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Effect of Translaminar Reinforcements and Hybridization on Damage Resistance and Tolerance of Composite Laminates

Permanent Link: http://ufdc.ufl.edu/UFE0044505/00001

Material Information

Title: Effect of Translaminar Reinforcements and Hybridization on Damage Resistance and Tolerance of Composite Laminates
Physical Description: 1 online resource (111 p.)
Language: english
Creator: Song, Min Cheol
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: composites -- damage -- fea -- hybridization -- translaminar
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The effects of translaminar reinforcements (TLR) and hybridization on impact damage resistance and tolerance of laminated composites were analyzed. The TLR increase the apparent fracture toughness of composite laminates and contribute to improved impact properties. Analytical and computational methods were used to investigate the damage of laminated composites with special focus on delamination. A non-dimensional analytical model for mode I delamination of z-pinned composites was developed and verified using finite element (FE) analysis. Then the analytical and FE models were compared with experimental results to evaluate the adequacy of the model in describing the role of translaminar reinforcements. The influence of the TLR on the apparent fracture toughness and bridging length was quantitatively investigated through parametric studies. The maximum allowable bridging force before inherent failure of the material was suggested as well. A 3D woven composite was chosen to study the effect of z-yarns on impact damage.  A detailed analysis was performed to understand the limitations of quasi-static analysis in low-velocity impact studies. The static equilibrium and shear stress evolution in an orthotropic material subjected to the quasi-static and the impact loadings with various speeds were compared. The maximum impact velocity for which static analyses are adequate was determined.  Short beam shear (SBS) test specimens of plain woven laminated composite and the 3D woven composites were analyzed for the purpose of evaluating the effect of z-yarns on delamination. The FE model considering both intralaminar and interlaminar fracture demonstrated proper damage patterns exhibited in the tests which initiated from the crack tips of transverse cracks. The z-yarns in the composites increase damage tolerance by interrupting crack propagation.  Finally, the advantage of hybridization of laminate composites was evaluated using the FE models. Hybrid composites reduce the maximum interlaminar shear stress in the beam-type specimens therefore enhance the damage resistance of composite laminates
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Min Cheol Song.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Sankar, Bhavani V.
Local: Co-adviser: Subhash, Ghatu.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044505:00001

Permanent Link: http://ufdc.ufl.edu/UFE0044505/00001

Material Information

Title: Effect of Translaminar Reinforcements and Hybridization on Damage Resistance and Tolerance of Composite Laminates
Physical Description: 1 online resource (111 p.)
Language: english
Creator: Song, Min Cheol
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: composites -- damage -- fea -- hybridization -- translaminar
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The effects of translaminar reinforcements (TLR) and hybridization on impact damage resistance and tolerance of laminated composites were analyzed. The TLR increase the apparent fracture toughness of composite laminates and contribute to improved impact properties. Analytical and computational methods were used to investigate the damage of laminated composites with special focus on delamination. A non-dimensional analytical model for mode I delamination of z-pinned composites was developed and verified using finite element (FE) analysis. Then the analytical and FE models were compared with experimental results to evaluate the adequacy of the model in describing the role of translaminar reinforcements. The influence of the TLR on the apparent fracture toughness and bridging length was quantitatively investigated through parametric studies. The maximum allowable bridging force before inherent failure of the material was suggested as well. A 3D woven composite was chosen to study the effect of z-yarns on impact damage.  A detailed analysis was performed to understand the limitations of quasi-static analysis in low-velocity impact studies. The static equilibrium and shear stress evolution in an orthotropic material subjected to the quasi-static and the impact loadings with various speeds were compared. The maximum impact velocity for which static analyses are adequate was determined.  Short beam shear (SBS) test specimens of plain woven laminated composite and the 3D woven composites were analyzed for the purpose of evaluating the effect of z-yarns on delamination. The FE model considering both intralaminar and interlaminar fracture demonstrated proper damage patterns exhibited in the tests which initiated from the crack tips of transverse cracks. The z-yarns in the composites increase damage tolerance by interrupting crack propagation.  Finally, the advantage of hybridization of laminate composites was evaluated using the FE models. Hybrid composites reduce the maximum interlaminar shear stress in the beam-type specimens therefore enhance the damage resistance of composite laminates
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Min Cheol Song.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Sankar, Bhavani V.
Local: Co-adviser: Subhash, Ghatu.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044505:00001


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1 EFFECT OF TRANSLAMINAR REINFORCEMENTS AND HYBRIDIZATION ON DAMAGE RESISTANCE AND TO LERANCE OF COMPOSITE LAMINATES By MIN CHEOL SONG A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2012

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2 2012 M in C heol S ong

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3 To my family

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4 ACKNOWLEDGMENTS I would like to express my deepest gratitude to my advisor Dr. Bhavani V. Sankar for his full support and excellent guidance through my years in graduate school. I would l ike to thank Dr. Ghatu Subhash for his supervision at my project meeting every week. I also would like to acknowledge my committee member s Dr. John J. Mecholsky Jr. and Dr. Peter G. Ifju for their advice on my research and dissertation. I would like to ac knowledge Dr. Chian Fong Yen at the U.S. Army Research Laboratory for the financial support for this project I would like to thank Dr. Timothy R. Walter and Dr. Madhwapati Prabhakar Rao for their discussion and many suggestion s I also thank my fellow gra duate students at the Center for Advanced Composites at the University of Florida I sincerely thank my parents for their love, support and encouragement. Finally I am very grateful to my beloved wife Jiyeon Sung my son Dongkun Song and my daughter Lily C haerin Song for their devotion and sacrifice.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 7 LIST OF FIGURES ................................ ................................ ................................ .......... 8 ABSTRACT ................................ ................................ ................................ ................... 11 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 13 Motivation ................................ ................................ ................................ ............... 13 Review of R esearch A pproach ................................ ................................ ............... 16 Representation of B ridging Z one ................................ ................................ ...... 17 Delamination M odeling using FEM ................................ ................................ ... 19 Application of the B ridging and D elamination M odel for a Z pinned C omposite ................................ ................................ ................................ ..... 21 Static I ndentation and L ow V elocity I mpact ................................ ...................... 22 Damage Behavior in Woven Composites during Short Beam Shear Test ........ 22 2 ANALYSIS OF MODE I DELAMINATION OF Z PINNED COMPOSITE USING A NON DIMENSIONAL ANALYTICAL MODEL ................................ ...................... 29 N on D imensional A nalytical M odel ................................ ................................ ......... 30 Verification ................................ ................................ ................................ .............. 35 A S ample P roblem for V erify ing the N on D imensional A nalytical M odel .......... 35 A FE S imulation for V erifying the N on D imensional A nalytical M odel .............. 37 Discussion of R esult s from the N on D imensional M odel ................................ ........ 39 Parametric S tudies using the N on D imensional A nalytical M odel .................... 39 Maximum A llowable T ranslaminar R e inforcement ................................ ........... 40 Effect of T ri L inear B ridging L aw ................................ ................................ ...... 42 Summary and C onclusions ................................ ................................ ..................... 43 3 A PPLICABILITY OF QUASI STATIC ANALYSIS FOR LOW VELOCITY IMPACT ................................ ................................ ................................ .................. 52 Finite E lement A nalys e s ................................ ................................ ......................... 53 Results ................................ ................................ ................................ .................... 54 4 ANALYSIS OF DELAMINATION IN 3D WOVEN COMPOSITES UNDER QUASI STATIC LOADING USING FINITE ELEMENTS ................................ ......... 59 Finite Element Model for 3D W oven C omposites ................................ ................... 59

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6 Geometry and B oundary C onditions ................................ ................................ 61 Damage M odel and M odeling S trategy ................................ ............................ 63 Mater ial P roperty and D ata S election ................................ ............................... 65 Finite Element M odel for 2D P lain W oven L aminates ................................ ............. 67 Results and D iscussions ................................ ................................ ......................... 68 3D W oven C omposites (SY3W and DY3W) ................................ ..................... 68 Plain W oven L aminated C omposites (2DPL) ................................ ................... 69 Summary an d Conclusion ................................ ................................ ....................... 72 5 DAMAGE RESISTANCE OF HYBRID COMPOSITES WITH FUNTIONALLY GRADED MATERIALS ................................ ................................ ........................... 83 Effect of H ybrid C omposites on R educing the M aximum S hear S tress .................. 84 Finite Element M odel for Maximum Shear Stress ................................ ............ 84 Results and Discussion ................................ ................................ .................... 85 Delamination B ehavior of H ybrid C omposites under a Quasi static Indentation ..... 86 Finite Element Model for Delamination ................................ ............................. 86 R esults ................................ ................................ ................................ ............. 87 Conclusions ................................ ................................ ................................ ............ 89 6 CONCLUSIONS AND FUTURE WORK ................................ ............................... 101 Conclusions ................................ ................................ ................................ .......... 101 R ecommendations for Future Work ................................ ................................ ...... 102 LIST OF REFERENCES ................................ ................................ ............................. 104 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 111

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7 LIST OF TABLES Table page 1 1 Summary of results obtained by various researchers and present study .......... 28 2 1 The various dimensions and properties of the DCB used in the numerical simulation ................................ ................................ ................................ ........... 51 3 1 Material properties of the DSBS specimen for FEA ................................ ............ 58 4 1 Dimen sions of geometric model of RVE ................................ ............................. 82 4 2 Material properties of the constituents the 3D woven composites ...................... 82 4 3 Elastic properties for plain woven composites ................................ .................... 82 4 4 Parameters for cohesive element in baseline composite model ......................... 82 5 1 Various hybrid composite laminates and FGM ................................ ................... 98 5 2 Elastic properties for composite materials in the FEM ................................ ...... 100

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8 LIST OF FIGURES Figure page 1 1 Schemati cs of composites reinforced by the TLR.. ................................ ............. 23 1 2 NASA and Boei ................................ .............. 24 1 3 Inserting of z pin ................................ ................................ ................................ 24 1 4 3D weaving machine ................................ ................................ .......................... 24 1 5 Schmatic s of 3D woven composites ................................ ................................ ... 25 1 6 Hybrid composites ................................ ................................ .............................. 25 1 7 Schematic of bridging zone developed during delamination ............................... 25 1 8 Bridging laws of TLR. ................................ ................................ ......................... 25 1 9 Spring model for bridging zone.. ................................ ................................ ......... 26 1 10 Traction separation law for the cohesive element. ................................ ............ 26 1 11 Quasi static SBS test setup ................................ ................................ ............... 26 1 12 Damages after SBS test ................................ ................................ .................... 27 2 1 Schematic of initia l geometry and development of bridging zone by z pins in the composite DCB ................................ ................................ ............................ 45 2 2 Idealization of bridging zone using a beam model ................................ ............. 45 2 3 Force displacement relation of the z pin. ................................ ............................ 46 2 4 Flowchart of the procedures for solving the non dimensional governing equation. ................................ ................................ ................................ ............. 46 2 5 DCB specimen reinforced by z pins ................................ ................................ ... 47 2 6 Load and crack length variation as a function of DCB deflection. ....................... 47 2 7 Bridging leng th and crack length as a function of DCB deflection. ..................... 48 2 8 Variation of apparent fracture toughness during crack propagation .................. 48 2 9 Traction separation law for the cohesive element. ................................ ............ 48 2 10 Cohesive and spring elements in the FE model of the DCB. .............................. 49

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9 2 11 Non dimensional steady state bridging le ngth. ................................ ................... 49 2 12 Apparent fracture toughness as a function of maximum friction force and interlaminar fracture toughness ................................ ................................ ......... 49 2 13 Comparison of strain with different loading types ................................ .............. 50 2 14 The tri linear bridging law. ................................ ................................ .................. 50 2 15 Load deflection curve from various bridging laws ................................ .............. 51 3 1 Short beam shear test ................................ ................................ ....................... 56 3 2 2D FE Model for the SBS test ................................ ................................ ............ 56 3 3 2D FE Model for the SBS test ................................ ................................ ............ 57 3 4 Load displacement curves at the indenter ................................ ......................... 57 3 5 Load displacement curves at the indenter ................................ ......................... 58 3 6 Shear stress profiles.. ................................ ................................ ......................... 58 4 1 3D woven composites for FE model.. ................................ ................................ 73 4 2 Geometric parameter of RVE ................................ ................................ ............ 73 4 3 Prediction of matrix cracks .. ................................ ................................ ............... 74 4 4 Traction separation law of cohesive element. ................................ ..................... 75 4 5 2D plain woven composite.. ................................ ................................ ................ 75 4 6 Implementation of the cohesive element on homogenized 2D plain woven composite ................................ ................................ ................................ .......... 75 4 7 Force displacem ent curves of the single/double z yarn 3D woven composites .. 76 4 8 Strain energy and damaged energy versus displacement curves of the SY3W and DY3W. ................................ ................................ ................................ ......... 76 4 9 Force displacement curves of the single and double z yarn 3D woven composites. ................................ ................................ ................................ ........ 77 4 10 Strain energy and damage energy versus displacement curves of the 2DPW. .. 77 4 1 1 Damage evolution of baseline composite Case I ................................ ................ 7 8 4 12 Damage evolution of baseline composite Case II. ................................ .............. 79

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10 4 13 Damage evolution of baseline c omposite Case III. ................................ ............. 80 4 14 Delamination patterns in the composites.. ................................ .......................... 81 5 1 FE model to determine the shear stress profile ................................ .................. 89 5 2 Shear stress profiles of the hybrid composites stiffened on the loading side ...... 90 5 3 Shear stress profiles of the hybrid composites stiffened on the free side ........... 90 5 4 Shear stress profiles of the hybrid composites stiffened on the top and bottom sides ................................ ................................ ................................ ....... 91 5 5 Shear stress profiles of homogeneous the FGMs ................................ ............... 91 5 6 The maximum shear stress of various composites ................................ ............. 92 5 7 The DSFGM specimen.. ................................ ................................ ..................... 92 5 8 Comparison of peak loads of various specimens with and without matrix cracks ................................ ................................ ................................ ................. 93 5 9 The GFC.. ................................ ................................ ................................ ........... 93 5 10 The CFC. ................................ ................................ ................................ ............ 93 5 11 The LS9Hybrid composite .. ................................ ................................ ................ 94 5 12 The LS18Hybrid composite .. ................................ ................................ .............. 94 5 13 Th e LS36Hybrid composite .. ................................ ................................ .............. 94 5 14 The LSFGM composite .. ................................ ................................ ..................... 95 5 15 The FS9Hybrid composite .. ................................ ................................ ................ 95 5 16 The FS18Hybrid composite .. ................................ ................................ .............. 95 5 17 The FS36Hybrid composite .. ................................ ................................ .............. 96 5 18 The FSFGM composite. ................................ ................................ .................... 96 5 19 The DS9Hybrid composite .. ................................ ................................ ................ 96 5 20 Load displacement curves .. ................................ ................................ ................ 97

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11 Abstract of Dissertation Presented to the Gra duate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy EFFECT OF TRANSLAMINAR REINFORCEMENTS AND HYBRIDIZATION ON DAMAGE RESISTANCE AND TOLERANCE OF COMPOSITE LAMINATES By Min C heol S ong December 2012 Chair: Bhavani V. Sankar Cochair: Ghatu Subhash Major: Mechanical Engineering The effect s of translaminar reinforcements (TLR) and hybridization on impact damage resistance and damage tolerance of laminated composites wer e analyzed The TLR increase the apparent fracture toughness of composite laminates and contribute to improved impact properties. A nalytical and computational methods were used to investigate the damage of laminated composites with special focus on delamin ation A n on dimensional analytical model for mode I delamination of z pinned composites was developed and verified using finite element (FE) analysis The analytical and F E models were compared with experimental results to evaluate the adequacy of the mod el in describing the role of translaminar reinforcements The influence of the TLR on the a pparent fracture toughness and bridging length was quantitatively investigated through parametric studies. The m aximum allowable bridging force before inherent failu re of the material was suggested as well. A 3D woven composite was chosen to study the effect of z yarns on impact damage A detailed analysis was performed to understand the limitations of quasi static analysis in low velocity impact studies. The static equilibrium and shear stress evolution

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12 in a beam made of orthotropic material subjected to quasi static and impact loading s under various impact speeds were compared The maximum impact velocity for which static analyses are adequate was determined S hort beam shear (SBS) test specimens of plain woven laminated composite and 3D woven composites w ere analyzed for the purpose of evaluating the effect of z yarns on delamination The FE model that consider ed both i ntralaminar and interlaminar fracture predict ed damage pattern s (transverse cracks and delaminations) observed in the tests The z yarns in the composites increase damage tolerance by i nterrupting crack propagation. Finally, t he advantage of hybridization of laminate composites w as evaluate d using th e FE models Some h ybrid composites reduc e the maximum interlaminar shear stress in beam type specimens and therefore enhance the damage resistance of composite laminates. Suggestions for continuing this study in the future are included at the end.

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13 CHAPTE R 1 I NTRODUCTION Motivation Composite materials are a combination of two or more materials with properties that cannot be achieved by any one of the constituent material s [Gibson 1994]. Composite laminates have been widely used in aerospace and automobile structures, sporting goods and military equipment because of their superior in plane stiffness and strength. However, fiber composites are anisotropic, showing relatively low stiffness and strength in directions other than the fiber direction. In the case of laminated composites delamination, i.e., separation of layers or plies, is one of the most significant damage modes, and is a key element of consideration in the design and operation of composite structures. Toughened resins can solve this problem to s ome extent. Rubber [ Kumar and Singh 2000, Yan et al. 2002] or thermoplastic toughened epoxy [Wong et al. 2010] is an examp le of such an approach. Recently n a n o paticles have been added to the resin phase to improve the fracture toughness [ Kalarikkal et al. 2006, Kim et al. 2008 Boesl et al. 2011 ]. However, the improvement in interlaminar fracture toughness due to the above mechanisms is not significant. Another method of improving interlaminar properties is the introduction of reinforcements in the thickne ss or z direction. These are called translaminar reinforcements (TLR) which seem to increase the fracture toughness and strength by an order of magnitude. Thus there seems to be a tremendous interest i n TLR for laminated composites. Stitching, z pinning a nd 3D weaving are some of the methods by which TLR can be provided in to composite laminates. Stitched composites can be manufactured by passing a sewing needle with a thread or yarn, looping or interlocking with bobbin

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14 thread, through laminate s ( Figure 1 1 A ) Several researchers have studied delamination resistance and damage tolerance of stitched composites under Mode I, Mode II and Mixed Mode loading [Sharma and Sankar 1997, Chen et al. 2005, Dransfield et al. 1998 Sankar and Zhu 2000, Jang and Sankar 20 05, Rys et al. 2010]. In general stitching reduces the in plane stiffness and strength properties [Dransfield 1994, Mouritz 1997, 2004]. Although stitching was found to reduce the impact damage resistance it significantly improved the impact damage toler ance [ Sharma and Sankar 1997 ] and compression after impact (CAI) [Farley et al. 1992]. Z pinning is a form of TLR similar to stitching in which rigid rods or pins are inserted through the thickness of the composite structure ( Figure 1 1B ) It is an alterna tive method to stitching and suitable for thick composites and sandwich structures [Wallace et al. 2001]. The z pinning process using an ultrasonic insertion device is efficient when pre pregs are used [Lander 2008]. Z pin increased the ultimate strength o f joints [ Byrd and Birman, Chang et al. 2006] while degrading in plane stiffness and strength [Mouritz 2007, Steeves 2006 Chang et al. 2007]. Increase in interlaminar fracture toughness in z pinned composites is due to the pull out mechanism of the pins a nd it depends only on the frictional and cohesive forces between the reinforcement and surrounding matrix material [ Cartie 2000, Cartie et al. 2004, 2006, Dai et al. 2004]. However, it is not as effective as stitching as the ends of the z pins are not anch ored as in stitching. Although stitching is a practical method of improving interlaminar properties, the reduction in in plane properties due to damage to the composite p re forms during the stitching process has become a matter of concern. At the same time developments in

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15 textile technology has lead to advanced weaving and braiding processes which can produce three dimensionally reinforced composites. Textile processes are well developed, fast and easy to implement. Complicated performs could be made using t extile processes where separate steps to implement TLR into 3D woven composites ( ( Figure 1 1C ) are not necessary. One single weaving process can texture warp yarns (0 degree tows) weft yarns (90 degree tows) and z yarns concurrently. Some of the 3D perfo rms are: 3D orthogonal weave, layer to layer interlock, and angle interlock ( Figure 1 5) From early on it has been recognized that 3D braided [Gong and Sankar 1991] and 3D woven composites [ Brandt et al. 1996, Byun et al. 1990, Walter et al. 2010] improve impact properties and damage tolerance of composites. However, there has been no systematic study to understand the effectiveness of various 3D architectures mentioned above on the impact properties of composites. This can be studied either by experimenta l testing by testing various woven architectures under various impact loading conditions or by simulating their behavior. The simulations could be performed using either analytical models or computational models. Analytical models are based on several si mplifying assumptions and hence tend to be approximate. However, they are useful in revealing the important physical mechanisms that are responsible for the improvements in the impact behavior. They are also useful in optimization studies which require tho usands of Monte Carlo simulations. Computational models use the finite element analysis. They tend to be more realistic. However they are expensive and time consuming. In this research we propose to use both analytical and computational models to understa nd the effect of TLR in various 3D woven composites.

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16 Hybrid composites also are capable of enhancing impact resistance [Wan g et al. 1991] and interlaminar fracture toughness [Hwang and Huang 1999]. Numerous combinations with a variety of fibers and resins are possible in improving composite properties ( Figure 1 6 ) The h ybrid composites using carbon and glass woven fabrics shows superior in plane properties [Pandya et al. 2011] and CAI [Naik et al. 2001] compared to glass woven fabric. And the carbon/glass hybrid composites ha ve increased failure strain [Stevanovic, 1992] C ombination of Kevlar fiber composites and c arbon fiber composites possess superior ballistic performance in armor applications [Grujicic et al. 2006] The present stud y is concerned with the effect of TLR on the damage resistance and damage tolerance of composite laminates. Damage resistance (specifically impact damage resistance in the present context) is the ability of the material to function without undergoing significant damage as exh ibited by the loss of stiffness or strength when subjected to a cert a in load. Damage tolerance pertains to material that is already damaged. Damage tolerance is the ability of the material that has sustained some damage to continue to perform and carry loa ds albeit with reduced stiffness of the structure. The advantages of hybrid composites b oth inter ply and intra ply hybrid composites and functionally graded c omposites will be evaluated Review of R esearch A pproach In the following we discuss the tech nical challenges in solving the proposed problem and the approach we will take to address those challenges. The major issues in the analytical model of TLR composites are to identify the interactions between the z reinforcement and the surrounding composit e and to represent them using appropriate mathematical models In computational model s,

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17 damage modes should be defined and appropriate numerical strategy to represent the damage needs to be developed. The way to model delamination between layers of composi tes is crucial since impact resistance and damage tolerance are significantly affected by delamination. At the same time intralaminar failure should be considered since this failure mode seems to interact with interlaminar delamination. In order to charact erize the effect of z yarn on impact resistance and damage tolerance, the mechanism of damage initiation and evolution will be investigated by simulation of the static short beam shear (SBS) test since static indentation tests are useful in understanding low velocity impact behavior. Experimental studies have indicated that damage mechanisms in static indentation and dynamic impact tests are similar and one can learn more from static tests about impact damage. Representation of B ridging Z one One of the big gest characteristics during delamination of composites reinforced by TLR is the development of the bridging zone denoted by length in Figure 1 7 The bridging zone can be regarded as a process zone where extra energy is dissipat ed due to the failure of TLR. In other words, the energy to create delamination in transversely reinforced composites is higher than that for original unreinforced composites. This increase in energy required increases the delamination resistance. This inc rease due to TLR can be quantified by comparing fracture toughness. The relationship between the two values of the fracture toughness that of reinforced composites with TLR and unreinforced composites is given by (1 1)

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18 wher e and respectively, are the apparent fracture toughness of reinforced composites with TLR and the fracture toughness of the original composite; and is the energy dissipation r ate due to the deformation and failure of TLR. So an analysis of the bridging zone can provide an understanding of the effect of TLR on delamination The behavior of the entire bridging zone must be characterized by failure mechanisms or a bridging law fo r each TLR. Experiments [Tan et al. 2010, Dai et al. 2004] have been conducted to measure the bridging law. It has been found that a simple force displacement relationship of the z reinforcement could describe the effect of TLR with sufficient accuracy. Mo reover bridging laws of TLR can be idealized as linear elastic and linear softening (linearly decreasing function) laws for stitching and z pinning, respectively. The use of spring element s is very adequate and efficient since their behavior can be define d by a force displacement relationship and makes it possible to simplify analysis. The s pring element s with these bridging laws are capable of being directly implemented into double cantilever beam specimens (DCB) or end notch flexure (ENF) specimens since the relative displacement at the two end nodes can be related to the deflection of beam. So the DCB and ENF specimens incorporating spring elements have been used for representing Mode I and Mode II fracture using analytical model or finite element analys is (FEA). Discrete spring model s and distributed traction model s are available for these applications. The d iscrete spring model s 2004] place spring elements discretely to represent TLR while the distributed traction model [Sankar an d Dharmapuri 1998, Sankar and Zhu 2000, Mabson and Deobald

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19 2000, Sridhar et al. 2002, Robinson and Das 2004, Byrd and Birman 2005 ] uses continuous elements obtained b y smearing the discrete forces. Delamination M odeling using FEM In order to demonstrate de lamination using FEM, J integral [Rice 1968] and virtual crack closure technique (VCCT) based on fracture mechanics [ Rybicki et al. 1977, Krueger 2002, Grassi and Zhang 2003] or the cohesive zone method (CZM) based on damage mechanics have been widely used The CZM has advantages in that it is not necessary to define a pre existing crack while the methods based on fracture mechanics require pre existing crack s with which strain energy release rate (SERR) can be calculated and used in a fracture criterion. H owever, The CZM is also related to fracture mechanics since dissipated energy of cohesive element s is the same as fracture toughness of the material, and this fact makes it possible to model delamination. The CZM was introduced by Barenblatt [Barenblatt 19 62] and Dugdale [Dugdale 1960]. The t raction separation law is a commonly adopted approach to describe behavior s of the cohesive zone. Several traction separation laws [ Tvergaard and Hutchinson 1992, Xu and Needleman 1994, Camacho and Ortiz 1996] were sugg ested. Among them, the b i linear traction separation laws [ Geubelle and Baylor 1998 ] and have extensively used for delamination problem s in Mode I, Mode II and Mixed mode problems. The b i linear traction separation law is depicted in Figure 1 10 and can be defined as

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20 (1 2) where is traction, is stiffness, is damage variable, is displacement, is the di splacement at damage initiation and is the final displacement [Davila et al. 2007] The damage behavior of the cohesive element incorporating the bi linear traction separation law can be defined by parameters cohesive strength, critical displacement and fracture toughness a mong which only t wo parameters are independent as the area under the stress displacement diagram should be equal to the fracture toughness. In addition to the above three parameters, the initial stiffness or t he penalty stiffness is also a crucial parameter that affects the compliance of the structure. The criteria for damage onset and propagation should be defined for progressive delamination modeling. Quadratic stress based failure criterion [Cui et al. 1992] given in Eq. ( 1 3) below has been used for delamination onset : (1 3) where and are the interlaminar normal and shear strength s Power law criterion [Wu and Reuter 1965] (Eq. 1 4) and B K criterion [Benzeggagh and Kenane1996] (Eq. 1 5) are well established for damage propagation

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21 (1 4) where is a material constant and (1 5) where is a material constant and Application of the B ridging and D elamination M odel for a Z pinned C omposite The FE models incorporating the cohesive element as well as the failure of TLR have been used for the simulation of progressive delamination behavior of DCB reinforced by TLR [ Ratcliffe and Krueger 2006, Dantuluri et al. 2007] The z pinned composite subjected to Mode I loading has been studied since the debonding of skin stiffener T jo int is one of the main issues for designers. T his analysis can provide direct insights in unders tanding the role of the TLR to damage tolerance by evaluating apparent fracture toughness. A summary of past work in this area is provided in Table 1 1 In the following we briefly describe the salient features of the work listed in that table. Cartie [ Cartie 2000 ] suggested a bi linear bridging law through experiments and conducted FE simulation using plain strain elements. Robinson and Das [ Robinson and Das 200 4 ], Mabson and Deobald [ Mabson and Deobald 2000 ] and Byrd and Birman [ Byrd and Birman 2005 ] used a [ Ratcliffe and 2004 ] proposed a discrete element analytical model. For delamination mod eling of z pinned DCB, Dantuluri et al. [ Dantuluri et al. 2007 ] used 2D cohesive elements

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22 incorporating beam elements and a bi linear bridging law. Ratcliffe and Krueger [ Ratcliffe and Krueger 2006 ] used 3D cohesive model with sol id elements. Grassi and Zh ang [ Grassi and Zhang 2003 ] used VCCT for delamination modeling. It is clear from the table that various researchers have used different approaches in modeling the stated problem. Static I ndentation and L ow V elocity I mpact Numerous studies have shown [Kwon and Sankar 1993, Nettles and Douglas 2000] that static indentation tests and simulations are useful in understanding low velocity impact behavior. As long as the mass of the impactor is higher than the target mass, beam mass in the present case, and the i mpact velocity is much less than the velocity of wave propagation in the target medium, the impact can be considered as low velocity impact. In such cases static tests and analyses provide good information on forces at which damage initiates and also the r esidual stiffness which is a measure of damage tolerance. Damage Behavior in Woven C omposites during S h ort Beam Shear T est SBS test is a test method used to determine interlaminar shear strength (ILSS). SBS tests for various 3D woven composites were conduc ted to investigate the effect of z yarns on ILSS and post damage behavior The test results revealed that delamination is a predominant damage mode and the damage tolerance became high as the volume fraction of z yarns increased However, with the increas ed amount of z yarns interlaminar shear strength decreased and the delamination s were contained leading to post damage behavior different from that of plain woven laminated composites [Walter et al. 2010].

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23 One of the significant attributes of damage evol ution was that matrix crack s occurred at the early stage of loading and induced interlaminar delamination. However, most matrix cracks developed inside of weft yarns (90 degree tows) as shown in Figure 1 12 where ellipses indicate the matrix phase This da mage pattern is a common feature in cross ply laminated composites [0/90], where transverse cracks initiate in 90 degree plies followed by delamination between 90 degree ply and 0 degree ply [ Geubelle and Baylor 1998, Ling et al. 2011] In fact each of the yarns in a 3D woven composite can be regarded as a unidirectional composite, which is weak in the transverse direction. Therefore the effect of parameters such as interlaminar shear strength, interlaminar fracture toughness and transverse strength of yarn s on damage initiation and evolution should be investigated. Figure 1 1 Schemati cs of composites reinforced by the TLR A ) Stitched B ) z pinned C ) 3D woven [Mouritz 2008]

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24 Figure 1 2 NASA and Boein Figure 1 3 Inserting of z pin [Cartie 2000, Mouritz 2007] Figure 1 4 3D weaving machine [Brandit 2001]

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25 Figure 1 5 Schmatic s of 3D woven composite s A ) 3D Orthogonal B ) Layer to layer interlock C ) Angle interlock [ Mahmood et al. 2011] Figure 1 6 Hybrid composites. A ) Intraply B ) Interply Figure 1 7 Schematic of bridging zone developed during delamination Figure 1 8 Bridging laws of TL R A ) Linearly increasing (stitching) B ) Linearly decreasing (z pinning)

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26 Figure 1 9 Spring model for bridging zone A ) discrete spring model B ) distributed traction model Figure 1 10 Traction separation law for the cohesive element Figure 1 1 1 Quasi static SBS test setup

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27 Figure 1 12 Damages after SBS test A) I n plain woven laminated composites B) I n 3D woven composites [Walter et al. 2010]

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28 Table 1 1. Summary of results obtained by various researchers and present study Reference Z p in in Analytic model Z pin model in FEM Crack propagation criterion in FEM Sub laminate model in FEM Comparison with experimental data Cartie 2000 Bi linear traction J integral Plane strain Yes Mabson and Deobald 2000 Distributed force (Non dimensiona l form 1) Byrd and Birman 2005 Distributed force (Dimensional form 1) Robinson and Das 2004 Distributed force (Dimensional form 2 ) Yes Grassi and Zhang 2003 Nonlinear spring ( B i linear function) VCCT Shell element Yes Ratcliffe 2004 Discrete force Yes Dantuluri et al 2007 Nonlinear spring ( B i linear function) 2D cohesive element Beam element Yes Ratcliffe and Krueger 2006 Nonlinear spring ( B i linear function) 3D cohesive element Solid element Present s tudy Distributed force (Non dimensional Form 2) Nonlinear spring (Linear softening function and tri linear bridging law) 3D cohesive element Shell element Yes

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29 CHAPTER 2 ANALYSIS OF MODE I D ELAMINATION OF Z PINNED COMPOSITE USI NG A NON DIMENSIONAL ANALYTI CAL MODEL Our goal here is to use both analytical and numerical approaches for studying the effect of z pins and compare them with available experimental data. In particular, we have developed a non dimensional model that will be useful in the design of t ranslaminar reinforce ments for a given application. In particular we have used a 3D shell model in conjunction with 3D cohesive elements. Both linear softening law s and more realistic tri linear bridging law s are used to model the z pins. In order to clari fy the role of z pins, a non dimensional analytical model is proposed. The solution of the non dimensional equation uses an iterative procedure. We have also derived an expression for the maximum density of z pins that can be allowed before the beam fails otherwise. The efficacy of the analytical model is verified by finite element simulation of the DCB specimen. We have used the example given in [ Cartie 2000 ] for this purpose. In the FE simulation, the ligaments of the DCB are modeled using shell elements. Cohesive elements are used to simulate the delamination and discrete nonlinear elements are used to model the z pins. The agreement between the analytical model and FE simulations is found to be excellent for various results such as load deflection, load crack length and effective fracture toughness. It was found that steady state bridging length and maximum apparent fracture toughness can be related to interlaminar fracture toughness of the composite and maximum frictional force supported by z pins. The r elationships among various parameters are non dimensionalized and the maximum pin friction that can be allowed before the composite beam itself fails is calculated. The non dimensional analytical

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30 model could be a useful design tool in selecting z pins for composite structures to improve interlaminar fracture toughness. N on D imensional A nalytical M odel Consider a z pinned composite DCB specimen of thickness 2 h with initial crack length of a 0 and subjected to Mode I loading as shown in Figure 2 1 A A pair of transverse forces F is applied at the tip of the DCB. Due to the applied transverse force F the initial crack tends to reach the current crack length denoted by a ( Figure 2 1 B ). When the crack reaches the region reinforced by z pins, a bridging zone of len gth c begins to develop ( Figure 2 1 C ). We can define an apparent crack length a p which is the length of the crack up to the beginning of the bridging zone. Current crack length ( a ) is the sum of the apparent crack length ( a p ) and the bridging length ( c ). W hen the z pins start to be completely pulled out of the composite as shown in Figure 2 1 D the bridging zone becomes fully developed and a new fracture surface is created in the z pinned zone. Therefore in the bridging zone the pull out of pins is partial. Beyond the current crack tip the pins are assumed to be intact. The relative deflection at the tip of the DCB is denoted by Figure 2 1 E Our goal is to determine the relationships among F a, c and Once the bridging zone length c is determined other parameters such as F a and can be found by solving the governing equations. In the following we describe th e non dimensional equation governing the bridging zone and procedures to determine the bridging length for a given load. We assume that the shear deformation is negligible and use Euler Bernoulli beam equations to model the ligaments of the DCB as shown in Figure 2 2 B We also assume that z pins are rigid and there is sufficient friction between the z pins and the composite

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31 material surrounding it. For simplicity, the relation between the frictional force ( f ) and pull out or slip distance ( d s ) is idealized as a linear softening function as shown in Figure 2 3. The validity of this assumption will be later verified in FE simulation. When the pins are intact they can exert a maximum friction force of f m As the pins pull out of the material, the loss of fricti on is proportional to the pullout distance d s When the pin is completely pulled out of the beam, the friction force reduces to zero. Thus the f d s relationship is given by (2 1) where h is half the thickness of DCB. Although the resistance offered by the pins on the beam is discreet, for the purpose of the analytical model we smear the discrete pin resistance as continuous distributed traction ( p ) acting on the crack surfaces as shown in Figure 2 2 C Then the traction can be d erived as (2 2) where N is z pin density expressed as number of z pins per unit area. The Euler Bernoulli beam equation for one of the ligaments, say upper ligament ( Figure 2 2 C ), of the DCB can be written as: (2 3) where b is the beam width. The effective bending rigidity of one of the ligaments of the DCB is represented by the term EI Note that the flexural rigidity for a laminated

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32 composite can be taken from the bending stiffness matrix in classica l plate laminate theory. S ubstituting for p from E q. ( 2 2 ) into Eq. ( 2 3) we obtain (2 4) The pullout length is equal to 2 w where w is the deflection of the top or bottom beam. Hence, the governing equation takes the form (2 5) The origin of the x coordinate is assumed to be the point at which the bridging zone begins. One should note that the origin moves as the crack propagates. The four boundary conditions (BCs) for th e bridging zone are: (2 6) In the above equation w 0 is a prescribed deflection at the origin of the coordinate system as shown in Figure 2 2 C This value will be less than h/2 in the beginning and will increase to a maximum valu e of h/2 as the DCB is loaded. Note that the slip distance of the z pin is equal to the total opening of the DCB at that location, i.e. 2w(x)=ds(x) As the crack propagates, the bridging zone will also move with the crack, but the crack opening displacemen t will remain as h at the origin with 2w(0)=ds(0)=h The terms V and M respectively, are the bending moment and transverse shear force on the beam cross section.

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33 One should note that the bridging length c is still an unknown. It can be determined from the fact that the strain energy release rate at the actual crack tip should be equal to the Mode I fracture toughness at the instant of crack propagation. The energy release rate can be determined from the equation derived by Sankar and Sonik [ Sankar and Soni k 1995 ] for beam like specimens containing delamination. (2 7) Thus the condition for determining c is (2 8) Before we solve the above equations we will non dimensionalize the equations and BCs appro priately. Normalizing the length dimensions by h and forces by Eh 2 the governing equation and the BCs take the following form: (2 9) (2 10) where and The equation for determining (Eq. ( 2 8)) takes the form (2 11)

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34 where the non dimensional fracture toughness is given by The solution for the governing Eq. ( 2 9) is (2 12) where The boundary condition at the point varies since deflection at this point increases from zero at the beginning of loading to 0.5 when the br idging zone is completely developed. Once the bridging zone is completely developed this value remains constant at 0.5 with additional increment of This is because the bridging zone is fully developed and it moves wit h the crack tip as it advances. The procedures to solve the above set of equations are shown in the flow chart depicted in Figure 2 4 The initial data includes the beam properties, characteristics of the z pins and the fracture toughness G IC The deflection at the b eginning of the bridging zone ( ) begins to i ncrease as the load is applied. When = 0.5, the bridging zone is fully developed and hereafter the deflection of the beginning of the bridging zone is constant. However, the apparent crack length increases during crack p ropagation. We need to use an iterative procedure as the bridging length ( ) is not known a priori. The strain energy release rate condition at the right end of the bridgi ng zone ( ) as given by Eq. ( 2 11) is then used to check for correct value of .When a given satisfies Eq. ( 2 11), then the procedure to determine is ter minated. After bridging zone is fully developed, bridging length corresponding to every increment of crack length can be determined.

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35 Verification A S ample P roblem for V erifying the N on D imensional A nalytical M odel In order to verify the analytical model a z pinned composites designed by Cartie [Cartie 2000 ] as shown in Figure 2 5 was selected. This z pinned composite was also used in many previous works [ Cartie 2000 Grassi et al. 2003 ]. The configurations and material properties are represented in Table 2 1 These values were used for both the analytical model and FE simulation presented in the next section. First, the load deflection curve ( Figure 2 6) was found for a p values ranging from 52.25 mm to 62.25 mm. Meanwhile the crack length as the summation of the apparent crack length ( a p ) and the bridging length ( c ) obtained from the procedure depicted in Figure 2 4 was computed. The variation of the bridging length as a function of DCB deflection is shown in Figure 2 7. The bridging length initially increases with loading until the bridging zone is fully developed, where the bridging length has the maximum value. In the beginning, the apparent crack length remains constant and the crack propagati on is only due to evolution of the bridging zone. Once the bridging zone is fully developed there is little change in the bridging length thus the crack propagation is almost due to the apparent crack length. In fact there is a slight decrease in the bridg ing length after it reaches the maximum as the bending moment due to the transverse loads applied is a function of the apparent crack length. If a pair of couple s is applied instead of a pair of forces as in the standard DCB test, then the bending moment w ill remain constant as the crack propagates, and one can see a steady state bridging length. The force F acting on the DCB and opening displacement at the end of the beam can be obtained using the following relations:

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36 (2 13) (2 14) The variations of transverse force and crack length with increments in opening displacement are shown in Figure 2 6. Initially the transverse force increases with opening displacement and begins to drop as interlaminar cra ck in unreinforced region propagates. However, the transverse force increases during the development of the bridging zone and decreases again with the movement of the fully developed bridging zone and the new crack surface. In order to estimate the appare nt fracture toughness during crack propagation we use the bending moment at the apparent crack tip as a measure of increased fracture toughness. This is similar to calculating the fracture toughness of the unreinforced beam using Eq. ( 2 7). Then the appare nt fracture toughness is defined as (2 15) The above relation can be non dimensionalized as : (2 16) The variation of apparent fracture toughness during crack propagation is shown in Figure 2 8 Th e a pparent fracture toughness increases during development of the bridging length since the bending moment required at the beginning of bridging zone to overcome the bridging force increases gradually.

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37 Maximum attainable apparent fracture toughness can be predicted by the relationship expressed in terms of interlaminar fracture toughness and bridging force due to z pins. The relationship is exactly the same as the equation based on energy balance. As the crack propagates it has to overcome the frictional fo rces in the z pins. The amount of extra work done is equal to the area under the load deflection diagram in Figure 2 3. Thus the maximum apparent fracture toughness can be derived as: (2 17) Multiplying throughout by (12/ Eh ) we obtain the above relation in a non dimensional form as: (2 18) This maximum value can be realized when a pair of couple s is applied instead of a pair of transverse forces. As shown in Figure 2 8 the apparent fracture toughness for transverse loading is always less than the maximum value. Furthermore, our definition of apparent fracture toughness in Eq. (2 16) is also slightly different from the energy based relation given in Eq. (2 18) making the apparent fracture toughness unde r transverse loading less than the maximum realizable value. This is useful in evaluating the increase in apparent fracture toughness for a given design of z pins. Moreover this value is related to strain exerted in the ligaments of DCB and will be discuss ed later. A FE S imulation for V erifying the N on D imensional A nalytical M odel For the sake of comparison with the analytical model, FE simulation of the same specimen using the finite element software, ABAQUS was performed. The FE model containing discret e z pins can verify the analytical model where z pins are smeared and

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38 represented by distributed traction. The three dimensional FE analysis was used to simulate crack propagation in the DCB specimen with z pins. The specimen was modeled using shell elemen ts (S4), and cohesive elements (COH3D8) were used to simulate progressive delamination. The behavior of cohesive elements can be characterized by a bi linear traction separation law ( Figure 2 9) [ Davila et al. 2007 ] given by ( 2 19) where is traction, K is stiffness, D is damage variable, d is displacement, d 0 is displacement at damage initiation and d f is final displacement. The above parameters were taken as: K =10 6 N/mm, m =35Mpa and G IC =0.258 N/mm [ Cartie 2000 Davila et al. 2007 ]. T he n onlinear spring element (CONN3D2) whose behavior is defined by a linearly decreasing force ( Figure 2 3) for each z pin was also implemented between the two ligaments of the DCB ( Figure 2 10) The load deflection curves and the variations of crack len gths with deflection are shown in Figure 2 6 Both the FE and analytical model results are presented with that from the experiment by Cartie [ Cartie 2000 ]. The agreement between the analytical model and the FE simulations is satisfactory for both load d efl ection and delamination length. The slight discrepancy in the initial slope of the load deflection curve between

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39 the analytical model and the FE model is due to the fact that the analytical model uses Euler Bernoulli beam theory whereas the FE model consid ered shear deformation as it occurs in test specimens. The good comparison between the analytical model and the results from FEA and experiments suggests that the discrete bridging force can be represented by a distributed traction. Discussion of R esults from the N on D imensional M odel Parametric S tudies using the N on D imensional A nalytical M odel Our goal was to study the effects of inherent interlaminar fracture toughness of the composite material and the non dimensional frictiona l force on (i) the maximum apparent fracture toughness and (ii) steady state bridging length Such relationships are extremely useful in the design process to evaluate the influe nce of design variables on performance of the composite. In this parametric study we assume that the DCB is loaded by end couples instead of transverse forces. This assures steady state crack propagation in the beam and the effect of increasing crack lengt h on the results is thus eliminated. In the simulations, is varied from 10 7 to10 4 and ranged from 10 10 (almost zero friction representing the case of a n unreinforced beam) to 0.1. Figure 2 11 shows t hat the steady state bridging length decreases with increasing and If the composite material is inherently tough, it will not allow a larger bridging length. Similarly a large friction force also will reduce the bridging length. The maximum apparent fracture toughness ( ) is another measure to evaluate the effect of translaminar reinforcement. Figure 2 12 shows that the maximum apparent

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40 fracture toughness, computed by Eq. ( 2 16) varies linearly with increasing and This result is in accordance with Eq. ( 2 18). Since the range of is wider than that of in r eality, is more sensitive to the value of In other words the apparent fracture toughness is dominated by translaminar reinforcement. From the results shown in Fig ure s 2 11 and 2 12, it is clear that hig her friction force exerted by the z pins provides enhanced fracture toughness and at the same time reduces the bridging length. The latter is important to maintain the stiffness of the structure for a larger bridging length leads to reduction in the stiffn ess of the structure. Maximum A llowable T ranslaminar R einforcement Even though large frictional force between the z pin and the surrounding matrix material is desirable for increased fracture toughness, a frictional force beyond a critical value will cause the beam to fail. The maximum normal strain in a beam cross section is given by (2 20) Note the strain is already non dimensional and the right hand side of the above equation can be written as (2 2 1) where is t he non dimensional curvature which has the maximum value at Using Eqs. ( 2 16) and ( 2 18) we obtain (2 22)

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41 Let us assume the allowable strain in the composite is given by Then (2 23) From the above equation one can derive (2 24) The above equation provides an upper limit on the z pin density which should be taken into consideration in the design of translaminar reinforcements. Thus the allowable density or maximum frictional force of z pins can be determined at a given geometry and material properties of a composite material. In order to verify the result for the maximum allowable t ranslaminar reinforcement given in Eq. ( 2 24), we performed finite element analysis of a specimen. The properties used were: [ Cartie 2000 ] The FE simulation was used to calculate the maximum strain in the ligaments of the DCB spe cimen at the instant of crack propagation, and it was compared with the u obtained from Eq. (2 24) Two loading cases, namely end couples and transverse forces, were considered. For the set of properties used Eq. (2 24) yields a maximum strain u = 4.510 3 Note that if then the maximum strain exceeds the ultimate strain. From the FE simulations we obtained max =4.52110 3 for the case of end couples and max =3.90610 3 for the case of transverse loading of the DCB ( Figure

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42 2 13) The results indicate the strains in the beam are higher for the moment loading compared to that due to the transverse loading. However, in a practical structure the delamination will experience a combination of shear and moment loadings, and hence the co nservative value should be used. That means the maximum allowable stitch density given in Eq. (2 24) should be used although it is applicable only to moment loading case, and hence conservative. Effect of T ri L inear B ridging L aw So far we have used the lin ear softening bridging law for developing the non dimensional analytical model and verifying the same using FEA. Although this bridging law is simple, it may not be realistic. Dai et al. [ Dai et al. 2004 ] performed pull out tests using z pins of various di ameters to determine the actual bridging law. They found that a high value of debonding force was reached before the debonding of the pins began. After the debonding was initiated, the pull out force dropped to a lower value before reducing linearly to zer o value as the pins were pulled out steadily against frictional force that seemed to have a constant coefficient of friction. They represented this pullout behavior by a tri linear bridging law as depicted in Figure 2 14. The tri linear behavior can be att ributed to an elastic deformation (from (0, 0) to (0.0185, 35.3) in Figure 2 14), debonding from surrounding matrix (from (0.0185, 35.3) to (0.17, 14.86) in Figure 2 14) and slip out of z pins (from (0.17, 14.86) to (1.6, 0) in Figure 2 14). The finite ele ment simulations of the DCB described above were repeated with the tri linear bridging law. We did not an attempt analytical solution as it will be very complicated due to the piecewise continuous bridging law. The numerical values for the tri linear law w ere chosen such that the area under the force displacement curve, which

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43 is related to the increase in fracture toughness, is equal to the linear softening law used in the earlier example. T he areas under the two force displacement curves in Figure 2 14 are equal to each other. The resulting load deflection diagram of the z pinned DCB specimen is shown in Figure 2 15. The load deflection behavior is almost identical to that obtained using the linear bridging law indicating that the apparent fracture toughnes s is same in both cases. It must be noted that the peak force in the tri linear case is almost twice as that of the linear softening law. The variations of crack length as a function of the DCB specimen opening (deflection) are also similar in both cases. This indicates that the details of the bridging law do not seem to affect the global behavior of the specimen as long as the energy dissipated by the pins is properly accounted for. Summary and C onclusions Mode I delamination propagation in DCB specimens containing z pins is studied. A simple analytical model based on a linear softening type bridging law for the z pins has been developed and suitable non dimensional parameters have been identified. The load deflection curve of the DCB specimen was calculat ed using the analytical model. It is seen that the bridging zone, wherein the pins are partially pulled out, develops as the crack propagates, but attains a steady state value. The length of the bridging zone is a function of the Mode I fracture toughness and the frictional force between the z pins and the surrounding material. An expression was derived for the apparent or effective fracture toughness values. Although increase in frictional force as the z pins increases the fracture toughness, there is an u pper limit to this friction as the DCB ligaments would break if the friction is very high. The limiting value of the pin friction is derived.

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44 The efficacy of the analytical model was evaluated by the simulation of the DCB specimen using finite element sim ulations. In the FE model the delamination propagation was simulated by cohesive elements and the z pins were modeled as discrete nonlinear elements. The results for the load deflection curve and the crack bridging zone length agreed quite well with the an alytical model. As an alternative to the linear bridging law, a more realistic tri linear bridging law was used in the FE simulations. It is found that the global delamination behavior of the specimen was not affected much as long as the energy dissipated by the pins is kept the same. The non dimensional model with few parameters will serve as a design tool when translaminar reinforcements such as z pins are selected for laminated composite structures in order to improve their fracture toughness. The analyt ical models will also be useful in optimization studies and simulation of large composite structures containing translaminar reinforcements.

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45 Figure 2 1 Schematic of initial geometry and development of bridging zone by z pins in the composite DCB F igure 2 2 Idealization of bridging zone using a beam model

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46 Figure 2 3 Force displacement relation of the z pin Figure 2 4 Flowchart of the procedures for solving the non dimensional governing equation

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47 Figure 2 5 DCB specimen reinforced by z pins [ Cartie 2000 ] Figure 2 6 Load and crack length variation as a function of DCB deflection

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48 Figure 2 7 Bridging length and crack length as a function of DCB deflection Figure 2 8 Variation of apparent fracture toughness during crack propag ation Figure 2 9 Traction separation law for the cohesive element

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49 Figure 2 10 Cohesive and spring elements in the FE model of the DCB Figure 2 11 Non dimensional steady state bridging length Figure 2 12 Apparent fracture toughness as a function of maximum friction force and interlaminar fracture toughness

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50 Figure 2 13 Comparison of strain with different loading types Figure 2 14 T he t ri linear bridging law is indicated by the solid line. The dotted line is the linear softening law in Figure 2 3. The areas under the force displacement diagram for both laws are the same.

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51 Figure 2 15 Load deflection curve from various bridging laws Table 2 1. The various dimensions and properties of the DCB use d in the numerical simulation [Carti e 2000 ] Dimension or Propert y Value B 20 mm H 1.6 mm E 1 138 Gpa E 2 11 GPa 0.34 G 12 4.4 GPa G Ic 258 N/m a 0 49 mm F m 18.43 N z pin density 0.5 % z pin diameter 0.28 mm

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52 CHAPTER 3 APPLICABILITY OF QUASI STATIC ANALYSIS FOR LOW VELOCITY IMPAC T Short Beam Shear (SBS) tests are widely used to characterize and quantif y impact resistance of the composites with various constituent materials and fiber architectures for both quasi static and dynamic loading THE SBS test is basically a three point be nd test with a much shorter span length The ratio of length to height of the specimen ( L/h ratio) is typically less than 10. The test designated as ASTM D2344 was originally designed for measuring interlaminar shear strength of laminated composites. Since it closely mimics low velocity impact response and damage, it has been used to measure impact resistance and impact damage tolerance of composite materials. The i n terlaminar shear strength (ILSS) is calculated from the maximum force P max before failure in it iate s in the specim e n The i nterlaminar shear strength is calculated as (3 1) The above equation is based on parabolic variation of transverse shear stress through the thickn ess of the beam. It is true only for homogeneous ma terials and also the cross section should be away from the loading region. The above equation which is based on static loading is also being used for understanding impact damage. It assumes that the specimen is in quasi static equilibrium and neglects any dynamic effects. However, the above assumption would be valid only below cer ta in impact velocity. Hence we perform finite element simulations of dynamic loading cases and calculate the detailed stress

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53 field and compare the results with corresponding stat ic solution in order to verify the validity of quasi static assumption for impact loading situations Finite E lement A nalys e s Equation ( 3 1) used in this study in conjunction with the measured impact force for estimating ILSS implicitly assumes that the s pecimen is in quasi static equilibrium neglecting any dynamic effects. In order to verify this assumption Finite Element Analysis (FEA) of the impact test was performed. The analysis was simplified by utilizing the symmetry of the specimen and therefore on ly one half of the SBS specimen was modeled as shown in Figure 3 2 In addition, the specimen was homogenized as an orthotropic elastic material with properties given in Table 3 1 [Xiao et al. 2 007 ] The use of fully elastic model removes any material rate dependency and focuses the study on the effect of inertia and wave mechanics. The FEA study included three rates of dynamic impact as well as quasi static loading to determine how loading rate affects the outcome of the analysis. The commercial FE software codes, Abaqus Standard and Abaqus Explicit were used for the quasi static and dynamic loading, respectively. The specimen was modeled using four node plane strain elements with reduced inte gration (CPE4R) and a thickness set at 20 mm. Approximately 32,000 elements were used for both static and dynamic cases. The indenter and the suppor t were modeled as rigid bodies. Dynamic loading was simulated using three different impact velocities, 11, 2 2 and 33 m/s. The maximum impact velocity in the experiments corresponds to the lowest rate, 11 m/s. The velocity profiles for the three impact simulations are depicted in Figure 3 3 The shape of the profiles was chosen such that they closely resemble th e

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54 experimentally measured profile discussed below. The duration of each profile was modeled to result in a final displacement of 2.5 mm. This displacement, which can be determined from the area under the v t diagram, is consistent with experimental results [Walter et al. 2012] Both the impact force (reaction force on the indenter) and the support reaction were calculated for each time increment. In the FEA the displacement of the rigid indenter was controlled by the velocity profile up to 2.5 mm under the displacement control. The time step for explicit analysis was calculated automatically by the FE program and it was in the order of 10 8 seconds. Results The impact force was determined from the reaction on the indenter. The impact force displacement curv es for various impact velocities are shown in Figure 3 4. The deviation of the dynamic loading curves from the quasi static results indicates the e ffects of inertia on the load displacement relation. From these curves it was observed that up to 22 m/s the effect of inertia on the results may be neglected. At 33 m/s this effect becomes much more significant reducing the accuracy of the results. It is clear from these curves that inertia limits the loading rates to approximately 22 m/s. Next the reaction for ce determined at the support is compared with the impact force at the indenter to verify quasi static equilibrium. The results for the each of the dynamic loading simulations are shown in Fig ure 3 5 (note that the results for quasi static loading were not analyzed as it is in static equilibrium). If the specimen is in equilibrium the reaction forces at the indenter and the total force at the supports should be equal. From Figure 3 5 A it can be noted that equilibrium is satisfied during impact at a velocity of 11 m/s. At 22 m/s impact velocity ( Figure 3 5 B ), the two forces varied at the beginning of impact event however after approximately 0.5

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55 mm of deflection, equilibrium is reached. It is observed from Figure 3 5 C that equilibrium is never fully satisfied w hen loaded at a rate of 33 m/s. Next the through thickness transverse shear stress profile was examined for the quasi static, 11 m/s and 22 m/s results. Since the results from 33 m/s impact were previously shown to be invalid, due to a state of non equilib rium, it was not analyzed for stresses These profiles were determined at a plane midway between the support and indenter (as shown in Fig ure 3 2 ). The purpose of this comparison is to evaluate the effect of loading rate on the response of the sample and c ompare the FEA profiles with theoretical profiles. The profiles were determined at a displacement of 1.5 mm and are presented in Figure 3 6. The theoretical profiles are determined using the indenter force and classical mechanics result for transverse shea r stresses where P /2 is the shear force, is first moment of area is moment of inertia and h is the thickness,) which is the basis for Eq. ( 3 1). The FEA profiles are shown as solid lines while the oretical profiles are shown as dashed lines. It is apparent from Figure 3 6 A that there is only a slight deviation between quasi static and dy namic shear stress profiles. This deviation is shown much more clearly in an enlarged view in Fig ure 3 6 B The the oretical results over predict the FEA shear stress (about 8% for quasi static and 11m/s impact velocity case, and 11% for 22m/s impact velocity case). The shear stress results for quasi static and 11 m/s impact velocity cases were quite similar; but the re sults for 22 m/s were about 3% higher than the quasi static results. As mentioned previously the actual shear stress is often much different then what is predicted using equation (1) and therefore is often used to compare different architectures. These res ults show that it is possible to

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56 compare results from static tests as well as dynamic tests at rates up to 11 m/s with little deviation and rates of 22 m/s with moderate error. From the above discussion it is clear that at an or below, impact velocity of 11 m/s the specimen may be considered to be in equilibrium, and the use of equation ( 3 1) for estimating the ILSS of the material could be considered valid. Figure 3 1 Short beam shear test Figure 3 2 2D FE Model for the SBS test

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57 Figure 3 3 2D FE Model for the SBS test Figure 3 4 Load displacement curves at the indenter

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58 Figure 3 5 Load displacement curves at the indenter Figure 3 6 Shear stress profiles. A ) Shear stress at the location of midplane between indenter and support B ) Magnified view Table 3 1 Material properties of the DSBS specimen for FE A [Xiao et al 2007] Material propert y Value D ensity ( k g/m 3 ) 2300 E 1 = E 2 (GPa) 27.5 E 3 (GPa) 11 .8 G 13 = G 23 (GPa) 2.14 G 12 (GPa) 2.9 13 = 23 0.4 12 0.11

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59 CHAPTER 4 ANALYSIS OF DELAMINA TI O N IN 3D WOVEN COMPOSITES UNDER QUASI STATIC LOADING USING FINITE ELEMENT S Delamination initiation and propagation in plain woven and 3D orthogonal woven composite laminates under three point bendin g were analyzed using f inite element analysis Both single and double z yarn 3D wo ven composites were considered. The models were guided by experimental observations of short beam shear (SBS) tests of specimens of same material systems. A series of mechani sms including creation of transverse matrix cracks and their interaction with delamination were modeled discretely The force displacement curves obtained from the FE simulation were compared with those from experiments. Further parametric studies were con ducted to understand the effect of z yarns in the 3D orthogonal woven composites on delamination The results from the FE simulations reveal that z yarns in the 3D woven composites can play a substantial role to impede the propagation of interlaminar crack s thus can remarkably enhance damage resistance and tolerance of the composite. Finite Element M odel for 3D W oven C omposites We selected three different types of specimens 3D woven composites with single z yarn and double z yarn (SY3W and DY3W) and 2D plain woven laminated composite (2DPL) for this study The effect of z yarns and inherent fracture toughness of the material on damage tolerance was investigated by comparing the SY3W and DY3W to the 2DPL specimen s and varying inherent interlaminar fract ure toughness of the 2DPL in a parametric study. It is well known that matrix cracks precede delamination when a laminated composite beam/plate is subjected to quasi static indentation or impact load. Walter et al. [ Walter et al. 2010 ] performed short bea m shear tests on 3D woven glass/epoxy

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60 composites to understand the damage initiation and propagation mechanisms under impact loads. The experiments showed that matrix cracks initiated in resin pockets between the z yarns and also in the weft yarns parallel to the y axis ( Figure 4 1 ). Optical micrographs also showed short delaminations and crushing failure beneath the indenter. However these failures were highly localized, but the delamination s emanating from matrix cracks significantly changed the global r esponse of the specimens In fact t he interaction of matrix cracks and delaminations is a common feature typically observed in cross ply laminated composites comprised of 0 and 90 layers. Woven composites, which consist of warp yarns (0) and weft yarns (90), are similar to cross ply laminates, and understanding of damage mechanism in the latter can be useful for woven composites. Although continuum damage models are able to capture the non linear behavior of composites in certain situations, they are no t suitable for the discrete characteristics of damage process observed in the present study [ Wisnom 2010 ] So the interaction between matrix cracks and delaminations was studied and used for modeling the damage discretely [ Hallett et al. 2008, Zhou et al. 2010 ]. Cohesive elements are widely used for modeling of delamination s and matrix crack s because of their versatility in dealing with damage initiation and evolution without defining any pre crack [ Nishikawa et al. 2011, Okabe et al. 2012 ]. Apart from dela mination which mostly occurs at the interface between layers, the location and direction of matrix crack may be arbitrary. The e xtended finite element method (X FEM) [ Belytschko et al. 2009 ] or the augmented finite element method (A FEM) [ Ling et al. 2009 ] has been used to deal with these arbitrary discontinuities due to cracks in the

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61 material system However it is difficult to predict exactly the creation of multiple crack s in complex material systems such as woven composites as observed in the experiment without considering uncertainties in geometric configurations, internal defects and material properties. Since our interest in the present study is focused on the investigation of delamination rather than exact prediction of the intra laminar damage attri butes in the given material system we use the results from experiments as a guide to predict the location of matrix cracks. The results from the SBS test s conducted by Walter et al. [ Walter et al. 2010 ] made it possible to predict the location of matrix c racks so that cohesive elements could be used in modeling of matrix cracks as well as delaminations T he potential locations for matrix cracks were assumed based on the SBS tests and found from principal stress directions for simplicity which will be desc ribed in detail in the following sections As a result of FE simulation, the force versus displacement at the indenter were obtained for the 3D woven composites and used to investigate the effect of z yarns as well as the interlaminar fracture toughness. T he relationship between damage patterns and the global response was fou nd and will be explained also. Geometry and B oundary C onditions The microstructure of the specimens and their boundary conditions in the FE models were similar to that in previous exper iments [Walter et al. 2010 ] The geometric parameters of the DY3W such as yarn dimension s and yarn spacing were determined from the micrographs of 3D orthogonal woven composite [ Walter et al. 201 0 ] (Table 4 1) As shown in Fig ures 4 1 and 4 2 the cross sec tion s of the yarns were treated as rectangles.

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62 Taking advantage of symmetry conditions, only a portion of the specimen is modeled. Due to symmetry about the yz plane passing through the mid span, only one half of the specimen in the x direction is modeled Since the specimen is assumed to be under plane strain normal to the y axis, one unit cell in the width direction ( y direction) should be sufficient. Further reduction can be made as the unit cell is symmetric about its mid plane parallel to the xz plane Thus only one half of the unit cell needs to be considered as shown in Figure 4 1 B The indenter and support of the SBS test setup were modeled as rigid bodies. T he span of the beam (distance between the supports) in the FE model was 40 mm. The thicknes s of z yarn ( t z ) in the SY3W was 0.4 mm while other geometric parameters remained the same as those in the DY3W including the configuration of crowns of z yarns in order to investigate the effect of the thickness of z yarn, hence the volume fraction of z y arns, on delamination of the 3D woven composite. Since the vertical part of z yarns play a significant role to interlaminar crack, this approach might be reasonable. In addition, the regions created by reducing the dimension of the z yarn were replaced wit h matrix since the micrograph from the experiments showed more matrix region in the SY3W [ Walter et al. 2010 ] Plane strain boundary conditions were assigned on the front and back surface of 3D woven composites ( x z planes in Figure 4 1) as well as the sym metry boundary conditions on the symmetry plane such that U x =0 (left y z plane in Figure 4 1). E ight node brick elements were used in modeling the yarns and matrix phase and eight node cohesive element s were used for damage modeling.

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63 Damage M odel and M odel ing S trategy D uring the beginning stage of SBS tests 3D woven composite specimens undergo elastic deformation. As the load increase s matrix cracks and delaminations occur in the specimen. The cohesive element associated with a bi linear traction separati on law was selected for modeling of both matrix crack s and delamination. This damage law enables the traction between two surfaces to be express ed in terms of relative dis placement and stiffness : where is the traction, is stiffness, is damage variable, is displacement, is the displacement at damage initiation and d f is the final displacement [ Camanho et al. 2003 ]. Quadratic stress based failure criterion and mixed mode strain energy release rate criterion were used for damage initiation and propagation respectively, f or both the transverse crack and delamination : where and are interfacial normal and shear strength and G Ic and G I I c are the Mode I and Mode II critical energy release rates Since the locations of cohesive

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64 elements which would act as potential crack paths should be defined a priori it is necessary to predict accurately the locations of expected cracks. The prediction for delamination is straight forward as it occurs between adjacent plies or layers, while that for matrix crack is complex Based on experimental observations [ Walter et al. 2010 ] several assumptions have been made to determine the locations of potential cracks for the efficient use of cohesive elements. The assumptions are as follows: i) Matrix cracks occur both within the fill tows and in the matrix pocket at the tensile bottom layer ii) Only a single matrix crack is created and allowed to evolve along the principal stress direction iii) The crack path is a straight line The above assumptions are consistent with experimenta l observations. iv) The effect of a matrix crack at the compressive top most layer on delamination can be negligible Formation of matrix cracks on the top layers of the beam needs special treatment. In the beginning of loading the top side is under compressio n and it is observed no matrix cracks form. However, after delaminations initiate, there is a redistribution of stresses and the delaminated top layers develop tensile stresses and they become sites for initiation of matrix cracks. These locations are iden tified from a preliminary FE analysis To begin with, the principal directions at the centers of fill tows in the tensile region using the specimen witho ut any damage ( Figure 4 3 A ) were found and used for further simulation to seek the principal stress dir ection over the entire region as seen in Figure 4 3 B In the following FE model, other cohesive elements for delaminations were

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65 placed between layers with the cohesive element for matrix cracks in the tensile region placed along the principal direction fou nd in the previous step, thus the principal directions in the compressive region after delamination c ould be found. In the second FE analysis using the specimen of Figure 4 3 B the load was applied until the moment that delamination occurred so that redistr ibuted stress filed could be obtained. In order to determine the principal stress directions in the weft yarns, stress values over the all elements of the weft yarns were checked and appropriate principal stress directions incurring tensile failure were ch osen. The principal directions determined are shown in Figure 4 3. Note that the angles shown in Figure 4 3 are measured from the z axis As a result of consecutive finite element simulation s the f inal configuration of Figure 4 3 C could be obtained and be came the final FE model for the study of delamination damage of the 3D woven composite materials. Material P roperty and D ata S election Material properties of the 3D woven composites used in the analyses are listed in Table 4 2. Elastic properties of the y arns were computed using the micromechanics formulas developed by Chamis et al. [ Chamis et al. 2007 ] The material system was S2 glass fiber / SC 15 epoxy with 62% fiber volume fraction. The strength and critical energy release rate values for the cohesive elements were selected from the literature The properties of cohesive element s for matrix crack and delamination were assumed to be same. A s Mode II shear strength value for the cohesive element s the apparent ILSS value reported by Walter et al. [ Walter et al. 2010 ] was chosen ( Table 4 2) This value was much lower than the range, 55 to103 MPa given by AGY manufacturer of S2 glass yarn, [ AGY ]. The ILSS could be much lower due to the voids and the type of binder

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66 used in the manufacturing process [ Wisnom e t al. 1996 Tanoglu et al. 2001 ]. T he specimens test by Walter et al. were manufactured using vacuum assisted resin transfer molding ( VARTM ) process [ Walter et al. 2010 ], the ILSS value could be lower because of inherent defects such as voids involved duri ng VARTM process. Based on the Mode II shear strength determined for the cohesive element, a Mode I normal strength value was estimated appropriately as listed in Table 4 2 In addition, the critical energy value for the cohesive element could be determined using available sources such as a Mode II fracture toughness value of quasi isotropic and cross ply composite with SC 15 reported by Huang [ Huang 2008 ], which was about 2000 N/m, as well as a Mode I fracture toughness of SC 15 epoxy provided by Applied Po leramic, the provider of SC 15 epoxy, which was 1000 N/m [ A pplied P oleramic ]. A bi linear traction separatio n law controlling the behavior of the cohesive element can be depicted by a triangle as shown in Fig ure 4 4. If the critical displacement denoted b y C in Figure 4 4 remains the same, the critical energy will be changed (from the area of OBD to the area of OAD ) with the change of the strength (from B to A ). This approach can explain the fact that fracture toughness tends to decrease due to t he lowered strength [ Madhukar et al. 1992 ]. Otherwise the lowered strength value can enhance damage resistance by increasing the critical distance of cohesive element under the assumption of same fracture toughness value (from the area of OBD to the area of OAE ) and might not be practical for this study. With these considerations, the critical energy value for the cohesive element was determined as listed in Table 4 2.

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67 Finite E lement M odel for 2D P lain W oven L aminates A plain woven laminate (2DPL) with stacking sequence was also analyzed The 2DPL was chosen for comparison with 3D woven composites thus investigating the effect of z yarns on delamination behavior For the sake of simplicity of the FE simulation each plain woven lamina was homogenized as an orthotropic material and one half of the beam was modeled using eight node plane strain elements ( Figure 4 5 ). The material properties of the homogenized plain woven layer are listed in Table 4 3 [ Xiao et al. 2007 ]. T he materi al properties for 45 pw could be obtained using coordinate transformation The creation of matrix cracks and placement of cohesive elements in 2DPL specimens were similar to the procedures used for 3D woven laminates. Our approach is confirmed by the exper imental observation that a single matrix crack tended to occur inside weft yarns of the 0 /90 pw [ Walter et al. 2010 ]. However, the tensile cracks in the matrix pocket of outer most layer seen in the 3D woven composites were not observed in plain woven lamin ates and hence they were not introduced in 2DPL specimens. Furthermore transverse shear stress vanishes at the free surface. Therefore cohesive elements were merely placed in the middle regions of the specimen only. Cohesive elements for possible matrix c racks were placed in the three inner homogenized layers of 0 /90 pw as shown in Figure 4 6. Principal stress directions were found in locations corresponding to the centers of fill tows through same procedure as 3D woven composites considering tensile region and compressive region after delamination

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68 individually. The possible matrix crack directions used in this simulation are shown in Figure 4 6. A parametric study varying Mode I/Mode II interlaminar fracture toughness (370/ 8 3 0 N/m, 1000/ 2000 N/m and 1000/ 33 00 N/m ) was performed in order to investigate the effect of interlaminar fracture toughness on the global behavior and damage pattern of the 2DPL specimens. These studies are referred to as Case I, Case II and Case II I ( Table 4 4 ), respectively Results an d D iscussions 3D W oven C omposites (SY3W and DY3W) The force and displacement at the indenter were recorded during the FE simulations. The force at a given displacement can be a proper measure to examine the macroscopic behavior the specimen. The results f or both SY3W and DY3W are shown in Figure 4 7. Two force displacement curves from the tests [ Walter et al. ] are also plotted for comparison. Note the two experimental curves of the SY3W type show slightly different behavior after damage initiation The so lid lines indicate the result from the FEM. Since the same properties for the cohesive element were used and most configurations remained the same except for the thickness of z yarns in the simulation it can be inferred that the difference between two FEM simulations ( solid curves in Figure 4 7 ) resulted from the effect of z yarns on delamination. The peak load value of DY3W is higher than that of SY3W. This reveals that z yarns can delay the damage initiation point. Besides the forces beyond the peak loads between two cases are quite different. T he SY3W showed that an abrupt force drop right after the peak loads whereas the force drop in DY3W is not noticeable or fairly small. Additionally the SY3W has another

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69 peak load followed by sudden force drop again w hich does not occur in the DY3W. These characteristics agree well with experimental results shown in Figure 4 7. The variation of strain energy and damage energy as a function of indenter displacement obtained from the FE analyses are shown in Figure 4 8. The strain energy represents the recoverable elastic energy stored in the specimen at any given instant. The damage energy is the unrecoverable energy dissipated by the cohesive elements. It is seen that the sudden force drop in the load deflection diagra m corresponds to sudden increase in damage energy. One can note that the damage energy in the DY3W specimen increases steadily without any abrupt damage process or loss of stiffness From these figures it is clear that z yarn can enhance damage resistance and tolerance. H owever optimum amount of z yarn should be used so that there is no deterioration in the composite properties [ Rao et al. 2009 ]. Plain W oven L aminated C omposites (2DPL) Unlike 3D woven composites, only one configuration but with various c ohesive parameters was considered in studying of the 2DPL. Although the same properties for cohesive element as 3D woven composites were initially used for the 2DPL, interlaminar fracture toughness values seems to be different since woven laminated compos ite s have higher fracture toughness than unidirectional fiber composite due to undulation of yarn or crimp [ Kalarikkal et al. 2006 ] The force displacement curves of both SY3W and 2DPL obtained from the tests [ Walter et al. 2010 ] showed different behavior during unloading The force drop of the SY3W was very rapid while that of the 2DPL was gradual. This could be explained by the higher fracture toughness values of woven laminates compared to tape laminates Out of the three sets of interlaminar fracture to ughness values used t he force displacement curve of Case III with the

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70 highest fracture toughness exhibited load displacement behavior similar to experimental results as shown in Figure 4 9 L ower fracture toughness values (cases I and II) resulted in dif ferent damage pattern and produced different load displacement curve s The force versus displacement curves at the indenter with respect to three different interlaminar fracture tough ness values are plotted in Figure 4 9 The difference among the force dis placement curves due to different fracture toughness values is apparent. As the fracture toughness values were increased, the peak loads were increased and the rate of force drop just beyond the peak load became moderate. If the interlaminar cracks propaga te mainly without other delaminations the forces will decrease rapidly. As seen in Figure 4 9 the rapid force drops in the Case I and Case II were attributed to the damages that occurred with very small increment of displacement and can be identified with the damage energy variation in Figure 4 10 Sudden increase of damage energy was accompanied by sudden loss of strain energy as well. But the damage energy of Case III increases gradually along with the increase of the strain energy. These variations of global responses are highly dependent on how damage evolves By observing the history of damage development, it is possible to gain insight in to understand ing delamination damage behavior of composites. Once delamination occurs from the matrix crack it wi ll propagate and reach the free end of specimen (b 1 and b 2 in Figure 4 11 ). The propagation of the crack to the free end causes the force to decrease abruptly After a complete delamination developed over the region from matrix crack to

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71 the end of the s pecimen, the force increases with displacement and strain energy is stored until the state of stress in another interlaminar region satisfies the criterion for crack initiation and propagation. Then the force will decrease suddenly again as shown in the Ca se I of Fig ure s 4 9, 4 10 and Figure 4 11 C Case II showed that the resistance to crack propagation increased with the interlaminar fracture toughness. Further increase in interlaminar fracture toughness (Case III) totally changed the tendency of crack pro pagation. High resistance against cracks kept the interlaminar delamination from propagating. Before a delamination reach ed the end of specimen, another delamination occurred This caused a gradual force drop instead of an abrupt drop iqn Cases I and II ( F ig ure s 4 9, 4 10 and 4 13 ) The relationship s between the damage pattern in Fig ure s 4 10 through 4 12 and the force displacement curve s in Figure 4 9 are similar to the test results of Walter [ Walter 2011 ]. It could be observed that the crack propagation of the SY3W reached the end of the specimen and resulted in the force displacement curve in Figure 4 7 while 2DPL and DY3W didn t allow delamination to propagate to the end of the specimen and other delaminations occurred at the same time they resulted in gradual force drop as shown in Figure 4 14 It should be noted that Figure 4 14 has been obtained from a dynamic test but exhibited similar damage pattern as quasi static test demonstrating that quasi static indentation tests can provide insight in to dama ge development due to impact loading in laminated 3D composites. From these results, it is possible to identify that the role of z yarns on interlaminar delamination. Z yarns provide a constraint to reduce the relative displacement between

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72 two layers thus enhanc ing the resistance against crack propagation Thus the z yarns result in higher apparent in ter laminar fra c ture toughness. Summary and Conclusion The effect of z yarns on damage tolerance of 3D woven composites was studied using the SBS tests It is found that tensile or transverse matrix cracks develop in the matrix pocket between z yarn crowns and the center of wept yarns Cohesive elements associated with bi linear damage model were placed along the principal stress direction for the matrix cracks. These elements are used in modeling of interlaminar delamination as well. The RVE model and simplified plain strain FE model provide d good result s to understand the role of z yarn s and inherent interlaminar fracture toughness From the FE simulation of SY 3W and DY3W specimens it is found that proper amount of z yarns can enhance damage resistance and tolerance. This conclusion agrees well with the result from the end notch flexure (ENF) specimen of 3D woven composites by Pankow et al. [ Pankow et al. 2011 ] It was found that the z yarns preven t two neighboring layers from being separated. In the case of 2DPL crim p or undulation of the yarns seems to interrupt the propagation delamination crack providing better damage tolerance than single z yarn 3D woven co mposites. However, double yarn 3D woven composites (DY3W) have superior damage tolerance compared to 2DPL and SY3W As it turned out, the prevention of crack growth along an entire interface is a key eleme nt to enhance damage tolerance. When damage occurs in different layers before the delamination propagates in the entire specimen, more energy is diss i pated in various interlayer damage mechanisms increasing the damage tolerance of the laminated co mposite structure.

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73 Figure 4 1 3D woven composites for F E model. A ) Orthogonally woven 3D composite with SBS setup B ) Representative Volume Element (RVE) for FE model Figure 4 2 Geometric parameter of RVE A ) left side view B ) front side view

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74 Figure 4 3 Prediction of matrix cracks The lines indicate potential crack paths A ) Cross section for exploring principal stress directions B ) Principal stress directions in the tensile region C ) Principal stress directions in the entire regions

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75 Figure 4 4 Traction separation law of cohesive element. F igure 4 5 2D plain woven composite. A ) Cross section of 2D plain laminated composite B ) Homogenized FE model Figure 4 6. Implementation of the cohesive element on homogenized 2D plain woven composite

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76 Figure 4 7. Force displacement curves of the s ingle/double z yarn 3D woven composites Figure 4 8. Strain energy and damaged energy versus displacement curves of the SY3W and DY3W SE and DE indicate strain energy and damage energy, respectively.

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77 Figure 4 9 Force displacement curves of the sing le and double z yarn 3D woven composites Figure 4 10 Strain energy and damage energy versus displacement curves of the 2DPW. SE and DE indicate strain energy and damage energy, respectively

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78 Figure 4 11 Damage evoluti on of baseline composite Case I. A ) d=0.8mm B ) d=1.1mm C ) d=1.7 mm

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79 Figure 4 12 Damage evolution of baseline composite Case I I. A) d=1mm. B) d=1.5mm. C ) d=1.75 mm

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80 Figure 4 13 Damage evolution of baseline composite Case I II. A ) d=1mm B ) d=1.5mm C ) d=1.75 mm

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81 Figure 4 1 4 D elamin a tion pattern s in the composite s. A ) 2DPL. B ) SY3W. C ) DY3W. [Walter 2012].

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82 Table 4 warp yarn, weft yarn and z yarn, respectively. (unit: mm) Table 4 2 Material properties of the constituents the 3D woven composites Yarns, Epoxy and Cohesive element Yarns (Transversely isotropic) E 1 ( G Pa) E 2 ( G Pa) E 3 ( G Pa) 12 13 23 G 12 ( G Pa) G 23 ( G Pa) G 31 ( G Pa) 54.9 11.4 11.4 0.26 0.26 0.29 4.4 4.4 4.4 Epoxy Cohesive element E ( G Pa) 2.7 0.3 Mode I/II Strength (MPa) 23/33 Mode I/II Fracture toughness (N/m) 370/830 Table 4 3 Elastic properties for plain woven compos ites E 1 =E 2 E 3 G 13 =G 23 G 12 13 = 23 12 ( GP a) ( GP a) ( GP a) ( GP a) 27.5 11.8 2.14 2.9 0.4 0.11 Table 4 4 Parameters for cohesive element in baseline composite model L s w z t x t y t z w y t g 1 .8 0. 4 0.5 0.6 0. 8 2.8 0.075 max G Ic max G IIc ( M P a) ( N/m ) ( M P a) ( N/m ) Case I 23 370 33 830 Case II 23 1 000 33 2 000 Case III 23 1 000 33 3 300

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83 CHAPTER 5 DAMAGE RESISTANCE OF HYBRID COMPOSITES WITH FUNTIONALLY GRADED MATERIALS Along with the attempts to enhance material properties such as strength and fracture toughness a way to reduce stress or strain ener gy release rate (SERR) subjected by external loading can be a significant consideration in the design and operation of composite structures. That is because the stress or strain energy release rate values in the material should be less than the strength or fracture toughness of the material i n order to prevent failure of a material. For laminated structure s interlaminar shear stress can be a proper measure to evaluate the damage resistance of materials. As discussed in previous chapters the beam like speci mens subjected to quasi static or low velocity impact loading exhibit a typical s hear stress distribution wherein the maximum value occurs beneath the contact surface Although t he shear stresses at this location create delami nation, they do not propagate due to rapidly diminishing shear stresses away from the loading point. The shear stresses are almost negligible near the top surface of the beam away from the point of contact There is another location where the shear stresses are high enough to cause del aminations. This is at the mid plane of the beam where the parabolic shear stress profile attains a maximum. In fact this region of maximum shear stress is substantially large Hence, delaminations can propagate all along the length of the specimen until t hey reach the ends as the shear force is constant in SBS specimens extending the region of maximum shear stress. In general the delamination due to an impact loading initiates when interlaminar shear stress exceeds interlaminar shear strength. As a method to reduce the maximum interlaminar shear stress, hybrid composite s comprised of more than two materials or functionally graded material s (FGM) with

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84 continuously va rying material property can be considered. In the present study i mpact damage resistance of hybrid laminated composites and functionally graded laminated composited are investigated. The motivation for this approach comes fr o m some earlier studies on functionally graded beams [Sankar 2001 ] and hybrid laminates [Sankar 1989 ]. It was found that the transverse shear stresses get redistributed in FG beams and hybrid laminates. The specimens of hybrid composites studied have various combinations based on the volume fraction of carbon fiber composite (CFC) and glass fiber composite (GFC) stacking seque nce while the functionally graded composite have three different aspects. The descriptions of those specimens are shown in Table 5 1 E ffect of H ybrid C omposites on R educing the M aximum S hear S tress Finite E lement M odel for Maximum Shear S tress The s hear stress profiles including maximum values in various specimens under an assumed force were found using FE M since t he evaluation of damage resistance of various materials c ould be achieved by comparing the maximum interlaminar shear stress The force in the FE model was assumed as a Hertzian contact force given by The FE model to determine the shear stress profiles is shown in Figure 5 1. Due to symmetry one half of the specimen was modeled using plane st rain elements. The dimension s of the half specimen was 100 mm 15.4 mm. Symme t ric conditions were implemented at x =0, and the vertical displacements at x =100 mm were constrained. T he shear stress profiles at the location of x= 50 mm (50 mm from the symmetry plane ) were obtained from the FE model

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85 The material properties used in the FE model are given in Table 5 2. The properties for the FGM were found using the weight ed average based on volume fraction of the CFC and the GFC This approach was based on the resent results obtained by Banerjee [ Banerjee 2012 ] using micromechanics methods for hybrid composites. Results and Discussion Hybrid composites w ith loading on the harder (stiffer) side Figure 5 2 shows the shear stress profiles of hybrid composites wit h carbon fiber laminates placed on the loading side In the same figure shear stress results for FGM with linearly decreasing stiffness from the loading side to the free side are also presented The maximum shear stress increases with the amount of carbon fiber layer. The maximum shear stress value of the LS36Hybrid is almost same as that of homogeneous composite such as the GFC or the CFC. It should be noted that the shear stress profile of the GFC is exactly the same as that of the CFC. The maximum shear stress in the LSFGM is higher than that of the GFC or CFC. Interestingly the location of the maximum share stress is shifted from the mid plane toward the loading side and thus can affect the point where delamination will occur. Hybrid composites with loa ding on the softer side Figure 5 3 shows the shear stress profiles of the hybrid composites with carbon fiber laminates placed on the bottom ( free ) side. As the volume fraction of carbon fiber is increased, the maximum shear stress become higher. The FSFG M has higher maximum shear stress than the homogenous material. T he location of the maximum sh ear stress is shifted from the middle toward the free side direction.

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86 Hybrid composites stiffened on the top and bottom side s Figure 5 4 shows the shear stress p rofiles of the hybrid composites with carbon fiber laminates placed on the both loading and free sides. The locations of the maximum shear stress values in all the hybrid composites stiffened on both sides are at the middle. The maximum shear stress values of both the hybrid composites and the FGM are lower than those of the composites made of a single material. T he DS18Hybrid composite has the lowest value while the DSFGM has the value between the hybrid composites and the composites of a single material. F unctionally Graded Composites Figure 5 5 shows the shear stress profiles of the FGMs. T he DSFGM has lower maximum shear stress level compare d to homogeneous composites wh ile the LSFGM and the FSFGM have higher maximum shear stress values. Like the hybri d composites the locations of the maximum shear stress in the LSFGM and the FSFGM are away from the midplane Comparison of maximum shear stress es Figure 5 6 shows the comparison of the maximum shear stress values. T he hybrid composite DS18Hybrid has the lowest value which is about 11 percent less than the homogeneous beam ( GFC or the CFC ) while both the LSFGM and the FSFGM have the highest value which is about 2.8 percent more. It can be expected that the specimen with lower shear stress level has higher damage resistance because an extra force or energy would be required to reach the shear strength value. Delamination B ehavior of H ybrid C omposites under a Q uasi static I ndentation F inite E lement M odel for D elamination It can be expected that the hybrid co mposite stiffened on both sides would show higher peak load over the one material composite specimens since lower shear stress

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87 has more margin to reach the strength value. Two FE models to determine peak loads of various specimens described in Table 5 1 un der 3 points bending were considered. ( Figure 5 7 C ) One has only delaminations ( Figure 5 7 A ) T he other has matrix cracks and delaminations ( Figure 5 7 B) All specimens in the FE models consist of 22 layers. C ohesive elements were placed between layers as mentioned in Chapter 4 thus delaminations were allowed to be included in the FE models. In the case of the specimen with matrix cracks, both the matrix cracks and the delaminations are modeled using the cohesive elements. The matrix cracks were placed at t he location of x=37.5 mm from the symmetry line. Plane strain elements were used for the half model due to symmetry. Both the indenter and the support were modeled using rigid bodies. The displacement of indenter was defined as a boundary condition and the load corresponding to each displacement was found. The peak loads from both FE models were compared so as to analyze the effect of matrix cracks on damage resistance. It is found that the matrix cracks altered the layer in which delaminations occurred R e sults Figure 5 8 shows the peak loads of all hybrid specimens considered in this study Figures 5 9 through 5 20 show the load deflection curves and the damage patterns in various specimens. Obviously the specimens with matrix cracks exh i bit lower peak loa d s This is because there is stress concentration at the tip of the matrix cracks enabling the delaminations to propagate at lower loads. As the matrix cracks were introduced the location of delamination shifts from the maximum interlaminar stress location to the interface that contains the crack tip. The DS18Hybrid composite has the highest peak load and hence highest damage resistance. Detail ed discussions about

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88 the location of delamination as well as load displacement curve for individual specimen s are g iven in the following Figure 5 9 and Figure 5 10 show the load displacement curve and the deformations after delamination of the GFC and the CFC, respectively. The location of delamination changed from the 11 th interface (the middle of the specimen) to th e 8 th i nterface As long as the specimen has a symmetr ic configuration delamination occurs at the middle since the shear stress is maximum at the center H owever the matrix cracks cause the stress field to be redistributed and change the delamination site. Figure s 5 11 through Figure 5 14 show the load displacement curve and the deformations after delamination of the LS9Hybrid composite, t he LS18Hybrid composite, the LS36Hybrid co mposite and the LSFGM composite (Loading side was stiffened) Figure 5 15 thro ugh 5 18 show the load displacement curve s and the deformations after delamination of the FS9Hybrid composite, t he FS18Hybrid composite, the FS36Hybrid composite and the FSFGM composite (Free side was stiffened). Figure 5 19 shows the load displacement cur ve s and the deformations after delamination of the DS9Hybrid and Figure 5 20 shows the load displacement curves of other cases in which both sides were stiffened. In the case of the specimens without matrix cracks the location of delamination can be determ ined as the location of the maximum shear stress. T herefore the location of delamination varied based on the configurations. However in all cases with matrix cracks the delamination occurred at the 10 th interface for the case of loading on the stiffer sid e at the 6 th interface for the case loading on the softer side, and at the 8 th interface when both sides were stiffened (symmetric beam )

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89 Conclusion s Hybrid composites including FGM composites can reduce the maximum shear stress and thus can enhance dama ge resistance. However the un symmetric FGM composites results in higher maximum shear stress. The effects of reducing the maximum shear stress become noticeable when stiff material s are symmetrically used in both sides. T he locations of delamination of th e doubly stiffened specimens are identical with the homogeneous composite. However, the delaminations of the singly stiffened specimen occur at the interface toward the stiffened side. Matrix cracks alter the locations of delamination and decrease damage r esistance. Figure 5 1 FE model to determine the shear stress profile

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90 Figure 5 2 Shear stress profiles of the hybrid composites stiffened on the loading side Figure 5 3 Shear stress profiles of the hybrid composites stiffened on the free side

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91 Figure 5 4 Shear stress profiles of the hybrid composites stiffened on the top and bottom sides Figure 5 5 Shear stress profiles of homogeneous the FGMs

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92 Figure 5 6 The maximum shear stress of various composites Figure 5 7 The DSFGM specimen. A) Without matrix cracks B) With matrix cracks. C) FE model to demonstrate the SBS test.

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93 Figure 5 8 Comparison of peak loads of various specimens with and without matrix cracks Figure 5 9 The GFC. A ) L oad displacement curve B ) Delamination behav ior of the specimen without matrix crack (d=6mm) C ) Delamination behavior of the specimen with matrix crack (d=6mm) where d denotes the displacement of the indenter Figure 5 10 The CFC. A ) L oad displacement curve B ) Delamination behavior of the spec imen without matrix crack (d= 4 mm) C ) Delamination behavior of the specimen with matrix crack (d= 4 mm)

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94 Figure 5 11 The LS9Hybrid composite A ) L oad displacement curve B ) Delamination behavior of the specimen without matrix crack (d=6mm) C ) Delaminatio n behavior of the specimen with matrix crack (d=6mm) Figure 5 12 The LS18Hybrid composite A ) L oad displacement curve B ) Delamination behavior of the specimen without matrix crack (d=6mm) C ) Delamination behavior of the specimen with matrix crack (d= 6mm) Figure 5 13 The LS36Hybrid composite A ) L oad displacement curve B ) Delamination behavior of the specimen without matrix crack (d=6mm) C ) Delamination behavior of the specimen with matrix crack (d=6mm)

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95 Figure 5 14 The LSFGM composite A ) L oa d displacement curve B ) Delamination behavior of the specimen without matrix crack (d=6mm) C ) Delamination behavior of the specimen with matrix crack (d=6mm) Figure 5 15 The FS9Hybrid composite A ) L oad displacement curve B ) Delamination behavior of the specimen without matrix crack (d=6mm) C ) Delamination behavior of the specimen with matrix crack (d=6mm) Figure 5 16 The FS18Hybrid composite A ) L oad displacement curve B ) Delamination behavior of the specimen without matrix crack (d=6mm) C ) D elamination behavior of the specimen with matrix crack (d=6mm)

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96 Figure 5 17 The FS36Hybrid composite A ) L oad displacement curve B ) Delamination behavior of the specimen without matrix crack (d=6mm) C ) Delamination behavior of the specimen with matrix crack (d=6mm) Figure 5 18 The FSFGM composite A ) L oad displacement curve B ) Delamination behavior of the specimen without matrix crack (d=6mm) C ) Delamination behavior of the specimen with matrix crack (d=6mm) Figure 5 19 The DS9Hybrid composit e A ) L oad displacement curve B ) Delamination behavior of the specimen without matrix crack (d=6mm) C ) Delamination behavior of the specimen with matrix crack (d=6mm)

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97 Figure 5 20 Load displacement curves A ) DS18hybrid B ) DS36hybrid C ) DSFGM : The l ocations of delamination in the symmetric specimens are same as those of the DS9hybrid shown in Figure 5 19

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9 8 Table 5 1. Various hybrid composite laminates and FGM Name Description Glas s fiber composite Consists of unidirectional glass fiber composites (GFC). Carbon fiber composite Consists of unidirectional carbon fiber composites (CFC). LS9Hybrid composite Consists of 9 percent CFC and 91 percent GFC. CFC was placed on the loading side. LS18Hybrid composite Consists of 1 8 percent CFC and 82 pe rcent GFC. CFC was placed on the loading side. LS36Hybrid composite Consists of 36 percent CFC and 64 percent GFC. CFC was placed on the loading side. FS9Hybrid composite Consists of 9 percent CFC and 91 percent GFC. CFC was placed on the free side ( opposite to the loading side).

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99 Table 5 1. Continued Name Description FS18Hybrid composite Consists of 1 8 percent CFC and 82 percent GFC. CFC was placed on the loading side (opposite to the loading side). FS36Hybrid composite Consists of 36 percen t CFC and 64 percent GFC. CFC was placed on the loading side (opposite to the loading side). DS9Hybrid composite Consists of 9 percent CFC and 91 percent GFC. CFC was placed on the top and bottom faces (symmetric). DS18Hybrid composite Consists of 18 percent CFC and 82 percent GFC. CFC was placed on the top and bottom faces (symmetric). DS36Hybrid composite Consists of 36 percent CFC and 64 percent GFC. CFC was placed on the top and bottom faces (symmetric). LSFGM composite Volume fraction of CF C is varied from 100 percent at the loading side to 0 percent at the free side in 11 steps.

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100 Table 5 1. Continued Name Description FSFGM composite The volume fraction of CFC at layers is varied from 100 percent at the free side to 0 percent at the lo ading side in 11 steps. DSFGM composite The volume fraction of CFC is varied from 100 percent both at the loading and free side to 0 percent at the center in 11 steps (symmetric). Table 5 2 Elastic properties for composite materials in the FEM E 1 = E 2 ( GP a) E 3 ( GP a) G 13 = G 23= G 12 ( GP a) 13 = 23 12 Glass fiber composite 26.9 8.6 3.1 0.28 0.16 Carbon fiber composite 70.0 6.8 3.3 0.15 0.13

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101 CHAPTER 6 CONCLUSIONS AND FUTU RE WORK Conclusions It was shown that the damage resistance and tolerance of laminated composites can be enhanced by the employment of translaminar reinforcements (TLR) such as stitching, z pinning and 3D weaving and also by h ybrid composites A non dimensional analytical model focused on Mode I delamination was developed to u nderstand the role of the TLR on delamination behavior. An explicit formula for the apparent interlaminar fracture toughness was derived in terms of the inherent fracture toughness of original materials and the bridging force due to z pins. This model is c apable of estimating the apparent fracture tou g h ness, the bridging length and allow able bridging force thus can be useful in the design of TLR for composite laminates Along with u nderstanding advantage s of TLR in increasing the damage tolerance of lamina ted composites, the damage behavior of laminated composites subjected to low velocity impact loading was studied. Based on the similarity in damage development between quasi static and dynamic loadings observed through the short beam shear ( SBS ) test s the FE analyses of the SBS specimens for quasi static indentation and at several rates of low velocity impact loadings were performed The results reveal that inertia effects in the typical velocity range of the striker in SHPB, around 10m/s, can be negligibl e and hence the quasi static analysis is useful and valid in the study of damage in composite specimens under low velocity impact loading. The delamination behavior of 3D woven composites was investigated focusing on the effect of z yarn. T he 3D woven com posites containing both single and double z

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102 yarn s were chosen and compared with the 2D plain woven laminate. The double z yarn woven composite exhibited enhanced damage tolerance compared to the single z yarn and the plain woven laminate. T he relative slid ing motion between two layers is constrained by z yarns thus the crack propagation of delamination is suppressed This mechanism increases the apparent interlaminar fracture toughness of the composites. The interlaminar shear stress profiles in the various hybrid composites were obtained from linear FE analyses and compared within the framework of damage resistance. Some hybrid configurations resulted in r educed maximum shear stress value for a given contact force thus demonstrating higher damage resistance Although matrix cracks may change the location of delamination and decrease damage resistance, hybrid composites can still be superior in structural applications R ecommendations for Future W ork It is expected that more studies in these areas would prov ide a good design tool and insights in to application of composite materials i n structures subjected to impact loading. First of all, methodologies for accurate prediction of intralaminar failure in yarns and matrix regions are needed and their implementati on in the FE model s should be studied. S ome useful experimental data provided several assumptions for using cohesive elements in this study However a n analysis scheme that does not require a priori knowledge of crack location would be desirable In addit ion to matrix cracks and delamination, debonding of yarns should be considered although this effect is less than delamination in orthogonal woven composites. The study including debonding can be used in modeling of different types of materials such as angl e interlock composites.

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103 Finally hybrid woven composites should be analyzed. Each layer and individual yarns can be made of various materials. The synergi stic effect due to z yarn s and hybridization should be evaluated for the best performance of composite materials subjected to impact loading

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104 L IST OF REFERENCES AGY. http://www.agy.com/technical_info/graphics_PDFs/Advanced_Materials.pdf Applied Poleramic Inc. http://www.appliedpoleramic.com/specs/vartm_rtm.php Barenblatt GI. The Mathematical Theory of Equilibrium Cracks in Brittle Fracture. Advances in Applied Mechanics. 1962; 7: 55 129. Belytschko T, Cra cie R Ventura G A R eview of E xtended/ G eneralized F inite E lement M ethods for M aterial M odeling. Modelling and Simul ation in Mater ials Sci ence and Eng ineering. 2009; 17 (4) : 043001 Benzeggagh ML Kenane M Measurement of Mixed Mode Delamination Fracture To ughness of Unidirectional Glass/Epoxy Composites with Mixed Mode Bending Apparatus. Composites Science and Technology 1996; 56: 439 449. Boesl BP, Bourne GR, Sankar BV. Insitu Multiscale Analysis of Fracture Mechanisms in Nanocomposites. Composites Part B 2011; 42: 1157 1163. Brandt J, Drechsler K Arendts FJ Mechanical P erformance of C omposites B ased on V arious Three D imensional W oven F ibre Pre forms Composite Science and Technology 1996; 56: 381 386 Brandt J, Drechsler K Filsinger J Advanced Texti le Technologies for the Cost Effective Manufacturing of High Performance Composites 1996 Byun JH, Gillespie Jr. JW. Chou TW Mode I Delamination of a Three Dimensional Fabric Composite Journal of Composite Materials 1990 ; 24: 497 518 Byrd LW Birman V The E stimate of the E ffect of Z P ins on the S train E nergy R elease rate, F racture and F atigue in a C omposite C o C ured Z Pinned D ouble C antilever B eam Composite Structure 2005; 68: 53 63 Byrd LW Birman V Effectiveness of Z P ins in P reventing D elamination of C o C ured C omposite J oints on the E xample of a D ouble C antilever T est Composites Part B, 2006; 37: 365 378 Camacho GT Ortiz M Computational M odeling of I mpact D amage in B rittle M aterials. Internatoinal Journal of Sol ids and Structure 1996; 33: 2899 2938 Camanho PP Dvila CG de Moura MF Numerical Simulation of M ixed M ode P rogressive D elamination in C omposite M aterials J ournal of Compos ite Mater ials. 2003; 37(16) : 1415 1438 Carti DDR Effect of Z fibres TM on th e Delamination Behaviour of Carbon Fibre/Epoxy Laminates. PhD thesis. Cranfield University, UK. 2000

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105 Carti DDR Cox BN Fleck NA Mechanisms of Crack Bridging by Composite and Metallic Rods Composites Part A 2004; 34: 1325 1336. Carti DDR, Anno GD, Pou lin E Partridge IK 3D reinforcement of S tiffener to Skin T J oints by Z P inning and T ufting Eng ineering Fract ure Mech anics. 2006 ; 73: 2532 2540 Chamis C C Handler LM Manderscheid J Composite Nanomechanics: A Mechanistic Properties Prediction. NASA/TM 2007 214673, Glenn Research Center, Cleveland, OH. 2007 Chang P Mouritz AP, Cox BN Flexural P roperties of Z P inned L aminates Composites Part A 2007; 38: 244 251 Chang P, Mouritz AP, Cox BN Properties and F ailure M echanisms of P inned C omposite L ap J oints in M onotonic and C yclic T ension Composites Science and Technology 2006; 66: 2163 2176 Chen L, Sankar BV Ifju PG Analysis of Mode I and Mode II T ests for C omposites with Translaminar R einforcements Journal of Composite Materials 2005; 39: 1311 1333 Chen L, Sankar BV, Ifju PG A N ew M ode I Fracture T est for C omposites with Translaminar R einforcements. Compos ites Sci ence and Technol ogy. 2002; 62: 1407 1414 Cui W, Wisnom MR, Jones M A Comparison of Failure Criteria to p redict Delamination of Uni directional Glass/Epoxy Specimens w aisted t Composites 1992; 23(3):158 66. Dai SC, Yan W, Liu HY Mai YW. Experimental S tudy on Z P in B ridging L aw by P ullout T est. Composites Science and Technology. 2004; 64: 2451 2457 Dantuluri V, M aiti S, Geubelle PH, Patel R Kilic H. Cohesive M odeling of D elamination in Z P in R einforced C omposite L aminates. Composite s Science and Technology 2007; 67: 616 631 Davila CG, Camanho PP, Turon A. Cohesive Elements for Shells. NASA TP 2007 214869. 2007. Dransfield KA, Baille C, Mai YW Improving the Delamination Resistance of CFRP by Stitching A Review Composites Science and Technology 1994; 50: 305 317 Dransfield KA, Jain LK Mai YW On the E ffects of S titching in CFRPs I. M o de I D elamination T oug hness, Composites Science and Technology 1998; 58 : 815 827

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106 Dransfield KA, Jain LK Mai YW On the E ffects of S titching in CFRPs II. Mode II D elamination T oughness Composites Science and Technology 1998; 58: 8 29 8 3 7 D ugdale DS Yielding of steel sheets containing slits Journal of the Mechanics and Physics of Solids 1960; 8: 100 108 Farley GL, Smith BT, Maiden J C ompression Response of Thick Layer Composite Laminates w ith t hrough the Thickness Reinforcement Reinforced Plastics and Composites 1 992 ; 11: 787 810. Geubelle PH, Baylor J. The I mpact Induced D elamination of L aminated C omposites: a 2D S imulation. Composi tes Part B 1998 ; 29: 589 602. Gibson RF. Principles of C omposite M aterial M echanics. NewYork: McGraw Hill, Inc. 1994 Gong JC Sankar BV. Impact Properties of Three Dimensional Braided Graphite/Epoxy Composites Journal of Composite Materials 1991 ; 25: 7 15 731 Grassi M Zhang X Finite E lement A nalyses of M ode I I nterlaminar D elamination in Z F ibre R einforced C omposite L aminates. Compos ites Science and Technology 2003 ; 63: 1815 1832 Grujicic M, Pandurangan B, Koudela KI, Cheeseman BA. A Computational A nalysis of the Ballistic Performance of Light Weight Hybrid Composite Armors. Applied Surface Science. 2006; 253: 730 745 Hallett SR Jiang WG Khan B Wisnom MR Modelling the I nteraction between M atrix Cr acks and D elamination D amage in S caled Q uasi I sotr opic S pecimens Compos ites Sci ence and Technol ogy. 2008 ; 68: 80 89 Huang HJ In P lane R esponse and Mode II F racture R esponse of Z P in W oven L aminates. PhD thesis. Univ. of Michigan 2008 Jain LK, Mai YW. Analysis of S titched L aminated ENF specimens for I n terlaminar M ode II F racture T oughness International Journal of Fracture 1994 ; 68 (3): 219 244 Jain LK, Mai YW. On the E ffect of S titching on Mode I D ela mination Toughness of L aminated C omposites. Compos ites Science and Technology. 199 4 ; 5 1(3) : 331 345 Jang I Sankar BV. Analysis of a C omposite D ouble C antilever B eam wit h St itched R einforcements under M ixed M ode L oading: Formulation (I) Journal of Mechan ical Science and Technology 2005 : 19(2): 567 577

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107 Kalarikkal S, Sankar BV, Ifju PG Effect of Cryogenic Temperature on the Fracture Toughness of Graphite/Epoxy Composites. ASME Journal of Engineering Materials and Technology 2006; 128(2):151 157 Kim BC, Pa rk SW Lee DG Fracture T oughness of the N ano P article R einforced E poxy C omposite Composite Structures 2008 86: 69 77 Krueger R. The V irtual C rack C losure T echnique: History, A pproach and A NASA/CR 2002 211628. 2002 Kumar P, Singh RK. Impact D amage A rea and I nterlaminar T oughness of FRP 2000; 9: 77 88. Kwon YS Sankar BV. Indentation Flexure and Low Velocity Impact Damage in Graphite/Epoxy L aminates", ASTM Journal of Composites Technology and Research 1993 ; 15:101 111 Lander JK Designing with Z P ins: L ocally R einforced C thesis. Cranfield University, UK. 2008 Mabson GE Deobald LR. Design C urves for 3d R einforceme nt L aminated D ouble C antilever B eams. In Mechanics of Sandwich Structure ASME 2000. 2000; 89 99 Madhukar MS, Drzal LT Fiber M atrix A dhesion and I ts E ffects on C omposite M echanical P roperties: IV. Mode I and Mode II F racture T oughness of G raphite/ E poxy C o mposites. J ournal of Compos ite Mater ials 1992; 26 (7) ; 936 968 Mahmood A Wang X, Zhou C. Modeling S trategies of 3D W oven C omposites: A R eview Composites Structure 2011 ; 93: 1947 1963 McBeath S. Safety Pins International Journal of Racecar Engineering 2002 ; 12: 56 62 Mou r itz AP Leong KH, Herszberg I A R eview of the E ffect of S titching on the I n P lane M echanical P roperties of F ibre R einforced P olymer C omposites. Composites Part A 1997 ; 28A: 979 991 Mouritz AP Fracture and T ensile F atigue P roperties of S titched F ibreglass C omposites. Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials Design and Applications 2004 2004; 218: 87 Mouritz AP. Review of Z P inned C omposite L aminate. Composites Part A 2007 ; 38: 2383 2397 Mou r itz AP Tensile F atigue P roperties of 3D C omposites with through T hickness R einforcement Composites Science and Technology 2008 ; 68: 2503 2510

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108 Nettles AJ Douglas MJ. A Comparison of Quasi Static Indentation to Low Velocity Impact NASA/TP 2000 2104 81 2000 Ling DS, Fang XJ, Cox BN Yang QD. Nonlinear Fracture Analysis of Delamination Crack Jumps in Laminated Composites Journal of Aerospace Engineering 2011 ; 24: 181 188 Ling DS, Yang QD, Cox B An A ugmented F inite Element Me thod for M odeling A rbi trary D iscontinuities in C omposite M aterials. International Journal of Fracture 2009; 156 : 53 73 Nishikawa M, Okabe T, Takeda N Numerical S imulation of I nterlaminar D amage P ropagation in CFRP C ross P ly L aminates under T ransverse L oading International Journal of Solids and Structures 2007 ; 44: 3101 3113 Okabe T, Yashiro S Damage D etection in H oled C omposite L aminates using an E mbedded FBG S ensor Compos ites Part A 2012; 43: 388 397 Pankow M, Waas AM, Yen CF, Ghiorse S Resistance to D elamination o f 3D W oven T extile C omposites E valuated using End Notch Flexure (ENF) T ests: Cohesive Z one B ased C omputational R esults Compo sites Part A 2011; 42(12):1863 1872 Rao MP, Sankar BV, Subhash G Effect of Z yarns on the S tiffness and Strength of T hree D imens ional W oven C omposites Compo sites Part B 2009; 40:540 551 Ratcliffe JG Fiber Reinforced DCB Specimens. NASA/TM 2004 213019, ARL TLR 3190 2004 Ratcliff e JG Kruger R A Finite Ele ment Analysis for Predicting Mode I Dominated Delamination Growth in Laminated Structure with t hrough Thickness Reinforcement. America Society for Composites 21 th annual technical conference, Dearborn, Michigan. 2006 Rice JR. A P ath I ndependent I ntegral a nd the A pproximate A nalysis of S train C oncentration by N otches and C racks Journal of Applied Mechanics 1968 ; 35: 379 386 Robinson P, Das S. Mode I DCB T esting of C omposite L aminates R einforced with z D irection P ins: a S imple M odel for the I nvestigation of D ata R eduction S trategies. Eng ineering Fract ure Mech anics. 2004; 71: 345 364. Rugg KL, Cox BN Massab R. Mixed M ode D elamination of P olymer C omposite L aminates R einforced through the T hickness by z fibres. Composites 2002 ; 33: 177 190 Rybicki EF, Kan ninen MF. A F inite E lement C alculation of S tress I ntensity F actors by a M odified C rack C losure I ntegral. Engineering Fracture Mechanics 1977 ; 9: 931 938.

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109 Rys T, Sankar BV Ifju PG Investigation of Fracture Toughness of Laminated Stitched Composites Subje cted to Mixed Mode Loading Journal of Reinforced Plastics and Composites 2010 ; 29 (3) : 422 430. Sankar BV. Interlaminar Shear Stresses in Composite Laminates Due to Static Indentation. Journal of Reinforced Plastics and Composites. 1989; 8(5): 458 471. S ankar BV, Hu S. Dynamic Delamination Propagation in Composite Beams, Journal of Compos Materials. 1991; 25(11): 1414 1426. Sankar BV Sonik V Pointwise Energy Release Rate in Delaminated Plates. AIAA J ournal 1995; 33 (7) : 1312 1318. Sankar BV, Sharma SK. Mode II D elamination T oughness of S titched G raphite/ E poxy T extile C omposites. Compos ites Sci ence and Technol ogy. 1997 ; 57 : 729 737 Sankar BV Zhu H. The effect of S titching on the L ow V elocity I mpact R esponse of D elaminated C omposite B eams. Composites Science and Technology 2000 ; 60: 2681 2691 Sankar BV Dharmapuri SM. Composite Materials 1998 ; 32: 2204 2225 Sankar BV. An elas ticity solution for functionally graded beams. Composites Science and Technology 2001; 61: 689 696. Sharma SK Sankar BV. Effect of Stitching on Impact and Interlaminar Properties of Graphite/Epoxy Laminates J ournal of Thermoplastic Composite Materials 1997; 10(3): 241 253. Sridhar N, Massabo R, Cox BN Beyerlein IJ. Delamination D ynamics in through T hickness R einforced L aminates with A pplication to DCB S pecimen D elaminated C omposite B eams. International Journal of fracture. 2002 ; 118: 119 144 Steeves C A FleckIn NA In P lane P roperties of C omposite L aminates with through T hickness P in R einforcement International Journal of Solids and Structures 2006 ; 43: 3197 3212 Tan KT, Watanabe N Iwahori Y. Experimental investigation of bridging law for single st itch fibre using Interlaminar tension test 2010; 92(6): 1399 1409. Tanoglu M, Robert S, Heider D, McKnigth SH, Brachos V Gillespie Jr JW. Effects of T hermoplastic Performing B inder on the P roperties of S2 G lass F abric R einforced E poxy C omposites. Interna tional Journal of Adhesion & Adhesives. 2001; 21(3): 187 195

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110 Tvergaard V, Hutchinson JW. The Relation between C rack G rowth Resistance and F racture P rocess P arameters in E lastic P lastic S olids. Journal of Mechanics and Physics of Solids 1992 ; 40 : 1377 139 7. Wallace BT, Sankar BV Ifju PG Pin R einforcement of Delaminated Sandwich B eams under A xial C ompression Journal of Sandwich Structures and Materials 2010 ; 3(2):117 129 Walter TR, Subhash G Sankar BV Yen CF. Monotonic and C yclic S hort B eam S hear R es ponse of 3D W oven C omposites Composites Science and Technology 2010 ; 70 : 2190 2197 Walter TR Characterization of D elamination in 3d W oven C omposites under S tatic and D ynamic loading. PhD thesis. Univ ersity of Florida 2011 Walter TR, Subhash G, Sankar BV, Song MC, Yen CF. A Novel Method for Dynamic Short Beam Shear Testing of 3D Woven Composites. 2012; Submitted Wisnom MR, Reynolds T, Gwilliam N Reduction in I nterlaminar S hear S trength by D iscrete and D istributed V oids. Compos ites Sci ence and Technol o gy. 1996; 56 : 93 101 Wong DWY, Lin L, McGrail PT, Peijs T, Hogg PJ. Improved F racture T oughness of C arbon F ibre/ E poxy C omposite L aminates using D issolvable T hermoplastic F Part A 2010 ; 41: 759 767 Wu EM Reuter Jr. RC Crack Extension i Report No. 275, University of Illinois. 1965 Xiao JR, Gama GA, G illespie Jr JW Progressive and D elamination in P lain W eave S 2 G lass/SC 15 C omposites under Q uasi S tatic P unch S hear L oading. Composite structures 2 007; 78 : 182 196 Xu XP, Needleman A. Numerical S imulation of F ast C rack G rowth in B rittle S olids Journal of Mechanics and Physics of Solids 1994 ; 42: 1397 1434. Yan C, Xiao K, Ye L Mai YW Numerical and E xperimental S tudies on the F racture B ehavior of R ubber T oughened E poxy in B ulk S pecimen and L aminated C omposites. Journal of Material Science 2002 ; 37: 921 927. Zhou ZQ, Fang XJ, Cox, BN, Yang QD The E volution of a T ransverse I ntra ply Crack C oupled to D elamination C racks. Int ernational J ournal of Fra ct ure. 2010 ; 165: 77 92

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111 BIOGRAPHICAL SKETCH Min Cheol Song was born in 1975 Seoul, South Korea. He attended Sungdong High S chool. He ear ned his Bachelor of Science in m echanical engineering in February of 1994 from the Chung Ang University in Seoul, South Korea. He e arned his Master of Science in m echanical e ngineering February in 2001 from the Korea Advanced Institute of Science and Technology (KAIST) in Daejeon, South Korea. During this period he conducted research on b ending collapse of Glass Fiber Rein forced Plastic (GFRP) and Aluminum co cured square tube. In January of 2001 he joined Agency for Defense Development in Daejeon, South Korea During this period, he was working on the topic of damage process and phenomena in various materials. He came to t he United of States and started his PhD at the University of Florida, Gainesville in the fall of 2008 under the supervision of Prof. Sankar. Mr. Song defended his D octor of P hilosophy in 2012.