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I. uni-[:4], W- ' I" ‘ ..‘-‘MJ'I- 1".) } 30°42 II: Ifi'l' {31‘ [Tina \- ’ 14:9"7‘ r4 . 1‘ ’33:}. “I". 1""?! (4" 1:" I‘I'm . .54 . "VAL": ‘ f . | 13:") I‘: ILL-4,5450? ‘1')!“ lg}? "‘H H . '.?’<'IIS) ‘52,: It obi-Ia I.- . It") ' 1/";9 4 1, 314;. NFL,” 375.1,! 5",!» ’1.,/‘. I U A, Iii/W1Iii/[fiii”“I“mm:m l ’50; 5 ”W I 00578 9502 ”E‘s LIBRARY g Michigan State University This is to certify that the thesis entitled FINITE ELEMENT ANALYSIS ON STITCHING TECHNIQUES FOR COMPOSITE JOINING AND REINFORCEMENT presented by Chienhom Lee has been accepted towards fulfillment of the requirements for Master Engineering degree in Mechanics Major professor Date W 8, /?~§@ 0-7639 MS U is an Affirmative Action/Equal Opportunity Institution PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before due due. DATE DUE DATE DUE DATE DUE MSU Is An Affirmative Action/Equal Opportunity Institution FINITE ELEMENT ANALYSIS ON STITCHING TECHNIQUES FOR COMPOSITE JOINING AND REINFORCEMENT BY Chienhom Lee A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Metallurgy, Mechanics and Materials Science 1988 ABSTRACT FINITE ELEMENT ANALYSIS ON STITCHING TECHNIQUES FOR COMPOSITE JOINING AND REINFORCEMENT BY Chienhom Lee Stitching has been found to be an efficient way in improving the interlaminar strengths of composite laminates. Its application in the composite joining and through-the—thickness reinforcement has also been explored experimentally. In this study, numerical methods were used to simulate the stitching techniques and to verify the experimental results. Two displacement finite element programs were developed for stress calculations. One was based on the orthotropic plane-stress elasticity and plane truss theory and the other combined the three- dimensional, orthotropic elasticity and space truss theory. Numerical results showed that by increasing the stitching density and span and using the overlap joint in conjunction with stitching, the strength of a stitching,joint could be improve. In the study of through-the- thickness reinforcement, different stitching densities resulted in identical stress distributions. It is believed that stitching can improve the interlaminar strength rather than reduce the interlaminar stresses in the composite interfaces. In memory of my great-uncle, Yun-Chiu Pai, who died of cancer on October 27, 1988 iii ACKNOWLEDGMENTS I would like to express the deepest gratitude to my advisor, Dr. Dahsin Liu, for his true friendship, constantiencouragement, and professional criticisms made during my master's studies. Also, I wish to extend my sincere appreciation to Dr. Nicholas J. Altiero who gave me the basic and complete knowledge about finite element method. Furthermore, I would like to acknowledge the financial support from the State of Michigan Research Excellence/Economic Development Fund. Especial thanks are dedicated to my dear father and mother; without their support, nothing would have become reality; and to my lovely wife, Yufangn who shares her patience, understanding and endless love with me. In addition, I am deeply indebted to Dr. John S. Tu for his wise advice on life and computer facilities. Finally, I would like to thank my colleague, X. Lu, for his friendship, and Seongho I-Iong for his excellently experimental works which helped me a lot in this thesis. Mr. Hang is also my best friend. iv TABLE OF CONTENTS Page LIST OF TABLES .................................................... vii LIST OF FIGURES ................................................... viii CHAPTER 1 INTRODUCTION ........................................ 1 1.1 Background and Objective ............................ l 1.2 Composite Joining ................................... 2 1.3 Stitching Joint ..................................... 3 1.4 Literature Review ................................... 6 1.5 Organization of Thesis .............................. 7 CHAPTER 2 FORMULATION OF FINITE ELEMENT ANALYSES .............. 8 2.1 Orthotropic Plane-Stress Elasticity ................. 9 2.2 Three-Dimensional Elasticity ........................ 12 2.3 Two-Dimensional Finite Element Formulation .......... 16 2.4 Three-Dimensional Finite Element Formulation ........ 19 2.5 Plane Truss and Space Truss Theories ................ 23 2.6 Combination of Elasticity and Truss Theories ........ 24 CHAPTER 3 STITCHING JOINING ................................... 27 3.1 Finite Element Modelings ............................ 27 3.2 Numerical Results ................................... CHAPTER 4 STITCHING REINFORCEMENT ............................. 4.1 Finite Element Modeling ............................. 4.2 Numerical Results ................................... CHAPTER 5 CONCLUSIONS ......................................... APPENDIX A THE DEFINITION OF MATRIX [O] FOR THREE- DIMENSIONAL ORTHOTROPIC MATERIALS ................... LIST OF REFERENCES vi 29 34 34 35 39 9O 93 LIST OF TABLES Table Page 1 Experimental results for different stitching parameters .... 41 2 Mechanical properties of the composite and stitching thread ............................. 4S 3 Effect of stitching pattern on matrix and stitching thread ................................ 58 4 Effect of stitching pattern an overlap joint ............... 64 5 Mechanical properties of 3M's 1003 glass/epoxy ............. 65 vii LIST OF FIGURES Figure Page 1 The relationship between delamination area and stitching density ...................................... 42 2 The relationship between material principal coordinates and reference coordinates .................................. 43 3 Finite element plane stress plus plane truss analysis ...... 44 4 Finite element model ....................................... 46 5 Stitching parameters ....................................... 47 6 Finite element model for overlap joint ..................... 48 7 Three-dimensional finite element model for overlap joint (bold face lines represent truss elements) ........... 49 8 Normalized stresses vs. stitching density for zigzag stitching ............................... 50 9 Percentage of load bearing vs. stitching density for zigzag stitching ............................... 51 10 Normalized stresses vs. stitching density for V-type stitching ............................... S2 11 Percentage of load bearing vs. stitching density for V-type stitching ............................... 53 12 Normalized stresses vs. stitching span for zigzag stitching .................................. S4 13 Stress concentration around stitching points ............... 55 viii 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 3O 31 32 33 34 Normalized stresses vs. stitching span for square stitching .................................. Normalized stresses vs. stitching span for V-type stitching .................................. Normalized stresses vs. stiffness of stitching thread for zigzag stitching ................................ Peel stress distribution in different overlap lengths ...... Shear stress in different overlap lengths .................. Effect of stitching location on peel stress ................ Effect of stitching location on shear stress ............... Stitching pattern for different stitching densities ........ Three-dimensional finite element model for the study of through-the-thickness stitching ............... 0 XX O Y? 'xy rr 00 r0 rr 00 rd in in in in in in in in in in in in the top interface of an unstitched laminate ......... the the the the the the the the the the the top interface of top interface of bottom bottom bottom bottom bottom bottom bottom bottom bottom interface interface interface interface interface interface interface interface interface an an of of of of of of of of of unstitched laminate ... unstitched laminate ... an unstitched laminate a 4s-stitched laminate an unstitched laminate an unstitched laminate an unstitched laminate a 4s-stitched laminate a 4s-stitched laminate a 4s-stitched laminate ...... 56 57 59 60 61 62 63 66 67 68 69 7O 71 72 73 74 75 76 77 78 79 35 36 37 38 39 4O 41 42 43 44 22 T zr 20 22 zr 22 2! 20 22 zr in in in in in in in in in in the the the the the the the the the the bottom interface of an unstitched laminate ...... bottom interface of an unstitched laminate ...... bottom interface of an unstitched laminate ...... top interface of an unstitched laminate ......... top interface of an unstitched laminate ......... bottom interface of a 4s-stitched laminate ...... bottom interface of a 4s-stitched laminate ...... bottom interface of a 4s-stitched laminate ...... top interface of a 4s-stitched laminate ......... top interface of a 4s-stitched laminate ......... 80 81 82 83 84 85 86 87 88 89 CHAPTERl Introduction 1.1 Background and Objective Among the developments of modern materials, the fiber-reinforced polymer matrix composite materials have grown rapidly in the last three decades. Their high stiffness/weight and strength/weight ratios and excellent design flexibility are the primary reasons for their popularity in many structures and structural components in the automotive, marine, aerospace, and military industries. However, being different from the conventional metals, the composite materials are made up of at least two different phases of materials, namely fiber and matrix. Consequently, the behaviors of the composite materials are quite complicated and are still not completely understood. There are many questions needed to be answered in order to take full advantage of the composite materials. In this course of study, one of the unique characteristics of composite materials - stitching technique - was of major concern. Two topics related to the feasibility of using the stitching technique in strengthening composite structures were studied. One was to study the efficiency of stitching joining in composite structure assembly and the parameters involved in the optimum design of joining strength. And the other was to analyze the effect of through- the-thickness stitching on the stress distribution in composite laminates subjected to out-of-plane loading. Some preliminary testings regarding the stitching joining and stitching reinforcement were performed by some other investigators. The objectives of this study were to use numerical techniques to simulate stitching technique and to verify the experimental results. Two finite element programs were 1 developed for stress calculations. One was based on the orthotropic plane-stress elasticity and plane truss theory and the other combined the three-dimensional orthotropic elasticity and space truss theory. The former was used to examine the roles of stitching parameters in woven composite laminates subjected to in-plane tension while the latter was used to examine the stitching effect on the stress states of thin composite laminates subjected to out-of-plane loading. 1.2 Composite Joining The size of a composite structure can be small or very large. For a large structure such as the wing of an airplane, it is necessary to divide the structure into several components, not only for manufacturing convenience but also for different requirements for fiber orientations at different locations of the structure. These components are then connected by joints to form a complete structure. The purpose of a joint is to transfer loads from one component to another in a structure. Because of the discontinuities in.both geometry and material involved in the structure joining, the joining area is usually a potential weak spot in the structure and can dominate the durability of the structure. There are two major types of joint, namely mechanical joint and adhesive joint, used in structure joining. A mechanical joint is created by fastening the components with belts or rivets while an adhesive joint uses adhesive to bond the components together. The advantages and disadvantages of each type of joint can be found in many references, e.g. [1-3], and are briefly listed below, respectively. 1. Advantages: A. Mechanical joints a. can be used in composite laminates of any kinds of thickness, b. can be disassembled for repair or maintenance, c. require little or no surface preparation, d. can be inspected visually for joint integrity, and e. have no residual stress problem. B. Adhesive joints a. can have a more uniform load distribution than mechanical joints, b. do not damage the integrity of the composite components, c. are relatively lighter than mechanical joints, and d. can give gas-tight and liquid-tight seals for structures. 2. Disadvantages: A. Mechanical joints a. can discontinue the fibers and reduce the composite strength, b. can create high stress concentrations around the joints, a. can add weight to the structures, and d. may create a potential corrosion problem around the joints. B. Adhesive joints a. cannot be disassembled without damaging the structures, b. have difficulty in curing the adhesive when high temperature, pressure, and vacuum are required, c. are difficult to be inspected visually, d. have potential corrosion problem, and e. have residual stress problems when different materials are joined together. 1.3 Stitching Joint Being different from mechanical joint and adhesive joint, a third 'kind of joint, stitching joint, was examined in this study. A stitching joint can be used in composite laminates made of prepreg tape or woven fabric. Although it is more difficult to apply stitching in the prepreg tape than in the non-impregnated woven fabric due to the high viscosity of the matrix, the curing technique for a stitched tape is identical to that of an unstitched one. However, the manufacturing process for the stitched non-impregnated woven fabric is much complicated because of the difficulty in wetting the fibers by high viscous matrix. The regular procedures used in fabricating the stitched woven composite laminates are as follows: 1. Two pieces of fabric are stitched together by manual or stitching machine. 2. Matrix is then applied to the stitched composite fabric with the help of high pressure to eliminate air bubbles. 3. The impregnated composite fabric is then cured in an autoclave. There are two visual differences between the stitching joint and the other two types of joint. The mechanical joint and the adhesive joint can be performed after the composite laminates are cured, however, as mentioned above, the stitching joining has to be applied to a composite material before the matrix becomes solid. Another major difference is in the joining area. It is necessary to have a third lap of material or an overlap area in performing mechanical joint and adhesive joint. However, these are not necessary for stitching joint. This is one of the significant characteristics of stitching joining. Except for these differences, the stitching joint, from the load bearing viewpoint, is very similar to the mechanical joint and adhesive joint. The stitching thread in a stitching joint works like a rivet or a bolt of a mechanical joint while the matrix has the same function as that of adhesive. In summary, while compared with the mechanical joint and adhesive joint, a stitching joint has the following disadvantages: 1. The stitching joint has to be applied to a composite laminate before the matrix is cured. 2. 3. Similar to an adhesive joint, a stitching joint can not be disassembled. Although the stress concentration around a stitching point is not as high as that of a mechanical joint, the stitching point is still a potential spot for crack initiation. The disadvantages of stitching joint may limit its success in becoming one of the major techniques in composite structure assembly. However, there also exists some positive reasons which make it worthwhile to further investigate the feasibility of using the stitching technique in composite joining and reinforcement. 1. With the use of stitching joint, the integrity of a continuous composite material can be recovered to some extend. And a relatively uniform stress distribution around the joining area can be achieved. This is especially true when the stitching joint is used in woven fabric. . In a stitching joint, the composite materials and stitching threads have to be cured together. Consequently, an excellent smooth surface can be obtained. . In the absence of an overlap area or a third lap required for joining purpose, the stitching joint is superior to both mechanical joint and adhesive joint in.weight saving. It is also needed to point out that the stitching threads are much lighter than bolts and rivets used in the mechanical joints. . In addition to being a pure connector for composite joining, stitching can also be used to improve the interlaminar strength. Experimental results have revealed that through-the-thickness stitching can help to reduce the delaminations caused by free-edge effect [4] and impact loading [5]. 1.4 Literature Review In recent years, numerous investigations made contributions to the stress analyses of mechanical joint and adhesive joint for composite materials. However, only very few publications are available for the study of the stitching technique. Holt [6] was one of the pioneers to present the idea of using stitching in composite joining. Sawyer [7] performed some testings to study the strength of stitching joint under static and fatigue loadings. Based on his experimental results, he concluded that a chain stitching with sufficient overlap length could largely increase the strength of a single-lap joint. Liu, Kim, and Hong [8] also presented experimental results of some stitching joints with different stitching parameters. The summary of their experimental results is shown in Table 1. It is one of the objective of this study to use finite element analysis to simulate the stitching technique and to understand the role of each stitching parameter in the joining efficiency. Another major application of stitching is to reinforce the composite laminate. Mignery, Tan, and Sun [4] used stitching to improve the resistance of the free-edge delamination in a composite laminate. In addition to experiments, a finite element method based on plane strain and bar theory was used to study the interlaminar normal stress and strain energy release rate along the free edge. It was found that the introduction of stitching caused very little change in interlaminar normal stress. However, the strain energy release rate was reduced as the delamination crack approached the stitching line. Ogo [10] investigated experimentally the effects of stitching an in-plane and interlaminar properties. He concluded that stitching could slightly decrease the tensile and compressive moduli but significantly increase the fracture toughness of mode I. The stitching's ability to moderately improve the fracture toughness of mode II was also identified. Liu [5] used stitching as a through-the-thickness reinforcement for composite laminates subjected to low-velocity impact. Up to 40% of reduction in delamination area was reported. The relationship between delamination area and stitching density is shown in Figure 1. It is another objective of this research. to study the effect of stitching on the stress distribution by finite element analysis. 1.5 Organization of Thesis In this thesis, first of all, the studies on stitching joint are reviewed. The experimental results from Liu and his associates are also included for further comparison. Since finite element method is the major technique for this course of research, details of the formulations of two-dimensional and three-dimensional elasticity are presented in Chapter 2. However, only brief derivations regarding the plane truss and space truss theories and the combinations of the elasticity and truss theories are included in this chapter. Chapter 3 gives the numerical results of the study of stitching joining while those of stitching reinforcement go to Chapter 4. The comparisons between the experimental results and finite element analyses are also discussed in these two chapters. Chapter 5 summarizes the conclusions from both studies. Chapter 2 Formulations of Finite Element Analyses The objectives of this research were to use finite element methods to verify the experimental results of stitching joining and stitching reinforcement in composite laminates. As mentioned in Chapter 1, two- dimensional stitching was performed to study the efficiency of stitching joining while three-dimensional stitching was used to investigate the feasibility of stitching reinforcement. Consequently, both two- dimensional and three-dimensional finite element programs were needed for the stress analysis of stitching joining and stitching reinforcement, respectively. In studying stitching joining, glass fabrics of plane weave manufactured by J.P. Stevens was used with Maraset 658/558 epoxy for the fabrication of composite laminates. Because the fibers were aligned either parallel or perpendicular to the uniaxial loading directicur, the relationship between strain and displacement of the composite material was assumed to be linear. This linearity was also used in the study of stitching reinforcement in which crossply laminates made of 3M glass/epoxy prepreg tapes were used. For stitching threads, Mingrey, Tan, and Sun [4] used one-dimensional structure members - bars - to simulate their behaviors. Similar technique was also adopted in this research. Therefore, for two-dimensional stitching, plane truss theory was used for stitching thread while linear, orthotropic plane-stress elasticity was used for two-dimensional composite lamiantes. In three- dimensional stitching, however, space truss theory and three- dimensional, orthotropic, linear elasticity were used. The formulations of the finite element analyses are expressed in the following sections. 2.1 Orthotropic Plane-Stress Elasticity The equilibrium equations for plane-stress elasticity can be expressed as do + 81 + fx _ 0 3x 8y 6’31 + 3011 + f - O 3:: 8y 3' (1) where o and 'xy are two-dimensional stress components, and fx and , 0 xx Y? fy denote the body forces along x and y directions, respectively. Figure 2 shows a unidirectional composite lamina with a set of axes coincided with material principal directions, namely x1, x2, and x, axes. x, is an in-plane coordinate in the fiber direction. x5 is another in-plane coordinate and perpendicular to x1. x8 denotes the coordinate in the thickness direction. Figure 2 also shows a set of reference axes, at, y, and z axes, in which z-axis coincides with xs-axis. The angle between the material coordinates and reference coordinates, 9, is measured positive from x-axis to xl-axis counterclockwise. The constitutive equations for two-dimensional orthotropic materials can be expressed in material coordinate system as 011 Q11 Q12 0 ‘11 022 _ Q12 Q22 0 ‘22 712 o o Q66 112 where E11 Q11 ' 1 ' V12 V21 ”12 E22 Q12 ' 1 ' V12 V21 10 E22 1 ’ V12 V21 Q22 ' st ' G12 and c“, :22, and 112 denote the strain components. However, an orthotropic lamina is often constructed in such a manner that the material principal directions do not coincide with the referential directions. Therefore, in order to have the stress-strain relations in xy coordinate system, the coordinate transformations for stress and strain are required. The following equations are the transformed stress- strain relations Q11 Q12 Q18 ‘ xx _ _ xx ayy ' Q12 Q22 92s eyy (2) 'xy Q1: Q23 st ny in which Q11 - Qllcos‘o + 2(ng+2Q.,)sin20cos20 + szsin‘o 4 2 2 4 Q22 - Q1181n 9 + 2(Q12+ZQ..)sin aces 0 + szcos 9 2 2 4 4 Q33 - (Q11+Q22-2Q12-2066)sin does 9 + Q.°(sin 0+cos 0) 2 2 4 4 Q12 ' (Q11+Q22'4Qog)81n 9608 0 + 012(sin 0+cos 0) s s Q23 ' (Q11+Q12'4Q¢¢)81n 06080 + (Q12-Q22+2Q,,)cos Osina (Q11+Q12-4Q..)C088081n0 + (le-Q22+2Q..)sin30coso O ..a (I I In this course of study, linear strain-displacement relations are assumed. They are: e__&L xx ax __QY_ 3 en 8y H 7 __és_+_iv_ xy 3y 8x where u and v are the displacements in x and y directions, respectively. Substituting Equations (3) into Equations (2), the relations between stresses and displacements can be expressed as axx Q11 Q12 Q13 #35 0),), ' Q12 Q22 62s a}. (4) f 0 Q 0 .33.. + ..QY— xy 1s 23 as 6y 8x The boundary conditions of a two-dimensional surface can be specified either by essential boundary conditions v-v or by natural boundary conditions a n +1 11 -t xxx xyy x rn+an-t (5) where A n - (nx . ny); A A u and v are specified displacements in x and y directions, respectively; and A A tx and ty are specified boundary forces (tractions) in x and y directions , respectively . Substituting Equations (4) into Equations (1), the equilibrium equations become l2 §;[Qu%§+du§-§+Qu<§§+§§>1 + gamete“ ”n+0” Xuglyhgfin + fx - o (6) %;IQ1sax+st%§+st(g§+g§)I + g§IQ12gfi+Q22ay+Q2s(g§+%§)l + fy ' 0 In addition, by substituting Equations (4) into Equations (5), the natural boundary conditions can be expressed as [Q1 1g§+Q12g+Qxdfl+fl>1nx + [Q1sgil4'Qza'gT'st(fl+fl)]ny - t dy dx dy dx x [Q1agif‘I'Q2sa;"'st(gig-F3?)]Ilx + [0128x+0223y+023(g§+%§)]ny _ ty (7) Equations (6) are the governing differential equations for linear, orthotropic plane-stress elasticity and can be used in finite element formulation. 2.2 Three-Dimensional Elasticity By using the same fashion as described in Section 2.1, the equilibrium equations for a three-dimensional, linear, elastic body can be written as do dr dr _xx_+_xy_+_1£z_+3_o dx dy dz x dr do dr dx dy dz y dr dr do -—-zz- + ...!Z. + ___22_ + B - 0 dx dy dz 2 where o , o , o , r , r and r are the stress components, and B , xx yy 22 xy xz yz x By and B: are the body forces in x, y, and 2 directions, respectively. The stress-strain relations for an orthotropic body in xyz coordinates are expressed as l3 r axx ‘ . Q11 Q12 Q13 0 0 Q1s 1 r ‘xx 1 Uyy Q12 Q22 Q23 0 0 Q20 ‘yy 022 Q1: Q2: Q3: 0 0 Q30 ‘22 ‘ 'yz " 0 0 0 Q“ (2.; 0 J 1Y2 r (9) 'x2 0 0 0 Q45 Q55 0 7x2 L rxy J L Q16 Q23 Q30 0 0 Qse . L TxY , The definitions of 011...Q., are shown in Appendix A, and ‘xx’ cyy’ ‘zz’ 7 , 7x2 and 1 are the strain components. yz xy The strain-displacement relations for a three-dimensional, linear, elastic body are xx dx yz dz dy -i __a.u_ Jy. eyy dy 7x2 dz + dx (10) 22 dz xy dy dx in which u, v and w are displacements in x, y, and z directions, respectively. Substituting Equations (10) into Equations (9), the stresses in terms of displacements can be expressed as l4 'axx‘ "on 0.. (2.. o o c'z..‘ r 43;;— ‘ ayy (2.. a... a... o o (2.. 31;— an (2.. 62.. 62.. o o 62.. -§;L "y..." 0 0 o 62.. 62., o bah—g: (11> Ga 0 o o (2.5 Q... 0 -§§- + £1;- L rxy J L Q10 <32. (2.. 0 0 (2.. _ L-g-EYL + “'3:— J The boundary conditions of a three-dimensional body can be specified by either essential boundary conditions, i.e. A 1.1-1.1 < l 4 > > w-w or natural boundary conditions, i.e. on+rn+rn-t XY)’ fxynx + ayyny + ryznz - ty (12) fun): + 'yzny + 02er2 - tz where n-(nx , ny , nz) u, v and w are specified displacements in x, y and 2 directions, respectively; and A A A tx’ ty and t2 are specified boundary (traction) forces in x, y, and 2 directions, respectively. 15 Combining Equations (8) and Equations (11), the equilibrium equations can be expressed in terms of displacements u, v, and w -3;-tQ.,—§§- + Q12-%;'— 4» Qu-gf- + Q,,(—g-;— + ‘31—” + aleu-gfi- + (22,-?)5- + Q’s-gil— + Q..<-33- .31.” + (13a) ‘32—I°«s<-3‘,'-+ -3-"—> + Qs.(-§§-+ -%V—)] + Bx — o “mu—g:— + Q2173";L + (28...?!21. 4' st(_g'L+ ‘ngH + _:L[Q12.g1:_ + 022-3—3— + st'gg— 4, ch(.g3_+ 'gLH + (13b) _L.[Q“(_gzzz_ .331.) + Q“??? _L)] + By _ Q _%x_[Q¢s(_a%'+ i) 4‘ st(-%“2L+ "2L” + .:L_[Q“(.2L .22.) + Q“(_%szr.+ .22.” + (13c) dz [Qis_g':— + QzangyL + Qas-g§_ + Qaa(_g$y‘—+ “I‘m” + 32 " 0 Besides, by substituting Equations (11) into Equations (12), the natural boundary conditions can be written as [fin—3':— + (312.32va + Q13‘3Z— + 01.(-g§-+-gx-)]nx + [on—gi— + Q,,—§-;-’— + (236.32. + Q“(_a_u_ _av_+)]ny + (14a) [Q‘5( dz y) + st(a “')]nz ' tx 16 [filo-g:— + Qu‘fi'y— 4’ Qso’gz— + Qu(_g'L+ 'gy—an + [(212%- + Q22.g;yL. + (223% + Q“(.gia. _2v_+)]ny + (14b) [Q44(_gzL+ _%g—) + Q45('%L+ -gw—)]nz " t), [Q.s(-312L+-3-"—)+st<-3-§-+-§"—>Jn + [Qs.<-§JzL+ -§¥—>+Q.s(-§§-+ -§-"—>1n + (14°) [st-gi— + st'g—y— + (Eu-g":— + é“(-§L+ innz " tz Equations (13) are taken to be the governing differential equations of a three-dimensional, orthotropic, linear, elastic body and can be used for finite element formulation. 2.3 Two-Dimensional Finite Element Formulation The formulations of the finite element analyses in this research are precisely in line with the approach presented in Reference [9] . For the two-dimensional case, the formulation is based on the governing equations, Equations (6), to construct an associated variational form over an element as by multiplying the first equation with a test function VI and the second one with another test function V, and then integrating the results by part. Equations (6) then become 3w1 awl _ - _ 0 ' Joe {6—}: [Q11%§+Q12%+Q13(g§+ +2119] + FIQ13%:+Q23%§+Q33(%§+%)H<1XC1Y - I e w1{IQu§§+Q1-.~§;+le<3§+ +§§>Jn + Ile§§+stLy+st<§§+ +3;>Jn Ids - Joe wlfxdxdy (15a) 17 avg - _ - aw? - - - 0 " Joe {KIngimzsayQu(g§+gin "' ‘aT[Q12g§+szay+st(g§+g'§)]}dXdY ' Ire W2{ [éugi'i'ézsaf'ésdgfi'fiflxnnx + [szaxfizngzs(g§+g§)lny}ds - Joe wzfydxdy (1%) By further substituting boundary conditions (14) into Equations (15) , the above equations can be written as aw, _ _ 6"1 0 " Jae {K'IQ11g§+sz%"'Qu(g;+g§)l + gthugfiQasg§+6uu3 - av: 3.» av» a¢ _ a»; w _ 61,6 a¢ + (0125‘#+st3£&1+st3;‘3;1+st3# a?” 1‘1de J 18 - fre fiitxds + I 0e pifxdxdy (178) n _a¢a¢_a¢a¢_a¢a¢_a¢a¢ j§1 Joe I (Q1s-'1"'1 + Q23"1 "1 + Q12'_1 "1 + Q2s__1 -—1 8x 3x 8: 3y 6y 6x 8y 8y )uj _ 3¢ a¢ . a¢ a¢ _ 8v 3¢ - as aw + (Q285;1 5;1 + Q223;1 3:1 + (2225';1 3;1 + Q235§1 5;1)Vj ]dxdy - ffe ¢ityds + I'oe pifydxdy (17b) or it can be written in a short form as 11 12 1 [K ] { u } + [K ] { v } - ( f ) 21 22 2 [K]{u}+[K]{v}-lf} ”8) [K11] (i,j - 1,2) are so called stiffness matrices and are defined to be 11 - a¢ 3* _ 3‘ 3* - 6¢ 3* _ 3* a¢ K13 ' Jae (Q113;1'§;1 + Q135;1'3;1 + Q133;1'5;1 + Q335;1‘5;1)dxdy 12 , 3* 3* _ 3* 3* _ 3* 3* _ 3¢ 3¢ K11 ' Jge ((2125;1 5;1 + Q135;1'§;1 + Q223;1'3;1 + Qas§§1 3:1)dXdY av. aw: __i __19dxdy 8x 6x + 8’8x 6y + Q126y 6x + Q236y 6y K2:_J,Qm:fi 611911 31121 13 a 13 22 , 3* 3* - 3¢ 3* - 3* 3¢ - 3¢ 3$ K11 ' JE3 ((2233;1 5;1 + Qae§;1'§;1 + Q223§1'3§1 + Q223§1 5:1)dxdy 1 f1 - ffe pitxds + I 0e pifxdxdy 2 A fi - §Pe pityds + I 0e fiifydxdy Apparently, it can also be concluded from the above equations that 12 21 Kij - Kji Once the element equations, Equations (18), for every element in the domain of interest are obtained, by assembling the equations and 19 applying numerical techniques, the displacements u and v at the points of interest, namely nodal points, can be found. 2.4 Three-Dimensional Finite Element Formulation Using the same procedure as described in Section 2.3, the governing equations of a three-dimensional, linear, elastic body become 0- eta—Em h+Qn+Q ”H2 (23+ 333)] 0 6x 11ax 12ay 1332 1a :3 + —[Q1oax + Q2oay + Qsoaz + Qoo(%g+ 31)] as + h1lQ4sch+ g!) + st(g§+ g!)] ' 313x}dxdydz ' ffe 81{[Q11%: + 612%; + Q1333 + Q1o(gg+ 32)]nx + [Q1agfi + Q203§ + Q3032 + Qoo(%g+ 32)]ny * IQ.,(§§+ L) + (255%: +§§nn1ds (19a) 882 O ' I69 {8x 2lQ1s§§ + Q2e§§ + Qsegg + Qoo(gg+ 3;)] +382 —[Q12ax + Q22ay + Q2332 + Q2o(%u+ 21)] +:g, ' -z-’lo.1<§1+ 3;) + <‘2.;<§§+ 335)] - 323 mdydz ‘ ffe 82{[Q1og§ + Q2sg§ + Q2033 + Qan<§g+ %X)]nx + [Q12gi + 622%? + Q2sgg + Q20<§§+ +%X)Jny + IQ..(§;+ 53-) + (2.5%“ +§§Hn )ds (1%) 20 0 'If {Z—fK-[Quch'f i!) 4‘ Q55<§§+ Q!” +85 45—Q11<§3§+ay)+ Q11<§§ + 335)] 8g, _ _ _ _ . 4' 3:;[Q1sgfi 4’ Q2533,I + Qsag’j + Qua"; + g” ' gsBxdeydz ' fr 9 SHIQ45<§X+ g!) 4' (255(QB+ LZHUX + [Q11(§;+ ‘21) + Q11<§§+ ginn + [62113: + Q1123; + (2.1%: + Q..(§§+ 311112111 (19c) where g1, g2, and g, are test functions. Introducing Equations (14), Equations (19) become 681 0 "Jlne {-3—}! [Q11g§+Q12%+Q1sg§+Q1a(g§+ +31” :31 —[Q1o%: 4’ Q2113"; 4' Q3023: + Qu<fl;+ 3.)] +33 81 —[Q‘5 (21+ '3'?) 4” Q55(gJ:+ g.) ] )dxdydz - if ngxdxdydz - ire gltxds (20a) 632 0 " I03 {a—x 2[Q1o%§ "" Q2og; + Q3533: 4' Qu -* #0 SYMMETRIC (t) b u b J L J Finally, by inserting Ki?) (i,j — 1, 2, 3, and h) in the corresponding locations of K6,), the combined stiffness matrix consisting a plane 13 truss element and a plate element is 26 sauna» K93’+K$§’ .9,» .9.» .g.>..<:> KM? .5.» .9.» 1.5.9.1.“) 1.9;) K513) x§§>+x K§§)+K(t) K(p) RIP) 22 28 24 27 28 .99 .9.» .5.» use) .9,» K9.” x22) .5.» .21.” .9,» x2? SYMMETRIC K511») K(p) d The combined stiffness matrix for three-dimensional elasticity and space truss theory can also be obtained with the same manner. Chapter 3 Stitching Joining 3.1 Finite Element Modelings According to Reference [8], the composite beams tested were made of 12-ply woven fabrics. Each ply contained two pieces of plane-weave fabric stitched together at the center. The stitching threads were the fill or the warp of the woven fabric. The mechanical properties of the woven composite, the matrix, and the stitching thread are listed in Table 2. The modulus of the unidirectional composite, i.e. stitching thread, is not higher than that of woven composite. It is believed that this is due to the curvature of the fiber tow. In order to study the effects of different stitching parameters on joining strength, three types of finite element modelings were used in this study. They are listed as follows: 1. The objective of this study was to investigate the efficiencies of different stitching parameters in composite joining. Because the composite beams were very thin and the stitching was applied in every ply, plane-stress elasticity could be used in the finite element formulation. One of the finite element models used for stress analysis is shown in Figure 4. It contains three zones and stitching lines. The side zones, which have dimensions of 18.48 mm by 25.44 mm, are for woven composites. These zones are slightly larger than the largest stitching area used in the tests. Finite element analysis has verified that they are large enough to be exempted from end effect. The central zone represents for the boundary layer. It is made up of pure matrix. And 0.5 mm is an average length between the woven composites measured from teSts. This type of mesh was used to simulate a single ply (out of 12 27 28 plies) of the composite beams. In this study, nine-node isoparametric elements were used. Each node had two degrees of freedom, i.e. u and v. It was believed that this type of element could be used to simulate versatile stitching patterns. As mentioned in Chapter 2, the stitching lines were represented by either plane truss or space truss members. The ends of the members were then fixed at the nodes of the finite elements and were called stitching points. It was obvious that there was no interaction between the members and the elements except for that at the stitching points. It was then expected that the stress singularity could occur in the stitching points. With the use of the plate element and plane truss member, several stitching parameters such as stitching density, stitching span, stitching pattern, and the stiffness of stitching threads were studied. Details of these parameters are illustrated in Figure 5. 2. The stitching technique described above is an efficient joining method for woven composites. However, it may not be suitable for unidirectional prepreg tape in which overlap area in the joining region is required. In this study, the effect of overlap area on the joining strength for woven composites was also of interest. It was recognized that if the two pieces of fabrics at each ply overlapped each other and then were stitched within the overlap area, the dominant stresses of the stitching strength were the peel stress and shear stress at the edge of the overlap area. Consequently, another type of finite element model was required to study the interlaminar stresses. Figure 6 shows the side view of a single-lap joint in which the top and bottom zones represent for the woven composites while the central zone represents for pure matrix. The thickness of the central zone was assumed to be one-tenth of the thickness of the woven composite. Although each composite beam 29 contained 12 single-lap joints, only one joint was analyzed because of the complexity of the finite element modeling in the multi-layered composite beams. The finite element formulation of the single-lap modeling was identical to that described in the previous section. In this study, the effects caused by the change in overlap length and the distance between stitches were investigated. The definitions of these two parameters can also be found in Figure 6. 3. Compared with the two-dimensional models, a three-dimensional model may be more suitable for the analysis of interlaminar stresses. Figure 7 shows the three-dimensional finite element mesh for a single- lap joint. Eight-node isoparametric elements were used in the finite element formulation. The stress distributions in the unstitched, zigzag- stitched, and chain-stitched composite beams were examined. 3.2 Numerical Results From experimental observations, the failure of the stitching joints was always initiated by matrix cracking and then completed oy‘breakages of stitching threads. Accordingly, it is important to examine the stress states in matrix and stitching thread. Due to the short span of the pure matrix zone, the complexity of material mismatching close to the boundary of the zone, and the stress singularity caused by stitching thread, the stresses in matrix obtained from finite element analysis were not uniform. The stresses in the central part of matrix zone were of interest. They were averaged across the width of a composite beam. Similarly, the stresses in the stitching threads also changed slightly from one thread to another. They were averaged by the number of stitching line. Both the averaged stresses were then normalized by the stress applied to the composite beam for comparison. 30 The numerical studies of the stitching parameters can be summarized as follows: 1. Figure 8 shows that the stresses in a zigzag stitching change with the number of stitch per unit length, namely the stitching density. If the stitching length remains 11.3 mm, which was used in experimental study, and the stitching density increases, the stress in the matrix decreases while the stress in the inclined stitches increase. However, the reduction of the stress in the straight stitches are not significant. It is obvious that as the stitching density increases, the angle of the inclined stitching reduces and results in carrying more loads form the matrix. It then is clear that the stresses in the straight stitching and inclined stitching approach to each other as the stitching density increases. The percentages of load bearing of the matrix and stitching threads is also shown in Figure 9. The loading percentages of the stitching threads are pretty much parallel to the increase of volume fraction of fiber. Since the stitching threads are the major load carriers in composite joints, the increase in stitching density indicates the increase in joining strength. This result is consistent with the experimental observations shown in Table 1. In addition, it is also necessary to point out that the stitching threads in the zigzag stitching are not symmetric to the centerline of the composite beam. However, the non-zero in-plane bending moment did not cause significant non-uniform stresses in the stitching threads. A symmetrical V-type stitching was further examined and similar results are shown in Figures 10 and 11. 2. Another important stitching parameter is stitching span, which is defined to be the length between two stitching points. As shown in 31 Figure 12, with the density equals to nine stitches per inch, the stress in the straight stitches decrease while in the inclined stitches increase as the span of a zigzag stitching increases. It is apparent that the angle between the straight and the inclined stitches reduces when the span becomes longer. However, all the stresses in the three components remain constant when the stitching span is longer than 18 mm. It is believed that, in short stitching span, these three components can be highly affected by the stress concentration caused by the stitching point as shown in Figure 13. And 18 mm may be the minimum length to avoid the stress concentration effect for this type of composite material. Experimental results in Table 1 also show the same conclusion. And similar results for square and V-type stitching are shown in Figure 14 and Figure 15, respectively. 3. The study of stitching pattern, in fact, is a study of stitching density. Four different stitching patterns with the same stitching span were examined in this study, as shown in Figure 5. With the same number of stitching point, cross pattern has the highest stitching density followed by zigzag stitching. And square stitching has the lowest stitching density. Therefore, the joining strength of cross pattern should be higher than that of zigzag, which should be higher than that of square. The numerical calculations, Table 3, and the experimental results, Table 1, both indicate similar trend. 4. Because the stitching thread is the major load bearer in the joining area, the stiffness of stitching thread also plays an important role in joining strength. As shown in Figure 16, if the stitching density and span remain the same in a zigzag stitching, the increase of the stiffness in stitching thread can result in higher stress in the straight thread and lower stress in the matrix. This result implies the 32 advantage of using high-stiffness threads. However, it is also necessary to keep in mind that the bonding compatibility between the stitching thread and the matrix has to be considered. 5. As mentioned in a previous section, in order to examine the interlaminar stresses in an overlap area, a single-lap joint shown in Figure 6 should be used. The numerical results regarding the peel stresses and shear stresses at the Gaussian points in the matrix layer are shown in Figures 17 and 18, respectively. There are three sets of data points representing for three different overlap lengths in the diagrams. Apparently, the shorter the overlap length, the higher the peel stresses and shear stresses. Since the failure mode of the composite beams of overlap joining is the shear-out of the joint, the experimental results shown in Table 1 can be interpreted as the result of the high interlaminar stresses. In addition, Sawyer [7] has demonstrated experimentally that there exists a 'critical' overlap length beyond which additional overlap length will not decrease the peel stress. Moreover, it is also necessary to bear in mind that the increase in overlap length can cause the increase in structure weight and processing labor. 6. From Figures 17 and 18, the ends of the overlap area can cause high stress concentrations. In this study, the stitching technique was applied to the critical areas to improve the stress states. Figure 19 and Figure 20 show the numerical results for stitched single-lap joints. Both the peel stress and shear stress are reduced as the stitching gets close to the ends of the overlap area. 7. In addition to two-dimensional models, a three-dimensional finite element model was also used in the study of stress distribution. Three different cases were examined. They were: unstitched, zigzag-stitched, 33 and chain-stitched woven composites. The unstitched one was exactly of a single-lap joint while the stitched ones added stitching lines to it. However, the zigzag stitching had stitching lines along the axis of the composite beam while the chain stitching had the stitching lines perpendicular to the axial direction of the beam. The comparisons among the unstitched, zigzag-stitched, and the chain-stitched beams are shown in Table 4. There is no significant difference in stresses between the zigzag stitching and the unstitched one. However, the interlaminar stresses in the chain-stitched one are lower than those in the unstitched one. This result indicates the superiority of the chain stitching to the zigzag stitching in improving the stress state of an overlap joint. Experimental results also show the inefficiency of the zigzag stitching though no chain stitching has been performed. Chapter 4 Stitching Reinforcement 4.1 Finite Element Modeling This study was to use a three-dimensional finite element model to examine the effect of stitching density on the through-the-thickness reinforcement. The numerical work was to simulate a previous testing [5] on the delamination resistance of 3H glass/epoxy laminates subjected to low-velocity impact. However, a static force instead of a dynamic loading was used in this study. This was based on the conclusion of another previous investigation [10] which indicated that the behavior of a thin composite laminate under low-velocity impact was very similar to that caused by global bending. The stacking sequence of the composite laminates tested was [05/905 /Q ]. The mechanical properties of the laminates are listed in Table 5. Since the stitching threads were also obtained from the same composite prepreg tape, they have the same properties as those of a unidirectional lamina. The stitching patterns are illustrated in Figure 21. That is, the ls stitching has one net stitch per quarter inch square while the 28, 35, and 4s stitchings have two, three, and four net stitches per quarter inch square, respectively. And Os represents for the laminates with no stitching reinforcement. The finite element model used in this study is shown in Figure 22. There are five layers of element in the thickness direction. The odd layers represent for the unidirectional composite laminae, namely 0, 9O , and 0 layers. However, the even layers represent for the interfacial layers, i.e. pure matrix layers, which have thickness about 34 35 one-tenth of a composite lamina. In the finite element formulation, eight-node, isoparametric elements were chosen. There were three degrees of freedom, u, v, and w at each node. In addition, due to the geometrical symmetry, only one quarter of the laminate was examined. The boundary conditions can also be found in Figure 22. 4.2 Numerical Results The objective of this study was to look into the effect of stitching density on the delamination resistance. Since both interlaminar normal stress and interlaminar shear stresses are responsible for the delamination, the stresses on the interfacial layers are of major interest. In addition, because the composite laminates studied were square plates, a rectangular coordinate system was suitable for finite element modeling. However, since the concentrated force applied to the composite laminate is axisymmetric with respect to thickness direction, a cylindrical coordinate system would be significant for data presentation. Accordingly, the transformation laws were used to transform the numerical results from the rectangular coordinates to the cylindrical coordinates in some discussions. As shown in Figure 21, a stitch was performed right at a laminate's center, to which a concentrated force was also applied. Because of the coincidence of the force and stitching, the stress distribution around the stitching point was changed significantly. However, the overall stress state in the laminate was not influenced by the local disturbance. Therefore, in .the following discussions, focus will be exercised on the area close the central point of the laminate. l. The in-plane stress components axx’ ayy , and 'xy for top and bottom interfacial layers are shown in Figures 23 to 28. As expected, 36 the normal stresses on the top layer are mostly in compression while those on the bottom layer are mostly in tension. Due to the geometrical symmetry, the in-plane shear stresses are very small. The comparisons between the stresses on the top and bottom layers indicate that central loading can only influence the stress distribution locally. In addition, it is necessary to point out that the in-plane stresses are not the dominant stresses to cause delamination in the composite laminates. 2. The effect of stitching density on the in-plane stresses are of interest. Numerical results again show that the stitching effect is very localized. With the stitching pattern studied, the increase in stitching density does not cause any change in the stress distribution. That is, the stress distributions in ls, 28, 3s, and 4s stitchings are identical. 0 a 00’ and 7 r0 at the bottom layer of 0s and 4s stitching are rr’ presented in Figures 29 to 34. 3. Delamination is the major damage mode in composite laminates subjected to out-of-plane loadings such as low-velocity central impact and quasi-static central bending. Shown in Figures 35 to 37 are the interlaminar normal stress 022 and interlaminar shear stresses 'zr and r on the bottom interface of a [0/90/0] laminate. Because of the 20 axisymmetry of the concentrated force, 7 is very small. The 20 distribution of rzr has a peanut shape with long axis aligned in the 0° fiber direction. This result is consistent with the delamination measurements from the previous tests. In addition, it is noted that the negative 022 can help to improve the delamination resistance around the loading area to some extend. 37 4. The stress distributions of 022 and frz on the top interface are depicted in Figures 38 and 39. The overall distribution of ’zr at the top layer is similar to that of bottom layer though the former are slightly higher around the stitching point. The 022's on the top layer, especially close to the loading point, are very large in compression. It is believed that these high compressive stresses can significantly influence the interlaminar shear stresses and result in a much smaller delamination area than that in the bottom layer. 5. The effects of stitching on the interlaminar stresses at the bottom and top layers are shown in Figures 40 to 44. Comparing the stresses ”:2 and 'zr between Os and 4s, it shows that stitching only cause small change in the distribution of interlaminar shear stress. However, because of the coincidence between the force and the central stitching, the interlaminar normal stresses around the central point of the composite laminate are changed significantly. The compressive 022 at the central point of the top layer is reduced while that at the bottom layer is changed from negative to positive. These changes in 022 have potential in decreasing the delamination resistance and violate the experimental results which reveal that the delamination area can be reduced with the increase in stitching density. However, it is also necessary to bear in mind that, with the use of stitching, not only the stress state around the stitching point can be changed but also the interlaminar strength can be improved. Ogo [11] examined the effects of stitching on the interlaminar strengths. He concluded that stitching could increase the interlaminar normal strength up to 12 times greater 38 than the unstitched one while it only slightly improved the interlaminar shear strength. Chapter 5 Conc lus ions Some important conclusions from the studies of stitching joining and stitching reinforcement can be summarized in the following two sections: A. Stitching Joining: l. The strength of a stitching joint in a composite laminate can be increased by increasing the stitching density. 2. There exists a maximum length for stitching span beyond which the stitching strength cannot be improved. 3. The study of stitching pattern is similar to the study of stitching density. 4. The higher the stiffness of stitching thread, the higher the load bearing in the stitching thread. 5. The strength of an overlap joint is proportional to the overlap length. 6. In an overlap joint, the closer of a stitching applied to the ends of the overlap area, the lower the peel stress and shear stress. 7. Chain stitching is superior to that of unstitched and zigzag stitching in improving the joining strength. B. Stitching Reinforcement: 1. Both the in-plane and the interlaminar stresses on the top and bottom interfacial layers are very similar except in the neighborhood of loading point. 2. Different stitching densities result in identical in-plane and interlaminar stress distributions. 39 40 3. The distribution of interlaminar shear stress rzr at the bottom interface has a peanut shape which is very similar to the delamination shape observed in the tests. 4. Stitching can cause small change in interlaminar shear stresses but significant change in interlaminar normal stress, especially around the loading point. 5. It is believed that the significance of stitching reinforcement is in improving the interlaminar strength instead of reducing the interlaminar stresses in the composite interfaces. From the above study of stitching joining, it is concluded that the stitching joint is a feasible technique in composite joining. For woven fabrics, it is not necessary to have an overlap length in the joining area. Consequently, smooth joining and weight saving can be achieved. However, an overlap area is necessary in joining the unidirectional composite prepreg tape. In addition, due to the high viscosity of matrix, there is a higher possibility of damaging the fibers in the prepreg tape than in the woven fabrics. The efficiency of using stitching technique in improving the interlaminar strengths is also verified in the study of stitching reinforcement. However, it is also necessary to keep in mind that a stitching point can damage the fibers and results in high stress concentration around the stitching area. The technique of stitching reinforcement is feasible only if the improvement is higher than the damage. This may again suggest that the technique of stitching reinforcement may be better for woven fabrics. 41 Table 1 Experimental results for different stitching parameters Conditions Descriptions Ultimate Ultimate stress(HPa) stress per stitch(MPa) Densitya’e 5 173.80 34.76 7 190.00 27.14 9 212.00 23.56 b,e Span 11.29 212.00 23.56 16.93 261.70 29.08 22.58 267.40 29.71 Pattern square 174.40 19.37 zigzag,N-shape 212.00 23.56 cross 298.50 22.96 Overlap lengthb 5.64 169.00 -- ll.29 200.80 -- 16.93 263.30 -- Comparison stitching onlyc 212.00 -- overlap onlyd 263.30 -- stitchingc+ overlapd 258.70 -- a unit : No. of stitches per inch b unit : mm c 9 stitches per inch d overlap length - 16.93mm e pattern : zigzag NORMALIZED DELAMINATION AREA 42 1.06} O 8 C) ' C) I4Sb O 0.6 — O O 0 48a 0.4 *‘ 0.2 - J I l l J 0 l 2 3 4 5 NUMBER OF STITCH PER 0.25 in2 Figure 1 The relationship between delamination area and stitching density 43 Figure 2 The relationship between material principal coordinates and reference coordinates 44 P / f ::: ‘5. stitching y ’ V l/, line ’::: < plate / /// ¢ x, U :::j (a) Cantilever plate with stitching P node point 4 3 truss :1 'plate 1 2 (b) Finite element idealization Figure 3 Finite element plane stress plus plane truss analysis 45 Table 2 Mechanical properties of the composite and stitching thread Mechanical Properties Material E11098) E22(MP8) C12098) ”12 au(MPa) Matrix 3046.05 3046.05 1087.88 0.4 30.48 Maraset 658/558 Woven Composite 29429.34 29429.34 5265.39 0.145 405.40 Stitching Thread 28395.95 -- -- 0.335 538.50 46 __.‘ ..__ 0.5mm 25 .44 IIImleIIm 11111666111111 n— 13.43.. —u 18.4” —> stitching point stitching line Figure 4 Finite element model 47 stitching +7. span (b) Zigzag, V type (a) zigzag (7 stitches per inch) (c) Cross (d) Square overlap length (e) Overlap Figure 5 Stitching parameters 48 ‘0 ‘—1 I [i 1.6mm J 3.51- 13;“: FO+C+ 1.6mm 632:}? P = 150-N '"overlap length ‘1‘ l-—. distance -¢' between stitches Figure 6 Finite element model for overlap joint 49 |-— 8m —+—16.93mm+— 8mm 5' / (a) Dimensions and boundary conditions 1.6mm ¢¢/;nuzm' / 0111111111.... 1" V lulmmmalalwwzwww .mmmmmmll'wzwwW/ lmr.mmmzm.rz,z:W/ ‘7 II IIIII I ..zawwzwwwumnwnmuma-znuw/, Anmnmgggggwuzgl ___________ ””//€Z} "=ZEEE 41' flflllfllflfill!£gL!' (b) Zigzag stitching .mmmnar:zummnmnmumalmnzg .. ::..'.2':.5',zz,;;z%z”w ,l/ // lllll..'.'.'.'."' "nym’I’gngfiagw.fiamuw'zuunnm'zuunmnzv ,, lamnzmr' 'Aunmmmz'W/ I muur ..wzuyy .7 Am-zawwmr ..'...I'Il'llllllllll .mmmzawzzzumnmwmamawmnm¢ anmlrzuunmarzmnV'.Iz'zwmuw ”I -_:: =¢quw : AV ..lzrzzuuw .zuumumu Anzzzal'zzwuwnm0zflggumyv x? “Allmmaw' (c) Chain stitching Figure 7 Three-dimensional finite element model for overlap joint (bold face lines represent truss elements) 50 wcwnuuaum wwuwfiu you zuamcov wsfinuufium .m> mmmmmuum vmnwamsuoz m munwfim €05 L3 352% to .05 >2sz 02.185 on cu m. o. m o — p n b p — n u p P — p P p P F p n b L n. b b ”.0 x502 ole , 53:” 29.5.0. Te 1. :33» 85.2.. min 15.0 rmd 1.— 5.5.1: ” comm In; mouBN “ Eofiom r $83818 3'1ISNEIJ. OEZI‘WWBON 51 wcfinoufium wmuwfiu How muamcwv magnouaum .m> wawummn vwoa mo wwwucmuuwm m muswfim £9: .3 855m to .05 Emzmo 0258.5 mu ON mp r 0% L m _r n b b —lrb[}Plb DIE n PIP - IPL L- - .5 I 55» 35.2. T... goat. 2925 Ta 5.502 ole EEO: " coam 039». " Eaten. roé ONIGVO'T :lO BOVLNBOHBd 52 weanuufium wa%ul> How huamamv wawnoueum .m> mommmuum vmufiamauoz oH «woman 32: .8 358% to .05 Emzmo ozioEw mm ON 9 o— m c rp.Lpr.»pb|PL[.t{_I. PLLHLL .t x592 I ...ofim ale . T a! rad 1;; m/ EEO: ” coam f cab > .OONEN " Eofiom SSHELLS SHISNBl GBZI'IVWHON 53 wafinoufiuw mauul> wow haemamv wswnoufium .m> mcfiummn coca mo mwmuamoumm flu muawfim 205 ..3 $535 to .05 £628 ozioEm mm cm 9 o. m o 7-.P._...P....»—pprh_.p.b0.0 xtuoz olo :35 min u. r S I . \ [Nd \ 1 Inc no.0 E50: " comm . no.0 33 > 60%? H Eofiom a . r3 ONICIVO'1 :JO 39V1N3083d 54 moanoufium momma» no“ swam wcfinoufium .m> mommmuum voNHHmauoz NH muswwm AEEV 25m ozioEm mm cu m. o. o — n n b b — p p b n l— L [- p b n p - Noll. 5.6: I . 55» $295 I ..o.o :22” 85.2... mum . [Nd rte m” [W1 No.0 10!! l¢.l\||\\o s M”! L mad I... T Wu; 58523:» a ” £28 ...: mo~9~ " Eofiom um.— In; 88381.8 TIISNELL GBZI'WWEJON 55 19.08 636 9.24 28.22 Figure 13 Stress concentration around stitching points 56 wsfinoufium mumscm pom swam wGH£UuHum .m> mmmmouum wwNHHmauoz «a muswfim . AEEV z you zoom moanoufium .m> mommmuum vmufiamauoz ma madman AEEV z<..._m ozioEw mm om . mp . 1 0.9L m a P D D b .1 — I n b b b b - I b L 1.1 O I x30: Ola r N o cozum film 10.0 soc_\m£o§m m ” £88 cab > 630? " Eofiom 883818 3'1|8N3.l. G3Zl-IVW8ON 58 Table 3 Effect of stitching pattern on matrix and stitching thread Stitching pattern Normalized stress Square Zigzag Cross Matrix 0.80 0.75 0.68 Straight stitch 0.99 0.89 0.79 Inclined stitch -- 0.56 0.58 59 wsfisoufium wmumfiu How vmmusu wcanuufium mo mmocMMHum .m> mmmmouum vmufiamsuoz o~ muswfim .v 338 Emmi 02.185 to mmmztfi n N _. b ._ _ x50: 1. 53:» 292% T... not? 35.05 mlm Q1 101 ml fl 658.20% a H £28 ESQ: u comm moNEN " 5030... 0.0 ..INd “4.0 .... w... 10.9 IN... m... In; In. F ION 8838.18 3'H8N3i O3ZI'IVW8ON 6O mnuwsma amaum>o usouommfiv aw coausnfiuumfiv mmmuum doom EEonmd .Iaotgo 4 EENGNQ ..uaotgo o EEVana—Oto‘é n is ..stm 0.6 Nd! .. was de ....3 ...ms was j1: H..N... 1 ll 883818 133:! G3ZI'WW8ON 61 mauwcoa amHum>o uamummwfiv aw mommouum Hmonm 0H shaman o\x 04 0.0 0.0 P b p n b — 1P - - “00' 8 I00 4‘ d 4 ed 0 d 0 do: ¢qd¢”i..o D .u lmflnv ...-o D D fi 0 a sauna I¢0 rod on a a EEona.o_.laoto>0 4 j0.0 55337835 c . EE¢oéla0to>0 0 r0... 883818 8V3H8 C|3ZI"IVW8ON 62 on ammuum Homa so coaumooa waanuuwum mo nommmm ma muswfim 550 8.18.5 285mm mozfima _ ... ... m a . 855m 4 oo 352325 -... . 558.8165. 8:96 .3 13 190 <4 I . a 4 up 14.. Inc I: 883818 1338 O3ZI'IVIN8ON _. 63 ON mmwuum ummnm so coauwooa wsfinoufium mo uummmm ow wuswfim 550 818.5 zmmamm mozfima _ ... a m o . 855m 4 co 3:385 -... . EEnmdFflfimco. @2320 rad f ”Kevlar 1.. 19 lvd a T00 r90 883818 8V3H8 C13ZI'IVW8ON 64 Table 4 Effect of stitching pattern on overlap joint Stitching typea Unstitched Chainb Zigzagb Normalized 0.844 0.758 0.871 peel stress Normalized 0.444 0.426 0.454 shear stress a for overlap length - 16.93mm b distance between stitches - 16.53mm 65 Table 5 Mechanical properties of 3M's 1003 glass/epoxy Material Matrix Unidirectional Stitching Composites Mechanical Property Epoxy 3M's 1003 Thread Glass/Epoxy 811(MPa) 38600.0 4770.0 79950.0 E22(MPa) 8270.0 4770.0 79950.0 E33(MPa) 8270.0 4770.0 79950.0 012(MPa) 4140.0 1766.7 31730.0 G13(MPa) 4140.0 1766.7 31730.0 623(MPa) 3220.0 1766.7 31730.0 V12 0.26 0.35 0.20 ”13 0.26 ~ 0.35 0.20 u“ 0.28 0.35 0.20 H H---H—--. 18 ~o---~---‘ --H---#o-- “CC-H-.- --~ .-Iu—oo- /--—.--—.—.-. o———. /.——o--——‘ -.~--._ ---“--fioi --H-p -H---—---H-‘ 38 66 ---H --- -H--- .._.,/- .,__._ .. _. mi: --#- u- o H--. ~----~----Q H--~-fi-OH- ” 45 Figure 21 Stitching patterns for different stitching densities 67 ['76.2 mm+ 76.2mm<| \\\\\\\\\\\\\ 3 \ \ , \ 70.2mm § I f \ concentrated force i ' S \V """ W i 76.2mm \w \\\ \ \ J \ 31 \\\\\\\\\\\\\\\\ (a) boundary conditions (b) Three-dimensional finite element model of the shaded area of (a) Figure 22 Three-dimensional finite element model for the study of through-the-thickness stitching 68 USIGXOS (-8.1611 — 1.4024) . / 1 “9'9. 000 1 ¢ N . d 5030 - Q o- 9' 3! V/ G 0'00 25.40 50.80 76.20 Figure 23 axx in the top interface of an unstitched laminate 69 USIGYOS (-9.9131 - 0.9945) " \\ 1- \Qo\\~\_:———_— -O .8 -—-""""— 5030 - a M 0.4 ZSAO '- " V: O 0.8 0.x I I l I r I 0.00 25.40 50.80 76.20 Figure 24 ayy in the top interface of an unstitched laminate 70 UTUXYOS (-O.9885 — 0.4155) 76.20 25.40 - 0.00 0.00 Figure 25 f X 25.40 50.80 76.20 in the top interface of an unstitched laminate 71 BSIGXOS {-1.4062 — 3.3252) \ . __‘-‘ 76.20 50.80 >JIQ'9 / J .49 \ =__/ J». _./’ ' 030 I I I/S—' I I <;;::2 0.00 25.40 50.80 76.20 Figure 26 axx in the bottom interface of an unstitched laminate 72 BSIGYOS (—o.9992 — 4.8065) 76.20 \w -\1.4”__: l w ’0.2 K“; I— o’ I 25.40 - —0 6 --"‘ 0.00 l l l 4 l 0.00 25.40 50.80 76.20 Figure 27 ayy in the bottom interface of an unstitched laminate 73 BTUXYOS (-o.4197 — —o.0024) 76.20 / V 76.20 Figure 28 Txy in the bottom interface of an unstitched laminate 74 BSIGROSP(—0.0999/3.3436) §21fl ELB‘ 50.80 \ l 50.80 76.20 Figure 29 a r r in the bottom interface of an unstitched laminate 75 BSIGSOSP(O.6881 /4.7749) 76.20 U I \ M 3.9 I Q, .3 g: / '2 2? ‘ .... 2.7 2.4 \_____________.2.4 / ['0 2.7 f2" —"" F \1,8 f 1.8 -——— 7.5 1,2 \ I '2 $1.2 ‘— 50.80 1 50.80 76.20 Figure 30 a" in the bottom interface of an unstitched laminate 76 BTURSOSP(-0.0902/O.5868) 0/3 0 032/ '- o O-\ 50.80 4 50.80 76.20 Figure 31 r in the bottom interface of an unstitched laminate r0 77 BSIGR4SP(—0.0987/6.3149) 9\ 50.80\ 1 ‘ \Q9:\ \12‘ \24.——- \21‘ Eta—— \15\ 0.9 -——-—- 50.80 76.20 Figure 32 a r r in the bottom interface of a tut-stitched laminate 78 BSIGS4SP(O.6868/5.5162) 76.20 L 7&4: ...—— \ 3.3:; \§ 33:; ”v / // I 50.80 76.20 Figure 33 000 in the bottom interface of a Aa-stitched laminate 79 BTURS4SP(—0.0898/O.5980) 76.20 L 0.3 é 03v 0 / Q I \/ 05 ' o \I o O ‘— .80 l 50.80 76.20 Figure 34 fro in the bottom interface of a 1.3-stitched laminate 80 BSIGZOSP(-3.6179/O.2285) \. \Q ) .L °\ 0 o /\o/\o J °y./\ 5080 7620 Figure 35 azz in the bottom interface of an unstitched laminate 81 BTU RZOSP(O.3804/6.4961) 76.20 W 5 / 4.__..1 3 __._.a \ / ‘ —_-‘ l 7 W 50150 ' 50.80 76.20 Figure 36 'zr in the bottom interface of an unstitched laminate 82 BTUSZOSP(-O.9675/-0.0167) 7620 5030 7620 Figure 37 720 in the bottom interface of an unstitched laminate 83 TSIGZOSP(—20.0446/1.1204) 76.20 1 \\ ‘1 I \ '5. O '7 ~1.../\ 0 <7 0 on M 7.4 . SE 008 A 7.45E‘008 .\/———‘ 7.45E—008 M l 5080 76.20 Figure 38 a z in the top interface of an unstitched laminate 84 TTURZOSP(O.3777/9.9773) \Q XX; \W/I_ 50.80 4 50.80 76.20 Figure 39 'zr in the top interface of an unstitched laminate 85 BSIGZ4SP(—O.2620/3.5358) W Lo \/° \ b/’——/’ °\OmoA 00/__ ONO/\ok >( 5080 7020 Figure 40 oz: in the bottom interface of a hs-stitched laminate 86 BTU RZ4SP(O.3792/7.6956) 76.20 \k 5‘ 4 ...... 3 “...—J x 7 W 50 so ' ‘ ° 50.80 76.20 Figure 41 'zr in the bottom interface of a (Is-stitched laminate 87 BTUSZ4SP(—O.7662/—0.01 65) \-01 50.80 50.80 76.20 Figure 42 r in the bottom interface of a (Is-stitched laminate 88 TSIGZ4SP(—4.5720/O.3033) 7.... . LQ 508D £K180 7020 Figure I43 022 in the top interface of a 4s-stitched laminate 89 TTURZ4SP(O.3790/7.9841) \W/r .80 50.80 76.20 Figure 414 '2r in the top interface of a As-stitched laminate APPENDIX APPENDIX A The Definition of Matrix [0] for Three-Dimensional Orthotropic Materials in which 4 2 2 2 2 4 Q11 - C Q11 + 2C 8 (Q12 + 2Q..) + 408(0 01. + S Q,,) + S Q22 2 2 2 2 4 6 C S (Q11 + Q22 ' 4Qeo) ' 203(0 ' S )(Q1e ' Q23) + (C + S )Q12 0 ... N l 2 2 Q13 ' C Q13 + 3 Q2: + ZCSQse Q15 ' 0 2 2 2 2 2 2 2 2 2 2 C (C '33 )Q1e'CSIC Q11'S Q22'(C ‘3 )(Q12+2Qee)]+3 (3C '5 )Q26 0 ... O I 4 2 2 2 2 4 Q22 - C Q22 + 2C S (Q12 + 2Qee) - 4CS(C Q26 + S Q16) + S Q11 90 Q11 ‘ Q12 ' 91 2 2 - 0 ' Q33 - O 2 2 (C - S )Q36 + 2 C Q44 - 2CSQ‘5 2 2 (C - S )Q‘5 + - 0 2 ' C Q55 + 2CSQ45 - 0 2 2 C S (Q11 + Q22 coso sinfl 1 ' V23Vs2 E2 E3 A ”21 + V31V23 £2 E, A 2 2 2 2 2 2 2 2 2 2 C (C '33 )on'CSIS Qli'c Q22‘(C '5 )(Q12+2Q66)]+S (3C ’5 )le CS(Q23 ’ Q13) 2 + S Q55 CS(Q44 - Q55) 2 2 2 2 2 2 ' 2Q12) + 2CS(C -S )(Qze ' Q15) + (C ' S ) Q53 ”12 + V32V13 E1 Es A Q13 Q22 ' Q23 ' Q33 Q44 Q55 Q68 "31 + V21V32 92 ”13 + V12V23 1 ’ VlaVsI E1 E, A "32 + V12Vs1 ElEsA 1 ' V12V21 E1 E2 A " E1E2A "23 + V21V13 E1E2A 1 ' V12V21 ' V23Vs2 ' V31V13 ' 2V21”32V13 El E2 Es LIST OF REFERENCES 10. 11. LIST OF REFERENCES Mallick, P.K., - te ang_ng§1gn, Chapter 6, Marcel Dekker, Ins., New York, NY, 1988. Vinson, J. R. and Sierakowski, R. L., v S s , Chapter 8, Martinus Nijhoff Publishers, Dordrecht, 1987. Hoskin, B. C. and Baker, A. A., Editor, §2m29§1;g_fl§£grigl_jgg A1;g13flfl;_figzggjmuugi, Chapter 8, American Institute of Aeronautics and Astronautics, Inc. New York, NY, 1986. Mignery; L. A., Tan, T. M. and Sun, C. T:, "The Use of Stitching to Suppress Delamination in Laminated Composites," ASTM STP 876, pp.371-385, 1985. Idu, D., ”Delamination Resistance in Stitched and Unstitched Composite Plates Subjected to Impact Loading,” Proceeding of the 1987 Meeting, American Society for Composites, Newark, DE, September 22-25, 1987. Holt, D. J., ”Future Composite Aircraft Structures May be Sewn Together," Automotive Engineering, Vol.90, No.7, pp.46-49, 1982. Sawyer, J. W., "Effect of Stitching on the Strength of Bonded Composite Single Lap Joints," AIAA Journal, Vol.23, No.11, pp.l744— 1748, 1985. Liu, D., Kim, Y. C. and Kong, 8., "Stitching as Joint in woven Composite Plate," Proceedings of the Third Annual Conference on Advanced Composites, Detroit, MI, September 15-17, 1987. ReddY. J. N.. W. McGraw- Hill, Inc., 1984. Hong, S. and Liu, D., "On the Relationship Between Impact Energy and Delamination Area," Accepted by Experimental Mechanics. Ogo, Y., The Effect of Stitching on In-Plane and Interlaminar Properties of Carbon-Epoxy Fabric Laminates, M.S. Thesis, University of Delaware, Newark, DE, 1987. 93