ^ xlOO (3.27) 1" where 0-— T^t?,.! 6i = i measured angle, and th n - number of measurements. Angles are measured over the range of - 9 0 ° to +90° with the 0° being the assumed mean angle. This definition can be used to define flake alignment in experimental mat where flake angles can be established. For a real board, it is more difficult to peel off the flakes layer by layer to measure the angles. Alternately, non-destructive testing method can be used to extract the horizontal density distribution of a sample board and random field theory can be used to evaluate the degree of orientation as described below. 54 Chapter III: Random Field Representation of Horizontal Density Distribution As discussed earlier, the flake alignment can be random (k = 0 in the Von Mises distribution, &i=f in the uniform distribution) and perfectly aligned (k = oo in the Von Mises distribution, 6 = 0 in the uniform distribution). It is reasonable to consider the random X orientation case as zero degree of orientation, and perfectly aligned orientation case (all flakes parallel to long axis of the panel) as 100% degree of orientation. From Figures 3.3 and 3.4, it is easily seen that all the correlation coefficients lie within these two curves, random and oriented. Therefore, the area between any particular curve and the curve for the random case is indirectly related to the degree of flake orientation. This provided a base to define the degree of orientation of flake, Orient (%>). The integration of correlation function (a) is actually the characteristic area A for any given range of angles in the uniform distribution and any value of k in the Von Mises distribution. 0 Figure 3.3 20 40 60 Distance (mm) 80 100 Correlation coefficients for different values of concentration parameter k in Von Mises distribution (Flake length 100mm and width 20mm). 55 Chapter III: Random Field Representation of Horizontal Density Distribution 0 20 40 60 80 100 Distance (mm) Figure 3.4 Correlation coefficients for different ranges of angles 6\ (angles represent ±) in uniform distribution (Flake length 100mm and width 20mm). When k * 0, the flake orientation follows the Von Mises distribution with the lower bound of the random case at k = 0. When k = 0, the flake orientation is uniformly distributed within the range of -G\ to +0\ with the lower bound of the random case at#, = •§-. By applying the results from Equations 3.25 and 3.26, the degree of orientation (Orient) can be obtained: Orient(k) = A(X,Q),f,k)-A(X,co,f,0) A(X,co,f,co)-A(X,co,^,0) xl00 = jA(A,co,j,k)-Xco | \n2}-Aa) xl00 (3.28) Orient(0.) =, \ A(X,co,e fi)-A(X,co^,Q) [ i A(A,6),0,0)-A(A,G),f,0) xl00 = A(A,co,6 ,0)- Xco i \TTA - XCO 2 xl00 Here, the square root is applied to make it compatible to the percent alignment because the percent alignment is a linear transformation of the mean angles. 56 Chapter III: Random Field Representation of Horizontal Density Distribution In the case of the mean angle other than zero (9o * 0 in Equation 3.2), the Cartesian coordinate can always be rotated at angle Go in counterclockwise direction. Thus the above derivations for correlation coefficients, variance functions and characteristics areas are all valid for any mean angles. In the case of mapping the degree of orientation with respect to a principal direction either x or y direction, the following equations may be used. Orient (9 , k) = Orient(k) • (1 x Q 29 n °-) (3.29) Orient (9 ,k) = 100-Orient (0 ,k) 0 x O By using this mapping technique, if Oo = 0, nothing changed. But if 9 = f , the degree of 0 orientation in the x direction is zero. Similar mapping can be applied to the uniform distribution in terms of OQ and 9\. 3.3. Evaluation of Density Image Autocorrelation A two dimensional autocorrelation function (ACF), variance function and degree of orientation for a commercial panel can be readily evaluated, provided that the density measurements of the panel in the horizontal plane D(x, y) can be obtained through some nondestructive testing methods. To determine the A C F of a digitized image, the autocovariance is calculated first as a function of various offsets and is given by Agterberg (1974) as: C ( r ' =7^ 5 ) S^-^Ws-S) h^r (N -r)(N -s) x y ^ (3-30) ^ where N and x are the number of pixels of the image in the x- and v-directions, 57 Chapter III: Random Field Representation of Horizontal Density Distribution r and s are the offsets in the x- and v-directions, x and y are the pixel coordinates in the image, and D is the average value of density in the image. It is aware that C(0, 0), which corresponds to zero offset, is the variance of density in the image. Therefore, the autocorrelation, Autoir, s), is obtained by dividing C(r, s) by the C(0,0) (Agterberg 1974, Pfleiderer, et al 1993) as: Auto(r,s) = ^fl C(0,0) V (3.31) J For zero offset, Auto(0,0) = 1, and with increasing offset, Autoir, s) gradually decreases as pixels become more and more independent or statistically uncorrelated. However, this decrease is anisotropic if flakes of the panel are aligned (or partially oriented) along a preferred direction (Figures 3.6c to 3.6d). Therefore, by visualizing the 2-D distribution of A C F values, the degree of orientation can be calculated by Equation 3.28. Functions for evaluating the A C F , the variance and the degree of orientation of a density image have been added to the density module of Winmat® simulation program (Lu, et al 1998). 3.4. Results and Discussions 3.4.1. Autocorrelation coefficient of density It can be noted that the autocorrelation coefficients of density for any two points in a mat domain are bounded with the random case as the lower bound and perfectly aligned orientation as the upper bound. The autocorrelation coefficient of density of the perfectly aligned orientation can be characterized by any two points lying in a line (0 < r <X) 58 Chapter III: Random Field Representation of Horizontal Density Distribution (Equation 3.20). The random case was well understood from Dodson's work (1971) (Equation 3.19). Increasing the concentration parameter k from 0 to infinity (in the Von Mises distribution) or decreasing the range of orientation angle 6\ from f to 0 (in the uniform distribution) moves the autocorrelation coefficient of density curve from its lower bound towards its upper bound (Figures 3.3 and 3.4). Flake length and width are the two other factors influencing the shape of the autocorrelation coefficient of density curves since any two points covered by one larger flake may not be covered by a smaller flake. If flake length is much greater than the width (A » density autocorrelation coefficient is simplified to only one case co), the (Equation 3.16). When the mean angle (do) changes from 0 to f at k = oo (k =700 in our case), the zero autocorrelation coefficient is also changed from r - A to r - co (Figure 3.5). Figures 3.6a to 3.6h present the autocorrelation coefficients of density for four simulated mats. It is obvious that the density autocorrelation coefficient is one when x and y coordinates (the distance between any two points) are zeros, and zero when x and/or y go to very large. For the random mat (Figures 3.6a and 3.6b), the autocorrelation coefficients form many concentric cycles around the origin. Figures 3.6c and 3.6d represent the autocorrelation of a partially oriented flakeboard mat (range of angles from -45 to +45 degrees). For the perfectly aligned mat (Figures 3.6e and 3.6f), the autocorrelation coefficient of density is a linear function of the distance between two points along either the flake length direction or the flake width direction. This agrees well with the findings in Figure 3.5. Figures 3.6g and 3.6h are another representation of the autocorrelation of perfectly aligned flakeboard mat, whose flakes are oriented at 45 degrees. 59 Chapter III: Random Field Representation of Horizontal Density Distribution \ v Mean direction \ 0 \ . - \ o X \ 10 \ \ \ 3 0 \ \ 4 0 \ 0.2 . 20 \ ^ > > ^ X . ^ - s ^ 90V 20 40 60 80 100 Distance between two points (mm) Figure 3.5 Correlation coefficient between two points in the mat with various mean direction ( c % ) in Von Mises distribution (Flake length 100mm and width 20mm, k = 700). Figure 3.6a Correlation coefficient between two points in a mat (3D graphical representation) in completely randomized distribution of flake location and orientation (Flake length 100mm and width 20mm). 60 Chapter III: Random Field Representation of Horizontal Density Distribution 0 20 40 60 80 100 X axis (mm) Figure 3.6b Correlation coefficient between two points in a mat (contour map) in completely randomized distribution of flake location and orientation (Flake length 100mm and width 20mm). Figure 3.6c Correlation coefficient between two points in a mat (3D graphical representation) with partial orientation of flakes (range of angles: 45 to +45 degrees, flake length 100mm and width 20mm). 61 Chapter III: Random Field Representation of Horizontal Density Distribution 0 20 40 60 80 100 X axis (mm) Figure 3.6d Correlation coefficient between two points in a mat (contour map) with partial orientation of flakes (range of angles: -45 to +45 degrees, flake length 100mm and width 20mm). Figure 3.6e Correlation coefficient between two points in a mat (3D graphical representation) in perfectly aligned flake orientation (0°) and random location (Flake length 100mm and width 20mm). 62 Chapter III: Random Field Representation of Horizontal Density Distribution 0 20 40 60 80 100 X axis (mm) Figure 3.6f Correlation coefficient between two points in a mat (contour map) in perfectly aligned flake orientation (0°) and random location (Flake length 100mm and width 20mm). Figure 3.6g Correlation coefficient between two points in a mat (3D graphical representation) in perfectly aligned flake orientation (45°) and random location (Flake length 100mm and width 20mm). 63 Chapter III: Random Field Representation of Horizontal Density Distribution 0 20 40 60 80 100 X axis (mm) Figure 3.6h Correlation coefficient between two points in a mat (contour map) in perfectly aligned flake orientation (45°) and random location (Flake length 100mm and width 20mm). Figure 3.7 Comparison of correlation coefficients between model prediction (lines) and computer simulation (markers) (Flake length 100mm and width 20mm). 64 Chapter III: Random Field Representation of Horizontal Density Distribution A comparison of the autocorrelation coefficients between model prediction and computer simulation is presented in Figure 3.7. The simulation results agree well with the theoretically predicted values. There is a slight deviation between the predicted and simulated values when the size of simulated mat is small because an infinite domain is assumed in the mathematical model. 3.4.2. Variance function of density The distributions of variance function are plotted in Figures 3.8 and 3.9. When the side length of square zones is less than the diagonal length of flake, the variance reduced dramatically as the side length of square zone increases. The variance reduction rates, which reflect the speed of variance reduction, shown in Figures 3.10 and 3.11 indicated that at the relative small sampling zones, the variance reduction rate is very high (absolute value). It becomes zero when the sampling zone size approaches infinite. It can be noted that the variance reduction rate is relatively constant at a certain level for perfectly aligned orientation at the side length of square zones less than flake length. The variance function at the concentration parameter k = 700 agrees well with the parallel case in the uniform distribution (Figure 3.12). Therefore, the k = 700 is sufficiently large to represent the perfectly aligned situation characterized by the Von Mises distribution. The model predicted and computer simulated variance functions agree perfectly as well (Figure 3.13). 65 Chapter III: Random Field Representation of Horizontal Density Distribution Side length of square zones (mm) Figure 3.8 The variance reduction with respect to different k values in Von Mises distribution and different side length of square zones (Flake length 100mm and width 20mm). Side length of square zones (mm) Figure 3.9 The variance reduction with respect to different ranges of angles (angles represent ± ) in uniform distribution and different side length of square zones (Flake length 100mm and width 20mm). 66 Figure 3.10 The variance reduction rate with respect to k values in Von Mises distribution (Flake length 100mm and width 20mm). Side length of square zones (mm) Figure 3.11 The variance reduction rate with respect to ranges of angles (angles represent +) in uniform distribution (Flake length 100mm and width 20mm). 67 Figure 3.12 The comparison of variance function for the perfect aligned flake by Von Mises distribution (k = 700) and uniform distribution (B\ =0) (Flake length 100mm and width 20mm). Figure 3.13 The comparison of variance function for the randomly aligned (±90) and perfectly aligned flakes by model prediction and simulation (Flake length 100mm and width 20mm). 68 Chapter III: Random Field Representation of Horizontal Density Distribution 3.4.3. Characteristic area Although it is clear that there is no variance reduction when the sampling zone reduces to a point and 100% variance reduction when the sampling zone is extremely large, the characteristic area of the correlation is not well understood. The product of variance function and the sampling zone area is constant when the sampling zone size approaches infinite. This parameter is called the characteristic area which equals to the area of a flake (ACQ) in randomly oriented flakeboard mat (k = 0 in the Von Mises distribution, or 0\=f in the uniform distribution) (Equation 3.26). Figures 3.14 and 3.15 show that the characteristic area increases when the flakes become more oriented. It can also be noted that the maximum characteristic area is approximately equal to the square of the flake length in perfectly oriented flakeboard mat because it is independent of flake width. The characteristic area indirectly reflects the degree of variance reduction as shown in Figure 3.16. Large characteristic area indicates high average variance reduction. 10000 § 2000 0 2 4 6 8 10 12 14 16 18 20 Concentration parameter k Figure 3.14 Characteristic area in relation to the concentration parameter k in Von Mises distribution of flakes (Flake length 100mm). 69 Figure 3.15 Characteristic area in relation to the ranges of angles (angles represent ±) in uniform distribution of flakes (Flake length 100mm). 0.15 0.00 1 0 1 2000 1 4000 6000 ' 1 8000 10000 1 12000 Characteristic area (mm ) 2 Figure 3.16 Characteristic area in relation to the average variance reduction from random orientation to perfect alignment (Flake length 100mm and width 20 mm). 70 Chapter III: Random Field Representation of Horizontal Density Distribution 3.4.4. Degree of orientation The degree of orientation of flakes was compared with the percent alignment calculated by Geimer's formula for the mean angle at zero degree in the Von Mises distribution (Figure 3.17) and in the uniform distribution (Figure 3.18). The degree of orientation started at 0 for random orientation case and ended at 100% for the perfectly aligned arrangement in both distributions. The predicted and simulated values from characteristic areas agree well with the percent alignment values. This suggests that the degree of orientation concept can also be used to estimate the flake arrangement of commercial panels if the horizontal density distribution of the panel can be evaluated through non-destructive testing. 3.5. Conclusions The degree of orientation of flakes in structural wood-based composites, such OSB, is an important processing parameter because it determines the degree of directional strength and stiffness of the panel. A random field theory representation of the variation of horizontal density in partially oriented flakeboard mats were presented in this study. The orientation of flakes in a mat was characterized by both the Von Mises distribution and the uniform distribution within a range of angles. Theoretical models for the correlation coefficients, variance functions, characteristic area and the degree of orientation were discussed. 71 Chapter III: Random Field Representation of Horizontal Density Distribution 100 1 1 80 a o •-C 43 60 s •B o 40 o — — «« <u WD « Predicted according to characteristic area Simulated according to Geimer's formula 20 i 0 0 20 40 60 80 100 i 120 i ! 140 ! 160 180 200 k values Figure 3.17 Degree of orientation of flakes with respect to concentration parameter k in Von Mises distribution (Flake length 100mm and width 20mm). Ranges of angles Figure 3.18 Degree of orientation of flakes with respect to the ranges of angles (angles represent +) in uniform distribution (Flake length 100mm and width 20mm). 72 Chapter III: Random Field Representation of Horizontal Density Distribution Theoretical analysis indicated that k = 700 is sufficiently large to represent the perfectly aligned orientation of flakes in Von Mises distribution. The correlation coefficients have a lower bound (random case) and an upper bound (perfectly aligned) in both distributions. Flake geometry and mean orientation angles have great influences on the correlation coefficient. The variance of density is considerably reduced as the sampling zone size increases within the length of flake and tends toward zero as the zone size goes to infinity. The characteristic area, another measure of correlation, has the minimum and maximum values, which are expressed as the area of a flake and the approximate square of flake length, respectively. It also indirectly indicated the degree of variance reduction in partially oriented flakeboards. The degree of orientation discussed in this study is an alternate way to estimate the flake alignment of OSB products. 3.6. References Agterberg, F.P. 1974. Geomathematics: mathematical background and geo-science applications. Elsevier Scientific Publishing Co. Amsterdam. 596p Dai, C. and P.R. Steiner. 1994. Spatial structure of wood composites in relation to processing and performance characteristics. Part 3. Modeling the formation of multi-layered random flake mats. Wood Science and Technology 28: 229-239 Dodson, C T . J. 1971. Spatial variability and the theory of sampling in random fibrous networks. Journal of the Royal Statistical Society, Series B (Methodological), 33(1): 82-94 Geimer, R.L. 1976. Flake alignment in particleboard as affected by machine variables and 73 Chapter III: Random Field Representation of Horizontal Density Distribution particle geometry. Research Paper 275. Madison, WL, U S D A , Forest Products Lab. Ghosh, B. 1951. Random distances within a rectangle and between two rectangles. Calcutta Math. Soc, 43:17-24 Harris, R.A. and J.A., Johnson. 1982. Characterization of flake orientation in flakeboard by the Von Mises probability distribution function. Wood and Fiber, 14(4): 254-266 Lang, E.M. and M.P.,Wolcott. 1996. A model for viscoelastic consolidation of wood strand mats. Part I. Structural characterization of the mat via Monte Carlo simulation. Wood and Fiber Science, 28(1): 100-109 Lau, P.W.C. 1981. Numerical approach to predict the modulus of elasticity of oriented waferboard. Wood Science, 14: 73-85 Lu, C ; P.R. Steiner; and F. Lam. 1998. Simulation study of wood-flake composite mat structures. Forest Products Journal, 48(5): 89-93 Mardia, K.V. 1972. Statistics of Directional Data. Acad. Press, London Pfleiderer, S; D.G.A. Ball; and R.C. Bailey. 1993. A U T O : A computer determination of the two-dimensional autocorrelation function of digital images. Computers and Geosciences, 19(6): 825-829 Shaler, S.M. 1991. Comparing two measures of flake alignment. Wood Science and Technology, 26: 53-61 Suchsland, O. and H. Xu. 1989. A simulation of the horizontal density distribution in a flakeboard. Forest Products Journal, 39(5): 29-33 74 Chapter III: Random Field Representation of Horizontal Density Distribution Triche, M.H. and M.O. Hunt. 1993. Modeling of parallel-aligned wood strand composites. Forest Products Journal, 43(11/12): 33-44 Vanmarcke, E. 1983. Random Fields: Analysis and synthesis. The MIT Press, Cambridge, Massachusetts, London. Xu, W. and P.R. Steiner. 1995. A statistical characterization of the horizontal density distribution in flakeboard. Wood and Fiber Science, 27(2): 160-167 75 Chapter IV: Study on the X-Ray Calibration and Overlap Measurements CHAPTER IV STUDY ON THE X-RAY CALIBRATION AND OVERLAP MEASUREMENTS IN ROBOT-FORMED FLAKEBOARD MATS* Abstract An X-ray based experimental study is presented for determining the overlaps and horizontal density distribution (HDD) in robot-formed wood flakeboard mats. The relationship between overlaps and the X-ray intensity ratio has been developed. The results show that the calibration materials have little effect on the overlaps obtained from X-ray scanning and the estimated overlaps highly agree with those from computer simulations. Robot formed flakeboard mats have higher repeatability with lower deviation in both overlaps distribution and HDD than hand-formed mats. 4.1. Introduction Several papers were recently published discussing the modeling and simulation of flakeboard mats. Steiner and Dai (1993) developed a rationale for predicting the spatial structure of wood composites in relation to processing and performance characteristics. Following the studies of paper formation (Kallmes et al. 1960, 1961 and 1963, Dodson 1971), a twodimensional model based on the Poisson probability theory was set up to determine the flake * A paper prepared from this chapter has been published in the Wood Science and Technology, 1999, 33(2): 8595. 76 Chapter IV: Study on the X-Rav Calibration and Overlap Measurements center distribution, flake overlap distribution, and HDD distribution in randomly formed mats (Dai and Steiner 1993, 1994a, 1994b and 1994c). Lang and Wolcott (1996) reported a model for viscoelastic consolidation of wood-strand mats through Monte Carlo simulation. These findings set up the foundation to characterize the structure of wood flake mats. However, experimental measurement techniques needed to determine the physical characteristics of flakeboard still pose a challenge in future model development especially in the determination of HDD and overlaps in the mat area. The traditional gravimetric method to measure H D D in a mat by drilling or cutting (Xu and Steiner, 1995) is an extremely timeconsuming and tedious procedure and it is impossible to get the same level of detail as the information available from computer simulation studies. The accuracy of the gravimetric technique is another factor of concern. Soft X-ray scanning technology, which is one of the nondestructive testing methods that could be applied to a flakeboard mat, is a promising approach for detecting physically based factors relating to structures, such as voids and overlaps. Luggage inspection systems commonly found at airports are examples of applications of soft X-ray in which 2dimensional line-scans are used. To date no literature has been found on the use of X-ray scanning for measuring HDD and overlaps in flakeboard mats. This studies presents an X-ray calibration method and measurement techniques for determining the X-ray intensities and overlaps of flakeboard mats based on the principle of scaling of X-ray attenuation to flake overlaps. Robot forming techniques have also been used to obtain samples with known structures and high repeatability. 77 Chapter IV: Study on the X-Ray Calibration and Overlap Measurements 4.2. X-ray Theory and Overlap Model 4.2.1. X-ray theory and calibration Beer's Law can be used to describe the X-ray absorption process in the scanned images. The value at any position defined by coordinate (x, y) in the mat area is given by the intensity kx,y)'(4.1) I , =h-e-" ^ rp (x y) where IQ = the intensity of the incident radiation, ju = the X-ray mass absorption coefficient (cm /g), 2 r= the panel thickness in which X-ray traversed (cm), and P(x, ) y = the panel density (g/cm ). 3 The X-ray scanning system consists of a soft X-ray source, a collimator, and a line array of detectors. What we get from X-ray scanning is the raw data (voltage levels) measured by the detectors which are meaningless unless properly calibrated. The X-ray system was calibrated with three aluminum plates (thickness for each plate 1.521 mm / density 2.700 g/cm ) and 3 three plexiglas (methyl methacrylate) plates (thickness of each plate 9.286 mm / density 1.194 g/cm ). Within each material, four image files containing the voltage levels of X-ray absorption characteristics for each pixel were obtained by placing three, two, one and zero plates in between the X-ray source and the detectors. The curve fitting program takes these four images and averages all the values for each pixel in each file, then simultaneously solves the equations to obtain four coefficients (ao, a\, at, ai) for a best fit third order curve: 78 Chapter IV: Study on the X-Ray Calibration and Overlap Measurements a +a v + a v +a v 2 0 l 2 3 (4.2) where v = values in the scanned images (voltage levels), cio, a\, ai, a-i = the cubic coefficients, and / = the intensity of the X-ray scanning. If we assume that the thickness (r) and the density (p) are uniform within each plate, then the X-ray intensity (/) reduces to kJo, where the subscript / = 0, 1,2,3 denotes the case which X-ray goes through zero, one, two and three plates, respectively and the is the scaling factor. During the calibration period, the coordinate y was kept constant; thus k- = e (4.3) For zero plate (ro = 0), k = e^ p 0 = 1.0 (4.4a) For one, two and three plates cases, the k\, &2, and kz are evaluated by the following equations: k, = e (4.4b) k2 -—p-PW -1c »v j 1 (4.4c) (4.4d) where t2 = 2r\ and TJ = 3T\ 79 Chapter IV: Study on the X-Ray Calibration and Overlap Measurements Therefore, the four coefficients (ao, a\, a% a?) can be solved as a function of IQ. The intensity ratio (/(//) will be used throughout the subsequent discussion. 4.2.2. Mass absorption coefficient The mass absorption coefficient, pi, consists of three components, the photoelectric effect, /u , e the Compton scattering effect, ju , and the pair production effect, n , i.e. c P M = Me + Mc + M (4-5) P At relative low energies (photon energy < 1.02 MeV (Selman 1994)), ju is mainly a mixture of photoelectric with Compton interactions. Cho (1975) and Tsai (1976) developed a model for calculating these coefficients and Lindgren (1991) successfully utilized this model to obtain the linear absorption coefficients for wood constituents at 73 keV. From their studies, the mass absorption coefficients can be summarized as follows: NZ M\=Me+Mc = K—ii NZ + k —f( ) m E 2 (4.6a) n Z~ m M2=Ve+J c= l-^nU k x + 2 of( ) k n E (4.6b) E where JU\ is the mass absorption coefficient for single element, //2 is the mass absorption coefficient for compound substances, k\ and ki are constants, no is the electron density (electron/g), A is the atomic weight, Z is the effective atomic number, N is the Avogadro's number, m is a constant which equals to 4.4 for all biological substances, and the E is the energy of the incident photon in keV. According to Selman (1994), an X-ray beam having a 80 Chapter IV: Study on the X-Ray Calibration and Overlap Measurements particular kVp (peek voltage) has a quality resembling a monoenergetic X-ray beam of about one-third to one-half the peak energy, i.e., the maximum monoenergetic X-ray energy for 74 kVp is 37 keV. The effective atomic number Z and the electron density no can be obtained according to the following equations: (4-7) Z=&a rf iZ and P,Z,/A, I(-PZ,/A.) where a, is the relative electron fraction of element i, and P, is the percentage weight of element i.f(E) is a function of photon energy E, which can be expressed as: 1 ^ p 2 where 2 0 + ^ ( 1 1 + 2/1 + 21) p h ( l + 2 f l _ ip M l . ( 4 . 1 0 ) (\ + 2p) 2 p=E/5\\. The above equations (Cho 1975) make it possible to calculate the mass absorption coefficient for any material as long as its chemical composition is known. According to Fengel and Wegener (1984), aspen (Populus tremuloides) consists of about 42.7% cellulose, 36.0% hemicellulose, 20.9% lignin and 0.4% ash on an extractive-free basis. The cellulose, 81 Chapter IV: Study on the X-Ray Calibration and Overlap Measurements hemicellulose and lignin can be chemically regarded as C6H10O5, C53H7.0O4.6, and C10H13O2, respectively. The calculated mass absorption coefficients for aluminum, plexiglas and aspen are tabulated in Table 4.1. Table 4.1. Calculated mass absorption coefficients Substances Chemical formula Aluminum Al Plexiglas C H 0 5 8 2 Volume fraction (%) M 100 0.6400 100 0.2391 Aspen 0.2395 Cellulose C6H10O5 42.7 0.2489 Hemicellulose C5.3H7.0O4.6 36.0 0.2441 Lignin C10H13O2 20.9 0.2169 4.2.3. Relationship between overlaps and intensity ratio (IQIT) Equation 4.1 can be rewritten as P(x,y) = —ln(-^-) = —ln(k ) () (4.11) where k\ ) = the intensity ratio (Io/I(x, )). Xty y 82 Chapter IV: Study on the X-Ray Calibration and Overlap Measurements The mass in a unit area in the panel equals to xp^ y It is also equal to the total flake height y 0( y)Tf times flake density p/. In mathematical form, it is Xi V<*.,) = <W//>/ (4.12) where 0( , j = overlaps in a unit area, X y tf = flake thickness, and Py = flake density. Therefore, the overlaps have the following relationship with X-ray intensity ratio, flake thickness and density 0 , =-^—ln(k^ ) pr {x y) y) fPf =^ - r (4.13) fPf It is evident that the number of flake overlaps has a linear relationship with panel density. Therefore, the flake overlap distribution within the mat can represent density distribution in the horizontal plane. 4.2.4. Image filter The application of a binomial filter is needed to remove the undesirable random noise during X-ray scanning and to smoothen the sharp points in the images. The values for the coefficients of an odd-sized binomial mask can be expressed directly using the following binomial distribution with mean of Qp and variance of Qp(l - p) (Bernd 1991). 83 Chapter IV: Study on the X-Ray Calibration and Overlay Measurements P, = q\(Q-q)\ P a-P) q (4.14) Where Q denotes the order of binomial levels, for p = Vi, the above equation becomes P, = 1 2 Q Q\ q\(Q-q)\ (4.15) When the order of the binomial Q = 2, the filter mask is V4[l 2 1]. The smallest mask of this kind in two-dimensional is a 3 x 3 binomial filter. The application of 3 x 3 binomial filter is referred to by Bernd (1991). 4.3. Materials and Methods Two different structures of 250mm x 250mm testing panels were prepared using phenolformaldehyde resin powder (Figure 4.1). Uniform aspen flakes (measured average density 0.435 g/cm ) were used with the dimension of 75 mm length, 19.1 mm width, and 0.61 mm J thickness in Structure I (Wang and Lam 1998) and 100 mm x 20 mm x 0.72 mm in Structure II. Flake centroid positions in the mat area (0 < x < 250, 0 < y < 250) are randomly generated based on the Poisson distribution for both structures of mats. All the flakes are perfectly aligned in each layer in Structure I with 90° in two adjacent layers and completely randomized flake orientation in Structure II. The robot forming techniques have been applied to obtain samples with the above-defined structures. The flake position (x, y) and orientation angles (0) are the inputs for a robot driver program to form the experimental mat. 84 Chapter IV: Study on the X-Ray Calibration and Overlap Measurements Structure I: Defined orientations and random positions Structure II: Completely randomized orientations and positions Figure 4.1 Schematic representation of the mat structures. Figure 4.2 Schematic diagram of the X-ray scanning system. 85 Chapter IV: Study on the X-Ray Calibration and Overlap Measurements During X-ray scanning, the distance between any two adjacent scans was selected as small as 2 mm in x direction. The X-ray attenuation profile, P(y), on the y axis is a function of cumulative X-ray absorption with X-rays projected normally to that axis in the z direction (Figure 4.2). As the panel moves along the x axis (X-ray source and detectors fixed) in the scanning process, the attenuation profile becomes two-dimensional. P(x, y) is then typically digitized in the resolution of 1.5mm x 2mm. The image analysis techniques mentioned above are applied to extract signals which could be used to characterize the presence of overlaps or/and voids in the panel. 4.4. Results a n d Discussions 4.4.1. X-ray calibration Two steps are used to obtain the flake overlaps from the X-ray images. First, the raw scanned (voltage levels) images were converted to X-ray intensity images by fitting the third order curve given in Equation 4.1. Then panel density and flake overlaps in the position (x, y) in the mat area can be obtained from the Equations 4.11 and 4.13. The scaling factors (i = 0, 1, 2, 3) used in the curve fitting process obtained from considering the X-ray absorbed by the two kinds of calibration materials (aluminum and plexiglas) are tabulated in Table 4.2. When there is no plate between the X-ray source and the detectors, the scaling factor is 1.0 for intensity which can be regarded as the incident radiation (7o) detected by the X-ray system and constant for both calibration materials. Since the flake overlaps are fixed at the location (x, y), the converted X-ray intensity ratio (IQ/T) 86 Chapter IV: Study on the X-Ray Calibration and Overlap Measurements should also be kept constant no matter what material was used in the calibration. However, due to the different voltage levels obtained from the two materials, we could not expect the calibration coefficients (a , a\, a , 03) 0 2 (Table 4.3) to be the same for both cases. In single byte storage (range from 0 to 255), the intensity ratio (IQ/T) decreases monotonically as the voltage level increases (Figure 4.3). Table 4.2 Scaling factors for X-ray intensity under different number of calibration plates No. of plates 0 1 2 3 Aluminum (/u =0.6400) 1.0000 0.7695 0.5921 0.4556 Plexiglas 1.0000 0.7672 0.5886 0.4516 (// = 0.2391) Table 4.3 Calibration coefficients for one pixel ( x I ) 0 fl2 #3 ao «i Aluminum 0.1240 2.2429E-03 2.2688E-05 -6.5513E-08 Plexiglas 0.0556 4.4953E-03 1.9411E-06 -9.8752E-09 87 Chapter IV: Study on the X-Ray Calibration and Overlap Measurements 8.0 t § 5.0 ^ 4.0 - 0.0 0 50 100 150 200 250 300 Voltage level (raw data) Figure 4.3 The relationship between X-ray intensity ratio and voltage levels. 4.4.2. Relationship between X-ray intensity ratio and flake overlaps In order to verify the relationship between X-ray intensities and flake overlaps, another 6 measurements were made on the different flake stacks (5, 10, 15, 20, 25, and 30 flakes). The results are as shown in Figure 4.4. The experimental measurement is in good agreement with the model predicted. As stated in Equation 4.13, the overlaps are a function of natural logarithm of the inverse intensity. However, at small overlaps (less than 50) a linear relationship was observed between flake overlaps and the X-ray intensity ratio (Figure 4.4). The reason is from Equation 4.13, l+ for small values of MT p O , f f (x y) (4.16) ^ fPf°{x, ). T y 88 Chapter IV: Study on the X-Ray Calibration and Overlap Measurements 1.30 0 5 10 15 20 25 30 35 Flake overlaps Figure 4.4 Flake overlaps in relation to X-ray intensity ratio Hence, Equation 4.13 can be rewritten as: q x , , * ^ ^ t*TfPf 2 (4-17) The commercial OSB panels with 10 mm thickness usually have less than 30 flakes in a column on average, while the laboratory made flakeboards have 24.5 flake overlaps for structure I and 21 flake overlaps for structure II. This will introduce an error in X-ray intensity ratio of 2.2% for commercial OSB panels and 1.1% - 1.5% for laboratory made samples by applying the approximation in Equation 4.17. 4.4.3. Flake overlaps from X-ray scanning images Figure 4.5 shows the X-ray scanning images of density of three replicated robot-formed 89 Chapter IV: Study on the X-Ray Calibration and Overlap Measurements panels and one simulated density image for each of the mat structures (Figure 4.1). Increase in darkness of the map denotes high density. The statistical results for flake overlaps are listed in Table 4.4. The simulated mats have higher standard deviation of the means due to higher resolution and accuracy. From the image pattern, the simulated mats match the experimental panels quite well and the three experimental panels in each structure have high repeatability. This means that few replications are required by using a robot in our experiment so that both time and costs will be reduced in the future studies. One particular scanning line from each of these images is drawn in Figure 4.6 for closer comparison. The variation of overlaps inside the panel at an arbitrarily scanning line agrees well with the simulation line at the same location. The overlap information in 240mm x 240mm of mat area was extracted and subdivided into 144 sampling zones (20mm x 20mm) (see Figure 4.5) to investigate the variation of local overlap averages among these zones. As shown in Figure 4.7 (only the first 48 zones were plotted), the corresponding sampling zones vary accordingly. Therefore, a quantitative characterization of the overlaps may lead to predictions of the variability of panel properties by nondestructive testing method and simulation process. 90 Chapter IV: Study on the X-Ray Calibration and Overlap Measurements Figure 4.5a Horizontal density distribution images from X-ray measurements in Structure I. 91 Chapter IV: Study on the X-Ray Calibration and Overlap Measurements Figure 4.5b Horizontal density distribution images from X-ray measurements in Structure II. 92 Chapter TV: Study on the X-Ray Calibration and Overlap Measurements Table 4.4 The statistical results from the X-ray analysis of three images (121 x 163 pixels) of each mat structure as compared to computer simulation Source Mean Maximum Minimum Std. dev. of Corrected density (overlap) (overlap) (overlap) the mean (g/cm ) Structure I-A 24.3 39.9 5.1 4.1 0.595 Structure I-B 23.1 36.7 4.8 4.3 0.565 Structure I-C 24.9 37.1 9.2 4.0 0.608 Simulation I 24.5 38.6 6.6 5.3 0.599 Structure II-A 19.3 30.4 6.3 3.4 0.552 Structure II-B 21.0 30.9 7.4 3.5 0.599 Structure II-C 20.7 30.2 5.9 3.3 0.591 Simulation II 21.1 33.3 8.4 4.2 0.609 3 93 Chapter IV: Study on the X-Ray Calibration and Overlap Measurements 40 30 a 1 > 20 'Simulation I c • Structure 1-A 10 -Structure I-B • Structure I-C 40 80 120 160 200 240 Scanning line length (mm) Figure 4.6 Flake overlaps in a particular scanning line in the mat area for the simulation and X-ray measurements in structure I. 0.4 Sampling zone size: 20 x 20 (mm) •Simulation I Structure 1-A - Structure I-B Structure I-C 0.0 12 24 36 48 Zone number Figure 4.7 Comparing the normalized standard deviation of density for the simulation and X-ray measurements in structure I. 94 Chapter IV: Study on the X-Ray Calibration and Overlap Measurements 4.5. Conclusions In modeling the overlaps and/or densities in the mat area, the mat is considered as a grouping of multi-stacked flake columns (Dai and Steiner, 1993). The overlap distribution contributes to the variation of the panel properties. This paper developed a model to predict the overlaps by an X-ray scanning method. A close relationship was found between number of overlaps and X-ray intensity ratio. The results also show that the calibration materials have little effect on the number of overlaps obtained from X-ray scanning and the overlaps are in good agreement with those from computer simulations. The robot formed flakeboard mats have higher repeatability with lower deviation in both overlap distribution and horizontal density distribution. A robot can play an active role in reducing replications of the experiment in future studies. 4.6. References Bernd, J. 1991. Digital Image Processing: Concepts, Algorithms and Scientific Applications. Springer-Verlag. Berlin Cho, Z. H.; Tsai, C. M. and Wilson, G. 1975. Study of contrast and modulation mechanisms in X-ray/photon transverse axial transmission tomography. Phys. Med. Biol. 20(6): 879-889 Dai, C. and Steiner, P. R. 1993. Compression behavior of randomly formed wood flake mats. Wood and Fiber Science, 25(4):349-358 95 Chapter IV: Study on the X-Ray Calibration and Overlap Measurements Dai, C. and Steiner, P. R. 1994a. Spatial structure of wood composites in relation to simulation of a randomly formed flake layer network. Part 2. Modeling and simulation of a randomly formed flake layer network. Wood Science and Technology, 28:135-146 Dai, C. and Steiner, P. R. 1994b. Spatial structure of wood composites in relation to processing and performance characteristics. Part 3. Modeling the formation of multi-layered random flake mats. Wood Science and Technology 28: 229-239 Dai, C. and Steiner, P. R. 1994c. Analysis and implication of structure in short fiber wood composites. Second Pacific Rim Bio-Based composites Symposium, Nov. 6-9, Vancouver, Canada, pi7-24 Dodson, C.T.J. 1971. Spatial variability and the theory of sampling in random fibrous network. J. Roy. Statist. Soc. Series B. 33(1): 88-94 Fengel, D. and Wegener, G. 1984. Wood : Chemistry, Ultrastructure, Reactions. Berlin ; New York. Kallmes, O. and Bernier, G. 1963. The structure of paper. IV. The free fiber length of a multiplanar sheet. Tappi 46(2): 108-114 Kallmes, O. and Corte, H. 1960. The structure of paper. I. The statistical geometry of an ideal two dimensional fiber network. Tappi 43(9): 737-752 Kallmes, O.; Corte, H. and Bernier, G. 1961. The structure of paper. II. The statistical geometry of a multiplanar fiber network. Tappi 44(7): 519-528 96 Chapter IV: Study on the X-Ray Calibration and Overlap Measurements Lang, E. M. and Wolcott, M. P. 1996. A model for viscoelastic consolidation of wood-strand mats. Part I. Structural characterization of the mat via Monte Carlo simulation. Wood and Fiber Science 28(1): 100-109 Lindgren, L. O. 1991. Medical CAT-scanning: X-ray absorption coefficients, CT-numbers and their relation to wood density. Wood Science and Technology 25:341-349 Selman, J. 1994. The fundamentals of X-ray and radium physics. Eighth Edition. Charles C Thomas Publisher. Illinois, U S A Steiner, P. R. and Dai, C. 1993. Spatial structure of wood composites in relation to processing and performance characteristics. Part 1. Rationale for model development. Wood Science and Technology, 28: 45-51 Tsai, C. M. and Cho, Z. H. 1976. Physics of contrast mechanism and averaging effect of linear attenuation coefficients in a computerized transverse axial tomography (CTAT) transmission scanner. Phys. Med. Biol. 21(4): 544-559 Wang, K. and Lam, F. 1998. Robot-based research on a three-layered oriented flakeboard. Wood and Fiber Science, 30(4): 339-347 Xu, W. and Steiner, P. R. 1995. A statistical characterization of the horizontal density distribution in flakeboard. Wood Fiber Sci. 27(2): 160-167 97 Chapter V: Relationship between Thickness Swelling and Mat Structures CHAPTER V RELATIONSHIP BETWEEN THICKNESS SWELLING AND MAT STRUCTURES IN ROBOT-FORMED FLAKEBOARD MATS Abstract The relationship between thickness swelling and mat structures in robot-formed flakeboard mats under humid conditions with consideration of the density/overlaps in the flakeboard and the absorbed moisture has been studied. Results show a highly correlated linear relationship between thickness swelling and relative moisture content. The relationship between moisture absorption coefficient and the density/ overlaps is also linear. 5.1. Introduction Oriented strand board (OSB), a reconstituted wood product with oriented flakes in both face and bottom layers and random flakes in the core layer, is hygroscopic and dimensionally unstable when exposed to a humid environment or immersed in water. It absorbs or releases moisture when subjected to an increase or decrease, respectively, in relative humidity at a given temperature. Thickness swelling influences composite panel performance, both visually and functionally. Johnson (1956) investigated the dimension changes in 36 types of commercial hardboard by exposing the specimens to water soaking for 24 hours and to a relative humidity of 90% for 8 months. A linear relationship between percentage of water absorption and thickness swelling was found for most panels in which the water was absorbed at approximately the same rate. Within 10 days after exposure to high relative 98 Chapter V: Relationship between Thickness Swelling and Mat Structures humidity conditions, most of the boards reached approximately the same moisture content. Johnson (1964) later reported that the increase in moisture and thickness swelling for specimens of commercial particleboard exposed to 90% relative humidity were at least twice the increase obtained at exposure to 65% relative humidity in the same time period. Halligan (1970) thoroughly reviewed the effects of board density, wood species, flake geometry, resin type and content, blending efficiency, pressing conditions and special treatments on the thickness swelling in particleboard/ flakeboards produced under conventional pressing. In terms of influence of flake geometry on thickness swelling, two factors: flake slenderness ratio (ratio of length to thickness) and flake thickness have been considered (Mottet 1967). The larger the slenderness ratio, the higher dimensional stability and the smaller the flake thickness, the smaller the swelling. In higher density boards the flakes are more severely compressed resulting in potential swelling as stresses are released. Lehmann (1972) found that thickness swelling was directly related to the amount of moisture absorbed. Under conventional press schedule, the maximum thickness increase for commercial particleboard was 9.7% under 10.29% moisture absorption, 16.2% for commercial three-layer particleboard under 11.62% moisture absorption, and 17.6% for laboratory-made particleboard under 11.08% moisture absorption. In some cases, functions of time were more highly correlated with thickness swelling than the amount of absorbed moisture. The logarithm of time, instead of square root of time stated in classical diffusion theory, provided the most highly correlated relationships with water adsorption. Gatchell et al (1966) related the board density to thickness swelling from exposure to high humidity environment. Suo (1991) approached a wood composite as a multilayer system and proposed an equation to predict the behavior of thickness swelling (TS) based on the properties of each 99 Chapter V: Relationship between Thickness Swelling and Mat Structures layer, TS = a +a (MC) + a p 0 x 2 (5.1) where do, a\, ci2 = constants, MC = moisture content, and p = board density. Xu and Winistorfer (1995) proposed a method to determine the thickness swelling distribution across the board thickness and to relate it to the vertical density distribution. The results showed that an average of 7.82% and 15.05% thickness swelling were obtained in a 24-hour soak test for OSB and particleboard, respectively. The thickness swelling at the surface region of the OSB and particleboard was higher than that in the center. Approximately a linear relationship was found between layer density and layer thickness swelling. The layer distribution of water absorption in OSB and particleboard was also believed to be positively correlated to layer density and to layer thickness swelling (Xu et al 1996). An average of 16.5% and 32.1% of water absorption were measured after 24-hour water soak exposures for OSB and particleboard, respectively. Geimer et al (1998) conducted research to investigate the effect of steam injection pressure and duration on the thickness swelling of resinless flake mats soaked in water and concluded that thickness swelling was greatly reduced by steam injection as compared to the thickness swelling of conventionally pressed mats. In liquid water or in high humid environment, thickness swelling consists of three major components: natural dimensional changes of wood material, release of residual stresses 100 Chapter V: Relationship between Thickness Swelling and Mat Structures imparted to the panel during pressing, and non-uniform board structure in terms of density variation (Song and Ellis 1997). Dimensional stability could be improved by decreasing board density (Vital et al 1979, 1980); therefore, the mat structure plays an important role because it controls the density/overlaps in the local area and the compression behavior during the manufacturing process. Thus the thickness swelling is not only time dependent and moisture dependent, but is also structure dependent. So far all the thickness swelling studies of OSB or particleboard are mainly based on empirical experimental data. There is a need to describe the relationship among the thickness swelling, moisture absorption and the structural properties of the panel. The thickness changes in wood composite panels in end-use applications are usually a result of two cases: 1) due to absorption of liquid water during construction, and 2) due to absorption of moisture in service in high humidity climates. For each of these cases, specific tests can be conducted to simulate the end-use conditions. The standard 24-hour water soaking test specified in A S T M (1994) would simulate the first one. A high humidity chamber test would simulate the second one. This chapter focuses on the dimensional changes in the thickness direction, due to absorbing moisture in service. The objectives of this study are: 1) to develop a relationship between the thickness swelling and relative moisture content; and 2) to predict the thickness swelling under high humidity test condition when the distribution of board density or overlaps is known. 101 Chapter V: Relationship between Thickness Swelling and Mat Structures 5.2. Model Thickness swelling is defined as the percentage of thickness increase over the original thickness. According to the definition of the Canadian Standard Association (CSA, 1993), the percentage of thickness swelling of a panel, TS, is calculated with the following formula: 75 =-5^2. x 1 0 0 (5.2) where T = the thickness measured at time t after testing (mm), and t To = the original thickness measured before testing (mm). 5.2.1. Strain and stress relationship The orthotropic nature of wood leads to complex relations between stresses and strains, especially when time dependence and moisture dependence are considered. The understanding on the behavior of wood elements subjected to humid condition is of great value when it concerns the quality of structural flakeboard in service. Martensson (1994) proposed a general constitutive model to represent the behavior of wood. According to this model, the time derivatives of the strain are functions of stresses and moisture conditions, i.e., £ = £ +£ +£ '+£ e C S m S (5.3) where £ = — is the elastic strain rate, E e £ is the pure creep strain rate, c 102 Chapter V: Relationship between Thickness Swelling and Mat Structures s = (or - Aa)M is the shrinkage and swelling strain rate, and s> s = mo-rn is the mechanosorptive strain rate. ms Here, M = moisture content, cr = current stress, a = absorption coefficient, E = elastic modulus and m = mechanosorptive parameter. Aa = kps +k(s e c +s ) ms is also a function of the elastic strain, the creep strain, and the mechanosorptive strain. When the specimens absorb moisture in a high relative humidity condition without external load, there is no external stress applied. Therefore, all the terms related to stress vanish and Equation 5.3 simplifies to: £ = s '=aM (5.4) s Since the strain, s, has the same definition as thickness swelling, TS, therefore, the following relationship also holds true: d(TS) dt de dM =— =a dt dt (5.5) Integrating both sides of Equation 5.5. TS(t)-TS(t ) 0 = a[M(t)-M(t )] Q (5.6) where TS(t), TS(to) = the thickness swelling at time t and to, respectively, and M{t), M(to) = the moisture content at time t and to, respectively. Assuming TS(to) = 0 (no thickness swelling before the specimens were put in a humidity chamber), Equation 5.6 can be expressed as: TS(t) = a[M(t)-M(t )] 0 = a-AM(t) (5.7) where AM(t) = absolute moisture absorbed from time to to time t. 103 Chapter V: Relationship between Thickness Swelling and Mat Structures Equations 5.5 shows that thickness swelling rate is a linear function of the rate of moisture content and Equation 5.7 indicates that thickness swelling is a linear function of the moisture absorbed. 5.2.2. Effect of mat structures The absorption coefficient, a, is a structure-related parameter. In general, it is believed that the increase of board density resulted in increase of thickness swelling (Wang and Lam 1998). Suchsland (1989) indicated that the dimensional stability of flakeboards is controlled by the higher density regions. Assuming the absorption coefficient a varies linearly with respect to the board density, a can be expressed to take on the following form: (5.8) a = f(p) = k +k p 0 x where p = local board density (g/cm ), and 3 ko, k\ = constants. Thus the Equations 5.5 and 5.7 can be expressed as a function of density as follows, ^ - at ( ^ p ) ^ dt TS(t) = (k +k p)AM(t) 0 l (5.9, (5.10r Lu et al (1998) showed that the density has the following relationship with the number of flake overlaps, O. OT p P = — ^T f f fJ (5.11) where O = local average number of flake overlaps, T/= flake thickness (mm), 104 Chapter V: Relationship between Thickness Swelling and Mat Structures •3 Pf= flake density (g/cm ), and T= board thickness (mm). Substituting the Equation 5.11 into Equations 5.9 and 5.10, the thickness swelling can be expressed as a function of flake overlaps as im^ ^L . ko+k;o) dt (5 12) dt TS(t) = (k + k*O)AM(t) 0 (5.13) where Tp f f k = &j ——— constant. r t 5.3. Materials and Methods Nine experimental flakeboards of size 240mm (length) by 240mm (width) by 10mm (thickness) were prepared by using a robot-forming technique. Uniform aspen flakes of size 100mm (length) by 20mm (width) by 0.8mm (thickness) with average density of 0.35g/cm 3 were used in each mat. All flakes were dried to an equilibrium moisture content of 10-12% and mixed with 6% (based on the oven-dry flake weight) powder phenol-formaldehyde resin (CASCOPHEN W91B) before the mat forming process. No wax was applied. The flake centroid positions (JC, y) and orientation angles (6) of the mat were randomly selected by a simulation program Winmat® (Lu et al 1998) assuming uniform distributions of x, y and 0. Six hundred and forty six (646) flakes were used in each mat for a target board density of 0.6 g/cm with an average of 19 overlaps of flakes in a column and 34 flakes in a 3 105 Chapter V: Relationship between Thickness Swelling and Mat Structures layer. The parameters (x, y, 6) of each flake were saved to a deposition data file and used for all nine mats. A robot mat-forming program Robot® loaded the data file and converted the parameters of each flake into the format of robot controller commands. These commands consist of seven variables for each flake. Three of them (Xm, y m z ) indicate the position of m flake over the mat area. Another three (xb, yb, Zb) specify where the robot arm should pick up the flakes from flake holder bins and the last one for the orientation angle (6) of a flake on a mat. The whole mat-forming process was controlled by a computer, which was used to send the robot commands one by one to robot controller. Thus a loose mat was formed one flake by one flake. The nine replicated mats were made from the same database by the robot mat forming system in this way. A laboratory hot press (300mm x 300mm) equipped with an MTS computer data acquisition system was then used to press the formed mat. Efforts were made to load the formed mat into hot press without flake movement. The origin of Cartesian coordinate was clearly identified on the pressed panels. The loose mats were hot-pressed at 180°C to 10 mm stops using 0.5 min. closing time, 5 min. at stops and 0.5 min. opening time. The temperature, pressure and pressing time can be accurately recorded in a file stored in a PC during pressing. The schematic diagram of the mat structure is illustrated in Figure 5.1, the simulated local density averages are shown in Figure 5.2 and the local density measured by gravimetric method is illustrated in Figures 5.4a and 5.4b. 106 Chapter V: Relationship between Thickness Swelling and Mat Structures X axis (mm) Figure 5.2 Contour map of simulated horizontal density distribution. 107 • Chapter V: Relationship between Thickness Swelling and Mat Structures 16 10 09 •Ul IB 05 M 06 08 04 03 01 -~ X 0 J 8 of these specimens tested under condition of 95% RH 8 of these specimens tested under condition of 90% RH Figure 5.3. Specimen cutting pattern corresponding to each square in Figure 5.2. 0.8 0 2 4 6 8 10 12 14 16 Specimen location number Figure 5.4a Local density averages of robot-formed flakeboard mats (TS1, TS2, TS3) as compared to the simulated mat (density for robot-formed mats were measured from 50mm x 50mm specimens and density for simulation was predicted (Si by Winmat at 50mm x 50mm sampling zone). 108 Chapter V: Relationship between Thickness Swelling and Mat Structures & 0.6 1 — 0.5 0.4 0 2 4 6 8 10 12 14 16 Specimen location number Figure 5.4b Local density averages of robot-formed flakeboard mats (TS4, TS5, TS6) as compared to the simulated mat (density for robot-formed mats were measured from 50mm x 50mm specimens and density for simulation was predicted by Winmat® at 50mm x 50mm sampling zone). The formed panels were placed in ambient conditions for more than 6 months before testing specimens were cut. Specimens cut from six panels were tested under 95% and 90% relative humidity conditions in which half of the specimens was used for model setup and the other half was used for the model verification. Specimens cut from the remaining three panels were tested under 24-hour water soaking. The 50mm x 50mm specimen size was chosen in this study because the sample board size is limited. Sixteen specimens from each board were prepared and 8 of them tested under condition of 95% relative humidity and the other 8 tested under condition of 90% relative humidity (Figure 5.3). For water soaking test, 16 specimens from each panel were divided into two groups of 8 each, which were tested separately. The initial thickness values of each specimen were measured at four corners 5 mm away from each edge using a micrometer (0.01 mm) and the initial weight was recorded by an electronic 109 Chapter V: Relationship between Thickness Swelling and Mat Structures balance (0.0001 g). The initial moisture content of the specimen was 5.2% on average. A humidity chamber was used to condition the specimens during humidity conditioning. The absorbing moisture tests were carried out under 95% and 90% relative humidity conditions at 60°C, and thickness measurements were taken at 2, 6, 12, 24, 48, 72, 96, 120, 144 and 168 hours at the same location of a specimen. The absorbing water tests were done by emerging specimens in water at temperature of 20 + 1°C and the thickness measurements were taken at 0.5, 1.5, 3, 5.5, 8, 14.5, 19.5, and 24 hours. Determination of thickness swelling was obtained from each measuring point and the average thickness swelling for a specimen was calculated by taking the average from four thickness measurements. Determination of the absorbed moisture was obtained from weight measurements taken at the same intervals as the measurement for thickness. The relative thickness swelling (the increase of thickness swelling between two adjacent measurements) and relative moisture (the increase of moisture absorbed between two adjacent measurements) were also calculated for each time interval during the tests. 5.4. Results and Discussions 5.4.1. Relationship between moisture content and time Flake based wood composites, such as flakeboard and OSB, contain both between-flake voids and within-flake micro-pores. Water absorbed into the between-flake voids does not directly contribute to the dimensional changes in the flake network; therefore, the volume changes are the results of expansion in the cell walls of the flakes. In order to simulate this situation, a humidity chamber was chosen to investigate this particular case. After measuring 110 Chapter V: Relationship between Thickness Swelling and Mat Structures thickness changes and weight changes with time, the thickness swelling and the moisture absorbed were obtained. The results show that the absorbed moisture of the flakeboard samples increased rapidly in the first 24 hours of test (Figure 5.5). Then it gradually leveled as testing time increased and reached a relative moisture content of 11.8% (5.2% initial moisture content excluded) under 95% relative humidity (Figure 5.5a) and of 9.8% (5.2% initial moisture content excluded) under 90% relative humidity (Figure 5.5b). According to the classical diffusion theory, absorbed moisture is expected to vary linearly with the square root of time. Figure 5.6 shows that the classical diffusion theory can be used to approximate the absorbed moisture versus time relationship at the initial absorption stage within the first 24 hours in flake-based wood composites. Time (hours) Figure 5.5a Absorbed moisture and thickness swelling in relation to test time under 95% relative humidity test condition. Ill Chapter V: Relationship between Thickness Swelling and Mat Structures 42 Absorbed moisture (90% RH) -o— Thickness swelling (90% RH) H 36 30 "l l l l 24 18 M 12 24 48 72 96 120 144 168 Time (hours) Figure 5.5b Absorbed moisture and thickness swelling in relation to test time under 90% relative humidity test condition. 14 Figure 5.6 Absorbed moisture of flakeboard in relation to square root of time under 95% and 90% relative humidity test conditions. 112 Chapter V: Relationship between Thickness Swelling and Mat Structures In the 24-hour water soaking test, approximately two-thirds (69.24%) of the total amount of water (99.18%) absorbed in 24 hours was taken up during the first half-hour (Figure 5.7), resulting in 24.56% (79% of total) thickness swelling. However, this fast-absorbed water does not necessarily saturate the fiber completely because most of the water stays as free water inside the specimen. The thickness continues to expand to an approximately constant level of 31% thickness swelling after 5 hours soaking. At this point, the fibers are fully saturated, and the thickness swelling keeps approximately constant no matter how much water is absorbed thereafter. 120 60 50 ^ x 40 g % 30 £ Vi Vi A 20 £ -< — Absorbed water u - e — Thickness swelling 10 § 0 8 12 16 20 24 Time (Hours) Figure 5.7 Water absorption and thickness swelling in relation to soaking time during 24-hour water soaking test. 5.4.2. Relationship between thickness swelling and moisture absorbed It is known that the thickness swelling and moisture absorption of flakeboard is timedependent both in high relative humidity conditions (Figure 5.5) and water soaking test 113 Chapter V: Relationship between Thickness Swelling and Mat Structures conditions (Figure 5.7). The thickness swelling and absorbed moisture are correlated in both 95% and 90% relative humidity conditions (Figures 5.8a and 5.8b). Similarly the relationship between the relative thickness swelling and relative moisture absorbed in each time interval is well correlated (Figures 5.9a and 5.9b). However, in water soaking test, the above relations are poorly correlated (Figures 5.8c, 5.9c, and 5.10c). This is because in the humidity chamber almost all the absorbed moisture contributed to thickness swelling. In water soaking test, a large amount of water absorbed by the specimens entered into the voids between flakes and pores inside the flakes as free water. Only partial absorption contributed to thickness swelling. When the fiber saturation point is reached, no further dimension changes occur. From the slope of the regression lines in Figures 5.8a and 5.9a, one can estimate that on average, a 1% moisture change of the flakeboard would result in a 2% change in thickness swelling. This phenomenon is further demonstrated in Figures 5.10a in terms of the rate of thickness swelling changes and the rate of moisture changes. A similar relationship was found in Figures 5.8b, 5.9b and 5.10b. 45 [ 40 I 0 I 1 0 2 1 4 1 i i i I 6 8 10 12 14 Absorbed moisture (%) Figure 5.8a The correlation between thickness swelling and absorbed moisture under 95% relative humidity condition. 114 Chapter V: Relationship between Thickness Swelling and Mat Structures 45 40 V 0 2 4 6 10 8 12 14 Absorbed moisture (%) Figure 5.8b The correlation between thickness swelling and absorbed moisture under 90% relative humidity condition. 45 , , 0 I 1 1 0 20 40 I I ! I 60 80 100 120 Absorbed moisture (%) Figure 5.8c The correlation between thickness swelling and absorbed moisture under 24-hour water soaking test. 115 Chapter V: Relationship between Thickness Swelling and Mat Structures 12 0 1 2 3 4 5 Relative moisture (%) Figure 5.9a The correlation between the relative thickness swelling and relative moisture absorbed at each time interval under 95% relative humidity condition. 10 0 1 2 3 4 5 Relative moisture (%) Figure 5.9b The correlation between the relative thickness swelling and relative moisture absorbed at each time interval under 90% relative humidity condition. 116 Chapter V: Relationship between Thickness Swelling and Mat Structures 35 30 # •• • 25 R = 0.9367 2 20 > •a 15 "3 10 • y = 0.3686x • 5 0 • • •• • «|*J^r^' = ° - j^gP> HP* 20 40 • 60 2557x R = 0.534 2 100 80 Relative moisture (%) Figure 5.9c The correlation between the relative thickness swelling and relative moisture absorbed at each time interval under 24-hour water soaking test (the bottom right chart is obtained without the first measurement). 0.9 0.8 0.7 0.6 ••TJ 0.5 0.4 0.3 0.2 y = 1.8429x 0.1 R = 0.9399 2 0 * 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 cLM/dt Figure 5.10a The correlation between the rate of thickness swelling and the rate of moisture changes under 95% relative humidity condition. 117 Chapter V: Relationship between Thickness Swelling and Mat Structures y = 1.5898x R =0.9564 2 0.2 0.4 0.6 0.8 dM/dt Figure 5.10b The correlation between the rate of thickness swelling and the rate of moisture changes under 90% relative humidity condition. dM/dt Figure 5.10c The correlation between the rate of thickness swelling and the rate of moisture changes under 24-hour water soaking test (the bottom right chart is obtained wihtout the first measurement). 118 Chapter V: Relationship between Thickness Swelling and Mat Structures 5.4.3. Relationship between density and absorption coefficient By nature of the random deposition process of the flakeboards, the density varies in different locations in the mat area. Along with the normal swelling of wood cell wall in solid wood, the residual compression stress is another factor to be considered in the strain release process during moisture absorption. This factor is handled by considering the absorption coefficient as panel density dependent in the model. It is common sense that wood swells only when moisture content is below fiber saturation point. More absorbed free water does not further contribute to swelling. Therefore, the development of a relationship between density and absorption coefficient is essential and critical in defining dimensional stability at high relative humidity conditions. Figure 5.11 shows the linear relationship between moisture absorption coefficient and panel density. Both higher density and higher relative humidity conditions result in a higher a. From Figure 5.11, the two regression lines for 95% and 90% relative humidity conditions are statistically parallel (slopes are 4.1839 and 4.2193) to each other. If the average slope of these two lines is taken for consideration, the resulting error is less than 0.5%. This means that the relative a changes are the same in both test conditions at the same density changes. 5.4.4. Verification of the model The model verification was carried out using the other half of the specimens. Tests follow the same procedures under 95% and 90% relative humidity conditions as described before. Figures 5.12a and 5.12b show that the measured thickness swelling agrees well with that of model predictions at three different density levels (0.66, 0.62 and 0.56 g/cm ). From Figure 5.13, the measured and predicted absorption coefficients versus density also agree well. 119 Chapter V: Relationship between Thickness Swelling and Mat Structures Figure 5.11 The relationship between absorption coefficient and density of flakeboard under 95% and 90% relative humidity test conditions. 35 Density (g/cm ) 3 r> 30 S — • 25 in 15 £ 10 Measured (0.66) Predicted (0.62) 20 Y 3 Predicted (0.66) • Measured (0.62) • Predicted (0.56) Measured (0.56) 5 4 6 8 10 12 14 Absorbed moisture (%) Figure 5.12a The predicted and measured thickness swelling in relation to absorbed moisture for three density levels (TS404 - 0.66, TS408 - 0.62 and TS406 - 0.56 g/cm ) under 95% relative humidity test condition. 3 120 Chapter V: Relationship between Thickness Swelling and Mat Structures 21 Density (gA 19 £ 17 Ml Predicted (0 • CA CA CA CD C -< o e 13 ii Measured (0 Predicted (0.< .S 15 • Measured (0. • Predicted (0 Measured (0 9 7 5 4 6 10 Absorbed moisture (%) Figure 5.12b The predicted and measured thickness swelling in relation to absorbed moisture for three density levels (TS512 - 0.66, TS410 - 0.62 and TS407 - 0.56 g/cm ) under 90% relative humidity test condition. 3.5 Figure 5.13 The predicted and measured absorption coefficients in relation to density under 95% and 90% relative humidity test conditions. 121 Chanter V: Relationship between Thickness Swelling and Mat Structures 5.5. Conclusions From this study, the following conclusions can be drawn: The thickness swelling was found to be time-dependent, moisture-dependent and densitydependent during moisture absorption test under high relative humidity conditions. The relationship between moisture absorption and time conforms to the classical diffusion theory. A linear relationship was obtained between the relative thickness swelling and relative moisture absorbed and between the rate of thickness swelling changes and the rate of relative moisture changes. The moisture absorption coefficient is a linear function of the board density and the localized panel density plays an important role in absorption since the moisture is mostly absorbed by wood cell wall components. 5.6. References American Society for Testing and Materials. 1994. Standard test methods for evaluating properties of wood-base fiber and particle panel materials. A S T M D 1037-93. Philadelphia, PA. Canadian Standard Association. 1993. Standards on OSB and Waferboard. 0437 Series-93. Ontario, Canada Gatchell, C. J.; B.G. Heebink and F.V. Herry. 1966. Influence of component variables on the properties of particleboard for exterior use. Forest Products Journal, 16(4): 46-59 Geimer, R.L.; J.H. Kwon and J. Bolton. 1998. Flakeboard thickness swelling. Part I. Stress 122 Chapter V: Relationship between Thickness Swelling and Mat Structures relaxation in a flakeboard mat. Wood and Fiber Science, 30(4): 326-338 Halligan, A . F. 1970. A review of thickness swelling in particleboard. Wood Science and Technology. 4(4): 301-312 Johnson, J. W. 1956. Dimension changes in hardboard from soaking and high humidity. Rep. Ore., Forest Prod. Lab. No. T-16, 1 lpp Johnson, J. W. 1964. Effect of exposure cycles on the stability of commercial particleboard. Forest Products Journal, 14: 277-282 Lehmann, W. F. 1972. Moisture-stability relationships in wood-base composition boards. Forest Products Journal, 22(7): 53-59 Lu, C ; P.R. Steiner and F. Lam. 1998. Simulation study of wood-flake composite mat structures. Forest Products Journal, 48(5): 89-93 Martensson, A . 1994. Creep behavior of structural timber under varying humidity conditions. Journal of Structural Engineering, 120(9): 2565-2582 Mottet, A . L . 1967. The particle geometry factor in particleboard manufacture. Proceedings, 1 Washington State University., Particleboard Symposium Pullman, 23-73 st Song, D. and S. Ellis. 1997. Localized properties in flakeboard: a simulation using stacked flakes. Wood and Fiber Science, 29(4): 353-263 Suchsland, O. and H. Xu. 1989. A simulation of the horizontal density distribution in a flakeboard. Forest Products Journal, 39(5): 29-33 Suo, S. 1991. Computer simulation modeling of structural particleboard properties. Ph.D. thesis, University of Minnesota, St. Paul, M N . 158pp 123 Chapter V: Relationship between Thickness Swelling and Mat Structures Vital, B.R. and J.B. Wilson. 1979. Factors affecting the water adsorption of particleboard and flakeboard. Symposium on wood moisture content - temperature and humidity relationships, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, Oct. 29, 1979. 97101 Vital, B.R; J.B. Wilson and P.H. Kanarek. 1980. Parameters affecting dimensional stability of flakeboard and particleboard. Forest Products Journal, 30(12): 23-29 Wang, K. and F. Lam. 1998. Robot-based research on three-layer oriented flakeboards. Wood and Fiber Science, 30(4): 339-347 Xu., W. and P.M. Winistorfer. 1995. A procedure to determine thickness swell distribution in wood composite panels. Wood and Fiber Science, 27(2): 119-125 Xu, W.; P.M. Winistorfer and W.W. Moschler. 1996. A procedure to determine water absorption distribution in wood composite panels. Wood and Fiber Science, 28(3): 286-294 124 Chapter VI: Model Application: A Case Study CHAPTER VI MODEL APPLICATION: A CASE STUDY In this chapter, the following properties of partially oriented flakeboard mats and oriented strand board (OSB) mats were simulated using Winmat® and analyzed using models developed in the previous chapters: density and overlap profiles, void size distribution, autocorrelation and variance functions, degree of orientations, and thickness swelling. The simulated density and overlap profiles and degree of orientation were compared with those in a commercial OSB panel. In the commercial OSB panel, the density profile was measured by an X-ray scanning technique and the corresponding overlap profile and degree of orientation were estimated from the experimentally determined density profiles. The usefulness and prospective applications of the models were also presented. Future research work and modifications with respect to this model were suggested. Finally, the limitations of the current model are discussed. 6.1. Simulation Parameters The size of the mats simulated here was chosen to be as close as possible to the standard size of commercial panels (2440mm x 1220mm). The flake positions were randomly generated based on the Poisson distribution. The orientation angles of the flakes are also randomly generated as follows: for the partially oriented flakeboard mats, the angles are restricted within -20° and +20°; for OSB, a three-layer structure is constructed with randomly oriented flakes (-90°<6!<+90 ) in the core and partially oriented flakes (from -20° to +20°) in both face o layers. Details of the simulation parameters are shown in Table 6.1. In total 100 mats were 125 Chapter VI: Model Application: A Case Study simulated for each of the structures by providing different random seeds to the pseudorandom number generator while keeping all the other input parameters unchanged. The results presented hereafter are mean results of all the simulations unless specified otherwise. Table 6.1 Simulation parameters Parameters Properties Partially oriented flakeboard OSB Target board size 2440mm x 1220mm x 11mm 2440mm x 1220mm x 11mm Target board density 0.56g/cm 0.56g/cm Flake size 100mm x 15mm x 0.7mm 100mm x 15mm x 0.7mm Flake density 0.4g/cm 0.4g/cm Flake location Random Random Flake orientation Random (-20° to +20°) Faces: random (-20° to +20 °) 3 3 3 3 Core: random (-90° to +90°) Random number seed 0-99000 (step 1000) 0 - 99000 (step 1000) Total number of flakes 43670 43670 Total number of layers 22 22 (top 4, core 14, bottom 4) Total flakes in one layer 1895 1895 Replications 100 100 126 Chapter VI: Model Application: A Case Study Four pieces of 200mm x 150 mm specimens were cut from a commercial OSB board and the density profile for each specimen was measured by X-ray scanning technique as described in Chapter TV. Based on the experimentally measured density profiles, the overlap profiles and the degree of orientation were obtained. The flake geometry as visually viewed from the panel surfaces is roughly rectangular with length in the range of 80 -110 mm and width in the range of 15-30 mm. 6.2. Densities and Overlaps The density and overlap results for the three types of mats were summarized as in Table 6.2. From Table 6.2, it can be seen that the mean densities for partially oriented flakeboard, simulated OSB, and commercial OSB agreed well. Compared to the commercial OSB, the standard deviation and coefficient of variation of density/overlaps of the simulated mats are higher. Figure 6.1 illustrates the comparison of maximum, minimum, mean and standard deviation of density between the simulated and the commercial OSB panels. Although reasonable agreement was noted, there are many unknown assumed variables for the commercial OSB, such as the exact flake geometry, flake density and compression ratio during pressing, which can affect the properties of the final product. Table 6.2 also indicates that the overall degrees of orientation for simulated partially oriented flakeboard, simulated OSB and commercial OSB are 66.7%, 40.6% and 42.3%, respectively. Again good agreement was observed in the degrees of orientation of the simulated OSB and the commercial OSB panels. 127 Chapter VI: Model Application: A Case Study Table 6.2 Basic properties of the mats Simulated partially Properties Simulated OSB Commercial OSB oriented flakeboard Density Overlap (g/cm ) Density Overlap (g/cm ) 3 Density Overlap (g/cm ) 3 3 Maximum 1.144 44.933 1.172 46.038 1.005 39.5 Minimum 0.124 4.885 0.119 4.678 0.156 6.129 Mean 0.560 21.989 0.560 22.006 0.556 21.852 Standard deviation 0.117 4.581 0.116 4.558 0.077 3.032 Coefficient of variation 20.8% 20.8% 20.7% 20.7% 13.9% 13.9% Degree of orientation 66.7% 40.6% 42.3% Figure 6.1 Density variation of the simulated OSB and the commercial OSB. 128 Chapter VI: Model Application: A Case Study 6.3. Void Area and Distribution 0 40 80 120 160 200 X Axis (mm) Figure 6.2 Void measurement and distribution in a part of a randomly formed layer. The void areas are determined layer by layer in a loosely formed mat. For example, the voids in 240mm x 240mm sample area cut from one of the randomized layers are counted in Figure 6.2 (using Winmat ). The void areas in every layer of simulated partially oriented and simulated OSB mats are listed in Table 6.3. It is noted that the voids in the mats are evenly distributed in every layer with very small coefficients of variation (1.1% for partially oriented mat and 1.2% for OSB mat). The total void area in one layer accounts for 37% of the total mat area, which is consistent with the result found in Chapter II. In fact, the proportion of voids can be theoretically calculated using the Poisson distribution (Dai and Steiner 1993). By substituting zero overlap (i = 0 means void area) and mean overlap (X - 1 means one layer) into the Poisson distribution H^- = e~ , a 36.8% of voids is found in x 129 Chapter VI: Model Application: A Case Study randomly formed flake layer. However, voids in each layer vary in size from 1 mm to a 2 maximum of 172,259 mm in a partially oriented flake layer and 56,027 mm in a random 2 2 flake layer. In the layers of randomized flake position and orientation, the average void size was smaller than those in the partially oriented layers. This may indicate that the voids are more evenly distributed in the randomized mat than those in the partially oriented mat. Table 6.3 Void areas in the simulated OSB mat and simulated partially oriented mat OSB Layer Partially oriented board No. of Total Largest Average No. Of Total Largest Average Voids (mm ) (mm ) (mm ) Voids (mm ) (mm ) (mm ) 1 832 1086970 105910 1306 832 1086970 105910 1306 2 771 1122689 144486 1456 771 1122689 144486 1456 3 879 1096632 57344 1248 879 1096632 57344 1248 4 800 1104965 65490 1381 800 1104965 65490 1381 5 1444 1103205 34550 764 826 1104529 60733 1337 6 1483 1096443 27936 739 771 1094785 103771 1420 7 1376 1125753 26494 818 731 1113459 140936 1523 8 1529 1075769 38526 704 855 1093203 78140 1279 9 1495 1104380 36090 739 773 1111367 91969 1438 10 1393 1113009 28638 799 782 1111345 111791 1421 11 1502 1081404 42131 720 846 1082779 83494 1280 12 1474 1099458 56027 746 818 1088159 75255 1330 13 1513 1081172 22820 715 771 1084854 133774 1407 14 1467 1094861 28767 746 876 1082058 145619 1235 15 1477 1114037 35486 754 834 1108856 113862 1330 16 1535 1083756 28044 706 916 1085997 72403 1186 2 2 2 2 2 2 130 Chapter VI: Model Application: A Case Study 17 1450 1105746 31300 763 756 1109466 125464 1468 18 1398 1099027 27208 786 873 1089183 64598 1248 19 111 1113008 172259 1432 777 1113008 172259 1432 20 814 1100792 83241 1352 814 1100792 83241 1352 21 745 1110078 107459 1490 745 1110078 107459 1490 22 804 1097242 91678 1365 804 1097242 91678 1365 Mean 1467* 1100473 33144* 750* 811 1099655 101348.9 1361 50* 12930 8465* 34.2* 48 11675 31252 89 Cov(%) 3.4* 1.2 25.5* 4.6* 5.9 1.1 30.8 6.6 Propor- - 37.0 - - Std Dev 36.9 - tion (%) Note: * These figures evaluated from the 14 core layers of the simulated OSB mat (random orientation). For the face layers, refer to the partially oriented board. 6.4. Autocorrelation Functions and Variance Functions The autocorrelation functions and variance functions can indirectly predict the approximate flake size and flake orientaion in the mats according to the correlation patterns (Figures 6.3 and 6.4). This is extremely useful when the information contained in the commercial panels is needed to predict other properties of the mats. For example, the size and orientation of flakes in OSB products are important indicators of key panel strength properties. Mats with shorter flakes have lower strength than mats with longer flakes. The axial strength in a highly oriented board is higher than that in a randomly formed board. Figures 6.5 and 6.6 show the autocorrelation functions and variance functions of the simulated partially oriented flakeboard, simulated OSB and commercial OSB in relation to the side length of sampling 131 Chapter VI: Model Application: A Case Study zones. From Figures 6.5 and 6.6, the first part (side length of sampling zone is less than 15 mm) of the autocorrelation function from the commercial OSB is below the random case. This is because all the mathematical predictions and simulations were made based on the assumed uniform flake size of length 100 mm and width 15 mm. There are a lot of fines contained in the commercial OSB panel which are ignored in our models. The autocorrelation curve for commercial OSB also suggests that the flake width may not be the same as that which was used in the simulation process. The prediction and simulation results agree well in both cases. Figure 6.3 Autocorrelation function for the simulated partially oriented flakeboard 132 Chapter VI: Model Application: A Case Study 0 20 40 i i i i i i 60 80 100 X axis ( m m ) Figure 6.4 Autocorrelation function for the simulated OSB 1.0 I - Partially oriented (predicted) a o 0.8 • c 0.6 c u 1 < OSB (simulated) OSB (commercial) ** A \ o ^ 0.4 0.2 Partially oriented (simulated) •OSB (predicted) s Parallel Random*" . rjTi% 0.0 20 40 60 pTj .1,1 80 \ 100 Side length of sampling zone (mm) Figure 6.5 Autocorrelation functions of the simulated partially oriented flakeboard, the simulated OSB and the commercial OSB. 133 Chapter VI: Model Application: A Case Study Figure 6.6 Variance functions of the silmulated partially oriented flakeboard, the simulated OSB, and the commercial OSB. 6.5. Prediction of Thickness Swelling By applying the results from Chapter V, the thickness swelling of the simulated partially oriented flakeboard, and simulated full-size OSB without wax can be predicted, assuming that the compressed mats have 5% initial moisture content and 12% additional moisture absorption under 95% relative humidity condition and 10% additional moisture absorption under 90% relative humidity condition (Table 6.4). The 12% and 10% moisture absorption is chosen because it is close to the equilibrium moisture absorption in the corresponding conditions. Based on the predicted results, it can be noted that the thickness swelling within a mat is also randomly distributed in the horizontal direction (Figures 6.7a, 6.7b, 6.8a and 6.8b). The predicted average value of thickness swelling under 95% R H condition is 25.75% 134 Chapter VI: Model Application: A Case Study for the simulated partially oriented flakeboard and 24.79% for the simulated OSB, respectively (Table 6.4). The thickness swelling is about 23% less when the samples are considered under 90% RH than under 95% R H condition. Table 6.5 lists the average values of thickness swelling of simulated OSB in each sampling zone of 50mm x 50mm under the condition of 95% relative humidity exposure. Table 6.4 Predicted thickness swelling under 95% and 90% relative humidity conditions Properties Simulated partially (%) oriented flakeboard Simulated OSB Commercial OSB 95% R H 90% R H 95% R H 90% R H 95% R H 90% R H Maximum 31.5 24.7 30.9 24.2 24.7 19.0 Minimum 19.3 14.5 18.3 13.7 20.2 15.2 Mean 25.8 19.9 24.8 19.1 22.7 17.3 Standard deviation 2.0 1.7 1.9 1.6 1.0 0.9 Coefficient of variation 7.8 8.4 7.7 8.4 4.6 5.1 135 Chapter VI: Model Application: A Case Study TS (%) 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 Length ( m m ) Figure 6.7b 2D representation of the predicted thickness swelling of the simulated partially oriented flakeboard under 95% relative humidity condition. 136 Chapter VI: Model Application: A Case Study Figure 6.8a 3D representation of the predicted thickness swelling of the simulated OSB under 90% relative humidity condition. TS (%) 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 Length ( m m ) Figure 6.8b 2D representation of the predicted thickness swelling of the simulated OSB under 90% relative humidity condition. 137 Chapter VI: Model Application: A Case Study Table 6.5 Distribution of thickness swelling of simulated OSB in horizontal plane (sampling zone: 50mm x 50mm)* ** 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1 20.2 20.9 22.4 23.7 23.2 22.7 24.1 24.2 21.1 18.4 18.3 19.3 20.4 21.2 21.5 22.3 23.3 23.1 22.0 21.6 21.1 20.2 19.7 19.9 2 21.3 22.4 24.2 25.1 24.3 24.2 25.9 25.8 22.5 19.9 19.5 19.7 20.5 22.0 23.1 23.8 24.4 24.1 23.7 23.5 22.6 21.2 21.0 21.6 3 21.2 22.8 24.9 25.9 26.0 27.0 28.9 28.1 24.3 21.6 20.9 20.5 20.9 22.6 24.1 24.7 25.3 25.6 25.9 25.7 24.2 22.6 23.0 24.3 4 19.9 21.2 23.1 24.6 26.2 28.6 30.7 29.4 25.0 22.3 22.0 21.8 21.9 23.1 24.1 24.8 26.0 26.8 27.0 26.5 24.9 23.9 25.0 26.4 5 19.9 20.4 21.3 22.9 25.2 27.7 29.4 28.2 24.7 22.8 23.3 24.0 24.2 24.6 25.0 25.9 27.1 27.4 26.4 25.4 24.7 24.8 26.0 27.1 6 21.5 21.7 22.0 23.4 25.1 26.3 26.7 25.8 24.1 23.7 25.1 26.6 27.1 27.1 26.9 27.2 28.0 27.5 25.4 23.8 23.8 24.9 26.2 26.9 7 23.0 23.6 24.7 25.9 26.7 26.3 25.0 24.0 23.6 24.3 26.2 28.1 29.0 29.3 28.8 27.9 27.9 27.3 25.3 23.7 23.9 24.9 25.7 26.2 8 23.0 24.4 26.8 28.5 28.6 26.7 24.0 22.5 22.9 24.1 26.1 28.2 29.7 30.3 29.6 28.2 27.5 26.7 25.0 24.2 24.5 24.5 24.6 25.0 9 21.5 23.4 26.6 29.2 29.2 26.6 23.3 22.0 22.7 24.1 26.3 28.9 30.3 30.0 29.1 28.5 28.0 26.1 23.8 23.7 24.5 23.8 23.4 23.9 10 19.6 21.6 25.0 27.9 28.2 25.8 23.1 22.7 23.8 25.3 27.8 30.5 30.9 28.9 27.7 28.8 29.5 26.8 23.5 23.2 24.0 23.4 22.9 23.1 19.0 20.8 23.8 26.2 26.8 25.2 23.5 23.6 24.9 26.3 28.4 30.5 30.1 27.4 26.4 28.6 30.0 27.7 24.4 23.5 23.9 23.7 23.3 23.3 12 20.0 21.3 23.4 25.4 26.4 25.8 24.7 24.4 24.9 25.7 27.1 28.1 27.5 25.9 25.8 27.5 28.2 26.4 24.3 23.5 23.6 23.7 24.0 24.5 11 13 21.1 21.7 23.2 25.2 26.8 27.0 26.0 24.9 24.4 24.6 25.3 25.5 25.0 25.0 25.9 26.3 25.2 23.6 22.7 22.5 22.6 22.9 24.2 25.6 14 21.2 21.5 22.7 24.8 26.7 27.0 26.1 24.8 23.8 23.8 24.6 24.8 24.5 24.9 25.9 25.6 23.7 22.1 21.9 22.1 22.1 22.3 23.5 24.9 15 21.4 21.7 22.9 24.6 25.8 25.8 25.4 24.6 23.6 23.6 24.8 25.6 25.3 25.0 25.2 24.9 23.6 22.8 22.9 23.0 23.0 23.0 23.2 23.6 16 22.6 23.0 24.0 24.6 24.3 24.1 24.6 24.8 24.2 23.9 25.0 26.0 25.8 24.8 24.2 24.0 23.8 24.1 24.5 24.4 24.2 24.2 23.7 23.3 17 24.0 24.7 25.3 24.4 22.8 22.4 23.6 24.7 24.6 24.0 24.5 25.4 25.4 24.5 23.8 23.8 24.6 25.8 26.2 25.2 24.6 24.7 24.3 23.7 18 24.3 25.3 25.8 24.4 22.3 21.6 22.5 24.0 24.6 24.2 24.2 24.9 25.2 24.7 24.0 24.0 25.5 27.4 27.2 25.4 24.6 25.1 25.0 24.4 19 23.1 24.5 25.5 24.8 23.3 22.4 22.6 23.8 25.0 25.2 25.0 25.4 25.7 25.5 24.6 24.0 25.2 27.1 26.9 25.4 25.1 25.9 26.0 25.7 20 22.1 23.6 24.9 25.0 24.6 24.2 24.0 24.5 25.8 26.5 26.2 25.8 26.0 26.0 25.3 24.5 24.8 25.9 25.9 25.4 26.0 26.9 27.1 27.0 21 22.2 23.8 24.8 24.7 24.9 25.5 25.5 25.1 25.7 26.7 26.5 25.8 25.6 25.7 25.6 25.4 25.5 25.3 24.9 25.0 26.0 27.1 27.4 27.2 22 23.4 24.8 25.4 24.4 24.4 25.6 25.8 24.6 24.5 25.7 26.2 25.9 25.7 25.6 25.7 25.9 25.8 24.9 24.0 23.8 24.6 25.8 26.3 26.2 23 24.6 25.8 26.0 24.6 24.3 25.3 25.1 23.5 23.0 24.2 25.3 25.8 26.1 26.1 26.1 26.2 25.5 24.1 23.0 22.6 22.7 23.4 24.1 24.3 24 25.4 26.3 26.2 25.0 24.7 25.2 24.7 23.1 22.3 23.1 23.9 24.9 25.9 26.1 26.1 26.3 25.4 23.4 22.2 21.7 21.1 21.3 22.1 22.6 25 25.5 26.2 26.0 25.1 24.8 24.9 24.7 24.0 23.5 23.6 23.7 24.3 25.3 25.5 25.3 25.5 25.0 23.4 22.3 21.5 20.5 20.5 21.5 22.1 26 24.9 25.4 25.3 24.6 24.1 24.2 24.8 25.3 25.5 25.4 24.7 24.3 24.9 25.1 24.5 24.4 24.5 24.4 24.1 23.0 21.5 21.2 22.0 22.5 27 24.9 25.1 24.9 24.0 23.3 23.5 24.5 25.5 26.0 25.8 24.8 24.2 24.7 25.1 24.4 23.9 24.4 25.4 26.0 25.1 23.4 22.7 23.0 23.1 28 25.0 25.5 25.4 24.4 23.4 23.4 24.0 24.5 24.6 24.2 23.7 23.8 24.4 24.7 24.1 23.7 23.9 24.9 26.0 26.1 25.2 24.2 23.6 23.5 29 24.4 25.3 26.0 25.5 24.5 24.0 23.9 23.9 23.2 22.5 22.7 23.4 23.8 23.6 23.3 23.2 23.1 23.4 24.7 26.2 26.4 25.0 23.7 23.3 30 23.7 24.9 26.0 26.1 25.4 24.8 24.6 24.2 22.9 21.6 21.9 23.1 23.8 23.7 23.5 23.5 23.1 22.9 24.2 26.3 26.8 25.3 23.6 22.8 31 23.3 24.3 25.1 25.2 25.1 25.1 25.1 24.8 23.5 22.1 22.2 23.6 25.2 25.7 25.2 24.7 24.3 23.9 24.7 26.4 26.9 25.5 23.9 23.2 32 22.1 22.8 23.6 24.0 24.5 25.1 25.2 25.1 24.7 23.9 23.6 24.6 26.4 27.1 26.3 25.7 25.9 25.6 25.4 26.1 26.6 25.8 24.9 24.5 33 21.1 21.7 22.8 23.9 24.8 25.1 25.2 25.7 26.3 26.0 25.1 24.9 25.9 26.6 26.2 26.3 27.3 27.3 26.4 26.3 26.6 26.1 25.6 25.5 34 21.8 22.4 23.3 24.4 25.1 24.9 24.6 25.8 27.5 27.5 25.7 24.2 24.3 25.4 26.5 27.5 28.2 28.3 27.9 27.4 26.9 26.1 25.2 24.8 35 23.6 24.3 24.6 24.8 24.9 24.2 23.7 25.1 27.3 27.3 25.3 23.4 23.1 24.5 26.7 27.8 27.8 28.3 28.8 28.1 26.9 25.7 24.3 23.1 36 24.4 25.3 25.8 25.5 25.1 24.5 24.0 24.6 25.6 25.3 23.9 22.8 22.8 23.9 25.5 26.1 26.1 27.0 28.1 27.7 26.6 25.5 23.4 21.5 37 23.2 24.5 25.9 26.4 26.4 26.3 25.7 24.9 24.3 23.5 22.6 22.7 23.5 24.0 24.3 24.6 24.7 25.0 25.8 26.3 26.3 25.5 23.1 20.9 38 21.9 23.0 24.7 26.2 27.4 28.0 27.1 25.7 24.8 23.8 22.9 23.3 24.4 24.3 24.1 24.7 24.8 24.1 24.0 24.9 25.8 25.5 23.5 21.7 39 21.5 22.0 23.0 24.8 27.0 28.2 27.6 26.4 25.8 25.3 24.1 23.8 24.3 24.1 23.8 24.6 25.0 24.0 23.3 23.9 25.2 25.6 24.3 23.2 40 22.1 22.0 22.2 23.6 25.9 27.5 27.6 26.8 26.1 25.8 24.8 23.6 23.4 23.5 23.5 24.3 24.9 24.4 23.5 23.6 24.9 25.9 25.2 24.3 41 23.4 23.2 23.1 23.7 25.0 26.5 27.2 26.6 25.7 25.4 24.6 23.3 23.3 24.1 24.6 25.4 26.2 26.0 24.8 24.0 24.8 25.9 25.5 24.6 138 Chapter VI: Model Application: A Case Study ** 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 24.3 24.7 25.2 25.1 25.0 25.7 26.4 26.0 25.1 24.8 24.2 23.4 24.3 25.9 26.6 27.5 28.5 28.1 26.4 24.8 24.0 25.3 26.7 26.6 25.6 25.4 25.7 25.3 24.7 24.4 23.8 23.5 24.9 26.8 27.6 28.5 29.4 29.0 27.3 25.3 44 23.3 25.0 26.9 27.1 26.1 25.6 25.7 25.5 25.2 24.8 23.7 23.0 24.4 26.6 27.9 28.7 29.1 28.7 27.5 25.5 45 23.0 24.6 26.5 27.2 26.9 26.7 26.8 26.7 26.1 25.1 23.4 22.2 23.1 25.5 27.5 28.5 28.5 27.8 27.1 25.9 21 22 23 24 42 24.5 25.1 25.0 24.3 43 24.0 24.2 24.3 23.9 46 47 48 23.8 23.6 23.8 23.4 24.2 23.3 23.4 24.8 26.4 27.2 27.7 28.3 28.7 28.2 26.8 24.9 22.9 21.3 21.4 23.7 26.1 27.5 27.4 26.3 25.9 26.1 25.3 23.6 24.2 25.7 26.8 27.0 27.7 29.1 29.9 28.9 26.4 24.2 22.6 20.8 20.5 22.2 24.5 25.9 25.9 24.9 24.7 25.9 26.0 24.0 24.6 26.3 27.2 26.8 27.4 29.1 30.0 28.4 25.6 23.6 22.5 21.0 20.5 21.9 23.8 24.9 25.1 24.3 24.3 25.7 26.0 24.1 23.2 23.0 22.7 22.6 22.5 22.4 22.5 22.3 The thickness swelling is predicted under the condition of 95% relative humidity exposure. * * The numbers in the first row and the first column in this table represent the location of each sampling zone by multiplying by 50. For example, the average thickness swelling in the sampling zone located at 500mm away from both the left and top borders (10 th row and 10 th column) is 25.3%, assuming additional 12% moisture absorption happened during exposure. 6.6. Degree of Orientation of Commercial OSB As discussed before, the predicted degree of orientation of commercial OSB is 42.3%, assuming uniform flake size of 100 mm x 15 mm x 0.7 mm. The assumed flake geometry was made based on the visual observation on a commercial OSB panel because we cannot simply break the OSB into individual flakes and measure their geometry. Figures 6.9 and 6.10 indicate that the degrees of orientation decrease as the flake length and/or width increase if a constant characteristic area is maintained according to the degree of orientation equation (Equation 30) derived in Chapter III. Thus if either length and/or width deviate from the initial assumption, the corrected predicted value can be obtained from these Figures. 139 Chapter VI: Model Application: A Case Study • , \ • Length: variable ; Width: 20 mm Length: 100 mm \ Width: variable 0 0 20 40 60 80 100 120 140 160 Flake length or width (mm) Figure 6.9 Degree of orientation of commercial OSB in corresponding to a constant characteristic area. o i 1 0 20 1 1 40 60 80 100 120 140 160 Flake length (mm) Figure 6.10 Degree of orientation of commercial OSB in corresponding to a constant characteristic area. 140 Chapter VI: Model Application: A Case Study 6.7. Future Research Work The future work on expanding the present model could be as follows: • Large OSB panels could be made by using a larger robot mat-forming system and an effective testing method to extract density information both horizontally and vertically can be developed. • Strength property models, such as tension strength and modulus of elasticity could be developed, by incorporating the autocorrelation function and variance function into the models. Since the degree of orientation of the flakes can be determined from the autocorrelation function and the longitudinal strength of the panel is related to the orientation of flakes in the mat, a relationship between the M O E and autocorrelation may exist. • The current simulation program could be expanded to incorporate the above strength prediction models. The next generation of the simulation will probably perform the opposite function, i.e., input density and strength requirements to intelligently compute what kind of raw materials and structure are needed to achieve our target properties. • The flake shift or movement could be considered during the mat forming and compressing processes because this will most likely happen in the real situation. • The current model provides a methodology to be used to predict the thickness swelling in high relative humidity conditions. Further work needs to be done to 141 Chapter VI: Model Application: A Case Study expand this model to allow the prediction of thickness swelling for the commercial OSB panels 6.8. Limitations There are a couple of limitations that need to be considered in the future studies: • Wax and other additives could be added to the experimental mats to make them close to the commercial panels to reduce the moisture and/or water absorption during thickness swelling tests. • Although the simulation program (Winmat®) allows the input of mixed flake sizes, the mathematical model can only take uniform flake geometry. Furthermore, the irregular shape of flakes could be considered in both simulation program and mathematical models. 6.9. References Dai, C. and P. R. Steiner. 1994. Spatial structure of wood composites in relation to processing and performance characteristics. Part 3. Modeling the formation of multi-layered random flake mats. Wood Science and Technology. 28:229-239 142 Chapter VII: Conclusions CHAPTER VII CONCLUSIONS In this thesis, the structural properties of partially mathematically oriented flakeboard mats were investigated according to the random field theory and probability distribution. Computer simulations, robot mat formation, and X-ray scanning techniques were also used to verify the model. Through extensive review of the past research, the author believes that the contribution of this thesis to the field of wood science could be concluded as follows: 1. A mathematical model based on random field theory was developed to characterize the horizontal density distribution in partially oriented flakeboard mats. In this model, the flake position was considered to be random because there is no intention to control it in commercial products. The orientation angle of the flakes was assumed to follow either the Von Mises distribution function characterized by concentration factor k or the uniform distribution function within a range of angle ±9 (0 < 9 < 90). Under this assumption, the autocorrelation function and variance function of the horizontal density distribution were investigated at different values of k and 9. Results indicate that when the concentration parameter k is greater than 700 in the Von Mises distribution, the orientation of all flakes can be regarded as parallel to each other, which is equivalent to the case when all the flakes have the same orientation angles in the uniform distribution. The upper and lower limits of the autocorrelation functions and variance functions for any given value of k and range of angle 9 can be identified as the perfectly oriented and 143 Chapter VII: Conclusions completely randomized cases, respectively. 2. The characteristic area concept from random field theory was first introduced to evaluate the degree of orientation of the flakes in a mat according to its autocorrelation function. For known flake geometry, the maximum characteristic area is roughly equal to the flake length squared and the minimum characteristic area to be equal to the area of the flake, which also correspond to the perfectly oriented and completely randomized flake orientations, respectively. The predicted and simulated degree of orientation agrees well based on the characteristic area concept, but differ slightly from the degree of alignment as defined by Geimer (1976). 3. The estimation of the degree of orientation for commercial panels could be a tedious task if the orientation of each flake needs to be measured. The model developed here allows estimating the degree of orientation non-destructively if the autocorrelation function and its characteristic area could be evaluated from the horizontal density distribution. For this reason, a non-destructive method, X-ray scanning technique, was used to extract the density profiles from flakeboard mats. A model that calibrates and maps X-ray voltage levels to overlaps and/or density was presented and discussed. The density and/or overlap were found to be a logarithm function of the X-ray intensity ratio (IQ/I: the intensity of the incident radiation to the intensity of radiation at location (x, y) in a mat). 4. Wood composites may suffer from two kinds of swellings in two different application stages: the swelling due to absorbing moisture in normal application conditions, and the swelling due to absorbing water during exposure to severe climate (e.g., direct exposure to rain water during construction). A study on the relationships between thickness 144 Chapter VII: Conclusions swelling and mat structures in robot-formed flakeboard mats was conducted under 95% and 90% relative humidity conditions and 24-hour water soaking tests. A model was established such that the thickness swelling of a flakeboard mat can be predicted, provided that the amount of moisture absorbed and the density distribution of the mat are known. The thickness swelling is also horizontally distributed due to the horizontal distribution of density and the amount of moisture absorbed in a particular location. Further studies are needed to determine the relationship between the thickness swelling and the amount of water absorbed in soaking tests because the water absorption is less predictable than the moisture absorption due to the sample becoming saturated in a relatively short period of time. A simulation program Winmat , based on the Monte Carlo technique, was written to compute all the statistics discussed in the thesis. The inputs to the program are: the simulated mat geometry, target mat density, the distribution of flake geometry, flake density and the distribution of the flake orientation angles. The output from the program includes: the horizontal distribution of overlap and density, free flake length and its distribution, number of flake crossings, the location and distribution of void sizes, the autocorrelation function, variance function and the degree of orientation of flakes in a mat. The simulation program can also determine the effect of sampling zone size on the density/overlap distribution. Similar analysis could also be performed on experimentally designed and robot-formed mats. 145 Appendix A: Pseudo-Code for Variance Calculation Program APPENDIX A PSEUDO-CODE FOR VARIANCE CALCULATION PROGRAM A.l. PDF for two Points in a Square with Side Length of A / ( x l , y\, r, A) = f -^- (1 - - c o s 0)(l -—sin 0)d0 A A A 1 A l fl = {0<r<A, 2 O<0<0l} f2 = ^A<r<42A, acos(—)<0<asin(—)j A.2. Correlation Coefficient for Two Points in a Rectangle with Side Length L and Width W r a(x2,y2,k,r,L,W,0\) 2 ( 1 _L =— s i _L n m C0S ^ ^ e ^d<p f 1 ey™vm d0 - kM J-8, al = {0<r<W, O<0<0l} a2 = \w<r<L, 0 < 9 < a sin(—)| a3 = jl < r < JL +W , 2 a4 = {0<r<L, 2 O<0<0\) [ a5 = <L<r< W 1 sin 01 , a6 = \ <r<^L [sin<91 W acos(-) < 0 < asin(—) J 2 a7 = Jz,<r<—^—, cosr91 L 1 ocos(—) <0 <01 > V +W , 2 J acos(—)<f?<asin(—)l r r J acos(—)<0<0\\ r 146 Appendix A: Pseudo-Code for Variance Calculation Program A.3. Variance Function Variance(W, L, A, Q\) = (/ (xl, yl, r, A) • alpha(x2, y2,k,r, L,W,0\)dr A.4. Variance Calculation for the von Mises Distributed Flake Orientation (k * 0 and 9\ = -) 2 / Inputs to the program are: flake length (L) and width (W), side length of square zones (A), and concentration parameter (k). Equations used to implement the Visual Basic program TA Sl(L,W,A,k) := f (0, ^ , r, A j • alpha 10, ^ , k, r, L, W j dr 0 S2(L,W,A,k) : = fjacos |—J, asin |—j, r, A j • alpha \0 A k, r, L, W j dr v a r l ( L , W , A , k ) := S1(L, W , A , k ) +• S 2 ( L , W , A , k ) •w S3(L,W,A,k) fjacos |—j, asin |—j, r, A j • alpha \0, —, k, r, L, W ] dr A 'A/2-A S4(L,W,A,k) : = f|acos|—|,asin(—],T,A\-alpha(o,asin[— j,k,r,L,W] dr v a r 2 ( L , W , A , k ) : = S l ( L , W , A , k ) + S 3 ( L , W , A , k ) +• S 4 ( L , W , A , k ) S5(L,W,A,k) f|acos | — j,asin | — j ,r,Aj-alpha|o, asin |—] , k , r , L , W ] dr W '^2-A S6(L,W,A,k) : = 147 Appendix A: Pseudo-Code for Variance Calculation Program v a r 3 ( L , W , A , k ) : = S l ( L , W , A , k ) •+• S 3 ( L , W , A , k ) +- S 5 ( L , W , A , k ) +• S 6 ( L , W , A , k ) L -r-W 2 S7(L,W,A,k) : = 2 t acos —Lasin — L r , A -alpha acos — Lasin — , k , r , L , W dr var4(L,W, A , k ) : = S l ( L , W , A , k ) -t- S 3 ( L , W , A , k ) +• S 5 ( L , W , A , k ) +- S 7 ( L , W , A , k ) rw S8(L,W,A,k) f(0,^,r,AJ-alpha(o,^,k,r,L,W) dr 0 S9(L,W,A,k) : = rA f(0,^,r,Aj-alpha[0,asinl— | , k , r , L , W ) dr w , /A\ /A\ \ I IW\ , f acos— ,asin— , r , A -alpha O . a s i n — , k , r , L , W dr S10(L,W,A,k) : = v a r 5 ( L , W , A , k ) : = S8(L,W, A , k ) -t- S 9 ( L , W , A , k ) -t- S10(L,W, A , k ) 'L Sll(L,W,A,k) f|acos | — | ,asin|—| ,r, A|-alpha|o,asin|— | , k , r , L , w | dr v a r 6 ( L , W , A , k ) :=S8(L,W,A,k) -t- S 9 ( L , W , A , k ) -t- S l l ( L , W , A , k ) +- S 6 ( L , W , A , k ) v a r 7 ( L , W , A , k ) :=S8(L,W,A,k) -t- S 9 ( L , W , A , k ) +- S l l ( L , W , A , k ) -t- S 7 ( L , W , A , k ) S12(L,W,A,k) f j O . ^ . r . A J - a l p h a l O . a s i n l — ) , k , r , L , W | dr fA S13(L,W,A,k) : = f|0,— ,r, Aj-alpha|acos |— j ,asin|— ] , k , r , L , W | dr ^2-A S 1 4 ( L , W , A , k ) := f acos —Lasin — L r , A -alpha acos — Lasin — L k , r , L , W dr 148 Appendix A: Pseudo-Code for Variance Calculation Program v a r 8 ( L , W , A , k ) : = S 8 ( L , W , A , k ) +- S12(L W, A , k ) -t- S 1 3 ( L , W , A , k ) -t- S 1 4 ( L , W , A , k ) ( L -t- W 2 2 S15(L,W,A,k) fjacos |—j ,asin|— j ,r, Aj-alpha|acos |—j ,asin|—j ,k,r,L,wj dr v a r 9 ( L , W , A , k ) : = S 8 ( L , W , A , k ) -t- S l ^ L . W . A . k ) +- S l ^ L . W . A . k ) +- S15(L,W, A , k ) 2 2 L +W S16CL,W,A,k) f 10, —, r, A j • alpha |acos |—j, asin | — ) , k, r, L, W ) dr v a r l O ( L , W , A , k ) := S 8 ( L , W , A , k ) +• S 1 2 ( L , W , A , k ) +• S 1 6 ( L , W , A , k ) Visual Basic pseudo-code i f (A<W) if (V2A<W) var = Varl = £f\-aldr+ elseif ^ f2-a\dr A (V2A<L) var = Var2 = §f\-a\dr+ elseif ( V2A<VL 2 +W 2 ^f2-aldr+ ^ fl-aldr ^f2-aldr+ ^f2-a2dr+ ) var = Var3 = ^ / l - a l j^f2-a3dr else var = Var4 = | * / l - a l r f r + £ / 2 - a l d r + ^f2-a2dr+ £ ^ f2-a3dr endif elseif if (W<A<L) ( V2A<L) var = Var5 = ^ f\-a\dr+ ^f\-a2dr+ ^ fl-aldr 149 Appendix A: Pseudo-Code for Variance Calculation Program elseif (V2A<VL +W ) 2 2 var = Var6 = ^ f\-a\dr+ ^f\-a2dr + £f2-a2dr + ^*f2-a3dr else fl-aldr + £fl-a2dr + }J2-a2dr+ £ f2-a3dr endif jlseif if (L<A<^L 2 +W 2 (V2A<Vl +W ) 2 2 var = Var8 = | " / l - a l r f r + ^f\-a2dr+ ^f\-a3dr+ ^ f2-a3dr A else var = Var9 = £ fl-aldr+ £fl-a2dr + £fl-a3dr+ f2-a3 dr endif else var = VarlO = £ fl-aldr+ ^fl-a2dr+ f\-a3 dr endif A.5. Variance Calculation for the Uniform Distributed Flake Orientation (k = 0) Inputs to the program are: flake length (L) and width (W), side length of square zones (A), and range of angles (0\). Equations used to implement the Visual Basic program rA T1(L,W,A,91) : = f(0 - r AJ-alpha(0,9l,ei,r,L,W)dr ) ) ) 0 T2(L,W,A,61) : = ffacosf— j.asinf—),r,A)-alpha(O,ei,01,r,L,W) dr \ \r/ \r 150 Appendix A: Pseudo-Code for Variance Calculation Program v a r l ( L , W , A , 6 1 ) : = T 1 ( L , W , A , 0 1 ) + T2(L, W , A , 0 1 ) W sin(01) f(acos [—j.asinf—),r,A|-alpha(0,ei,ei,r,L,W) dr T3(L,W,A,01) A '^2-A f acos — j.asinf— |,r,A]-alpha(o,asin(— ] , 0 1 , r , L , W ] dr T4(L,W,A,81) : = W Jsin(91) var2(L,W,A,61) : = T 1 ( L , W , A , 0 1 ) +- T 3 ( L , W . A . 0 1 ) T4(L,W,A,61) rL T5(L,W,A,91) : = fjacos | — | , asin |—j, r, A j -alpha |o, asin | — ) , 0 1 , r , L , W ) dr W sin(01) '^2-A T6(L,W,A,01) : = f (acos | — | , asin |— j, r, A j • alpha jacos |—j, asin | — ], 01, r, L, W ] dr var3(L,W,A,01) : = T 1 ( L , W , A , 0 1 ) +- T 3 ( L , W , A , 0 1 ) + T5(L, W , A , 0 1 ) 2 L + W T7(L,W,A,01) T6(L,W,A,01) 2 /A\ /A\ \ / /L\ /W\ l f acos — ,asin — , r , A -alpha acos — ,asin — ,01,r,L,W dr var4(L,W,A,01) : = T 1 ( L , W , A , 8 1 ) + T 3 ( L , W , A , 0 1 ) +-T5(L,W,A,81) +-T7(L,W,A,01) W sin(01) T8(L,W,A,01) f(O,^,r,A|-alpha(O,01,01,r,L,W)dr rA f(oAr,AJ-alpha|o,asin|—j.ei.r.L.w) dr T9(L,W,A,01) W sin(01) 151 Appendix A: Pseudo-Code for Variance Calculation Program 2-A f |acos |—j, T10(L,W,A,ei) : = asin |—j, r, A j • alpha |fj, asin | — ) , 01, r, L , W ] dr v a r 5 ( L , W , A , 6 l ) : = T 8 ( L , W , A , 0 1 ) -t- T 9 ( L , W , A , 9 l ) +• TIOCL.W,A,91) rL /A\ /A\ \ / /W f|acos|— ,asin — , r , A -alpha O.asin —] ,01,r,L, W ) dr r / \r / / \ \ r 1 Tll(L,W,A,ei) : = var6(L,W,A,01) : = T 8 ( L , W , A,01) -t- T9(L, W , A,01) -t- T11(L,W, A,01) ± T 6 ( L , W , A,01) var7(L,W,A,01) :=T8(L,W,A,01) + T 9 ( L , W , A,01) -t- T11(L,W, A,61) -t- T 7 ( L , W , A,01) •L f( 0 A r, A j • alpha jo, asin | — |, 01, r, L , W | dr T12(L,W,A,ei) W sin(01) •A T13(L,W,A,01) : = f j 0, ^ , r, A j • alpha |acos | - | , asin | — j , 91, r, L , W ) dr L T14(L,W,A,91) : = f|acos |— j,asin|— j ,r, AJ-alpha|acos |—) ,asin(—) , 9 l , r , L , W ) dr var8(L,W,A,91) : = T 8 ( L , W , A , 9 1 ) +• T12(L,W, A,91) + T13(L,W, A,01) +- T14(L,W, A.81) 2 2 L +W f acos —Lasin — , r , A -alpha acos —Lasin — , D l , r , L , W T15(L,W,A,01) dr var9(L,W,A,01) : = T 8 ( L , W , A , 0 1 ) +- T12CL.W,A,01) + T 1 3 ( L , W , A , 9 1 ) -t- T 1 5 ( L , W , A , 0 1 ) 2 2 L +W T16(L,W,A,9l) f (0 A r, A j -alpha |acos | - | , asin |—j, 01, r, L , W j dr v a r l O ( L , W , A , 0 1 ) : = T 8 ( L , W , A,01) + T12(L,W, A,01) +- T16(L,W, A , 0 1 ) 152 Appendix A: Pseudo-Code for Variance Calculation Program varll(L,W,A,ei) := v a r l ( L , W , A , 0 1 ) T17(L,W,A,61) : = •L f|acos |—j, asin |— j, r, A j • alpha( 0,01,01, r, L , W ) dr A /A\ /Al \ I IL\ f acos — L a s i n — , r , A -alpha acos — ,01,01,r,L,W T18(L,W,A,01) : = dr v a r l 2 ( L , W , A , 0 1 ) := T 1 ( L , W , A,01) -+- T17(L,W, A,01) + T18(L,W, A,01) W sin(Gl) f (acos (—1,01, r, A j -alpha (acos (—j, 01,01, r,L,W"] dr T19(L,W,A,01) L '^2-A fjacos |—j, asin — | j ,r, A j-alpha |acos — j j ,asin(— \ ,91,r,L, w) dr T2O(L,W,A,01) W sin(Ol) v a r l 3 ( L , W , A , 0 1 ) : = T1(L, W , A,01) + T 1 7 ( L , W , A , 0 1 ) •+- T 1 9 ( L , W , A , 0 1 ) -t- T20(L, W , A,01) L -t-W 2 2 r • A\ \ / /L\ . /W\ . . . r acos — ,asm — , r , A -alpha acos — Lasin — , 9 1 , r , L , W dr \ \r/ \r/ / \ \r A T21(L,W,A,01) : = Q 1 W J sin(91) v a r l 4 ( L , W , A , 9 1 ) : = T 1 ( L , W , A,01) -t- T 1 7 ( L , W , A , 0 1 ) -f T19(L,W, A,91) -t- T21(L,W, A,01) rL T22(L,W,A,01) : = f(O,^,r,A|-alpha(O,01,01,r,L W)dr ) 0 rA T23(L,W,A,01) : = f(O,-|,r,AJ-alpha|acos|-j ei,01,r L,W] dr ) ) 153 Appendix A: Pseudo-Code for Variance Calculation Program A/2-/ f |acos |—j, asin j — j , r, A j -alpha |acos | - j , 01,01, r, L, W j dr T24(L,W,A,ei) : = v a r l 5 ( L , W , A , 0 1 ) : = T22(L,W, A,01) -t- T23(L,W, A,01) +- T 2 4 ( L W , A , 0 1 ) ) W sin(01) fjacos | — j , asin |—j, r, A j • alpha |acos |—J ,01,01,r.L.wj dr T25(L,W,A,01) v a r l 6 ( L , W , A , 0 1 ) : = T 2 2 ( L , W , A 0 1 ) •+- T23(L,W, A,01) + T25(L,W A,01) + T2fXL,W, A,01) > > v a r l 7 ( L , W , A , 0 1 ) : = T22(L, W , A , 0 1 ) +- T23(L,W, A,01) -t- T 2 5 ( L , W , A , 0 1 ) +- T21(L,W, A,01) W sin(Ol) T26(L,W,A,01) : = f(0,^,r,Al-alphalacosl-|,01,01,r,L,W) dr JL T27(L,W,A,01) f[0,^,r,AJ-alpha|acos (-) ,asin(— | , 0 1 , r , L , W ] dr W sin(01) v a r l 8 ( L , W , A , 0 1 ) : = T22(L,W,A,01) + T26(L,W,A,01) -+- T 2 7 ( L , W , A , 0 1 ) + T 1 4 ( L , W , A , 0 1 ) v a r l 9 ( L , W , A , 0 1 ) : = T22(L,W, A,01) -+- T26(L,W, A,01) + T27(L, W , A,01) +- T15CL, W , A,01) f ((A, r, A j -alpha |acos j - j , asin | — ], 01, r, L, W) dr T28(L,W,A,01) W sin(01) var2O(L,W,A,01) : = T22CL,W,A,01) +- T26CL.W, A,01) •+- T28(L,W, A,01) var21(L,W,A,01) : = v a r l l ( L , W , A , 0 1 ) var22(L,W,A,01) := v a r l 2 ( L , W , A , 0 1 ) 154 Appendix A: Pseudo-Code for Variance Calculation Program cos(91) fjacos j — j , asin j — J, r, A j -alpha jacos ( - ) , 91,91, r, L, w) dr T29(L,W,A,ei) : = v a r 2 3 ( L , W , A , 9 l ) : = T 1 ( L , W , A,91) +- T 1 7 ( L , W , A , 9 l ) •+- T29(L,W, A,91) rA f[0,-|,r,A|-alpha|acos|-|,81,ei,r L,w| dr T30(L,W,A,ei) : = ) L fjacos j — j , asin |— j , r, A j • alpha |acos |—j ,01,01,r,L,W ) dr T31(L,W,A,01) : = JA var24(L,W,A,01) : = T 2 2 ( L , W , A , 0 1 ) -t- T 3 0 ( L , W , A , 0 1 ) -t- T 3 1 ( L , W , A , 0 1 ) L cos(01) T32(L,W,A,01) fjacos |— j ,asin|—j ,r, AJ-alpha|acos j—J ,01,01,r,L,W ] dr var25(L,W,A,01) : = T22(L,W, A,01) +- T30(L, W . A . 0 1 ) + T32(L,W, A,01) L cos(01) floAr.AJ-alphalacosl-j^l.gi.r.L.W] dr T33(L,W,A,01) : = var26(L,W,A,01) : = T 2 2 ( L , W , A , 0 1 ) + T33(L,W,A,01] Visual Basic pseudo-code if W (01>asin(—)) if A< if W sinful (V2A< W sin 01 155 Appendix A: Pseudo-Code for Variance Calculation Program Varl = elseif • al dr + j^* f2• al dr (V2A<L) Var2= [fl-aldr+ ^f2-aldr + | y / 2 - a 2 dr sin 01 elseif (V2A<VzF7w ) T . Var3 w = I fl-aldr+ ^f2-aldr+ f^f2-a2dr+ f f2-a3dr A sin (91 else Var4= §flaldr+ ^f2-a\dr+ ^ f2-a2dr w + f2-a3dr sin 01 endif W ( <A<L) sin 01 if V2A<L) elseif ( Var5 = p fl-aldr+ ^_fl-a2dr+ i | f2-a2dr sin 01 elseif (V2A<VL +W ) Var6 = 2 2 p^/l.aldr + | V _ / l - a 2 d r + f / 2 - a 2 d r + f * f2-a3dr 2 sin 01 else Var7 = p > / l - a l J r + j) _fl-a2dr+ v ^ f2-a2dr+ £ f2-a3dr sin 01 endif elseif if (L<A<VL +W 2 2 ) (V2A<VL +W ) 2 2 Var8= p > / l - a l d r + J ^ _ / l - a 2 J r + £ / l - a 3 J r + f2-a3dr sin 01 else 156 Appendix A: Pseudo-Code for Variance Calculation Proeram Var9= ^ f\-a\dr+ ^_fha2dr + £ fl-a3dr+ j^**' fl-ahdr sin(91 endif else w VarlO= p fl-aldr+ j^_fha2dr+ i _[ f\-a3dr sin 01 endif elseif W ( O > asm( , ylL +W 2 if =) x ( A < if L 2 ) (V2A<L) V a r l l = |*/l-a4<*r + r elseif (^ 2 A < j^f2-a4dr w ^~^r) sin 0\ Var\2 = f ' / l • «4 dr + £ / 2 elseif •a4dr+f^f2-a5dr (V2A<VL +W ) 2 2 w r- Varl3 = | * / l - a 4 d r + £ / 2 - a 4 d r + p / 2 - a 5 a V + )™ T < f2-a6dr sintfl else Varl4= | * / I . f l 4 d r + £ / 2 - a 4 d r + p > / 2 - a 5 a V + fyf^ f2-a6dr sin (91 endif elseif (L<A< ) sin 01 if (V2A<-^-) sm 01 Var\5= £fl-a4dr+ ilseif £fha5dr+ J^ A f2-a5dr (V2A<Vl +W ) 2 2 157 Appendix A: Pseudo-Code for Variance Calculation Program Varl6= j f / l - a 4 dr + £fl-a5dr + pf2-a5dr + \„ f2-a6dr sin 01 else w Vflrl7= j[/l-fl4rfr+ | / l - a 5 + / 2 - a 5rfr+ f2-a6dr sin 01 endif T,J/ elseif ( if <A<VL +W ) 2 sin 9\ 2 ( V2A<Vl +W ) 2 2 A w Varl8= [f\-aAdr+ pi f\. dr+ a5 ^fha6dr+ r f2-a6dr sin 01 else w | V f\-a6 dr + Var\9= ^fl-a4dr+pfl-a5dr+ / 2 - a 6 dr sin 01 endif else f\-aAdr+ pfl-aS dr+ f\-a6dr sin 01 endif else if ( A < L if ) (V2A<L) Var21 = |*/l-a4tf> + ^ fl-aA e l s e i f (V2A<—^—) cos (91 Var22= | / l - a 4 t f r + £ / 2 - a 4 d r + ^ 4 A fl-al dr else Var23= £f\-a4dr+ £ / 2 - a 4 dr + jp» / 2 - a 7 tfV 158 Appendix A: Pseudo-Code for Variance Calculation Program endif elseif if (L<A<—-—) cosf?l (V2A<— cos 61 Var24= j£fha4dr + £ fl-al dr + J ^ fl-al Varl5 = £fha4dr + £ fl-al dr + A dr else fl-al dr endif else Var26= £fl-a4dr+ ^ fl-al dr endif endif 159
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Organization of wood elements in partially oriented flakeboard mats Lu, Congjin 1999
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Title | Organization of wood elements in partially oriented flakeboard mats |
Creator |
Lu, Congjin |
Date Issued | 1999 |
Description | Partially oriented flakeboard mats play a significant role in commercial flake-based products, such as three-layered oriented strand board (OSB). The study presented in this thesis mathematically investigates the structure of partially oriented flakeboard mats. To better understand the nature of the structures of flakeboard mats, a simulation program Winmat®, based on the Monte Carlo technique, has been written to compute the horizontal distribution of overlap and density, free flake length and its distribution, number of flake crossings, the location and distribution of void sizes, the autocorrelation function, variance function and the degree of orientation of flakes in both simulated mats and experimental mats. This program can also determine the effect of sampling zone sizes on the density/overlap distribution. In the model development, flake position was considered to be random. The orientation angle of the flake was assumed to be random following either the Von Mises distribution or the uniform distribution. A mathematical model based on these distributions was developed. The autocorrelation function and variance function of the horizontal density distribution were investigated at different k values and θ angles. The characteristic area concept from random field theory was first introduced to evaluate the degree of orientation of the flakes in a mat. In the process of estimating the degree of orientation of a flakeboard, the horizontal density distribution is needed to compute the autocorrelation function and the characteristic area. A non-destructive method, X-ray scanning technique, was used to determine the density profiles from experimental flakeboard mats. A model that maps X-ray voltage levels to overlaps and/or density was presented and discussed. The density and overlap were found to be a logarithm function of the X-ray intensity ratio (Io/I: the intensity of the incident radiation to the intensity of radiation at location (x, y) in a mat). A study of the relationships between thickness swelling and mat structure in robot-formed flakeboard mats made without wax was conducted under 95% and 90% relative humidity conditions and 24-hour water soaking tests. A model describing such relationships was established for two relative humidity conditions. With this model the thickness swelling of flakeboard mats (without wax) can be predicted, provided that the amount of moisture absorbed and the density distribution of the mat are known. Finally, a case study was presented to demonstrate the application of the models developed in the thesis. Two kinds of mats, partially oriented flakeboard mats and OSB mats, of size 2440 mm x 1220 mm were simulated and characterized. Their density/overlap profiles and degree of orientations were then compared with a commercial OSB panel whose density profile was obtained by X-ray scanning technique. The thickness swelling values of these simulated mats were predicted and the degree of orientation of the commercial OSB panel was presented. |
Extent | 7153903 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2009-07-20 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0089865 |
URI | http://hdl.handle.net/2429/10942 |
Degree |
Doctor of Philosophy - PhD |
Program |
Forestry |
Affiliation |
Forestry, Faculty of |
Degree Grantor | University of British Columbia |
Graduation Date | 1999-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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