ANALYSIS OF BENDING RIGIDITY OF NONWOVEN FABRICS A THESIS Presented to The Faculty of the Division of Graduate Studies By- Wing Chi Lau In Partial Fulfillment of the Requirements for the Degree Master of Science in Textiles Georgia Institute of Technology August, 1976 ANALYSIS OF BENDING RIGIDITY OF NON-WOVEN FABRICS Approved; Jl L. /7 Aiiiad Tayebi, Chairman /c'j. 9, /^// L. Howard Olson Hvl^nd Chen Date Approved by Chairman 11 ACKNOWLEDGMENTS I would like to express my sincerest appreciation and gratitude to Dr, Amad Tayebi. Without his patience and guidance as my thesis advisor, this research would not have been possible. I am grateful to Dr. W. Denney Freeston, Jr., Director of the A. French Textile School, for providing the research assistantship which made my attendance of graduate school possible. I would also like to express my gratitude to Dr. L. Howard Olson for serving on my reading committee and providing valuable as­ sistance. I would also like to express my gratitude to Dr. Hyland Y. L. Chen for serving on my reading committee and providing valuable ad­ vice. Special thanks to my wife, for her patience and encouragement during ray year of graduate study. iii TABLE OF CONTENTS Page ACKNOWLEDGMENTS ii LIST OF TABLES iv LIST OF ILLUSTRATIONS v SUMMARY vii Chapter I. INTRODUCTION 1 II. THEORETICAL ANALYSIS 5 Point of Interest The Analysis Determination of Total Number of Fibers in a Unit Cell Bending Rigidity of Nonwoven Fabric with Complete Freedom of Relative Fiber Motion Bending Rigidity of Nonwoven Fabric with No Freedom of Fiber Motion Application of Theoretical Analysis to Nonwoven Fabrics with Different Fiber Orientation Distribution Functions III. EXPERIMENTAL VERIFICATION 35 Experimental Measurement of the Bending Rigidity of the Nonwoven Fabric Predicted Bending Rigidity of the Nonwoven Fabric IV. CONCLUSION AND RECOMMENDATION 44 BIBLIOGRAPHY 46 IV LIST OF TABLES Table Page 1. Bending Rigidity of the Nonwoven Fabric Measured Experimentally 36 2. Tensile Strength of the Nonwoven Fabric at Various Orientation Angle 38 3. Bending Rigidity of the Nonwoven Fabric with Complete Freedom of Relative Fiber Motion 40 4. Bending Rigidity of the Nonwoven Fabric with No Freedom of Relative Fiber Motion 42 LIST OF ILLUSTRATIONS Figure Page 1. Coordinates of Unit Cell 7 2. Different Orientation Angles and Fiber-to-Fiber Normal Distances of Fibers in the Unit Cell 9 3. Relationship between Fibers of Same Orientation Angle in the Unit Cell 10 4. Generalized Fiber Orientation Distribution Function <f> (B) 10 5. Deformation of a Nonwoven Fabric Segment in Bending 15 6. Strain ^j» of a Fiber with an Orientation Angle ^ 16 7. Structure of Reeraay (2.15 oz/sq.yd.) 19 8. Dimension of Unit Cell 21 9(a). Fiber Orientation Distribution Function—Random 23 9(b). Fiber Orientation Distribution Function-- Parallel Laid 24 9(c). Fiber Orientation Distribution Function--Cross Laid 25 9(d). Fiber Orientation Distribution Function—Ellipse 26 9 10(a). Bending Rigidity (lb-in~) with Special Fiber Orientation Distribution Function—Random (C.F.) 27 10(b). Bending Rigidity (Ib-in^ X 10"^) with Special Fiber Orientation Distribution Function--Parallel Laid (C.F.) 28 2 -5 10(c). Bending Rigidity (Ib-in X 10 ) with Special Fiber Orientation Distribution Function—Cross Laid (C.F.) 29 vi LIST OF ILLUSTRATIONS (Continued) Figure Page 1 -5 10(d). Bending Rigidity (Ib-in^ X 10 ) with Special Fiber Orientation Distribution Function-- Ellipse (C.F.) 30 2 11(a). Bending Rigidity (Ib-in ) with Special Fiber Orientation Distribution Function—Random (N.F.) 31 11(b). Bending Rigidity (Ib-in^ X 10"^) with Special Fiber Orientation Distribution Function-- Parallel Laid (N.F.) 32 2 -3 11(c). Bending Rigidity (Ib-in X 10 ) with Special Fiber Orientation Distribution Function- Cross Laid (N.F.) 33 2 -3 11(d). Bending Rigidity (Ib-in^ X 10 ) with Special Fiber Orientation Distribution Function- Ellipse (N.F.) 34 2 -3 12. Bending Rigidity (Ib-in X 10 ) Measured by Cantilever Bending Tester 37 13. Tensile Strength (lb) at Various Orientation Angle 39 14. Predicted Bending Rigidity (Ib-in^ X 10'^) from Actual Fiber Orientation Distribution Function (C.F.) 41 15. Predicted Bending Rigidity (Ib-in^ X lO""^) from Actual Fiber Orientation Distribution Function (N.F.) 43 vii SUMMARY Theoretical analysis of the bending rigidity of the nonwovens are carried out in this thesis. The effects of fiber diameter, fiber tensile modulus, fabric density, fabric thickness, and fiber orienta­ tion distribution function on the bending rigidity of the nonwovens are studied. A rectangular, planar unit cell model assuming complete freedom and no freedom of relative fiber motion are used in the ana­ lysis. For the case of complete freedom of relative fiber motion, moment equilibrium method is used and for the case of no freedom of relative fiber motion, energy approach is used. The bending rigidi­ ty is predicted for the spunbonded, Reemay (2.15 oz/sq. yd.), non- woven fabric. The predicted and experimentally observed bending ri­ gidities of the nonwoven fabric are compared. It is found that bend­ ing rigidity of the nonwoven fabric is accurately predicted by the rectangular unit cell model assuming no freedom of relative fiber mo­ tion. CHAPTER 1 INTRODUCTION Nonwoven fabrics noay be defined as structures produced by bond­ ing or the interlocking of fibers, or both, accomplished by mechanical, chemical, thermal or solvent means. The term nonwovens does not in­ clude fabrics which are woven, knitted, tufted or made by the wool felting processes. The basic difference between these nonwovens and woven fabrics is the greater bending rigidity of nonwovens. This is due to the dif­ ferent mechanical principles of deformations in the two structures. The present analysis of bending rigidity of the nonwovens will be based on the contribution of the fiber component rather than on the influence of the bond. Theoretical analysis of the effects of fiber diameter, fiber tensile modulus, fabric density, fabric thickness, and fiber orienta­ tion distribution function on the bending rigidity of nonwovens are carried out in this thesis. The bending rigidity in the two extreme cases of complete freedom and no freedom of individual fiber motion during fabric bending are analyzed. The predicted and experimentally observed bending rigidity of spunbonded Reemay (2.15 oz/sq. yd.), are compared. Also, the bending rigidity of the nonwoven is predicted for several fiber orientation distribution functions. Much has been claimed for the future of the nonwoven industry, but little has been known about the design criteria for the nonwovens. These analyses should provide a guide to effective utilization of fi­ bers in nonwovens. It is also shown here how bending rigidity in the two extreme cases of complete freedom and no freedom of individual fi­ ber motion during fabric bending can be predicted by a different ap­ proach. 1 Cox carried out an analysis on the effect of orientation of fi­ bers on the stiffness and strength of paper and other fibrous materials. It was shown that these effects might be represented completely by the first few coefficients of the distribution function for the fibers in respect of orientation, the first three Fourier coefficients for a pla­ nar matrix and the first fifteen spherical harmonics for a solid medium. He hypothesized that the behavior of a planar material tested at vari­ ous angles could be represented by a composition of sets of parallel fi­ bers in appropriate ratios. The analysis also showed that the effect of short fibers might be represented merely by use of a reduced value for their modulus of elasticity. 2 Backer and Petterson compared the aiechanical behavior of woven and nonwoven fabrics. In their study, they outlined the principal me­ chanisms by which woven fabrics deform and recover in manufacture and in end usage. They formulated parallel sequence of events in the distor­ tion process for nonwovens and showed how these actions influence the ability of nonwovens to fit smoothly on three-dimensionally curved sur­ faces, to hang and drape freely, and to recover from bends and tensile extension. It was demonstrated how knowledge of fiber properties and web structure can be used effectively to predict web behavior. o Abbot and Coplan"^ analyzed the fabric flexure, relating fabric behavior to fiber properties through the geometry of the structure, particularly as it is affected by fiber interactions. They developed equations for the flexural rigidity of the fabric, based on the degree of fiber association within the yarn and the effect of the yarn inter­ sections within the fabric in producing a cyclic fluctuation of rigidi­ ty along the length of the yarn. Petterson and Backer^ found out that the orthotropic theory de­ veloped for rigid materials predicted well the behavior of selected flexible textile-fiber nonwoven structures. The predicted properties agreed closely to the measured properties of the test samples. The orthotropic theory was useful in the characterization of nonwoven structures as engineering materials. Also the prediction of angular properties of nonwoven based on knowledge of principal-direction pro­ perties was made possible with this theory. Freeston and Piatt presented an analysis on the mechanisms de­ termining the bending rigidity of nonwovens. Equations were developed to predict the bending rigidity in the two extreme cases of complete freedom and no freedom of fiber motion during fabric bending. Effects of fiber orientation, fiber bending, and torsion were considered in the case of complete freedom of fiber motion. Effects of fiber orientation, tension, and compression were considered in the case of no freedom of fiber motion. Different methods of production were shown for making tex­ tile nonwovens more flexible. In this thesis, a rectangular, planar unit cell model assuming complete freedom and no freedom of relative fiber motion is used in the prediction of bending rigidity of the nonwoven fabric.