Online Measurement of Fabric Compression: Principles and Validation, Papers of Materials science

Download Online Measurement of Fabric Compression: Principles and Validation and more Papers Materials science in PDF only on Docsity! HUANG, WENSHENG. Online Characterization of Fabric Compressional Behavior. (Under the direction of Tushar K. Ghosh and Winser E. Alexander) Response of a fabric to applied forces normal to its plane is known as fabric compressional behavior. It is one of the important properties that determine fabric performance in many applications. The principle of a system used to measure fabric compressional characteristics, online, is proposed in this paper. A controllable nip formed by a pair of rollers is employed to apply compressional deformation to a moving fabric while the compression force and displacement are continuously recorded. The influence of various system parameters on the sensitivity of the system has been analyzed. By assuming a stepwise anisotropic behavior in the thickness direction, Incremental Differential Algorithm (IDA) is developed to calculate the pressure-displacement relationship from the measured force-displacement data obtained from the online system. A prototype online measurement system has been developed based on this principle. A number of woven and nonwoven fabrics have been evaluated using the online system as well as a number of other commercially available fabric compression testers. The compressional characteristics obtained from the online measurement system compare well with the same parameters measured using the other commercially available compressional testers. ONLINE CHARACTERIZATION OF FABRIC COMPRESSIONAL BEHAVIOR by WENSHENG HUANG A dissertation submitted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the Degree of Doctor of Philosophy FIBER AND POLYMER SCIENCE Raleigh 1999 APPROVED BY: Chair of Advisory Committee Co-Chair of Advisory Committee iv ACKNOWLEDGEMENTS The author would like to express his most sincere gratitude to Dr. Tushar K. Ghosh, Chairman of his Advisory Committee, for providing the opportunity to undertake a truly rewarding research project and for his guidance, inspiration, and support throughout the course of this work and his study. He would like to extend his appreciation to Dr. Winser E. Alexander, Co-Chairman of his Advisory Committee, for providing the chance for the author to co-major in Computer Engineering and for his direction and advice. The author would also like to thank his Advisory Committee members, Dr. Subhash K. Batra and Dr. Clay S. Gloster for their generous help and valuable suggestions for this study. Sincere thanks are extended to the faculty and staff of the College of the Textiles and the Department of Electrical and Computer Engineering, and his fellow graduate students and other friends for their encouragement and help. Special thanks are due to the National Textile Center for its financial support. In addition, the author would like to thank his family members back home in China, especially his parents, Chengfa Huang and Xiangyin Lu, for their long-time confidence, support, and encouragement. Finally and most importantly, the author wishes to express his utmost appreciation to his wife, Hong Zhang, for her constant love, understanding, sacrifice, and support, without which none of this would have materialized. v TABLE OF CONTENTS LIST OF TABLES……………………………………………………………………...vii LIST OF FIGURES……………………………………………………………………viii INTRODUCTION………………………...…..…………………………………………1 PART I: PRINCIPLE OF MEASUREMENT……………………………………........3 1 INTRODUCTION……………………………………………………………..……3 2 BACKGROUND……………………………………………………………….…...5 3 THE DEVELOPMENT OF ONLINE MEASUREMENT SYSTEM………….….10 3.1 The Principle of Measurement……………………………………………….10 3.2 System Sensitivity Analysis………………………………………..………..14 3.3 Computation of Fabric Stress-Strain Relation in Compression……………...19 4 CALCULATION OF STRESS-THICKNESS RELATIONSHIP: AN EXAMPLE ………………………………………………………………………………...……22 5 CONCLUSIONS.………………………………………………….………………23 PART II: EXPERIMENTAL VALIDATION………………………………………..28 1 INTRODUCTION…………………………………………………………………28 2 SYSTEM IMPLEMENTATION AND CALIBRATION…………………………29 3 DATA PROCESSING……………………………………………………………..33 4 EVALUATION OF THE MEASUREMENT SYSTEM………………………….37 5 EXPERIMENTAL RESULTS……………………………………………………..40 6 CONCLUSIONS AND RECOMMENDATION FOR FUTURE WORK……..….45 APPENDICES…...…………………………….……………………...………………...48 APPENDIX 1 ONLINE CHARACTERIZATION OF FABRIC COMPRESSIONAL BEHAVIOR………………………………………………………….48 APPENDIX 2 HARDWARE USED IN THE ONLINE MEASUREMENT SYSTEM……………………………………………………………113 APPENDIX 3 OPERATION AND CONTROL OF THE ONLINE MEASUREMENT SYSTEM……………………………………...114 APPENDIX 4 THE DETERMINATION OF NIP MOVEMENT………………....115 vi APPENDIX 5 DATA SMOOTHING METHODS………………...……………....119 APPENDIX 6 COMPARISON OF THE AVERAGE THICKNESS FOR A SPECIFIC SAMPLE..…..……………………………….…....…...122 ix Figure 23. Definition of coordinate system for a fiber assembly ............................................60 Figure 24. Parallelepiped formed by fiber ),( φθA and )','( φθB in contact .........................61 Figure 25. Forbidden length centered at the contact point of fiber elements E and E’ ...........65 Figure 26. Deformation of a fiber element due to a contact point load jC ............................69 Figure 27. Fabric moving between the nips...........................................................................101 Figure 28. The force experienced by the top roller while rollers contact and rotate with each other......................................................................................................................116 Figure 29. Force-displacement curve of top roller with bare nip ..........................................117 Figure 30. Change of roller opening (about 3 rotations of the bottom roller) .......................117 1 INTRODUCTION Response of a fabric to applied forces, normal to its plane, is known as fabric compressional behavior. The compressional property of a fabric is closely related to fabric handle, comfort, and warmth. The compression testers of KES-F (Kawabata Evaluation System for Fabrics) and FAST (Fabric Assurance by Simple Testing) systems are commonly used to measure fabric compressional behavior. These static offline instruments are generally used in laboratories to test a small number of samples that represent a large amount of products. Use of these instruments requires careful sample preparation and manual operation of the testers. Ability to test fabrics reliably and cost effectively in a continuous manner is key to quality assurance and process control. Online measurement system is the first step toward this. In an online measurement system, one or more of the key product or process characteristics are monitored continuously as a function of time. As a result, quality monitoring can be performed on the entire product on a timely basis that accurately reflects the variability within the product. Once a defect or unacceptable quality is detected, the system can either signal an alarm or shut down the production. Ideally, the information from an online measurement system can be used in process control. The general objective for this research is to develop an online measurement system for fabric compressional behavior. More specifically, the objectives are, 2 1) To develop a measurement principle that can be used to evaluate fabric compressional behavior online; 2) To design and develop a prototype measurement unit and instrumentation for online measurement of fabric compressional behavior; 3) To apply the principle experimentally so that the potentials and limitations of the proposed online measurement system can be studied. This dissertation includes two parts. In the first part the principle of online measurement is introduced. Implementation of the principle in designing a prototype system and its validation is presented in the second part. 5 2 BACKGROUND Fabric compressional behavior is described by the relationship between the applied force (normal to the fabric plane) per unit area and the resulting fabric thickness. The relationship is obtained by a simple test, in which the fabric specimen being tested is placed horizontally on a platten, and subsequently loaded by a presser foot of a known area. The thickness of the fabric, t, which is the distance between the presser foot and the platten, is recorded as a function of the applied pressure. This pressure-thickness relationship describes the compressional characteristic of a fabric. Fabrics (woven, knitted, and nonwoven) are assemblies of fibers and/or yarns with void-spaces (air) in between. In the process of fabric compression, the inter-fiber spaces decrease continuously. The force necessary to compress a fabric has to overcome the internal stresses of the fibers and the inter-fiber frictional force. A typical fabric pressure-thickness curve under compression is shown in Figure 1. The compressional deformation of a fibrous structure is generally composed of three stages. When a fabric is compressed, the compression plate first comes into contact with the protruding fibers on the fabric surface. This region of deformation could be approximated as an elastic region, where compressional force increases linearly with fabric deformation. During further deformation the applied compressional force has to overcome inter-yarn and/or inter-fiber static frictional force, and slippage of fibers/yarns at contact points will occur. As a result, the inter-fiber spaces decrease with reduction in thickness. If the void spaces decrease sufficiently, then lateral compression of the fibers begins, and the structure behaves more like an incompressible solid with “infinite” rigidity. 6 Often, fabric compressional behavior is studied by recording pressure-thickness in both loading and unloading. The loading-unloading curve typically exhibits hysteresis, as seen in Figure 1. This hysteresis loss is due to the inter-fiber friction and the viscoelastic nature of the fibers within the fabric. Figure 1. Typical pressure-thickness curve for a fabric in compression A number of characteristic parameters can be obtained from the pressure- thickness curve (Figure 1). Thickness of almost any fibrous structure is a function of applied normal pressure. Therefore, the determination of the thickness of a fabric consists of precise measurement of the distance between two plane parallel plates when they are separated by the fabric, with a known arbitrary pressure between the plates being applied and maintained. Fabric thickness at any given pressure can be obtained from the pressure- thickness graph of a fabric. Compressional modulus is the slope of the pressure-thickness strain curve. Often the deformation of fabrics in the thickness direction is expressed as a strain ε , that is, the change in thickness as a fraction of the original thickness. In that case, the local compressional modulus, cE , can be expressed in the derivative form, F t p )gf/cm( 2 B C D E mp 0p mt0t O 7 εddpEc /= (1) where p is the applied normal pressure as a function of ε . Other parameters, such as compressional resilience [21], etc. can also be used to characterize the compressional behavior of fabrics. The most commonly used approach to study fabric compressional behavior theoretically is the micromechanical method (see Appendix 1). In micromechanical methods, mechanical principles are employed to analyze the microstructure of fibrous assemblies, and the mass response of fiber assemblies to external forces is related and treated as the total of the responses of the basic fiber units. The analysis of deformation of fiber structures is not relevant to the development of online measurement system; thus, it is presented in Appendix 1. Unfortunately there is no generalized theoretical analysis for the exact description of fabric compressional behavior. Experimental studies of fabric compressional behavior have been a significant part of the research efforts in this area. Many empirical equations used to describe the fabric compressional behavior were set up based on the observation of experimental results (see Appendix 1). They are simpler and more direct compared with theoretical results, though caution has to be taken in applications. The development of instruments used to characterize fabric mechanical properties dates back to the classic paper by Peirce [30] in 1937. The fabric thickness tester was developed by Peirce, and this device was later incorporated in the Shirley Institute Fabric Thickness Tester [7]. In 1972, Kawabata [20] and his coworkers developed a set of instruments to measure the key mechanical properties of fabrics, commonly referred to as KES-F 10 3 THE DEVELOPMENT OF ONLINE MEASUREMENT SYSTEM 3.1 The Principle of Measurement The principle of the proposed system is based on compression of the fabric in a roller nip. The schematic of the proposed system is shown in Figure 2. The fabric being monitored is passed through the nip of compression rollers. The bottom roller is fixed while the top roller is moved up or down to deform the fabric. To facilitate the analysis that follows, a Cartesian coordinate system xy is setup. The origin o of the coordinate system is located at the center of the bottom roller. The center of the top roller, o’, is located on the y-axis. In Figure 2, 0t is the initial fabric thickness, t is the fabric thickness under compression, 1r and 0r are top and bottom roller radii, respectively. The half-length of contact between the bottom roller and the fabric along the x-axis is l. The fabric moves from the left to the right at speed 0v m/sec. Figure 2. Schematic for online compression measurement system A B o’ o y x l t 1r 0r 0t 0v h 11 The nip area is divided into two regions along the x-axis; one is the entry region from A to y-axis and the other is the exit region from y-axis to B. As the fabric enters the entry region, it undergoes a compressive strain that is dictated by the geometry of the nip region. For each increment of applied strain, there exists a given compressive stress within the fabric whose magnitude is determined by its compressive stress-strain curve of the fabric. The force necessary to sustain this stress is generated by the displacement of the top roller. When the fabric enters the exit region of the nip, the fabric, upon emerging from compression, begins to recover. In doing so (and under the condition that its rate of recovery is greater than the rate of the opening of the nip region [35]) it creates a pressure, which is also sustained by the load applied by the top roller. The magnitude of this pressure in this region is determined by the recovery part of the stress-strain curve of the fabric in compression. Because of the difference in the resultant strain in the contact region, the stress experienced by the fabric in the nip is non-uniform. The nonlinear stress distribution in the contact region, as shown in Figure 3, is postulated by Hintermaier [18] while working on a similar problem on moisture removal in the paper making process. Note that if the roller diameters are made sufficiently large, the stress distribution is likely to be more uniform. If the diameters of both rollers are infinitely large, then the nip geometry is similar to compression between two flat plates, and the stress distribution is likely to be uniform. These compressive and recovery stresses comprise the pressure in the nip region. 12 (Parameter l is the half-length of the contact region) Figure 3. Assumed stress distribution in the nip region In the present analysis, the fabric being tested is assumed to be homogeneous. Based on the geometry of the nip, the thickness of the fabric, t, at any point in the nip can be expressed as a function of its position along the x-axis, 22 0 22 1 xrxrht −−−−= (2) where h stands for the height of the top roller as seen in Figure 2. The half-length of the contact region, l, can be expressed as 10 01010 )(2 rr htrrrr l + −++ = (3) where 010 trrh ++≤ . In the following analysis, F(h) is the compressional force applied by the top roller at height h, )(1 tp is the compressional pressure in the entry region of the nip, )(2 tp is the compressional pressure in the exit region of the nip, and D is the length of the top roller. The total compressional force exerted by the top roller can be expressed as [18, 36, 37]: ∫∫ += − l l DdxtpDdxtphF 0 2 0 1 )()()( (4) x-l l σ o 15 [19], Bogaty et al. [6], Larose [26], van Wyk [33, 34], de Jong et al. [14], Batra et al. [3], and Kothati and Das [22, 23] have proposed various empirical as well as analytic pressure-thickness relationships. For convenience, the pressure-thickness relationship proposed by Bogaty [6] is used in the present analysis. According to Bogaty [6], ctp b at + += )( (7) where a, b, and c are constants specific to each fabric. The constant a corresponds to the thickness at high pressures. For a fabric, therefore, a is the limiting thickness of the compressed fabric as the pressure increases. As the pressure approaches zero, the thickness is given by a+b/c, and hence b/c represents the total deformation in compression for a fiber assembly. The constant c is a correction to the pressure, which accounts for the fact that fabrics do not become thicker without limit as the pressure decreases. It may be thought of as a force of attraction between the fibers/yarns that holds them together when there is no external pressure on the fabric. Equations 2, 6, and 7 can be combined to show that the total force F(h) is related to the position of the top roller, h, by, ∫ − −−−−− = l Dcldx axrxrh b DhF 0 22 0 22 1 22)( (8) In this equation, l is related to h by Equation 3, and thus )(hF can be solved as a function of top roller position, h. According to Bogaty [6], the parameters of the pressure-thickness equation (Equation 7) can be determined experimentally. He suggested that the value of a is defined as the thickness observed at a pressure of 2.0 2lbs./in (140.7 2cmgf/ ), and c is 16 assumed to be 0.05 2lbs./in (3.5185 2gf/cm ). In this sensitivity analysis, a is determined as the fabric thickness measured at 100 2gf/cm from FAST [12, 13] compression test results, and b is then estimated from Equation 7. A plain woven fabric made of polyester/cotton with weight per unit area of 1.8775x 210− 2g/cm is used as the sample for this study. From the FAST results, a is 0.4835 mm. The thickness of this fabric under the pressure of 2 2gf/cm is 0.7074 mm, thus, b can be calculated as 0.12356 g/cm . When constants a, b, and c are substituted into Equation 8, the force-displacement relationship of the fabric can be computed. Figure 5 shows the result for the test fabric, where 1r is 0.75 inches, 0r is 3.25 inches, and D is 2 inches. 0 50 100 150 200 250 0 0.05 0.1 0.15 0.2 0.25 h , Displacement(mm) F (h ), F or ce (g f) Figure 5. Simulation results of force-displacement relationship of the top roller As mentioned earlier, the sensitivity of this online system is determined by many factors, such as the characteristics of the load cell, the precision of positioning the compression roller, and the physical dimensions of the nip. In this system, the total normal force is closely related to the contact area between the fabric and the rollers in the 17 nip. The radius of the top roller, 1r , the length of the top roller, D, and the radius of the bottom roller, 0r , determine the total force applied by the top roller for a given deformation. Therefore, these three factors determine the system sensitivity and should be optimized in system implementation. By fixing two of the three parameters, we can analyze the effect of the change of each of these parameters to the sensitivity of the testing system. Bigger contact area will result in larger resultant force, thus it is desirable to have larger dimensions of 0r , 1r , and D. As mentioned before, another benefit of larger 0r and 1r is that they make the stress distribution in the nip more uniform. More uniform stress distribution in the nip also helps improve the Incremental Differential Algorithm (IDA) that is used to solve the pressure-thickness relationship from force-displacement data. However, from practical design and instrumentation perspective, the values of 0r , D, and 1r should be limited within ranges used in our calculation. Figure 6 shows the sensitivity curve for a range of 1r , the radius of the top roller, where the length of the top roller, D, is fixed at 2 inches and the radius of the bottom roller, 0r , is fixed at 3.25 inches. Figure 7 shows the computed sensitivity curve for various values of D, the length of the top roller, where the radius of the bottom roller, 0r , is fixed at 3.25 inches and the radius of the top roller, 1r , is fixed at 0.75 inches. 20 1 11 1 ' cos2 r hr DrA − = − (9) where h’ denotes the length of the segment dc, the normal deformation of the fabric at point c, and D is the length of the top roller. Figure 9. Top roller in contact with the fabric As discussed earlier, the stress experienced by the fabric is non-uniform along the nip. Strictly speaking, the levels of stress the fabric experiences in the entry region (ac, in Figure 9) and in the exit region (cb, in Figure 9) are different. To begin the analysis we assume symmetric stress distribution about σ axis (in Figure 3) and we further assume stepwise nonlinearity of the stress-thickness relation of fabrics in compression. In other words, we assume that the stress-thickness relationship is linear within an infinitesimal increment of deformation. This is similar to the assumption made by Carnaby et al. [9]. Following Carnaby, we have developed the Incremental Differential Algorithm (IDA) to obtain the stress-strain relationship in compression. Figure 10 shows the incremental movement of the top roller as it compresses the fabric. Let Position 0 denote the point at which the top roller comes into contact with the fabric during its downward movement. The top roller then moves down further in o d a c b 21 increment of ∆h to positions 1 and 2 and so on, until the predetermined maximum force is attained. Figure 10. Increment movement of the top roller If we use variable i as the step number of the movement of the top roller, with i = 0 for the point at which the top roller comes into contact with the fabric, then the area of contact between the top roller and the fabric at step i , )(iA , can be expressed as, 1 11 1 cos2)( r hir DriA ∆− = − , i = 0, 1, 2,… (10) thus, if the area of contact is increased by an amount of )(ia at every step, as shown in Figure 10, then )1()()( −−= iAiAia , i = 1, 2, 3, … (11) Hence the total compression force F of the top roller at step n, )(nF , is, ∑ = +−= n i inainF 1 )1()()( σ , n = 1, 2, 3 ,… (12) where )(iσ denotes the stress experienced by the fabric at step i (as shown in Figure 10). Obviously, according to stepwise nonlinearity assumption, the stress is assumed to be constant within a step. 0 1 2 Step 1 2 … … h∆ 22 The analysis thus far takes into account the contact geometry. The precision of the analysis depends largely on the step size ∆h. In the incremental method, smaller step size improves accuracy, while on the other hand finer step size results in more accumulative error, which makes the algorithm unstable. The optimum step size must be decided empirically. 4 CALCULATION OF STRESS-THICKNESS RELATIONSHIP: AN EXAMPLE In order to demonstrate the functionality of the system and the validity of the Incremental Differential Algorithm (IDA), an example of calculation of the pressure- displacement relationship from force-displacement data using the IDA is presented in this section. Figure 11 shows a typical force-displacement relationship of a test fabric measured from the online measurement system. 0 20 40 60 80 100 120 0 0.1 0.2 0.3 0.4 0.5 0.6 Displacement(mm) F or ce (g f) Figure 11. A typical force-displacement relation obtained from the online system 25 [6] Bogaty, H., Hollies, N. R. S., Hintermaier, J. C. and Harris, M., The Nature of a Fabric Surface: Thickness-Pressure Relationships, Textile Res. J. 23, 108-114 (1953). [7] Booth, J. E., “Principles of Textile Testing: An Introduction to Physical Methods of Testing Textile Fibers, Yarns and Fabrics,” New York, Chemical Publishing Company, Inc., 1969. [8] Caban, J. C., Denier Control – On-Line or Off-Line?, Fiber Producer 10, 40-47 (1982). [9] Carnaby, G. A., and Pan, N., Theory of Compression Hysteresis of Fibrous Assemblies, Textile Res. J. 59, 275-284 (1989). [10] Chase, L., Goss, J., and Anderson, L., On-Line Sensor for Measuring Strength Properties, TAPPI 82, 89-97 (1989). [11] Clapp, T. G., The On-Line Inspection of Sewn Seams, in “NTC Research Briefs,” 1995. [12] CSIRO, “Instruction Manual: Fabric Assurance by Simple Testing,” CSIRO and Australian Wool Corporation. [13] CSRIO, “User's Manual: The Fast System for the Objective Measurement of Fabric Properties, Operation, Interpretation and Application,” CSIRO and Australian Wool Corporation. [14] de Jong, S., Snaith J. W., and Michie N. A., A Mechanical Model for the Lateral Compression of Woven Fabrics, Textile Res. J. 56, 759-767 (1986). [15] Dorrity, J. L., and Vachtsevanos, G., On-Line Defect Detection for Weaving Systems, in “IEEE Annual Textile, Fiber and Film Industry Technical Conference,” Atlanta, GA, USA, 1996. [16] Graf, J. E., Enright, S. T., and Shapiro, S. I., Automated Web Inspection Ensures Highest Quality Nonwovens, TAPPI 78, 135-138 (1995). [17] Habeger, C. C., and Baum, G. A., On-line Measurement of Paper Mechanical Properties, TAPPI, 106-111 (1986). 26 [18] Hintermaier, J. C., Wedge Effects in a Plain Press Nip, TAPPI 46, 190-195 (1963). [19] Hoffman, R. M., and Beste, L. F., Some Relations of Fiber Properties to Fabric Hand, Textile Res. J. 21, 66-77 (1951). [20] Kawabata, S., “The Standardization and Analysis of Hand Evaluation,” 2nd ed., The Hand Evaluation and Standardization Committee, Text. Mach. Soc., Japan, 1980. [21] Kim, C. J., “A Study of the Physical Parameters Related to the Mechanics of Fabric Hand,” in Textile and Polymer Science: Clemson University, 1975. [22] Kothari, V. K., and Das, A., Compressional Behavior of Nonwoven Geotextiles, Geotextiles and Geomembranes 11, 235-253 (1992). [23] Kothari, V. K., Das, A., Compressional Behavior of Spunbonded Nonwoven Fabrics, J. Textile Inst. 84, 16-30 (1993). [24] Langevin, E. T., and Giguere, W., On-Line Curl Measurement and Control, TAPPI 77, 105-110 (1994). [25] Lantz, K. G. and Chase, L. M., On-line Measurement and Control of Strength Properties, TAPPI, 75-78 (1988). [26] Larose, P., Observations on the Compressibility of Pile Fabrics, Textile Res. J. 23, 730-735 (1953). [27] Ly, D. G., Tester, D. H., Buckenham, P., Roczniok, A. F., Adriaansen, A. L., Scaysbrook, F. and Jong, S. D., Simple Instruments for Quality Control by Finishers and Tailors, Textile Res. J., 61, 402-406 (1991). [28] Matsudaira, M., and Qin, H., Features and Mechanical Parameters of a Fabric's Compressional Property, J. Textile Inst. 86, 476-486 (1995). [29] Pallas-Areny, R., and Webster, J. G., “Sensors and Signal Conditioning,” John Wiley & Sons, Inc., USA, 1991. [30] Peirce, F. T., The “Handle” of Cloth as a Measurable Quantity, J. Textile Inst. 21, T377-416 (1930). 27 [31] Thomas, R. K., You Need Online and Offline Testing for Top Yarn Quality, Textile World 143, 50-51 (1993). [32] Vahey, D. W., An Ultrasonic-Based Strength Sensor for On-Line Measurements, TAPPI, 79-82 (1987). [33] van Wyk, C. M., Note on the Compressibility of Wool, J. Textile Inst. 37, T285- T292 (1946). [34] van Wyk, C. M., A Study of the Compressibility of Wool, with Special Reference to South African Merino Wool, Onderstepoort J. Vet. Sci. Anim. Ind. 21, 99-224 (1946). [35] Victory, E. L., High Speed Compression Properties of Textile Structures and Sheet-Like Materials, J. of Applied Polymer Science 3, 297-306 (1964). [36] Ye, X. F., A Study of Dynamic Properties of Fluid Removal from Fabrics by Squeezing, J. of China Text. University 15, 77-86 (1989). [37] Yih, C. S., and McNamara, S. J., The Crushing of Wet Paper Sheets, TAPPI 46, 204-208 (1963). [38] Zhou, N., and Ghosh, T. K., On-Line Measurement of Fabric Bending Behavior, Part I: Theoretical Study of Static Fabric Loops, Textile Res. J. 67, 712-719 (1997). [39] Zhou, N., and Ghosh, T. K., On-Line Measurement of Fabric Bending Behavior, Part II: Effects of Fabric Nonlinear Behavior, Textile Res. J. 68, 533-542 (1998). [40] Zhou, N., and Ghosh, T. K., On-Line Measurement of Fabric Bending Behavior, Part III: Dynamic Considerations and Experimental Implementation, Textile Res. J. 69, 176-184 (1999). 30 The prototype online compression measurement system designed and built in this study consists of a let-off and a take-up motion, a compression unit, and a data acquisition and control unit. Figure 13 shows a schematic of the online measurement system. A and A', B and B': Transport rollers; C: Bottom compression roller; D: Top compression roller; E: Load cell; F: Linear motor; G: Timing belt; H: Test fabric Figure 13. Schematic of the online compression measurement system The continuous movement of fabrics under constant in-plane tension through the online measurement system at a constant linear speed is controlled by the let-off and take-up mechanisms (not shown in Figure 13) as well as feed and take-up rollers (A, A’) and (B, B’), respectively. Ultrasonic sensors are used in the take-up mechanism to help maintain constant throughput. At the measurement head, the bottom roller C is aligned with rollers A’ and B’ such that the top surfaces of these three rollers are in the same plane in order to eliminate out-of-plane distortion of the fabric. The top roller D is attached to a load cell, E, at the F C BA E D G H A’ B’ From Let- off motion To Take- up motion 31 top. The load cell in turn is connected to a linear motor, F. The linear motor controls the displacement of roller D for loading and unloading. The diameters of rollers D and C and the length of roller D are determined based on the system sensitivity analysis presented earlier [10]. In choosing the load cell (E, Figure 13) we have considered the required sensitivity to detect small forces encountered in fabric compression as well as the weight of the top roller (D, Figure 13) that is attached to the load cell permanently. A reversible compression-tensile load cell with a range of 250 gf is selected. The data acquisition and control unit of the online measurement system is developed using LabVIEW (Version 4.1) [1-3]. LabVIEW is a graphical programming system for control, data acquisition, and data analysis. Figure 14 shows a functional diagram of the data acquisition and control unit. A general-purpose computer installed with the LabVIEW software controls the entire system. The Timers/Counters in the system generate the pulses that are fed into the various stepper motor drives. The signals from the digital I/O interface control the motors. The ultrasonic sensors detect the position of the fabric in the take-up system and ensure constant web tension in the fabric and the throughput of the system. The load cell signal is pre-processed and amplified by the strain gage signal conditioner and then processed in the computer. The hardware of this system, see Figure 14, consists of a computer with a plug-in digital/analog I/O board and a plug-in Timer/Counter board, three stepper motors (for let- off, take-up, and compression) and their drives, an ultrasonic proximity detection system, a load cell, and a strain gage signal conditioner. The hardware used in the system and their functional descriptions are listed in detail in Appendix 2. 32 Figure 14. Functional diagram of the online measurement system A flow chart and detailed descriptions of the operation and control of this system are given in Appendix 3. Parameters such as the maximum compression force, the compression rate, and the initial gap (between roller D and C in Figure 13) are input into the control software before test begins. The initial gap setting is important to compressional tests and it needs to be determined according to the thickness of the fabric, and needs to be set before testing. The control system then repositions the top roller according to the prescribed gap setting. The compression test is performed by moving the top roller (D, in Figure 13) step-by-step (step-size is programmable) to compress the fabric, while recording the resulting load continuously. After the maximum compression Computer Timer/Counter Digital I/O Analog I/O Linear Motor Drive Let-off Motor Drive Take-up Motor Drive Ultrasonic Ranging/Proximity System Strain Gauge Signal Conditioner Signal from ultrasonic sensors Load cell signal 35 In the present application, there is more than enough information in the over- sampled data (in terms of sampling rate as well as the step size of compression). Therefore, a modified version of the median filter has been developed and designated as a Static Median Filter (SMF). In the RMF, the window is moved through all the data points in the sample sequence, and thus the processed data sequence has the same number of samples as in the original data. In the modified version windows are placed at every N (an odd integer) samples, therefore there are n/N windows in a sequence containing n samples in the original sequence (in the RMF, there are n windows). Therefore, the output sequence of the SMF has only about n/N samples. Figure 16 shows a measured force-displacement relationship of a woven fabric, marked as “measured.” A SMF with a window size of 5 was applied to this experimental data, and the processed data is marked as “filtered” in Figure 16. After applying the SMF, the variation of the data is reduced, and the filtered sequence has 1/5 of the original number of points. However, there is still noticeable fluctuation in the filtered data. Cubic spline [6] Smoothing is further used to process the data before the IDA is applied. Note that the reduced number of samples in the filtered sequence makes the subsequent spline smoothing easier. The spline-smoothed force-displacement data is also shown in Figure 16, marked as “spline”. The IDA is subsequently applied to this processed force-displacement data to obtain the stress-displacement relation, as shown in Figure 17. The “calculated” data is the direct computation results by applying the IDA, which is based on the stepwise nonlinearity model described earlier [10]. As seen in the figure, the “calculated” data is not smooth. Spline smoothing is further used to process this data, which results in a 36 smoother stress-displacement relationship. From this stress-displacement curve, characteristics of fabric compressional behavior such as thickness and compression modulus can be obtained. Figure 16. Static Median Filtering and Spline smoothing on the raw force-displacement data Figure 17. The calculated stress-displacement relationship 37 4 EVALUATION OF THE MEASUREMENT SYSTEM In order to validate the measurement principle, data processing methods, and the accuracy of the online measurement system, 9 fabric samples (4 woven and 5 nonwoven) were tested. Details of the fabrics tested are presented in Table 1. The fabrics were evaluated on the online measurement system under both static and dynamic conditions. The static tests were performed on stationary fabric samples, while the dynamic tests were performed on moving fabrics. The same fabrics were also evaluated on the KES-F3 (Kawabata Evaluation System for Fabrics) [12], FAST-1 (Fabric Assurance by Simple Testing) [7, 8], Instron [4, 5], and a manual thickness tester1. Table 1. Fabric samples evaluated for system evaluation Sample ID Fiber-type Process/texture Weight/Area ( 22 10/ −×cmg ) N1 Polypropylene Spunboned 0.5425 N2 Polyester Spunbonded 0.6950 N3 Polyester Dry laid, point bonded 0.3063 N4 Polypropylene Spunbonded 0.4075 N5 Polyester Spunbonded 1.3975 W1 Polyester (filament) Woven 2.1750 W2 Polyester/cotton (staple/filament) Woven 1.8775 W3 Rayon (staple) Woven 2.6450 W4 Acrylon (staple) Woven 2.5513 1 . Analog thickness tester by Randall & Stickney. 40 5 EXPERIMENTAL RESULTS Experimental values of fabric low-pressure thickness, 0T , obtained from various instruments are shown in Table 3. All the results listed in the tables are averages of eight replications. For the purpose of discussion the results from online static and dynamic tests are designated as “Online-S” and “Online-D,” respectively. Experimental results for high-pressure thickness ( mT ) values are presented in Table 4. For clarity, the test results are presented in graphical form in Appendix 6. In both cases, each data point represents the mean of eight measurements of samples. Table 3. Low-pressure fabric thickness, 0T (mm) Sample KES-N Online-S Online-D FAST Manual Instron N1 0.095 0.282 0.271 0.277 0.233 0.241 N2 0.082 0.314 0.328 0.276 0.242 0.278 N3 0.124 0.317 0.259 0.282 0.244 0.207 N4 0.148 0.334 0.368 0.347 0.281 0.295 W1 0.426 0.576 0.582 0.622 0.565 0.559 W2 0.535 0.707 0.632 0.707 0.612 0.625 W3 0.564 0.778 0.678 0.659 0.587 0.525 W4 0.828 1.004 0.897 0.992 0.774 0.859 N5 1.792 1.903 1.879 1.857 1.763 2.015 The discrepancy in test set up for various testing instruments are apparent in the variation in data in Table 3 and Table 4. It should be noted that tests for fabric compression involve small load and displacement values, and therefore, very high resolution instrumentation and reliable calibration are needed to obtain reliable test data. The variation in sample size and rate of compression (see Table 2) between various 41 instruments also make it difficult to campare these values. Nevertheless, it is obvious that the data obtained from KES-F3 [12] measurements are significantly different from those obtained from other instruments. The difference is particularly pronounced for nonwoven fabrics, in fact, in the case of some fabrics (e.g. N1) the KES-F3 values are 50~70% lower. Table 4. High-pressure fabric thickness mT (mm) Sample KES-N Online-S Online-D FAST Manual Instron N1 0.032 0.203 0.204 0.203 0.215 0.158 N2 0.035 0.251 0.265 0.221 0.230 0.195 N3 0.014 0.197 0.179 0.172 0.181 0.119 N4 0.064 0.242 0.291 0.252 0.259 0.206 W1 0.300 0.391 0.469 0.456 0.467 0.426 W2 0.346 0.520 0.512 0.484 0.488 0.444 W3 0.396 0.535 0.494 0.513 0.527 0.435 W4 0.513 0.642 0.628 0.601 0.627 0.583 N5 1.341 1.367 1.326 1.313 1.387 1.435 Before carrying out further analysis of the data we decided to repeat the KES tests at another location2. Thickness data obtained from the repeated tests together with the earlier KES data are presented in Table 5. For convenience the KES data obtained from NCSU and Georgia Institute of Technology laboratories are described as KES-N (KES data from NCSU) and KES-G (KES data from Ga. Tech), respectively. The difference between the KES data measured at these two locations is significant and remains inexplicable. The pressure-thickness plots as well as other data (thickness, etc. produced by the KES system as part of standard output) from both instruments were checked for 42 errors. Unfortunately, no inconsistancies or errors were found. Comparing the two sets of KES data with others in Table 3 and Table 4, it seems KES-G data is of the same order of magnitude as others, particularly in the case of some nonwoven fabrics. Table 5. The KES test results from KES-G and KES-N (unit: mm) 0T mT Sample KES-G KES-N KES-G KES-N N1 0.298 0.095 0.234 0.032 N2 0.279 0.082 0.240 0.035 N3 0.290 0.124 0.212 0.014 N4 0.358 0.148 0.288 0.064 W1 0.596 0.426 0.484 0.300 W2 0.679 0.535 0.511 0.346 W3 0.589 0.564 0.508 0.396 W4 0.964 0.828 0.669 0.513 N5 1.968 1.792 1.455 1.341 The thickness values ( 0T and mT ) are plotted in Figure 18 and Figure 19 for comparison. It is clear that both thickness values measured by the online system, in either static or dynamic condition, are similar to all other testing methods except for the KES-N results for nonwoven fabrics. The online results are closest to those of the FAST system. Low-pressure thickness values measured by the manual tester are consistently smaller than the results from the FAST and Online methods, which may suggest that the “minimum” pressure for the manual tester may actually be greater than 2 2gf/cm . Interestingly, for most fabrics the low-pressure thickness values measured using online- dynamic method are smaller than those obtained by using online-static method. This may 2 Department of Polymer and Textile Engineering, Georgia Institute of Technology, Atlanta, GA 45 6 CONCLUSIONS AND RECOMMENDATION FOR FUTURE WORK An online measurement system for fabric compressional behavior has been developed. The system is controlled by a computer and various test parameters (deformation rate, fabric linear speed, etc.) can be easily controlled. The Incremental Differential Algorithm used in the study allows calculation of fabric pressure-thickness relationship from force-time relationship obtained by the online instrument. The results obtained from the evaluation of the online instrument compare very well with other traditional offline measurement instruments (KES, FAST, etc.). The real-time data obtained from the online instrument can be used to control various relevant textile manufacturing processes (finishing, etc.). The online system developed in this research can be improved in a number of ways. In the present system, the fabric may slip at the compression roller nip. This may cause local deformation of the specimen and thereby introduce experimental errors. To eliminate this potential for error, the bottom compression roller could be driven at the same linear speed as the transfer rollers. The movement of compression-roller nip is another potential source of error. This could be eliminated by precision machining and mounting of the bottom compression roller. The data processing techniques developed for the online system could include computation of compressional rigidity. The highly nonlinear pressure-thickness relationship could be assumed as a bilinear function. The slope of the initial part of the almost-linear relationship over which the inter-fiber voids are eliminated during compression is critical in determining the “softness” or “fullness” of the fabric. The slope of the final portion of the pressure-thickness curve is also of importance to characterize 46 the fabric structure. The transition region between these two portions of the curve can be identified from the second derivative of pressure with respect to thickness. Once this transition point (or region) is determined, it is relatively simple to determine the rigidity values of these two regions of interest. REFERENCES [1] “LabVIEW Data Acquisition Basics Manual,” National Instrument Corporation, 1997. [2] “LabVIEW Function and VI Reference Manual,” National Instruments Corporation, 1997. [3] “LabVIEW User Manual,” National Instrument Corporation, 1997. [4] “TestWorks: Operator's Guide,” MTS Systems Corporation, 1998. [5] “TestWorks: Reference Guide,” MTS Systems Corporation, 1998. [6] Conte, S. D., and Boor, C. D., “Elementary Numerical Analysis: An Algorithm Approch,” 3rd ed., McGraw-Hill Book Company, 1980. [7] CSIRO, “Instruction Manual: Fabric Assurance by Simple Testing,” CSIRO and Australian Wool Corporation. [8] CSRIO, “User's Manual: The Fast System for the Objective Measurement of Fabric Properties, Operation, Interpretation and Application,” CSIRO and Australian Wool Corporation. [9] Haddad, R. A., and Parsons, T. W., “Digital Signal Processing: Theory, Applications, and Hardware,” Computer Science Press, 1991. 47 [10] Huang, W., and Ghosh, T. K., Online Characterization of Fabric Compressional Behavior, Part I: Principle of Measurement, in process (1999). [11] Johnson, J. R., “Introduction to Digital Signal Processing,” Englewood Cliffs, NJ, Prentice Hall, 1989. [12] Kawabata, S., “The Standardization and Analysis of Hand Evaluation,” 2nd ed., The Hand Evaluation and Standardization Committee, Text. Mach. Soc., Japan, 1980. 50 classification [6], defect detection for weaving systems [21], etc. However, very little work on the online measurement of fabric mechanical properties has been reported. Recently Zhou and Ghosh [89, 90] reported development of an online measurement system to characterize fabric bending behavior. Dynamic loop shapes of fabric are measured online through ultrasonic sensors, which is used to calculate fabric bending rigidity or bending length from relationships developed using nonlinear models of fabric deformation. In paper industry, online measurement systems that are designed to measure tensile properties of paper [8, 33, 54, 77] have been reported. Unfortunately, however, this review did not reveal any published report on online measurement systems for the compressional behavior of fabrics or similar sheet-like materials. This report presents a critical review of the theoretical and experimental studies of the compressional behavior of fabrics reported in the literature. It also reviews the development of fabric compression testers and various test methods that are in use today. 2 FABRIC COMPRESSIONAL BEHAVIOR The compressional property of a fabric is one of the most important mechanical properties. It is closely related to fabric handle, i.e., the softness and fullness of the fabric and also with the fabric surface smoothness [63]. It is also closely related to comfort. The fabric thickness and compressibility has a linear relationship with thermal conductivity [5], the warmth of a fabric is largely a function of the airspace and its distribution in the structure [12]. 51 In the following section, the parameters that are of critical importance in describing fabric compressional characteristic are introduced. Subsequently, a review of the theoretical analysis of the compressional behavior of fibrous structures will be presented. Finally, results of experimental studies of fabric compression behavior will be discussed. 2.1 Characterization of Fabric Compressional Behavior Fabric compressional behavior is generally described by the relationship between the applied force (normal to fabric plane) per unit area and the resulting fabric thickness. The standard testing process, which is used in KES-F [43] and FAST [18] systems as well as in ASTM D-39 standard, usually consists of the following steps. The fabric specimen is placed horizontally on an anvil, in a relaxed state, see Figure 22. Then force F is applied on the presser foot (of area A) normally to the fabric surface. The force per unit area AFp /= and the resulting thickness t of the fabric (the separation of anvil and presser foot) are recorded. As the presser foot is lowered continuously at a predetermined speed, the force F increases gradually and the process is repeated until p attains the predetermined highest value. This is the compression cycle of the test. The recovery cycle of the test is performed in similar manner. In recovery the presser foot is gradually moved away from the fabric surface to its original position. Figure 22 is the typical fabric stress-thickness curve under lateral compression. As seen in Figure 22, the stress-thickness curve of textile fabrics in lateral compression is highly nonlinear. Furthermore, the loading-unloading plot exhibits hysteresis, i.e. energy 52 is lost during the deformation cycle. This hysteresis loss is due to the interfiber friction and the viscoelastic behavior of the fibers within the fabric. Figure 22. Typical stress-thickness curve for fabric in lateral compression. F is the force applied to the foot and t is the thickness of fabric. Curve BCD stands for compression cycle and DEB is recovery. During the testing of fabrics, it is important to select the maximum compressional pressure that is likely to reflect the values experienced in the actual use of the garment, normally 2/50 cmgf is selected [43]. The following characteristic parameters are used to describe the fabric compressional behavior: 1) The Thickness, t Thickness of almost any fibrous structure is a function of normal pressure. Therefore, the determination of the thickness of fabric consists of the precise measurement of the distance between two plane parallel plates when they are separated by the fabric, with a known arbitrary normal pressure between the plates being applied and maintained [29]. 2) Compressional Modulus, h F t p )/( 2cmgf B C D E mp 0p mt0t Othickness(m 55 Textile fabrics can be grouped into three categories: woven, knitted, and nonwoven. Woven and knitted fabrics are made from yarns. Obviously the compressional behavior of these fabrics is determined by the properties of yarns and the structure of the fabric. There are some studies of mechanics of yarns [15, 56-58, 71, 74, 79], but little work has been reported on the compression analysis of fabric based on the yarn properties [26, 85]. On the other hand, yarns are made from fibers, and that makes it possible to treat the woven and knitted fabrics as assemblies of fibers. The other textile structure, the nonwoven, are generally made of fibers, therefore these are indeed an assembly of fibers. A significant amount of research work on the compressional behavior of fiber assemblies has been reported in the literature. For the purpose of this review, we will treat all textile materials as assemblies of fibers and study the generalized theory of fabric compressional behavior, also known as micromechanical analysis. In micromechanical approach mechanical principles are employed to the analyzed microstructure of fibrous assemblies. In micromechanical method, the mass response of fiber assemblies to external forces are related and treated as the total of the responses of basic fiber units. The basic fiber unit is assumed to be the portions of fibers between adjacent contact points, which is called free fiber segments. This method is based on the fact that the forces applied on masses are transmitted to all constituent fibers by means of their contact points. As the result, each free fiber segment deforms in response to the loads externally applied. Hence the determination of number of contact point in a fiber assembly is very important in the study of the compressional behavior of fiber assembly. 56 The compression of a mass of fibers involves several processes, for example, bending, twisting, slippage and extension of the fibers. This is a very complicated mechanical system, but the problem is considerably simplified when only parts of the processes are considered. van Wyk [82] pioneered the work on the micromechanics of fiber assemblies in 1946. Since then various others contributed in improving this theory. The following are the common assumptions used in all the analysis included in this report to simplify the fiber deformation pattern as well as the structure of the fiber assembly: 1) All the fibers are assumed to be homogeneous, linearly elastic, with a circular cross section, and are of the same length; 2) The fibers are a collection of numerous independent segments bounded by two neighboring contact points; 3) Tensile and compressional deformation at contact points are ignored; 4) Fiber torsion can be ignored; 5) The fiber assembly is considered as a system of bending units of free fiber segment. To have a clear picture of the micromechanic study of fiber assemblies, we will divide the discussion into five sections, however these are closely related. In the first section, the pioneering work of van Wyk’s stress-volume theory [82], which is based on the isotropic assumption of fiber assembly, is introduced as a starting point. In second section, anisotropic behavior of fiber assemblies under compression is taken into account and number of fiber-to-fiber contacts is discussed. In section 3, we introduce the method to calculate the compressional modulus and Poisson’s ratio. Section 4 addresses the important work by Carnaby et al. [14] in which the compression hysteresis behavior is 57 revealed. This is followed by a discussion on the energy methods that have been used to study the compressional behavior of fibrous assemblies. 2.2.1 Stress-Volume Theory In 1946, in his theory of the compressional behavior of a fiber assembly, van Wyk [82] reduced the compressional process of fibers to that of simple bending of the fibers. van Wyk considered the fiber assembly as a system of bending units, which are the elements of fiber between adjacent contacts with other fibers. van Wyk also assumed that: 1) the fiber elements are orientated at random, and the mass of fibers is uniformly packed; 2) the frictional forces are negligible; 3) a fiber is as a rod that is supported horizontally at a large number of points equally spaced at a specified distance 2b apart, with equal downward forces, F, acting midway between the points of support; 4) All fibers are of constant length. For relative small deformation, the deformation y of a rod (fiber) and the force F are connected by the relation y b EI F ff 3 24 = (16) whrer fI is the moment of inertia of cross-section of the rod, fE is Young’s modulus of elasticity, and b is the free fiber length (the mean distance between fibers). 60 Figure 23. Definition of coordinate system for a fiber assembly Komori et al. assumed that the probability of finding the orientation of a fiber in the infinitesimal range of angle θ∼θ+dθ and φ∼φ+dφ is φθθφθ ddsin),(Ω , where θφθ sin),(Ω is the density function of orientation of fibers. They further assumed that there are N fibers of straight cylinders of diameter frD 2= , and length fl in the fibrous system of volume V. Komori et al. set up a geometrical model to calculate the probability in which two fibers are in contact with each other. They assumed that a fiber A of orientation ),( φθ is placed in an arbitrary position in a volume V first, and then another fiber, B, having an orientation )','( φθ be placed in the same volume, see Figure 24. The two fibers will then contact each other when the center of mass of fiber B is brought into the suitable region in the neighborhood of fiber A. Then B is slid over A from one end to the other, keeping the direction and the contact point on B unchanged. The same is done on the opposite side of fiber A. X, 1 Y, 2 Z, 3 φ θ 61 Figure 24. Parallelepiped formed by fiber ),( φθA and )','( φθB in contact The volume, v, of this parallelepiped is: χφθφθ sin2)',',,( 2fDlv = (20) where fl is the length, D is the diameter of the fibers, χ is the angle between the two axes of fibers A and B which is given by )'cos('sinsin'coscoscos φφθθθθχ −+= (21) It is obvious that whenever the center of a fiber of orientation (θ’, φ’) enters this region, it will necessarily contact fiber A. According to Komori et al., the probability p that fiber B is in contact with A when the former is arbitrarily placed in V, is also the probability that the center of mass of fiber B is found in the parallelepiped region v, which is given by, χsin 2 2 V Dl V v p f== (22) They further showed that when there are N fibers in volume V, the average number of the fibers which contact fiber A is: ),( φθA )','( φθB D2 fl fl χ 62 ∫∫ Ω−= ππ θφθφθφθ 00 'sin)','('')1(),( pddNn ),( 2 2 φθJ V DNl f≈ (23) where, ∫∫ Ω= ππ θχφθφθφθ 00 'sinsin)','(''),( pddJ (24) They also showed that the mean length b between the centers of two neighboring contact points on the fiber is DLI V b 2 = (25) and the total number of contacts in a fiber assembly is given by: I V DL n = (26) where V is the volume of the fiber assembly, D is the diameter of the fiber, L is the total length of the fiber in the assembly and I is given by ∫∫ Ω= ππ θφθφθφθ 00 sin),(),(JddI (27) Although the length of fiber fl is assumed to be the same for both (see Figure 24), Komori and Makishima showed that it is not necessary [46]. They gave some examples of the calculation of the density function of fiber orientation. In the case of random orientation of fibers in an assembly, π 1 =Ω . In the case of sheet-like assembly such as paper and certain kinds of nonwoven fabrics, ) 2 ( 1 ),( π θδ π φθ −=Ω , where Dirac’s delta function ( ()δ ) is used. 65 )/1( 2 ψLVDV DLI n + = (35) Motivated by Pan’s effort in 1994, Komori and Itoh [48] developed a formulation using steric exclusion of fiber contacts, which considers not only the forbidden length effect, but the forbidden volume effect between fibers as well. Komori and Itoh [48] treated fiber segment as statistical elements, which may be assumed as individuals independent of each other, and defined the length of individual fibers as the persistent length, λ. Figure 25 shows the steric exclusion concept. Figure 25. Forbidden length centered at the contact point of fiber elements E and E’ When a contact is formed by two fiber elements E and E’ directed in o (a unit vector, called the director) and o’ in the assembly, element E’ makes a forbidden length of )',(sin/2 ooχD on element E. Hence the expected value of the total forbidden length formed on element E by all other N-1(≅N) elements of length λ is given by ∫ Ω= ')'()',(sin/)',(2)(* ωχλ dpND oooooo (36) O’ E’ O ED χ 66 where p(o, o’) stands for the probability that two elements oriented in o and o’ directions make a contact in a mass of a large number of basic fiber elements. 'ωd is the infinitesimal solid angle around director o’ and is defined by φθθω ddd 'sin'= (37) Note that the formulation of Equation 24 is neither a rigorous nor a general one for the forbidden length. For example, the difference in the probability of the fiber to contact a pre-determined fiber due to different number of pre-existing fibers is ignored to simplify the derivation. Komori and Itoh proved that when an additional element AE directed in Ao was brought into the mass, it would touch the inhibited length )(* oλ formed on E, if the center of AE could enter a space around the contact point, whose volume is given by ),(sin)(*2),(* ooooo AA Dv χλλ= (38) from the same reason developed in the derivation of Equation 8. They further showed that the total forbidden volume, )(* AV o , against locating element AE , which is contributed by all pre-existing contact points, is thus given as ∫∫ ΩΩ= ')'()()',(),(*21)(* 2 ωωddpvNV AA ooooooo (39) Using the steric exclusion concepts, Komori and Itoh [48] introduced hindrance correction factor or simply the hindrance factor h to relate p(o, o’) and )',(0 oop , the idealized probability that the two fibers contacts. It is given by )',()',()',( 0 oooooo php = (40) 67 where h represents the deviation from the idealized case due to steric hindrance between fibers and, )',(sin 2 )',( 2 0 oooo χ λ V D p = (41) Komori and Itoh [48] further introduced the following notations ∫ Ω= ')'()',()( ωdhF oooo (42) and, ∫ Ω= ')'()',(sin)'()( ωχ dFG ooooo (43) and showed that ] )(1 )'(21 )'(1 )(21 [ 2 1 )',( 2222 o o o o oo GVq FqV GVq FqV h f f f f αα − − + − − = (44) where π/8=q , the aspect ratio of the basic element of persistent length D/λα = , and the fiber volume fraction V DN V f 4 2λπ = . Equation 32 is a recursive definition, which can be solved using an iterative method, with the initial value of )',(0 ooh specified ( )',(0 ooh =1 for example). Komori and Itoh [48] also showed that the direction-dependent mean free fiber length b(o), is given by )](/[)( oo JqVDb f= (45) where function J is defined as, ')'()',(sin)',()( ωχ dhJ oooooo Ω= ∫ . (46) 70 jjj kjk jk b b / / δ δ γ − = , )( kj ≠ (52) Thus Lee and Lee derived relationships for the initial compressional modulus as jj j jj m IK V LD BE 22 3 32 )(48= (53) and the initial compressional Poisson’s ratio kjj jjk jk Km Km =γ , )( kj ≠ (54) van Wyk [82] suggested that the correct beam deformation theory to use when calculating the deformation of middle point is that for a beam with built-in ends. This equation ensures continuity in curvature at the contact points, conservation of mass, and no slippage at the contact points, which Carnaby [14] also regarded as more appropriate. In that case, jk j jk mB bC 6 3 ±=δ , ( kjkj =−≠+ ,;, ) (55) should be used instead of Equation 38. 2.2.4 Compressional Hysteresis The studies of van Wyk [82] and Lee and Lee [60] postulated that the compressive strain of the assembly of fibers is translated directly into the bending strains in the individual fibers, and that the resistance of the assembly to an externally imposed stress arises solely form the resulting increase in bending energy in the fibers. In 1989, Carnaby and Pan [14] for the first time incorporated the fiber-fiber slippage in the 71 deformation of fibrous assembly in both compression and recovery cycle. This made possible the theoretical prediction of the full hysteresis curve for compression and recovery. As the response of the material is nonlinear, the compression modulus and Poisson’s ratio are presented as a tangent compliance terms dependent on the initial state of stress and strain. Carnaby et al. used all the assumptions previously used by Lee and Lee [60]. They made an additional assumption that the effect of slipping contacts on the nonslipping contacts can be ignored. Carnaby [14] argued that even in the absence of any external load applied to a mass of fibers, there were still significant nonzero contact forces between fibers in the assembly. The fibers are prevented form recovering to their lowest energy configuration because of mutual interference and frictional restraints to slippage. This interaction can be detected by single fiber withdrawal experiments. The withdrawal force fW per unit length of fiber is: 0' ff WpW += µ (56) where p is the external pressure applied to the fiber mass, 'µ is the proportionality factor with dimensions of length, and 0fW is the value of fW when p=0. Carnaby et al. used Figure 26 to illustrate the deformation of a fiber element due to a contact point jC in direction j. The force jC may be resolved into jnC , which is normal to the bottom fiber, and jpC , which is parallel to the bottom fiber. The top fiber will thus begin to slide along the bottom fiber if 0fjnjp bWCC +≥ µ (57) 72 where µ is the coefficient of friction between two contacting fibers and b is the length of fiber. If the unit cell is subject to a combined compressive load, the criterion for slippage given in Equation 45 remains the same, but the values used for jnC and jpC must also include the components of force due to pressure exerted in the other orthogonal directions. Carnaby et al. used 1C , 2C , and 3C to stand for net contact forces per contact point in 1, 2, and 3 directions. The components of jC normal and parallel to the arbitrary fiber section are [60]; 2 122 11 )cossin1( φθ−= CC n , φθ cossin11 CC p = 2 122 22 )sinsin1( φθ−= CC n , φθ sinsin22 CC p = θsin33 CC n = , θcos33 CC p = . (58) Equation 45 is thus equivalent for the general loading case to 0bWFCC j jn j jp +≥ ∑∑ µ (59) that is, θφθφθ cossinsincossin 321 CCC ++ 03 2 122 2 2 122 1 )sin)sinsin1()cossin1(( bWFCCC ++−+−≥ θφθφθµ . (60) No general analytical solution has been reported so far, although it is fundamental to the development of a general tangent compliance matrix. Although it is possible to give a numeric solution now, however, Carnaby et al. reported the solution for the special case of uniaxial compression. They selected the direction of uniaxial compression force as axis 3, and rewrote Equation 48 as 75 Lee and Lee’s theory [60] of compressional modulus and Poisson’s ratio of fiber assembly expressed in Equation 41 and 42 apply only for small strains of the bending element. In the incremental approach used by Carnaby et al. to accommodate the large deformations of the assembly as a whole, they assumed that the actual bending strains in the fibers remain small in each step. The incremental form of Equation 43 is jk jj jk mB bC 6 3'∆ ±=∆δ (72) and, ' jk jk jk b δ ε ∆ =∆ (73) and thus the compressional modulus jjE and the Poison’s ratio jkγ are expressed as: jj j jj p E ε∆ ∆− = , (74) and, jjjj kkjk jk b b / / δ δ γ ∆ ∆ = , (75) where jp is the compressional pressure in direction j. Using the conclusion by Lee and Lee [60], Carnaby et al. further showed that )1( 192 22 3 2 j sj jj jj ffjj C C SNSN m IK VEE ∆ ∆ +−= η π j = 1, 2, 3 (76) where η is the shape factor of the fiber related to bending and jkjj kjjk jk IKm IKm =γ j≠k, j=1, 2, 3, k=1, 2, 3 (77) 76 Carnaby et al. [14] used the incremental method to cope with the problem of large deformations typical to fibrous assemblies. They defined the algorithms that are to be used to update the system geometry on successive increments. Following the steps described above, the stress-strain curve of the compressional behavior of the fibrous assembly can be obtained. In the recovery stage, the external load is removed. In practice this is once again achieved by a succession of small decreases in the value of jC , but the behavior of the slipping contact points is subtly different during recovery. Referring again to Figure 26, the cause of the recovery now is the stored bending energy in the bottom fiber, but this applies only a normal force to the slipping fiber. Even when the value of the normal force jnC is reduced to zero, the top fiber will not slip back “up” the bottom fiber. Carnaby et al. [14] related the necessary return slippage with the recovery of the bent fibers. They assumed that a finite value of ijC pointing in the reverse direction is needed to overcome the resistance 0fbW at the contact. That is, 0cos fjij bWCC == θ . (78) Carnaby et al. [14] further showed that in recovery, the tangent modulus now becomes )1( 192 22 3 2 j sj jj jj ffjj C C SNSN m IK VEE ∆ ∆ −−= η π (79) but now, ∫∫ Ω= 2 0 0 0 secsin),(4 πξ ξξτξτξ jjjjjjsj bWFddC cj . (80) 77 As the recovery proceeds with reductions in jC , the critical angle also reduces, but the changing orientation function also ensures more fibers at a smaller polar angle. The path on recovery thus differs from the compression curve. When the external load jP is reduced to zero, the assembly still contains some strain energy in the bent fibers. This energy is now locked into the assembly by virtue of its own internal friction. 2.2.5 Energy Method The application of energy method in the problem of compressibility of fibrous assembly relates the increment of energy stored in the infinitesimal deformation of the assembly and the increment of stress in mass. This method is more lucid and concise than other theories in which recovery force of the bent fiber is directly related to the stress of the mass, and the mechanical equilibrium has to be considered in each spatial direction. 2.2.5.1 Straight-Beam Model In 1991, Komori and Itoh [49] pointed out that in the theory of compression of fiber assemblies proposed by van Wyk [82], Lee and Lee [60], and Carnaby and Pan [14], there was a question about the treatment of the elemental fiber units whose bending responses are totaled and related to the overall behavior of the mass. van Wyk, Lee and Lee, and Carnaby and Pan assumed that any bending element had, regardless of its direction, a length equal to the orientation-independent mean free fiber length, which was obtained by averaging the distance between two adjacent fiber contacts over all fibers in 80 Summing the local bending energy along all fibers contained in a unit volume of the mass leads to the total increment Uδ in the energy density of the mass. Komori and Itoh showed that ωδελδ doJBDU iii ),()(),(32 22232 oÂo Ω= ∑∫ . (86) By taking the partial derivative of the energy density Uδ with respect to iiδε , they showed that ∑= j jjijii OH δεελλδσ )()/'(/ 3' '0 (87) where '0λ is the initial fiber length per unit volume in the unstrained mass, and H and ijO are defined by 3' 0 63' 0 264 λπλ fEDBDH == , (88) ωdooJO jiij ),(),()( 222 oÂo Ω≡ ∫ (89) They also showed that the Poisson’s ratio is given by )/()( 2ijjjiijkijjjikik OOOOOOO −−=γ . (90) Using the assumption of proportional deformation, Komori et al. derived the basic equation that rules the change in orientation density due to the compression as 11 222 ]1cossin3)/ln(cossin)/ln(coscossin[)(ln δεφθφφφθφθθδ −+∂Ω∂+∂Ω∂−≡Ω 22 222 ]1sinsin3)/ln(cossin)/ln(sincossin[ δεφθφφφθφθθ −+∂Ω∂−∂Ω∂−+ 33 2 ]1cos3)/ln(cos[sin δεθθθθ −+∂Ω∂+ . (91) If the mass is axi-symmetric with respect to axis k, the direction in which the compression force is applied, jkik OO = and jjii OO = , thus the Poisson’s ratio becomes )/( ijiiikik OOO +=γ . (92) 81 Later, Komori and Itoh [51] extend the above theory on the compression mechanics of fiber assemblies to a general case including compression, elongation, and shearing [51]. They formulate the theory so that any mode of energy of deformation stored in the fiber segments may be introduced, if necessary, in place of the beam- bending model that is introduced here. This theory has theoretical value, and can be applied to the more general case. But the difficulty with this method is how to determine the function )( fFF UU ε= , the energy stored in a unit length of the deformed fiber segment, which the authors did not explain. 2.2.5.2 Curved-Beam Model In 1990, Lee, Carnaby, and Tendon [59] presented an energy analysis of compression in which a circularly curved beam was assumed as a bending unit. In 1992, Komori and Itoh [47] adopted this method and refined the straight beam theory. They assumed that fibers are not necessarily straight before deformation, but they all have the same curvature before deformation. Komori and Itoh [47] defined the straight-line segment connecting the ends of the free element as the skeleton of a fiber segment. They further assumed that all the fiber elements whose skeletons are oriented equally along vector o have the same length b, the orientation-dependent mean free fiber length, which is given by 1)](2[ −Λ= oJDb F , (93) where FΛ is the fiber length per unit volume of the mass. 82 Again, Komori et al. assumed the affine deformation for the elemental response of fibers. They showed that the change in skeleton length and direction cosine l and io are ij i j jioolle δεδδ ∑∑== / (94) and mn m n nmiminiii ooooooo δεδδ ∑∑ −=−= )(* . (95) where inδ is the Kronecker delta. Itoh and Komori [38] reported in their earlier work that the orientation density Ω of the skeleton must change with the deformation of the mass, i.e., ∑∑ Ω−+=Ω m n mnmnnmmn ooQ δεδ )()(ln (96) where mnΩ and mnQ were given in [38]. Komori and Itoh assumed that each free fiber segment is curled in a definite curvature κ, the curvature increment due to an infinitesimal deformation is eWb δβδκ )()/4(−= (97) where, the arc length b is assumed to remain unchanged during the incremental process, that is, the fiber slippage is ignored and function W is defined by )cot1(2 ββ β − =W (98) where, β is the crimp angle and is defined by 2/κβ b= (99) They showed that the increment (of second order) uδ in the bending energy per unit length of the skeleton is given by 85 Although a lot of assumptions and simplifications are included in the theory of the compression of fibrous assemblies, the above approaches have formed the framework of this theory. A lot of other workers, such as Dunlop [22-25], Stearn [76], Nachane et al. [64], Hearne et al. [35] etc. made significant contributions. More recently, Komori and Itoh [45] tried to take the fiber length variation into account, and Nechar [66] recently developed a theory which considers the uncompressed areas between contact points. However, a lot of work remains to be done to refine the theoretical study of the compressional behavior of fiber assemblies. For example, Carnaby [14] was successful in explaining the hysteresis of the compression and recovery process, but he ignored the orientation distribution effects of the fibers in the compression and recovery process. The challenge is to unify the theoretical analysis to overcome the various deficiencies discussed earlier. It is evident from the discussion that there is no generalized analysis for the exact description of the pressure-thickness relationship of fabric compressional behavior. The reason for the shortcomings of all these analysis to represent adequately the relationship between pressure and thickness of fabrics is not hard to find. Consideration of what must take place when a fabric is being compressed shows that the compressional process is a complex one which involves several factors, each of which plays a different role and that to a varying degree. When pressure is applied to a fabric as it is in a compression meter, several things may take place. There is undoubtedly a deformation of the fibers of the type discussed by van Wyk [82], where the fibers are regarded as cylindrical beams supported at regular intervals along their length. During compression, the fibers must take new positions, and 86 in doing so must slip past other fibers at certain points of contact, thus giving rise to frictional effects. The number of such contacts and the effort required to overcome the frictional forces will increase, as the fabric becomes denser. Some of the fibers will merely be bent during the process, and this will necessitate a force, which is a function for the angle through which the fiber is flexed. Some of the fibers, particularly if crimped, can be regarded as springs being compressed along their length. In addition, there will be some compression of the fibers in a radial direction where fibers are in contact and have a general direction perpendicular to the direction of the force applied. Since most or the fibers in fabrics do lie in such a direction (pile fabrics are an exception), this compressional effect is one which cannot be neglected and which becomes particularly important at higher pressures, when the density of the fabric is large. The relative importance of each of these factors depends on the type of fiber and the construction of the fabric. It is evident, then, that an equation that would represent adequately the relationship between pressure and thickness would be a complex one containing several terms with a number of constants which would vary with each fabric, and which would depend on the nature of the fibers, the size and twist of the yarns, the tightness and type of weave, the crimp of the yarn, and the crimp, size and shape of the fibers. Even the direction of twist or the relative direction of twist in the weft and in the warp yarn will affect the ease with which the fiber can move under the load applied. One must conclude from these considerations that the task of developing an equation of general application to express the compressibility of fabrics is likely to be a difficult one at best. 87 2.3 Empirical Equations of Fabric Compressional Behavior It is obvious that the theoretical analysis of compressional behavior of fibrous assemblies is quite complex and not readily useful in practice, especially for real-time applications. Therefore, experimental studies of compressional behavior have been a significant part of the research efforts in this area in addition to theoretical analyses reported earlier. This section is the summary of the effort and reported results in this domain. Thickness is a very important characteristic of fabrics, but this depends on the amount by which the material is squeezed. Since the thickness of a fabric is a variable, the measured thickness is determined by the thick places, and the larger the presser foot the greater is the chance of including thick spots, and therefore the greater the average thickness. So in tests, a standard size of the presser foot should be used in order to compare the testing results. It should be remembered further that the cloth under the presser foot in compression test consists of threads that continue beyond it, so that an edge effect is to be expected. The method that might eliminate or minimize the effects due to irregularity is to measure the thickness in multiple layers. In multi layer test, the random variations are likely to compensate each other. However, this method introduces other arbitrary factors and edge effects become pronounced. There is no absolute definition of the “thickness”. For comparison, it is sufficient to adopt the simplest reasonable conditions of measurement on single layer of fabrics [70].