P ERIDYNAMIC M ODELING AND IMPACT TESTING OF DYNAMIC DAMAGE, FRACT UR E , AND FAILURE PROCESS IN FIBER - REINFORCED COMPOSITE MATERIALS By Wu Zhou A DISSERTATION Submitted to Michigan State University i n partial fulfillment of the requirements for the d egree of Mechanical Engineering - Doctor of Philosophy 2018 ABSTRACT PERIDYNAMIC MODELING AND IMPACT TESTING OF DYNAMIC DAMAGE, FRACT UR E , AND FAILURE PROCESS IN FIBER - REINFORCED COMPOSITE MATERIALS By Wu Zhou This study focuses on developing a perid ynamics (PD) theory based model for the prediction of impact - induced fracture and failure process in laminated composites , and the impact testing of damage evolu t ion in composites. First, the 2D bond - based PD method was evaluated for the dynamic fracture p rocess in polymethyl - methacrylate (PMMA) simply supported beams. PMMA Single Edge Notched Bending (SENB) specimens were impacted with a drop - weight machine. The impact fracture process was recorded with a high - speed camera and the images were analyzed with the digital image correlation (DIC) method. The fracture path and crack velocities simulated with PD basically match the experimental results. However, as the peak crack velocity increases, the ratio of the simulated peak velocity over the experimental o ne also increases. This deviation was confined with the fitted failure criteria for impact fracture in composites with higher peak velocities in the next chapter. To capture the impact fracture process more accurately and apply it to composites, two major developments have been made to the PD theory - based models. Firstly, a bond - based mesoscale peridynamic model has been developed for orthotropic composite materials. The model defines a continuous in - plane material constant for orthotropic materials as the mesoscale off - axis modulus in the laminated composite theory. The changes continuously from the fiber direction to the transverse direction with an effective orthotropy. This treatment differs from the existing PD composite models which defin e a micro - modulus for fibers and for matrix. It is more efficient in simulations of large vo lume of materials. The mesoscale model was calibrated and employed to simulate the in - plane impact - induced fracture patterns in the unidirectional fib er composite beams. Secondly, the simultaneous crack - velocity - related dynamic strain energy release rate was introduced into the PD failure criteria. Besides the final failure of the composites, the fracture process and crack velocity can be predicted more accurately by using the fitted PD model . The simulation was validated with the comparison to the experimental results. The mesoscale PD model has been extended into three - dimensional for laminated composites. In the model, both the intralayer and interla yer material constants and critical bonds stretch were defined for laminated composites. The PD model was then employed to study the impact damage of the laminated composite plates subjected to out - of - plane impact loading. The matrix and intralayer damage, and the interlayer delamination have been simulated in the composite laminates with different fiber layouts. To improve the impact resistance, novel composite structures with reinforcement in the through - thickness direction were explored. A previously dev eloped quasi - three - dimensional (Q3D) braiding method was examined. In this work, the Q3D [0/ ± 60] 4 carbon fiber composite plates were fabricated. The in - plane tensile and the out - of - plane impact experiments were performed. The results showed that the Q3D co mposite limited the intra - layer damage and the inter - layer delamination while keeping the comp etitive in - plane stiffness and strength. iv ACKNO W LEDGEMENT S I would like to thank: --- My a dvisor Professor Dahsin Liu for his persistent support, guidance , and advice throughout my Ph .D. research work and life at MSU. --- M y committee members Professor Xinran Xiao, Professor Ronald Averill, and Professor Zhengfang Zhou for all their advice , help, and encouragement during my Ph .D. research. --- Dr. Danielle Ze ng for the funding support from Ford Motors. Thank Dr. Chian - Fong Yen for the funding support from U S Army Research Laboratory . Thank ME Department, College of Engineering , and Graduate School MSU for the funding support and their help . --- My labmates an d friends for their help and support in my PD modeling work. They are Miao Wang, Wei Zhang, Yile Hu, Kur t Lundeen, Fang Hou, Danghe Shi, Yi Sun. --- My labmates and friends for their help and support in my experimental testing, they are: Shutian Yan, Hoa Nguyen, Lingxuan Su, Anthony Wente, Xinyu Mao, Andy VanderKlok, Ben Swanson, Erik Stitt, Suhas, Dr. Tekalur, Todd , Dr. Craig Gunn , Dr. Guojing Li, Daniel Schleh, and Professor Yang Guo . --- M y parents, families, and friends for all their support and encouragement , and thanks as well to everyone else who helped complete this research work . v T ABLE OF CONTENTS LIST OF TABLES ................................ ................................ ................................ ........... viii LIST OF FIGURES ................................ ................................ ................................ ........... ix KEY TO SYMBOLS ................................ ................................ ................................ ........ xv Chapter 1. Introduction ................................ ................................ ................................ ... 1 1.1 Fiber - Reinforced Composite Materials ................................ ................................ 1 1.1.1 Basic characteristics ................................ ................................ ...................... 1 1.1.2 Failure of composites ................................ ................................ .................... 2 1.2 Peridynamic theory ................................ ................................ .............................. 3 1.2.1 Bond based peridynamics ................................ ................................ ............. 3 1.2.2 Damage criteria ................................ ................................ ............................. 6 1.2.3 Numerical algorithm ................................ ................................ ..................... 7 1.3 Scopes of the dissertation ................................ ................................ ..................... 8 1 .4 Outline of the dissertation ................................ ................................ .................... 9 Chapter 2. Literature Review ................................ ................................ ........................ 12 2.1 In - plane Fracture Propagation ................................ ................................ ............ 12 2.1.1 Experimental Study ................................ ................................ ..................... 12 2.1.2 Numerical Modeling and Simulation ................................ .......................... 14 2.2 Out - of - plane Im pact Induced Delamination and Plane Failure ......................... 16 2.2.1 Experimental Study ................................ ................................ ..................... 16 2.2.2 Numerical Modeling and Simulation ................................ .......................... 19 Chapter 3. Analyzing the Dynamic Fracture Process in Polymethyl - methacrylate (PMMA) Beams with Three - point - bending Impact Testing and Peridynamic Simulation .. ................................ ................................ ................................ ................... .. 20 3.1 Introduction ................................ ................................ ................................ ........ 20 3.2 Experimental Testing of the Impact Fracture in PMMA Beam ......................... 23 3.2.1 Impact experiment setup ................................ ................................ ............. 23 3.2.2 Impact force on PMMA beam ................................ ................................ .... 24 3.2.3 Digital image correlation (DIC) analysis setup ................................ .......... 25 vi 3.2.4 Impact fracture process with displacements fields from DIC analysis ....... 27 3.3 Peridynamic Simulation of the Impact Fracture in the PMMA Beam ............... 32 3.3.1 Peridynamic simulation settings ................................ ................................ . 32 3.3.2 Peridynamic simulation results and discussion ................................ ........... 33 Chapter 4. A Peridynamic Model for Fiber - reinforced Composite Materials and Its Capturing the Dynamic Fracture Process in the Composite Beams ................................ . 41 4.1 Introduct ion ................................ ................................ ................................ ........ 41 4.2 Peridynamic model for orthotropic composites ................................ ................. 42 4.2.1 Peridynamic micromodulus of bonds ................................ ......................... 42 4.2.2 Elastic deformation verification ................................ ................................ .. 47 4.2.3 Failure criteria ................................ ................................ ............................. 48 4.3 Problem setup ................................ ................................ ................................ ..... 50 4.4 Simulation of the Impact Fracture Patterns in Composites with the Model ...... 53 4.5 PD modeling of the dynamic fracture process ................................ ................... 57 4.5.1 Introducing the dynamic fracture criteria to fit the PD model .................... 57 4.5.2 Simulation of the dynamic fracture process with th e fitted PD model ....... 59 4.6 Conclusion ................................ ................................ ................................ .......... 68 Chapter 5. Peridynamic Modeling of Impact - induced Damage Evolution and Delamination in La minated Composite Materials ................................ ............................ 70 5.1 The PD Model for Laminated Composites ................................ ........................ 70 5.1.1 Micromodulus of bonds ................................ ................................ .............. 70 5.1.2 Failure criteria ................................ ................................ ............................. 71 5.1.3 Problem description ................................ ................................ .................... 73 5.2 Results and discussion ................................ ................................ ........................ 75 5.2.1 Impact - induced intralayer damage and delamination process in laminates [90/0/90] ................................ ................................ ................................ .................... 76 5.2.2 Impact - induced intralayer damage and delamination in laminates [90/15/90], [90/30/90], [90/45/90], [90/60/90], and [90/75/90]. ................................ ................. 79 5.3 Conclusion ................................ ................................ ................................ .......... 84 Chapter 6. A n Experimental Study of the In - plane Tensile Properties and the Out - of - plane Low - velocity Impact Damage Process of the Unidirectional Laminate, Two - dimensional Woven Laminate, and the Quasi - three - dimensional Carbon Fiber Composite Materials ................................ ................................ ................................ ..................... 85 6.1 Introduction ................................ ................................ ................................ ........ 85 vii 6.2 Fabrication of the composite plaques ................................ ................................ . 87 6.2.1 The f iber structures preparation ................................ ................................ .. 87 6.2.2 Fabrication process of the composite plaques ................................ ............ 88 6.3 In - plane quasi - static tensile testing ................................ ................................ .... 90 6.3.1 Specimen preparation ................................ ................................ .................. 90 6.3.2 Tensile testing set up ................................ ................................ ................... 92 6.3.3 Tensile testing load - displacement curve ................................ ..................... 93 6.3.4 ................................ ........................... 96 6.3.5 Tensile strength ................................ ................................ ........................... 98 6.3.6 Tensile failure analysis of UDL, 2DW, and Q3D composite materials specimens ................................ ................................ ................................ ................ 102 6.4 Out - of - plane low - velocity impact testing ................................ ........................ 103 6.4.1 Impact specimen preparation ................................ ................................ .... 103 6.4.2 Impact testing setup ................................ ................................ .................. 104 6.4.3 Impact loading, deflection, and energy absorption of the three composites ................................ ................................ ................................ ................... 106 6.4.4 3D DIC analysis setting in GOM and calibration with the loading - displacement curve ................................ ................................ ................................ .. 109 6.4.5 A comparison study of the impact failure mechanism of the three composite structures ................................ ................................ ................................ ................. 110 6.4.6 The impact failure process ................................ ................................ ........ 116 6.5 Conclusion ................................ ................................ ................................ ........ 121 Chapter 7. Conclusion and Outlook ................................ ................................ ........... 123 7.1 Conclus ion and Contributions ................................ ................................ .......... 123 7.2 Outlook of Future Work ................................ ................................ ................... 126 APPENDICES ................................ ................................ ................................ ................ 128 Appendix A Numerical Flowchart of a Peridynamic Program ................................ .... 129 Appendix B Impact algorithm in the Peridynamic modeling ................................ ...... 130 Appendix C Original notch definition in the Peridynamic modeling .......................... 132 Appendix D PD Simulated Laminates Damage Evolution Process ............................. 133 BIBLIOGRAPHY ................................ ................................ ................................ ........... 140 viii LIST OF TABLES Table 5 - 1 ................................ ......... 74 Table 5 - 2 .................. 74 ix LIST OF FIGURES Figure 1.1 Schematic of interaction in peridynamics. (a) Horizon and family. (b) Configuration deformation. ................................ ................................ ................................ . 3 Figure 1.2 Discretization of the horizon for material point x i . ................................ .......... 7 Figur e 2.1 Experimental setup of the three - point bending impact [24] . ........................... 13 Figure 2.2 A laminated composite panel subjected to transverse impact by a low - velocity impactor [47] . ................................ ................................ ................................ .................... 18 Figure 2.3 A schematic description of two basic Impact damage growth mechanisms of laminated composites. [47] ................................ ................................ ............................... 18 Figure 3.1 Drop weight impact experi mental setup. (a) schematic diagram, (b) lab setup. ................................ ................................ ................................ ................................ ........... 23 Figure 3.2 Drop weight impact experimental setup. ................................ ......................... 24 Figure 3.3 Impact Load ing history with the impact velocities of 2 m/s and 3 m/s. .......... 25 Figure 3.4 PMMA beam surface preparation for DIC ................................ ...................... 26 Figure 3.5 Crack initiation and propagation at different time steps, with the corresponding displacement fields (displacements in x - direction: (a), (b), and (c); displacements in y direciton (d),(e), and (f)) in the PMMA beam after the impact at the velocity of 2 m/s. (Th e crack tips are marked with the white arrows) ................................ ........................... 28 Figure 3.6 Crack initiation and propagation at different time steps, with the corresponding displacement fields (displacements in x - direction: (a), (b), and (c); displacements in y direciton (d),(e), and (f)) in the PMMA beam after the impact at the velocity of 3 m/s. (The crack tips are marked with the white arrows) ................................ ........................... 29 Figure 3.7 Cr ack initiation and propagation length history in PMMA beams subjected to impact with different impact velocities. ................................ ................................ ............ 30 Figure 3.8 Crack speeds after crack initiation in the PMMA beams subjected to impact with different impact velocities. ................................ ................................ ....................... 31 Figure 3.9 Verification of the peridynamic simulation of the crack propagation path. ... 34 Figure 3.10 Comparison of (a) the experimental observation and (b) the peridynamic simulation of the crack initiation and propagation at different time steps, with (c) the corresponding strain energy density, in the PMMA beam after the impact at the velocity of 2 m/s. ................................ ................................ ................................ ............................ 35 x Figure 3.11 Comparison of (a) the experimental observation and (b) the peridynamic simulation of the crack initiation and propagation at different time steps, with (c) t he corresponding strain energy density, in the PMMA beam after the impact at the velocity of 3 m/s. ................................ ................................ ................................ ............................ 37 Figure 3.12 Comparison of the experimental observation and the peridynamic simulation of the crack propagation velocity, in the PMMA beam after the impact at the velocity of 2 m/s and 3 m/s. ................................ ................................ .............................. 38 Figure 4.1 Peridynamic model of a lamina. (a) material constants for a lamina, and (b) biaxial loading state. ................................ ................................ ................................ ......... 42 Figure 4.2 Continuous peridynamic constant has an effective orthotropy for different / ratios. ................................ ................................ ................................ ..................... 45 Figure 4.3 Comparison of the off - composites theory. ................................ ................................ ................................ ............ 47 Figure 4.4 Peridynamic failure criteria. (a) Critical stretch of bonds in fiber direction and in matrix direction. (b) Schematic of damage for a material mode i . ............................... 49 Figure 4.5 Fiber bonds directions in (a) grid - friendly 0º, 45º, and 90º and bonds close to (b) arbitrary directions l ike 15º and 60º. ................................ ................................ ........... 50 Figure 4.6 Impact experimental setup of SEN orthotropic beam. ................................ .... 51 Figure 4.7 Comparison of impact fracture from (a) experimental result [26] and (b) PD computational results. ................................ ................................ ................................ ....... 53 Figure 4.8 Prediction of impact damage in unidirectional lamina with fiber oriented in (a) 0º, (b) 15º, (c) 30º, and (d) 90º w ith respect to the impact loading direction. .................. 54 Figure 4.9 Crack propagation patterns in different directions: (a) grid - friendly 0º, 45º, and 90º. (b) Other directions such as 30º and 75º. ................................ ................................ ... 55 Figure 4.10 Instantaneous dynamics energy release rate versus crack velocity for cracks along fiber orientation in unidirectional composites. ................................ ....................... 57 Figure 4.11 Crack in laminae with fibers oriented in 0º (first row), 15º (second row), and 30º (third row) orientations. First column: Experimental results [24] . Second column: The corresponding PD computational damage. Third column: The cor responding PD computational strain energy density. ................................ ................................ ................ 61 Figure 4.12 Crack propagation process (First row) in the composite beam with fiber in 0º with respect to the loading direction and the corr esponding strain energy density (Second row). ................................ ................................ ................................ ................................ .. 64 xi Figure 4.13 Crack velocities in a lamina with fibers in 0º orientation. Comparison of the experimental result [24] , PD simulations with ordinary fracture energy and the fitted simultaneous dynamic fracture energy. ................................ ................................ ............ 64 Figure 4.14 Crack propagation process (First row) in the composite beam with fiber in 15º with respect to the loading di rection and the corresponding strain energy density (Second row). ................................ ................................ ................................ .................... 65 Figure 4.15 Crack velocities in a lamina with fibers in 15º orientation. Comparison of the experimental result [24] , PD simu lations with ordinary fracture energy and the fitted simultaneous dynamic fracture energy. ................................ ................................ ............ 65 Figure 4.16 Crack propagation process (First row) in the composite beam with fiber in 30º with resp ect to the loading direction and the corresponding strain energy density (Second row). ................................ ................................ ................................ .................... 66 Figure 4.17 Crack velocities in a lamina with fibers in 30º orientation. Comparison of the experimental result [24] , PD simulations with ordinary fracture energy and the fitted simultaneous dynamic fracture energy. ................................ ................................ ............ 66 Figure 4.18 Percent of error of peak velocity value calculated with PD by usin g ordinary fracture energy and by using the fitted simultaneous dynamic fracture energy for fracture along fibers in 0º, 15º, and 30º orientations. And the percentage of mode II fracture in the corresponding fracture patterns [24]. ................................ ................................ ................ 67 Figure 5.1 Peridynamic failure criteria. Critical stretch of bonds in fiber direction and transverse direction. ................................ ................................ ................................ .......... 72 Figure 5.2 The intralayer material b ond interacts with the material points in the adjacent layer within the horizon. ................................ ................................ ................................ ... 73 Figure 5.3 The impact on the top surface of the laminated composite plate. ................... 74 Figure 5.4 Impact algorithm. (a) Body force added to the material point at the center of the top layer. (b) Body force. ................................ ................................ ............................ 75 Figure 5.5 Intralayer damage, delamina tion, and displacement field of the laminates [90/0/90] at damage initiation, after loading for 150 s . ................................ .................. 77 Figure 5.6 Intralayer damage, delamination, and displacement field of the laminates [9 0/0/90] after loading for 250 s . ................................ ................................ .................... 77 Figure 5.7 Intralayer damage, delamination, and displacement field of the laminates [90/0/90] after loading for 350 s . ................................ ................................ .................... 78 Figure 5.8 Intralayer damage, delamination, and displacement field of the laminates [90/15/90] after loading for 250 s . ................................ ................................ .................. 80 xii Figure 5.9 Intralayer damage, delaminat ion, and displacement field of the laminates [90/30/90] after loading for 300 s . ................................ ................................ .................. 81 Figure 5.10 Intralayer damage, delamination, and displacement field of the laminates [90/45/90] after loadin g for 300 s . ................................ ................................ .................. 81 Figure 5.11 Intralayer damage, delamination, and displacement field of the laminates [90/60/90] after loading for 250 s . ................................ ................................ .................. 83 Figure 5.12 Intralayer damage, delamination, and displacement field of the laminates [90/75/90] after loading for 250 s . ................................ ................................ .................. 83 Figure 6.1 Quasi - three - dimensional (Q3D) composite struc ture. (a) The braiding machine, (b) the braiding process, and (c) the braided Q3D glass fiber structure. .......................... 87 Figure 6.2 The carbon fiber braided structures. (a) UD laminate, (b) 2DW laminate, an d (c) Q3D fabric. ................................ ................................ ................................ .................. 88 Figure 6.3 The vacuum infusion process. (a) the layout of the releasing cloth and the fabric, (b) vacuum infusion setup. ................................ ................................ .................... 89 Figure 6.4 The three different composite plaques ................................ ........................... 90 Figure 6.5 specimens ................................ ................................ ................................ ....... 92 Figure 6.6 Tensile testing setup wit h MTS machine ................................ ........................ 93 Figure 6.7 DIC extracted longitudinal strain field for the UDL laminate 0 º specimen ... 94 Figure 6.8. Tensile testi ng loading - displacement curves ................................ .................. 95 Figure 6.9. The tensile modulus of the three composite specimens with different fiber orientations ................................ ................................ ................................ ........................ 96 Figure 6.10 Tensile modulus of the three composite specimens with different fiber orientations ................................ ................................ ................................ ........................ 98 Figure 6.11 The tensile strength of the three composite specimens with different fiber orientations ................................ ................................ ................................ ........................ 99 Figure 6.12 Specimens failure of the UDL, 2DW, and the Q3D composites in different orientations ................................ ................................ ................................ ...................... 101 Figure 6.13 T he three composite plaques with the corresponding painted impact plate samples ................................ ................................ ................................ ............................ 103 Figure 6.14 Out - of - plane impact testing setup ................................ .............................. 104 xiii Figure 6.15 The impact loading and deflection history ................................ .................. 106 Figure 6.16 The impact Loading and absorbed energy to deflection ............................ 108 Figure 6.17 Calibration of the DIC deflection curves with Dynatup machine recorded data ................................ ................................ ................................ ................................ .. 109 Figure 6.18 impact failure on both sides of the samples ................................ ................. 111 Figure 6.19 The impact failure on the bottom surface of the samples at different time steps ................................ ................................ ................................ ................................ . 113 Figure 6.20 The DIC analysis of major strain on the bottom surfaces of the samples at different time steps ................................ ................................ ................................ .......... 114 Figure 6.21 The DIC analysis of out - of - plane displacement fields during the impact process ................................ ................................ ................................ ............................. 116 Figure 6.22 The DIC analysis of displacement fields in x - direction during the impact process ................................ ................................ ................................ ............................. 117 Figure 6.23 The DIC analysis of major strain fields during the impact process ........... 119 Figure 6.24 The DIC analysis of shear angle (in - plane shear strain) fields during the impact process ................................ ................................ ................................ ................. 120 Figure A.1 The flowchart o f a peridynamic program ................................ ................... 129 Figure B.1 Impact algorithm in the PD modeling. ................................ .......................... 130 Figure C.1 Impact algorithm in the PD modeling. ................................ ......................... 132 Figure D.1 Intralayer damage, delamination, and displacement field of the laminates [90/15/90] at damage initiation, after loading for 150 s . ................................ .............. 133 Figure D.2 Intralayer damage, delamination, and displacement field of the laminates [90/15/90] at after loading for 350 s . ................................ ................................ ............ 134 Figure D.3 Intralayer damage, delamination, and displacement field of the laminates [90/30/90] after loading for 200 s . ................................ ................................ ................ 135 Figure D.4 Intralayer damage, delamination, and displacement f ield of the laminates [90/30/90] at after loading for 400 s . ................................ ................................ ............ 135 Figure D.5 Intralayer damage, delamination, and displacement field of the laminates [90/45/90] after loading for 200 s . ................................ ................................ ............... 136 xiv Figure D.6 Intralayer damage, delamination, and displacement field of the laminates [90/45/90] after loading for 400 s . ................................ ................................ ................ 136 Figure D.7 Intralayer damage, delamination, and displacement field of the laminates [90/60/90] at damage initiation, after loading for 150 s . ................................ .............. 137 Figure D.8 Intralayer damage, delamination, and disp lacement field of the laminates [90/60/90] at after loading for 350 s . ................................ ................................ ............ 137 Figure D.9 Intralayer damage, delamination, and displacement field of the laminates [90/75/90] at damage initiation, after loading for 150 s . ................................ .............. 138 Figure D.10 Intralayer damage, delamination, and displacement field of the laminates [90/75/90] at after loading for 350 s . ................................ ................................ ............ 138 xv KEY TO SYMBOLS material density u displacement horizon f pairwise bond force b body force relative position between material points relative displacement C micromodulus ( material constant ) of bond s micromodulus ( material constant ) in fiber direction micromodulus ( material constant ) in all other directions the homogenized continuous material constant material constant in the longitud inal (fiber) direction material constant in the transverse direction s stretch of bonds s t stretch of bonds in a lamina xvi s 1t critical stretch of bonds in fiber direction in a lamina s 2t critical stretch of bonds in the transverse direction in a lamina s c compression of bonds in a lamina s 1c critical compression of bonds fiber direction in a lamina s 2c critical compression of bonds in the transverse direction in a lamina micropotential W str ain energy density W PD str ain energy density in peridynamics W CL str ain energy density in continuum mechanics d t time step dV volume of material point f z (t) time - dependent body force (loading) added to material point i index of the current material point j index of any other material points within the horizon 1 Chapter 1. I ntroduction Fiber - reinforced p olymer matrix composite material s are wi dely used in advanced aircraft and automotive structures due to their excellent mechanical properties, such as high strength to weight ratio, high stiffness to weight ratio , and high fatigue resistance . The composite material s are made by combining light - weight polymer ic resin with stiff and strong reinforcing fibers , such as carbon fibers and glass fibers . By combining the t wo compo nents together, the composite material s can be strong , stiff , and lightweight [1] . In this proposal, the composite materials of interest are laminated fiber - reinforc ed composite materials. T he lamina e can be a unidirectional ply or woven fabric ply. 1.1 Fiber - Reinforced Composite Materials 1.1.1 Basic characteristics The most common form of fiber - reinforced composites for structural applic ation s is the laminated compos ite [2] . A c omposite l aminate is made by stacking several lamina s together while a lamina is a thin ply of made with reinforced fibers and matrix materials . The fibers generally can be unidirectional alignment or woven. T he matrix can be ceramics, metal, or polymers. For example, epoxy resins, the commonly used polymer matrix material . The fiber ori entation of each lamina and the sequence of different laminas contribute to various functional or structural properties of the laminated composites. Therefore, specific laminated composites can be designed by using different fiber or matrix materials, by f abricating laminates with different orientations of lamina plies [1] . 2 1.1.2 Failure of composites In this study , we will focus on the dynamic behaviors and failure analysis of laminated composites. Complicated deformation and failure processes occur in composites when they are applied as structural components due to the complex materials constructions of composites . The main failure modes can be divided as follows [3] . 1) Delamination -- One of the most common structural failures of laminated composites is delamination. L aminated composite s are more of structure s than material s. When subjected to out - of - plane loading, due to t he bending and transverse shear ing , delamination can occur . [4 6] . 2) M atrix Failure -- Direct matrix tensile failure occur s when composites are subjected to critical tensile loading [7] . The commonly referred matrix compression failure is associated with the matrix shear failure , which occurs at an angle such as 45° with respect to the compression loading direction. 3) F iber F ailure -- Catastrophic composite fa ilure occurs with a large amount of energy release due to the fiber tensile failure in composites [8] . For compression, w hen subjected to critical compression loa ding, fiber kink and buckling can occur in composites [9,10] , owing to microfiber bulking and f iber misalignment. 4) Other Failures -- F or composites under static loading, matrix cr eep can occur due to its viscoelastic property [11] . Fatigue damage initiation and fatigue failure can occur when composites are loaded with cyclic forces [12] . 3 1.2 Peridynamic th eory Damag e and fracture of composites have been studied experimentally as well as numerically , such as with Finite Element (FE) methods [13 15] . However, the FE method has its limitation in dealing with problems of discontinuity which occurs commonly during damage process because the equation of motion is in a partial differential form of di splacement fields. Re - meshing with a prior knowledge of t he dama ge path may be needed for the FE method t o study fracture and damage processes [16,17] . Peridynamics (PD), a nonlocal form of continuum mechanics, has been proposed by Silling from Sandia National Lab [18 20] . It is formulated with an integration approach rather than the derivation approach used in continuum mechanics. The PD method can avoid the difficulty of discontinuity when used to study fracture problems. 1.2.1 Bond based peridynamics Figure 1 . 1 Schemati c of interaction in peridynamics . (a) Horizon and family. (b) Configuration deformation . 4 Peridynamic theory defines that in a reference configuration B , each material point x has a subdomain H with a radius of , which is called the material horizon, as sh own in Figure 1 . 1 (a) . Point x interacts with all the point s x in its horizon through the pairwise force f , which has a unit as force per volume squared. T he equation of motion at any time t for material point x can be expressed as shown in Eq. 1 - 1 , where is the density and u is the displacement. The force density for point x in peridynamics is an integration of all pairwise forces between x and x in its horizon , which is different from the differenti ation in continuum theory . b is the body force. ( 1 - 1 ) The relative position between point x and x is , and the relative displacement is . ( 1 - 2 ) ( 1 - 3 ) For each bond, the relative elongation is presente d as the stretch s . 5 ( 1 - 4 ) The p airwise force can be defined as ( 1 - 5 ) where c is the peridynamic micromodulus ( material constant ) of bonds. For any material point outside the horizon of material point x , the pairwise force is zero. The material is defined to be micro elastic if the pairwise fo rce can be derived from a micropotential as in Eq. ( 1 - 6 ) below. ( 1 - 6 ) ( 1 - 7 ) ( 1 - 8 ) 6 Then the micropotential can be further expressed with the formulation containing micromodulus , stretch and relativ e displacement as shown in Eq. ( 1 - 7). T he str ain energy density in peridynamics can be calculated with integration , as shown in Eq. ( 1 - 8). Micromodulu s C can then be calculated by equating the strain energy density in PD and that from continuum theory. 1.2.2 Dam age criteria Material point x connects with any other point x in its horizon with a PD bond. The bond breaks when bond stretch s is over the critical stretch s 0 . To describe the connection of a bond, a history - dependent scalar function µ is defined as sho wn in Eq. ( 1 - 9). It equals one when the bond stretch is smaller than the critical stretch, which means the bond still work s . Otherwise, the bond is broken and the corresponding bond force becomes zero. Critical stretch can be determined exp erimentally or t heoretically [19] . ( 1 - 9 ) ( 1 - 10 ) The d amage of material point x is defined as the function in Eq . ( 1 - 10). If there is no bond broken for point x within its horizon, the damage value equals 0. And w hen all 7 bonds originally connected to a point are broken, the damage value of the material point is 1 , which corresponds to the whole material poi nt being peeled off. A c rack is considered to occur when the damage function reaches a value close to 0.5. 1.2.3 Numerical algorithm In peridynamics , the reference region is uniformly discretized into fi nite points (material points). Each point has a certain v olume in the configuration , as shown in 0 . Figure 1 . 2 Discretization of the horizon for material point x i . Then for material point i , the e quation of motion ca n be discretized numerically as ( 1 - 11 ) w here the sub script i is the current point number, j is the nonlocal point number within its horizon, and n is the integration st ep number. is the volume of point and is 8 represented by a square lattice area for 2D problems when the model is discretized as orthogonal uniform grids . is the displacement for point i at the time step n . T he displacement for the ne xt time step can be obtained explicitly from the cen tral difference formulations: ( 1 - 12 ) ( 1 - 13 ) 1.3 S copes of the dissertation In this work, dynamic damage, fracture, and failure process will be investigated experimentally with impact testing , and numerically with PD modeling and simula tion. The main challenges of this study are : 1) Adjust experimental conditions to simulate specific dynamic condition s of composites encountered in real life. Such as the in - plane impact fracture, and the out - of - plane impact damage in composites structures . 2) Develop a n effective constitutive PD mode l for composites and verify it. Details include building and testing the continuous PD meso - scale material constants for 9 the lamina, defining the bonds failure criteria in the model, adjusting the numerical integr ation and its stability etc. 3) Add specific criteria to the PD modeling to capture the dynamic fracture and failure process . For example, the application of instantaneous fracture toughness to the critical bond failure criteria in PD. 4) E xt end the lamina PD model to laminated composites PD model. D etails include defining the intralaminar and the interlaminar PD material constants and failure criteria for laminates, adjusting the numerical efficiency etc. 5) Design the composite materials with novel fiber layout and woven structures, to explore the composites with high stiffness, high strength, and impact resistance for vehicle light - weighting applications . 1.4 Outline of the dissertation The dissertation is organized as follows: Chapter 1 introduces the problem an d defines the scope of work. Chapter 2 provides a literature review on current research about fiber - reinforced composite materials and peridynamics , including the impact damage and failure of laminated composites, the peridynamic modeling of fracture and d amage in comp osi tes. Chapter 3 presents a study of the impact loading /energy s influence on the fracture process in the PMMA simply supported beams, analyzes the limitation of the traditional 2D bond - based PD method to capture the dynamic fracture process by simulating the 10 impact fracture. The limination will be confined with the fitted failure criteria for impact fracture in composites with higher peak velocities in the following chapter. Chapter 4 develops a bond - based mesoscale peridynamic model for orth otropic composite materials. The model defines a continuous in - plane material constant for orthotropic materials as the mesoscale off - axis modulus in the laminated composite theory. The changes continuously from the fiber direction to the tra nsverse direction with an effective orthotropy. This model differs from the other PD composite models which define a micro - modulus for fibers and for matrix. It is eligible for simulations of large vo lume of materials. By implementing the cra ck - velocity - related strain energy release rate into the failure criteria, the dynamic fracture propagation process and crack velocities can be captured more accurately with the model besides the final failure in the unidirectional fiber composite beams. Ch apter 5 extends the mesoscale model into three - dimensional, introduces the intra - layer and the inter - layer material bonds failure criteria . With the impact loading added to the laminated composite, the lamina damage and the delamination process will be sim ulated with the PD model. Chapter 6 investigated previously developed quasi - three - dimensional (Q3D) braiding method. Three composite structures were fabricated and tested in comparison. They are the Q3D [0 / ± 60] 4 carbon fiber composite plates, the [0/ ±60] 4 laminates made of unidirectional plies (UDL), and the two - dimensional triaxial braided plies [ 0/ ± 60 ] 4 (2D3A). The in - plane tensile and the out - of - plane impact experiments were performed 11 combining 3D DIC method . The in - plane properties and the out - of - plan e impact - induced intra - layer damage and the inter - layer delamination will be discussed . Chapter 7 presents the conclusions, contributions of the dissertation and the future work. 12 Chapter 2. L iterature Review The reliability study and failure prediction of composit es are critical issues during their application s , especially under dynamic loading conditions such as impact. The studies of d ynamic failure of composites will be reviewed in this chapter . They include in - plane fracture propagation, delamination due to in - plane compression, and delamination resulting from out - of - plane impact. 2.1 In - plane F racture P ropagation One of the typical failure mode s of the composites during the structural application is the in - plane dynamic fracture, which has been widely studied expe rimentally and numerically . 2.1.1 Experimental S tudy The experimental setup of the in - plane dynamic fracture studies is usually designed by adding the uni axial impact loading to a central or edge notched plate, or by adding an in - plane three - point bending impac t on an edge notched laminated beam [21] . The dynamic failure process is dependent not only on the critical strain energy release rate (toughness) of the material but also on the loading rate added and the crack velocity in the specimen. Shokrieh [22,23] studied the effects of strain rate s on properties and fracture of unidirectional glass fiber epoxy composi tes in the range of 0.001 100 s - 1 by using a servo - hydraulic testing apparatus. The e xperimental results show a significant increase in the tensile strength and shear strength by increasing the strain rate. Both unidirectional 13 and quasi - isotropic laminates were investigated with an increa sing strength with the loading strain rate. Another experimental setup to study the dynamic fracture is the there - point - bending impact, besides the uniaxial dynamic loading. Lee and Tippur studied the dynamic fracture propagation in unidirectional graphit e/epoxy composite s T800/3900 - 2 [24,25] , PETI - 5 and IM7/PETI - 5 [26,27] . Rectangular composite plate sample s with a single - edged notch are loaded with the impact at the center of the top surface , as shown in Figure 2 . 1 . Mode - I or mixed - mode (mode - I and - II) dynamic fracture s were observed with the fiber in a different orientation . The deformation fields and the rapid crack growth in fiber - reinforced composites were recorde d by using the digital image correlation method and high - speed camera photography . The dynamic fracture toughness values with fiber orientation angle s were extracted. There is a good experimental correlation between dynamic toughness and crack - ti p velocity histories for samples with fiber orientations in 0 °, 15 °, and 3 0 °, which means the dynamic toughness is crack - velocity - related in dynamic situations. Figure 2 . 1 E xperiment al setup of the three - point bendin g impact [24] . 14 2.1.2 Numerical M odeling and S imulation FEM and PD methods have been employed to study the in - plane fractur e of composites numerically [28,29] . Cahill [30] stud ied the crack propagation in the linear elastic unidirectional fiber reinforced composites with an enriched finite element method . The results sho w that the material orientation is the driving factor of crack propagation in the composites, the crack will predominantly propagate along the fiber direction, regardless of the specimen geometry, loading conditions or presence of voids. Pineda and Waas [31] proposed a thermodynamically - based work potential theory for modeling intralaminar progressive damage in laminated composites. The theory was implemented into a FEM for simulating the damage. The method a ssumes that the material fracture initiates and propagates from the matrix microdamage. By studying the uniaxial tension on the T800/3900 - 2 panels with a central notch and different fiber orientation and stacking sequences, the very good correlation was ac hieved quantitatively for g lobal load versus displacement . Recently peridynamics has been employed for pro gressive damage in composite materials due to the advantage of the method . Kilic and Madenci [32] investigated a PD model with fiber and matrix in separated material particles and predicted the matrix damage in laminated composites accounting for the inhomogeneous distinct p roperties of the fiber and matrix . Oterkus et al. [33,34] built the PD model for composites with four material constants in a lamina and predicted the deformation and damage in laminates. Hu and Bobaru [35] proposed a PD model to study dynamic crack propagation by applying the J - integral into PD . However, the application of dynamic fracture toughness 15 into the PD failure criteria has not been investigated yet, which will be in vestigated in this thesis. To study the impact compression of laminates under low and high strain rate, constitutive and failure models should firstly be developed [8] . Sun [36,37] developed a rate - dependent nonlinear constitutive model and a dynamic compressive strength model (fiber micro - buckling model) for the unidirectional carbon fiber composite. The model was established based on the low strain rate off - axis test data and it can predict the failure and micro - buckling of the composites in different compression strain rate. An analytical model was developed by K utlu [38] for simulating the compression response, fr om in itial loading to the final collapse of laminated composites containing multiple through - the - width delamination s . The model is comprised of three parts: a stress analysis, a failure analysis, and a contact analysis. Also, it was inputted into a nonline ar finite element code to simulate the compression on the graphite/epoxy composites. Good agreements were obtained between the predictions and the test data from the initial loading to the final collapse of the specimens. Other FEM modeling and simulations are summarized in the literature [39] , the relatively limited and future needed FEM research on laminated composite s modeling are summarized as [39] : 1) Material nonlinearity effects on the structural behavior of composite laminates. 2) Fa ilure and damage analysis under viscoelastic effects such as thermal and creep effects. 3) Failure and damage analysis under cyclic loading. 4) Micromechanical approach for damage analysis. 5) Analysis of the damage evolution in composite laminates. 16 6) Multiscale mode ling of crack initiation, propagation, and overall structural failure. Modeling a nd simulation of the compressive failure and structure buckling have also been studied limitedly with peridynamics [40 44] . Bond based PD was used by Kilic [40] to investigate the elastic stability of simple structures to determine the buc kling characteristics of the peridynamic theory . Also, the bond based PD theory w as used to simulate basic compression damage of concrete materials [41,42] . Extended non - ordinary state - based peridynamics was developed with the maxim um tensile stress criterion and the Mohr - Coulomb criterion by Wang [43] . The PD model was then used to simulate the crack initiation, propagation , and coalescenc e in the rocks subjected to compressive loads. For composite materials, only Hu [44] used the developed bond based PD model to simulate the compression behavior. 2.2 Out - of - plane I mpact I nduced D elamination and P lane F ailure One of the major weaknesses of the composites is the li mited transverse strength when subjected to the out - of - plane impact loading such as the bird strike, ballistic impact etc. Complicated deformation and failure occur in composites when subjected to impact loading, especially under different impact energy. T he impact force and energy can be metry and loading directions [45,46] etc . 2.2.1 E xperimental S tudy The experimental studies of the impact on composite materials are usually conducted with the drop - weight and Dynatup machines as shown in Figure 2 . 2 . Different material deformation and failures modes occur when t he laminated composites are loaded with 17 different energy of impacts. Under low - velocity impact, plane damage initiates and propagates until delamination happens . The basic damage mechanism resulting from line - loading impact can be summarized as shown in Figure 2 . 3 by Chang [47,48] : 1) Intra - ply matrix cracks (referred to as the shear or bending matrix cracks) are the initial damage mode. 2) Delamination initiate s from these matrix cracks which propagate into the nearby interface with the dissimilar materials. 3) Extensive multiple micro - matrix cracks will be generated along with the delamination propagation. 4) A shear matrix crack located in the inner plies of the laminates will generate a substantial delamina tion along the bottom interface and a small, confined delamination along the upper interface of the cracked ply. 5) A bending matrix crack located at the surface ply of the laminates will generate a delamination along the first i nterface of the cracked ply ( Figure 2 . 3 ). Topac [49] investigated the damage initiation and growth process during low - velocity impact on [0 7 /90 4 ]s and [90 7 /0 4 ]s cross - ply CFRP laminates . The two - dimensional damage progression and dynamic strain fields during impact w ere tracked and recorded by using an ultra - high speed camera and DIC technique. 18 Figure 2 . 2 A laminated composite panel subje cted to transverse impact by a low - velocity impactor [47] . Figure 2 . 3 A schematic description of two basic Impact damage growth mechanisms of laminated composites. [47] Three - dimensional studies of the delamination for laminated composites are also investigated experimentally [50 54] . Delamination results from the intra - plane damage propagation or the directly inter - plane shear or open loading [55] . The characterization of three - dimensional delamination can be conducted with the CT scan technique. 19 2.2.2 Numerical M odeling and S imulation FEM methods have been employed to study the deformation and failure of composit es due to out of impact loading [51,52,56 60] . To simulate the impact failure and delamination with FEM, limitations are summarized as [58] : 1) An interface element is necessary t o simulate matr ix cracks and delamination. 2) A coupling between the intra and inter - ply damages is needed and information must be exchanged between the interfaces elements of the matrix cracks and delamination. Peridynamics has been applied to investigate t he impact delamination of composites due to the advantage of simultaneous fracture mechanism of the method itself. Xu [61] firstly studied the impact delamination of laminates with PD. The inter - lamina bonds were set the same as the matrix bonds and only stretch bond failure was investigated. The development of proper interlaminar bonds is s ignificant for PD modeling of impact delamination of laminated composites. 20 Chapter 3. Analyzing the D ynamic F racture P rocess in P olymethyl - methacrylate (PMMA) B eams with T hree - point - bending I mpact T esting and Peridynamic S imulation 3.1 Introduction Dynamic fracture pr ocess in polymethyl - methacrylate (PMMA) beams have been investigated during the three - point - bending impact tests at different im pa ct velocit i es , conducted in a drop - weight impact tower instrument. The impact - induced crack initiation and propagation have be en recorded with a high - speed camera, to determine the instantaneous fracture length and crack velocity during the impact process. The beam deformation and displacement fields were extracted and analyzed by using the digital image correlation (DIC) techniq ue during the impact. The impact loading history has been recorded with a load cell attached to the dropping weight. The whole experimental study is a suitable technique to determine the influence of the impact velocit i es (impact energy) on the dynamic fra cture init i ation and propagation at different crack speeds. Dynamic fracture in structural materials is a significant issue because it concerns the failure of structural materials in their dynamic service. The i mpact is one of the most common dynamic loadi ng forms but complicated since the material properties and failure behaviors are complex in a dynamic situation. The dynamic fracture problems have been studied experimentally [1 3] and numerically [4,5]. Joudon [1] studied the dynamic stress intensity fac tor by using a strain gauge method associated with high - speed cinematography on a three - point - bending test with specimens made of M21 epoxy resins. Cramer [2] conducted dynamic fracture experiments using boron - doped silicon single 21 crystals followed by clea vage fracture with the propagation of a faceted crack front with amorphous materials. Owen [3] studied the critical dynamic stress over a range of loading rates of 2024 - T3 aluminum sheets ranging in thickness from 1.63 - 2.54 mm. The dynamic fracture process in three - point - bending beams made with an isotropic polymer [4] and orthotropic composite materials [5] have been numerically simulated with peridynamics . Fracture in PMMA ha s also been studied. Takahashi [6] investigated multiple dynamic fracture paramet ers such as the dynamic stress intensity at the crack tip as well as crack velocity and acceleration. They analyzed the initiation and propagation behavior of the crack of thin PMMA sheet under tensile load. Lataillade [7] studied the mechanical behavior o f PMMA under various loading rates as well as the properties of the polymer at yield stress and fracture toughness of PMMA and tensile loading rates. On the other han d, Loya [8] performed a quasi - static three - point bending test on PMMA beams and recorded the crack - front propagation process throughout the specimen thickness. The crack - length and the average steady crack propagation were extracted and studied. In a more recent study, Huang [9] adopted a different technique, dynamic semicircular bend testing, and performed fracture testing on PMMA specimen with a split Hopkinson pressure bar. Their study determined the fracture velocity under different loading rates as wel l as surface fracture toughness and its relationship with fracture energy. However, the impact - induced dynamic fracture process in PMMA with a precise record of crack propagation and speed has rarely been studied before, especially the fracture caused by impact with different velocities. In the former studies, the recording time step period is relatively long. For example, only the average crack velocity for the whole 22 fracture can be obtained. To better understand more detail dynamic fracture process, inc luding the beam deformation and crack propagation, the more precise experimental investigation in more precise time steps is essential. In this chapter , the impact - induced dynamic fracture process in PMMA beams has been investigated. The experiment was c onducted with drop - weight tower instruments . During the impact test, the impact loading history has been recorded by a load cell attached to the bottom of the dropping wei g ht . The impact process was recorded with a high - speed camera at the time resolu t ion of about 15 microseconds. The impact - induced crack initiation and propagation have been extracted from the images recorded with the high - speed camera, to determine the instantaneous fracture length and crack velocity during the impact process. The beam def ormation and displacement fields were calculated and extracted by using the digital image correlation (DIC) technique . The fracture in beams subjected to different impact velocities ha s been compared and analyzed . 23 3.2 Experimental Testing of the Impact F racture in PMMA Beam 3.2.1 Impact e xperiment setup Figure 3 . 1 Drop weight i mpact experimental setup . (a) schematic diagram, (b) lab setup. The experiment was conducted by performing a three - point - bending impac t test ing on a single - edge - notched PMMA beam specimen by using a drop - weight impact tower as shown in Figure 3 . 1 . Figure 3 . 1 (a) is the schematic diagram of the impact setup, Figure 3 . 1 (b) is the setup in the lab. The drop weight was located above the PMMA sample and set free to drop and impact at the center of the specimen top surface . Two different impact velocities (2 m/s and 3 m/s) were achieved by dropping the weight/impactor from different heights. To monitor the impact force applied to the PMMA beam, a load cell was attached to the bottom of the drop weight to record the loading signals during the impact process. The signals from the load cell were then amplified by an amplif ier, displayed and recorded with an electronic oscilloscope provided by National Instruments. 24 A high - speed camera was placed perpendicular to the vertical surface of the specimen to record the beam deformation and the fracture initiation and propagation du ring the impact process, as shown in Figure 3 . 1 . The recorded images were used to extract the crack propagation details and the corresponding displacement field contours with the DIC method at different time steps. The sample bea m is made with PMMA (purchased from McMaster - Carr) and prepared with the length of 140 mm, the width of 38 mm, and height of 25.4 mm. A notch of 16 mm was initially cut in the center of the bottom edge of the plate as shown in Figure 3 . 2 . The notch tip was then further scratched with the thinner knife to prepare the original micro crack tip. Figure 3 . 2 Drop weight impact experimental setup . 3.2.2 Impact force on PMMA beam The impact loa ding s were extracted from the signals recorded with the load cell. The loading recording resolution was set as 10 s . The loading history curves of the impact processes with different impact velocities (v = 2, 3 m/s) are presented in Figure 3 . 3 . The 25 loading/force curves initiate from zero before the moment when the impactor (loadcell) contact the top surface of the PMMA specimen. After reaching the peak value, the force then drops suddenly till even negative values, which indicate the loadcell recording of the reflection of the impact stress/strain wave. The peak values of the impact force at different impact velocities are different. The peak force for impact at 3 m/s is larger than that at 2 m/s. Figure 3 . 3 Impact Loading history with the impact velocities of 2 m/s and 3 m/s. 3.2.3 Digital image c orrelation (DIC) analysis setup The DIC method is an optical method of experimental mechanics that can be used to measure and calculate the displacement, deformation, and strain fields of the specimen surface in mechanical testing [62 64] . The DIC testing preparation steps were set as follows: Firstly, the specimen surface was cleaned and polished to keep smooth. Then the 26 white background painting with evenly lo cated black speckles painting w as sprayed on the surface of the specimens. The size of the black painting speckles and the appropriate distances between each speckle can be determined by the suggestions in [65] . A prepared specimen surface is shown in Figure 3 . 4 . The high - speed camera was then set to focus on the crack propagation region on the specim en. The resolution of the camera was 256 x 256 pixels, which correlated to the square area at the center of the specimen. A sequence of images w as extracted from the video recorded with the camera, with the time increment of 15 milliseconds. The images wer e then imported into the software GOM C orrelate for the DIC analysis of displacement, deformation fields, and the crack propagation process. Figure 3 . 4 PMMA beam surface preparation for DIC For the DIC a nalysis with GOM C orrelate software , a surface component was created at first . Emphasizing the granularity of the sample, a surface co mponent of 37 pixels was chosen, with a facet offset of 18 pixels. The area of interest was selected by using the Select/D eselect Polygon tool to include the crack propagation region. The original notch 27 tip and the crack tip at each time step can be located in the images with GOM, the absolute crack propagation distances were directly extracted with GOM . 3.2.4 Impact fracture proce ss with displacements fields from DIC analysis The impact fracture process in PMMA beam with the corresponding displacement fields ha s been presented in Figure 3 . 5 and Figure 3 . 6 , with the impact velociti es at 2 m/s and 3 m/s respectively. Figure 3 . 5 shows the fracture process in PMMA beam with the impact at a velocity of 2 m/s. The fracture initiates at 285 s after the dropping weight contact s the top surface of the PMMA beam. During that time, the dropping weight subject impact loading at the middle of the top surface of the beam, which cause s the beam bending with the increase of the stress concentration a t the crack top. The crack propagates from the initiation to 60 s , till 150 s , with the crack tip marked with the white arrows. During the fracture process, as the crack length increases, both displacements in x and y directions increase correspondingly. The detail displacement field contour with a color bar is shown in Figure 3 . 5 . Displacements fields are symmetric to the vertical line of the original notch. The change of the color in the displacement contour indicates the defor mation process of the beam during the impact process, which lasts during the time period as short as about 150 s . 28 Figure 3 . 5 Crack initiation and propagation at different time steps, with the corresponding displacement fields (displacements in x - direction : (a), (b), and (c); displacements in y direciton (d),(e), a nd (f)) in the PMMA beam after the impact at the velocity of 2 m/s. (The crack tips are marked with the white arrows) Once the dropping weight reaches the top surface of the beam, the beam is subjected to an impact loading and starts to bend due to the sim ply support ed boundary conditions at the bottom surface. During the impact bending process, the tensile stress concentration increases at the tip of the original notch. The crack initiates to propagate once the stress intensity factor reaches the critical value (fracture toughness). Figure 3 . 6 shows the fracture process in the PMMA beam with the impact at a velocity of 3 m/s. The fracture initiates at 110 s after the dropping weight contact s the top surface 29 of the PMMA beam. The crack propagations from the initiation, to 60 s , till 150 s , with the crack tip marked with the arrows. Similarly, during the fracture process, the crack length increases, and both the displacements in x and y directions increase correspondingly. The det ail displacement field contour with color bar s can be found in Figure 3 . 6 . Displacements fields are also symmetric to the vertical line of the original notch. The displacements contours indicate the deformation process of the beam during the impact process, which lasts during the time period as short as round 150 s too. Figure 3 . 6 Crack initiation and propagation at different time steps, with the corresponding displacement fields (displacements in x - direction : (a), (b), and (c); displacements in y direciton (d),(e) , and (f)) in the PMMA beam after the impact at the velocity of 3 m/s. (The crack tips are marked with the white arrows) 30 The crack initiation and propagation length history in PMMA beams subjected to impact at different impact velocities are shown in Figure 3 . 7 . During the impact process, the time step when the dropping weight contact s the surface of the beam is set as 0. The crack initiation time for PMMA beam subjected to impact at velocities of 2 m/s and 3 m/s are 265 s and 110 s , respectively. Obviously, the loading time before the crack initiates is much longer for higher impact velocity, shorter for lower impact velocity. The cracks propagate t o 20 mm within about 150 s , but with a different slope of the length cu rves, which means the crack velocities are different, as shown in Figure 3 . 8 . Figure 3 . 7 Crack initiation and propagation length history in PMMA beams subjected to impact with dif ferent impact velocities. 31 Figure 3 . 8 Crack speeds after crack initiation in the PMMA beams subjected to impact with diff erent impact velocities. The crack propagation speeds after crack initiation in PM MA beams subjected to impact with different impact velocities are shown in Figure 3 . 8 , in which the time is set as 0 at the crack initiation point. The crack velocities in beams subjected to different impact loading have the simil ar trend. Crack velocities start from a relatively lower value a round 100 m/s, rise to the peak value a round 200 m/s, then decrease till the fracture. The peak crack velocities for fracture in beams subjected to different impact are different. For fracture in the beam under the impact of 3 m/s, the peak crack velocity is highest as 212 m/s. The peak crack velocities in the beam under the impact of 2 m/s are as low as 195 m/s. 32 To numerically study the impact - induced fracture propagation, peridynamics was em ployed. The peridynamic simulation of the fracture propagation caused by the different impact energy/velocity is discussed in the coming sections. 3.3 Peridynamic Simulation of the Impact Fracture in the PMMA Beam The three - point - bending impact - induced fractur e process was simulated with the two - dimensional PD method . The experimental results of crack length and crack basically were compared with the PD simulation. 3.3.1 Peridynamic simulation settings The detail s about the peridynamics method derivation and the ma terial damage definition have been stated in the introduction in chapter 1. The PD simulation used for simulating the impact on PMMA beam was the two - dimensional PD method. The 2D material constants C and the critical stretch s of the material bonds are sh own as the equation s below [66] . ( 3 - 1 ) ( 3 - 2 ) 33 The PMMA beam/plate is discretized into the 280*76 orthogonal PD grids, with a square unit length size of dx = 0.5 mm . In the PD simulation, each material point has a certain horizon with the size of = 3.2*dx as suggested by Hu [67] and Silling [19] . The time increment (time step) for the explicit integration is specified as dt = 1 x 10 - 7 s , which is effective for the simulation in this case. The PMMA beam/plate is simply supported as the boundary conditions. In P D settings, the supported point and the family points in its horizon were set with the displacement and velocity in the y - direction with the value 0. The setting of the impact algorithm in PD is stated in the Appendix B. The definition of the original crac k/notch and the explicit impact algorithm in this study are described in Appendix C . 3.3.2 Peridynamic simulation results and discussion The PD simulated crack propagation length changing according to time overall matches well with that recorded from the impac t experiments, as shown in Figure 3 . 9 . The comparison of the crack propagation in a beam under the impact at 2 m/s is shown in Figure 3 . 9 (a). For the crack propagation before about 12 mm, the PD simulated curve has the slope as large as that of the experimental curve. After 12 mm, the PD simulated crack curve increases slower (lower crack velocity) than that of the experimental curve. The comparison of the crack propagation in a beam under the impact at 3 m/s is shown in Figure 3 . 9 (b). The comparison of the simulation and the experimental curves are different. For the crack propagation before about 10 mm, the PD simulated curve has the larger slope than that of the experimental cu rve. After 10 mm, the PD simulated crack curve increases at a similar pace with that of the experimental curve. More details about the crack propagation velocity are discussed in the following sections. 34 Figure 3 . 9 Verification of the peridynamic simulation of the crack propagation path. 35 Figure 3 . 10 Comparison of (a) the experimental observation and (b) the peridynamic simulation of the crack initiation an d propagation at different time steps, with (c) the corresponding strain energy density, in the PMMA beam after the impact at the velocity of 2 m/s. The PD simulation of the fracture propagation processes is shown in the Figure 3 . 10 and Figure 3 . 11 , which includes: The experimental observation of the impact fracture at the time steps from the fracture initiation, to 45 s , 105 s , and 150 s , (first row), the material damage contours from PD simulation at the corresponding dynamic fracture time steps (second row), and the corresponding PD - simulated strain energy density contours (third row). 36 The PD simulation of materia l damage (crack propagation) contours describe s the consistent crack propagation process in each beam specimen. The material damage is illustrated as the azure color line in the contours, which has a damage ratio value around 0.4 according to the color bar description. The PD simulation of the crack length matches well with the experimental observation at each time step as shown in the Figure 3 . 10 and Figure 3 . 11 . The strain energy density contours demonst rate the corresponding energy distribution status during the crack propagation process. The strain energy density contours are symmetrical due to the symmetrical beam deformation by the crack propagation in the vertical direction. The much higher strain en ergy density accumulated at the front side of the crack tip , which of the crack propagation. In the crack initiation stage, the strain energy density is severely concentrated at both the crack tip and loading areas. As the crack propagates, the less distribution of the energy contour explains that as the crack propagates, the newly generated fracture surface dissipated the strain energy in the sample. 37 Figure 3 . 11 Compar ison of (a) the experimental observation and (b) the peridynamic simulation of the crack initiation and propagation at different time steps, with (c) the corresponding strain energy density, in the PMMA beam after the impact at the velocity of 3 m/s. 38 Fig ure 3 . 12 Comparison of the experimental observation and the peridynamic simulation of the crack propagation velocity, in the PMMA beam after the impact at the velocity of 2 m/s and 3 m/s. 39 The comparison of the experimental crack propagation velocity and the PD simulation crack velocity is shown in the Fig ure 3 . 12 . The PD simulation results match well with the impact experimental results overall. The Fig ure 3 . 12 (a) describes the fracture in the PMMA beam subjected to the impact at 2 m/s. The figure shows that the PD simulated crack velocity curve has a lower peak value than the experimental crack velocity curve, while the crack velocity changing trends are co nsistent. The Fig ure 3 . 12 (b) describes the fracture in the PMMA beam subjected to the impact at 3 m/s. The figure illustrates that the PD simulated crack velocity curve has a slightly higher peak value than the experimental result s, and the crack velocity changing trends are the same. The PD simulation of the crack velocity in the PMMA beam under impact at 3 m/s matches better with the experimental results than that for the impact at 2 m/s. A V maxPD is defined as the peak crack vel ocity simulated with the bond - based PD method. The V maxEx p is defined as the peak crack velocity in the beam measured from the impact experiment. The comparison of the two crack velocity curves indicates that the ratio of V maxPD / V maxEx p increases as the V maxEx p increases. Both the PD simulated crack velocity curves decrease according to crack propagation time after the peak value. But the decreasing trend speed is different for the two crack velocity curves. The PD - simulated crack velocity curve for impac t at 2 m/s decreases slower than that for the impact at 3 m/s. A trend can be concluded from the two crack propagations with different peak velocities: For the impact fracture with higher peak crack velocity, the PD simulated crack velocity curve has a hig her ratio of V maxPD / V maxEx p , and a bigger absolute curve decreasing slope. More comparable studies of the PD fracture simulation and the fracture experiments need 40 to be conducted to determine whether the trend can be applied to all the impact fracture. F or example, the impact fracture with higher peak crack velocity. 41 Chapter 4. A P eridynamic M odel for F iber - reinforced C omposite M aterials and Its Captur ing the D ynamic F racture P rocess in the Composite Beam s 4.1 Introduction To investigate orthotropic materials, two independent material constants have mostly been used to describe the micro - modulus in peridynamics [35,61] . They include in the fiber direction and in all other directions. Based on commonly used composite theories , stiffness/modulus change s continuously with the fiber angle in unidirectional lamina e . Hu [67] investigated the quasi - static mechanical performance and damage of laminated composite materials with a peridynamic model . Gahjary [68] proposed a continuous model for orthotropic media with an eight h - orde r ed sinusoi dal function and studied the failure modes of anisotropic materials. In this Chapter , firstly, a new bond - based peridynamics model has been developed for orthotropic composite materials. The model has a homogenized continuous micromodulus . ch anges continuously from the fiber direction to the transverse direction with an effective orthotropy , lamina from the fiber direction to the transverse direction. Secondly , the impact dynamic fracture process has been investigated by inputting the simultaneous dynamic strain energy release rate into the proposed PD model, which has never been studied in former PD studies . The proposed peridynamic model has been employed to study the dynamic crack propagation in an orthotropic beam under impact induced three - point bending. Crack initiation and propagation velocities have been predicted and validated by being compared to the imp act ex perimental results from literature [24] . 42 4.2 Peridynamic model for orthotropic composites 4.2.1 Peridynamic micromodulus of bonds A t wo - dimensional peridynamic model has been devel o ped in this paper for a composite lamina. As show n in Figure 4 . 1 ( a ) , any material point i is connected with any other material point j with in its horizon with a bond. The bond is in an arbitrary direc t ion and has a unique material constant . In t he fiber direction (direc t i on 1 as shown in Figure 4 . 1 (a) ), bond micromodulus is defined as , and in the transverse direction, the material constant is . For any other bond with an angle to the fiber direc t i on, the material constant is defined as a dependant . changes continuously for bond s orientate from the fiber direction ( ) to the transverse direction ( ). Figure 4 . 1 Peridyna mic mode l of a lamina. (a) m ate r i al constants for a lamina, and (b) b iaxial loading state . 43 Two steps are used to calculate the mic romodule s for the material bond in any direction . Firstly, a continuous ratio is defi ned with an effective orthotrop y as a function of the angle to fiber orientation. Secondly, the strain energy density from continuum theory is equated with that calculated from PD by inputting the ratio obtained. When a two - dimensional isotropic media is loaded with bi axial strain as shown in Figure 4 . 1 (b) , the two - dimensional PD micromodulus can be calculated as described in papers [69,70] . Calculations of strain energy density from continuum theory and PD are shown in Eq. ( 4 - 1 ) and Eq. ( 4 - 2 ), respectively. Micromodulus C can be derived by equating them as shown in Eq. ( 4 - 3 ), where In bond - based peridynamics, the material points interact only through a pair - potential , which problems for isotropic and linear el astic material [19] . The material constant C is dependent with is dependent with ( 4 - 1 ) ( 4 - 2 ) 44 ( 4 - 3 ) T o presen t the orthotropy of in a unidirectional composite lamina, the spherical harmonic expansion of mate ri al constant C up to the eighth degree sinusoidal assumption has been used by Ghajary [68] . I n this paper, a ratio assumption R = was directly defined as shown in Eq. ( 4 - 4 ) based on the dependency of PD material constant C and properties in continuum theory . In Eq. ( 4 - 4 ), is the off - modulus and he ratio can b e reduced to the constant 1 when bond angle is in the transver se to fiber direction, and can be directly reduced as in the tra n sverse direction and it has a n effective orthotropy as the direction changes from fiber direct ion to the tranverse direction . T he ratio also indicates that material constant for bonds in arbitrary directions is dependent on , ( 4 - 4 ) [67,68] , the mate ri al constant in this study was directly linked with the material 45 propeties and in continuum mechanics of composite materials. The ratio assumption can be applied to define the change of micromodulus according to different orientat ions in any materials with the material properties and ratio assumption can be used to define the corespondin g PD micromodulus distributions for further modeling of the material. Figure 4 . 2 Continuous peridynamic constant has an effective orthotropy for different / ratios. The details of conti nuous micromodulus for orthotropic materials are described as shown in Figure 4 . 2 . changes continuously from the fiber direction to the trans verse direction with a significant orthotropy as the change of off - modulus in a lamina from the fiber direction to the transverse dir ection. Different ratios of off - axis micro modulus have been compare d with the corresponding o ff - axis rangin g from 4 to over 50, as shown in Figure 4 . 2 . Moreover, when the model is applied to isotropic materials, the assumption ratio is equal to 1 for bonds in all directions. The corresp o nding calculated micromodulus can be automatically 46 reduced to a constant C aga in. The corresponding reduced PD constant is the same as the one stated in the traditional bond - based peridynamics for isotropic materials [18,19] . I n the modeling of the composite lamina, o nce the material constants ratio and t he material co nstant in the transverse directi on is obtained , materi al constants in all other direc t i ons can be obtained by using the ratio in Eq. ( 4 - 4 ) . To calculate and all other material constants, the approach introduce d in paper [19] is e mployed. Suppose a square lamina is loaded with biaxial strain , the corresponding peridynami c bond stretch s then equals to . The PD strain energy density is calculated by using Eq. (4 - 5) , where is described in Figure 4 . 2 and numerically inputted into the integration. The strain energy density can be calculated from continuum mechanics , as Eq. (4 - 6) . can then be calculated by equating the strain energy density from peridynamics and co ntinuum theory , as shown in Eq. (4 - 7) , where R is the ratio defined in Eq. (4 - 4). for all other bonds in the horizon can be calculated by multiplying the key ratio to . ( 4 - 5 ) ( 4 - 6 ) ( 4 - 7 ) 47 4.2.2 Elastic deformation verification Figure 4 . 3 Comparison of the off - modulus from the PD model and composites theory. The model is verified b y comparing the off - and the one from laminated composite theory [71] . A square unidirectional composite plate with fiber in direction is loaded with tensile stress on both ends, as shown in Figure 4 . 3 . The plate is made of carbon fiber composite material T800/3900 - 2 . The 11 = 171.6 GPa , E 22 = 8.25 GPa . The shear modulus is G 12 = 6.21 GPa 12 = 0.344 [24] . The plate is meshe d into square grids with a size of 0.5 mm , and a horizon = 3.2*dx is applied in PD simulation. Comparison of the off - 48 the composite PD model and the one from laminated composite theory is demonstrated in Figure 4 . 3 . The modulus in the fiber direction E 11 is 170.71 GPa and in the transverse di rection E 22 is 8.46 GPa . The two are almost the same as the provided material to the value calculated from composite theory as shown in Figure 4 . 3 . The off - calculated from the PD model matches well with the one from laminated composite theory [25]. 4.2.3 Failure criteria For the two - dimensio nal (plane stress) problem, critical stretch can be calculated by equating the work needed to break all the bonds per unit fracture area in PD and the critical strain energy release rate of the material [66] . The 2D formulation [66] is used to calculate critical stretch in this study. Fo r the homogenized peridynamic model of an orthotr opic lamina, the fiber the matrix as shown in Figure 4 . 4 (a) [61] . For bonds in the fiber direction, bonds break if they are stretched over the critical stretch . For bonds in all other directions, the fa i lure is simply set as a matrix material failure. Bonds are defined broken if they are stretched over the critical stretches . The critical stretch is calculated by equating work required to break all the bonds per unit fracture area in a homogenized 1 and the strain energy re lease rate ( m ode I strain energy release rate for fracture in transverse to fiber direction [31] ) for a lamina. Critical stretch is calculated by equating the corresponding work and the strain energy re lease rate ( m ode I strain energy release rate for fracture in fiber direction [31] ) . Equations of and are presented in Eq. ( 4 - 8 ) . 49 ( 4 - 8 ) The d amage of a material point in PD is defined as the ratio of broken bonds to all the original bonds of the point . As shown in Figure 4 . 4 (b) , material point i is connecte d with bonds to any point j in its horizon. Once there is a broken bond for point i , it means that point when the ratio of broken bonds over all original bonds reaches 0.4 ~ 0.5. A crac k propagates when the bonds keep breaking along a damaged path [7]. Figure 4 . 4 Peridynamic failure criteria. (a) Critical stretch of bonds in fiber direction and in matrix direction. (b) Schematic of damag e for a material mode i . 50 Figure 4 . 5 Fiber bonds directions in (a) grid - friendly 0º, 45º, and 9 0º and bonds close to (b) arbitrary directions like 15º and 6 0º . grid - friendly ections 0º, 45º , and 90º as shown in Figure 4 . 5 (a) , there are fiber bonds in exactly the fiber direction with the largest value of material constant C 1 . During the crack propagation process, these bonds will be much more difficult to break than the bonds in other directions. For lamina with fiber directions in 15º, 30º, 60º and 75º as shown in Figure 4 . 5 (b) , not all fiber bonds are in the exact fiber direction. The bonds within a horizon on both sides of t he fiber direction are set as fiber bonds. The material constant C for these bonds has a value very close to C 1 because of the continuous micro modulus defined in this model. Therefore, no further defining of fiber bonds is needed once we set the fiber direction and material constant C . 4.3 Problem setup The developed perid ynamic model is employed to study dynamic crack propagation in an orthotropic plate under impact induced three - point bending. The plate is made with carbon fiber composite material T800/3900 - 2 . Its material properties are stated in the verification part in this paper and also in the literature [24] . The plate is 200 mm long and 51 50 mm high . It has a fiber orient ation of ( 90º, 105º, 120º , and 135º ) with respect to the plate length (correspondingly 0º, 15º, 30º , and 45º with respect to the opposite loading direction). A notch of 10 mm is initially assigned in the middle of the b ottom edge of the plate as shown in Figure 4 . 6 . The plate is discretized into orthogonal PD grids with a square unit length size of 0.5 mm . Grids density among the plate is 100*400. Each material point has a certain horizon with the size of = 3.2*dx as suggested by Hu [67] and Silling [19] . The time increment (time step) for the explicit integration is specified as dt = 1 x 10 - 8 s , which is sufficiently small in all cases based on the stability conditions analysis in the paper [19] . The lamina plate is simply supported by hinges as shown in Figure 4 . 6 . In PD boundary conditions, the supported point and its family points in its horizon were set with no displacement an d velocity in the y direction . Figure 4 . 6 Impact experimental setup of SEN orthotropic beam . The impact algorithm in this study is shown in the Figure B . 1 and dem o nstrated in the Appendix B. I mpact loading is added to the middle of the top surface as a boundary 52 condition too. For the impact algorithm in PD simulation, we set a spherical projectile with the same size, mass and a velocity of 4.8 m/s toward the plate, as the setup in the experiments [24] . The impactor is defined as a rigid body due to the hardness disparity between the imp actor material and the composite sample. As shown in Figure B . 1 (a), the impactor moves towards the sample in the beginning. Once the impactor contacts the sample, it penetrates inside and overlaps with the material point s as shown in Figure B . 1 (b) . To model the rigid impact, t he point s are forced to move to the surface of the impactor at the closest path Figure B . 1 (c) . Thus, the contact surface is defined between the impactor and the sample at the current time step. Displacements of points at the sample surface area result in the corresponding bond forces, which interact with the impactor explicitly. Similar impact algorithm is used in the peridynamics as described by Madenci [34] . 53 4.4 Simulation of the Impact F racture P atterns in C omposites with the Model Figure 4 . 7 Comparison of impact fracture from (a) experimental result [26] and (b) PD computationa l results. The i mpact - induced c rack initiation and propagation are simulated by the PD model. For composite material T800/3900, the critical strain energy release rate in the fiber direction is 179.68 KJ/m 2 and in the transver se direction is 0.418 KJ/m 2 [ 22, 25]. Corresponding critical stretch in PD can then be calculated and applied to the fracture s t imulation . The PD simulation result for the impact fracture on a lamina with fiber orientation in 45º is shown in Figure 4 . 7 (b) . Th e result shows that crack (PD material damage) propagates along the fiber direction until the final material failure. The corresponding impact fracture from the experimental result s by Lee [26] is shown in Figure 4 . 7 (a) . The fract ure is also in the fiber direction. The PD fracture simulation result match es well with the impact experimental result . 54 Figure 4 . 8 Prediction of impact damage in unidirectional lamina with fiber oriented i n (a) 0º, (b) 15º, (c) 30º, and (d) 90 º with respect to the impact loading direction . The PD model with a continuous micromodulus has been further applied to simulate the impact fracture on lamina beams with fiber in directions of 0º, 15º, 30º, and 90 º wit h respect to the impact loading direction. The PD fracture simulation results are shown in Figure 4 . 8 . Fracture (material damage) initiates from the tip of the original notch and propagates straight along the fiber directions. The PD crack size depends on the discretized grid size due to the material point damage definition in PD method. But the fracture direction is mesh size independent as the crack propagates along fiber directions. 55 To obtain a stable result during the computati on process, there is a horizon dependent limited maximum stable time step suggested by Silling and Askari [19] . Based on the suggestion, a stable time step dt = 1 x 10 - 8 s has been used in this study, which is suggested as sufficiently small for all PD simulations [19,70] . The s urface correction factors [33,34] of a c omposite lamina can be added to the e quation of motion (Eq. 11), resulting in more uniform distributed displacements fields at the boundary corn e r of the composite plate, but almost the same impact fracture patterns, which can be due to the strong materia l orthotropy of the unidirectional fiber composites. Figure 4 . 9 C r ack propagation patterns in different directions : (a) grid - friendly 0º, 45º, and 90º. (b) Other directions such as 30º and 75º. For lamina with fiber in 0º, 45º , and 90º directions, the PD fracture propagates straightly along the exact fiber directions. The damage ratio has a consistent value around 0.4 according to the damage contour color bar. However, for lamina with fiber in other arbitr ary directions such as in 30º and 15º, the PD fracture path is wider and some of the material point s even have a damage ratio value close to 1. This is due to the orthogonal 56 grid pattern of material point s that ha ve been used in PD. As shown in Figure 4 . 9 (a) , there are cyclic sufficient point s aligning in exact 0º, 45º and 90º directions. Once a crack - point s beside s the crack have almost the same ratio of broken bonds crossin g the crack, which is the damage ratio. However, in other directions like 30º and 75º as shown in Figure 4 . 9 (b) , not all material point s line up in the exact directions. Point s within a range of the fiber directions are defined as the point s of fiber directions ( Figure 4 . 9 (b) ), which contributes to the wider fracture path pattern in PD in these directions. Moreover, the damage ratios of material point s near the crack are not exactly the same. Some materia l point s on the crack path can be almost totally extracted out from the sample after the fracture , which results in a damag e value as high as close to 1. To reduce the mesh grid dependency of the PD fracture pattern in a lamina with any arbitrary fiber di - direction is aligned with the fiber direction. Therefore, a consistent PD simulated fracture pattern can be obtained for a lamina with arbitrary fiber direction. Or a bigger /dx (m ratio) can be used in PD simulation, which means there can be more material points within a horizon, and more possible points can be aligned in the fiber direction. While a bigger delta value brings a more expensive computation, the balance of accu racy and computational efficiency can be made according to specific problems and purpose. More ncreasing the m - ratio with a uniform grid and further adaptive refinement are discussed in the paper [72] . 57 4.5 PD modeling of the dynamic fracture process 4.5.1 Introducing the dynamic fracture criteria to fit the PD model Figure 4 . 10 Instantaneous dynamics energy release rate versus crack velocity for cracks along fiber orientation in unidirectional composites . For dynamic fracture problems, the fracture toughness (strai n energy release rate) is no t constant during the crack propagation process. Especially when the crack velocity reaches a certain high magnitude , the strain energy release rate will increase exponentially to any f urther increase of crack speed [73,74] . High - order polynomial functions can be extracted from experimental results to describe the instantaneous mode I dynamic s train energy release rate in relation to crack velocity in brittle polymers like PMMA and epoxy [73,74] . Therefore, to simulate the dynamic fracture process in peridynamics , a simultaneous crack velocity related strain energy density should be used 58 rather than a constant one . For a unidirectional composite lamina as studied above, the simultaneou s dynamic effective fracture energy [24] can be employed to simulate crack propagation. To obtain the inst antaneous mode I dynamic energy release rate in unidirectional composites with different fiber orientations, the experimental data of the drop weight impact on the u nidirectional composite plate [24] can be extrac ted and extended as polynomial formulations for composites with fibers in different orientations. Based on the experimental results [24] , t he dynamic energy release rate data are extracted and fitt ed into polynomial velocity dependent functions as described in Figure 4 . 10 for lamina with fiber in 0º, 15º , and 30º directions . The strain energy release rate keeps almost constant ( quasi - static critical energy release rate ) dur ing the relative lower crack velocity stage. As the crack velocity increases to certain critical value, the strain energy release rate increases sharply in accordance with the increasing crack velocity. This trend of dynamic strain energy release rate chan ging with crack velocity matches well with the experimental studies of isotropic and composite materials [74 76] . The corresponding polynomial functions of dynamic energy release rate for fibers in 0º, 15º , and 30º directions are described in Eq . (4 - 9 , 4 - 10 , and 4 - 1 1 ) . Where y is the variable of the simultaneous dynamics fracture energy and x is the variable of crack velocity. The detail curves of these functions are shown in Figure 4 . 10 . 59 ( 4 - 9 ) ( 4 - 10 ) ( 4 - 11 ) 4.5.2 Simulation of the dyn amic fracture process with the fitted PD model The fitted formulations of the simultaneous dynamic strain energy release rate have been applied to PD to simulate the impact crack propagation in a lamina beam with fiber orientated in 0º, 15º, and 30º direct ions. Crack propagation status at 28 after its initiation has been studied by the dynamic PD simulation. The PD simulation results are compared to the experimental Digital Image Correlation ( DIC ) results [24] to describe the accuracy of the simultaneous PD fracture criteria. Fracture in the composites with fiber in 0º , 15º and 30º is shown in Figure 4 . 11 . Based on the ex perimental ( DIC ) results (first column s of Figure 4 . 11 ) , crack length in lamina with fib er in 0º, 15º, and 30º are close to 15 mm , 12 mm , and 9 mm respectively. The corresponding PD damage is shown in the second column s of Figure 4 . 11 , which 60 matches the experimental results very well, especially for lamina with fiber in 0º and 15º direction s . Based on the experimental dynamic str ess intensity factor analysis [24] , mode I dominates in the fracture for lamina with fiber in 0º and 15º direction. F or lamina with fiber s in 30º and 45º directions, the mode I and mode II mixture increases. Strain energy density from PD simulation is described in the third column of Figure 4 . 11 correspondingly. For the lamina with fiber in 0º, strain energy is symmetric by the crack in the middle. On both sides of the crack far from the tip, the strain energy density is almost zero, which means the energy has already been released by the new ly generated fracture surface. For the part around the crack tip, strain energy is tremendously higher, especially in front of the crack tip; the dark color in contour tells the highest value, which demonstrates there is a high concentration of strain energy density at the crack tip and the crack is driven to propagate at the energy concentration of the material. 61 Figure 4 . 11 Crack in lamina e with fibers oriented in 0º (first row) , 15º (second row), and 30 º ( third row) orientations . First column: Experimental results [24] . Second column: The c orresponding PD computational damage. T hird column: The c orresponding PD computational strain energy density. 62 The same energy concentration is also described in contour for the other three fi ber directions. In contours of the lamina with fiber in 15º and 30º directions, the strain energy density on the lower right side of the crack has a higher value than on the up left side . This clearly describes the loading status of the whole beam during i mpact fracture: bending occurs on the outer side part and it bears almost all the loading from the impact, which causes higher strain energy on this side from the large bending deformation. While on the upper left side of the crack, it is almost a free bou ndary after the crack propagated, which makes the strain energy density much lower than that on the lower right side. Obviously, the PD simulation can accurately describe the crack propagation length at a certain time and the strain energy density distribu tion of the beam during the impact process. Matrix damage in the dynamic fracture process (first row) and the corresponding strain energy density status (second row) in the sample are simulated by using the simultaneous dynamic PD fracture energy. The res ults are described in Figure 4 . 12 , Figure 4 . 14 and Figure 4 . 16 , correspondingly for lamina with fiber in 0º, 15º, and 30º directions. The contours accurately describe the consist ent crack propagation along the fiber direction in each sample. Results for lamina with fiber in 0º directions is more meticulous than the one in 15º and 30º directions, which is because the PD discretization arrangement of material points is orthogonal. F iner PD mesh grid size or grid orientation can be studied for it. Strain energy density contours demonstrate the corresponding energy distribution status during the crack propagation process. The extremely high strain energy density at the front of the cr 63 at the crack tip for the crack propagation. In the crack initiation and early propagation stage, strain energy is highly concentrated and tremendously distributed around both the crack tip and loading areas. As the crack propagates, strain energy distributes less and less all over the sample, which explains that as the crack propagates, more generated new surface from the fracture dissipated the strain energy in the sample. Theref ore, the PD strain energy contours can explain the fracture mechanism consistently with fracture mechanics theory. A c omparison of the experimental crack velocities [24] and those calculated from PD is described in Figure 4 . 13 , Figure 4 . 15 , and Figure 4 . 17 , correspondingly for lamina with fiber in 0º, 15º, and 30º with respect to the impact loading direction. For 0º fiber lamina, the velocity calculated from the simultaneous dynamic fracture energy criteria matches the experimental results better than the one calculated from constant fracture energy. The velocity from constant fracture energy is much higher than the experimental velocity starting from the crack initiation, especially at the velocity peak value. However, the velocity from the simultaneous PD fracture energy brings down the higher value and matches with the experimental result very well, especially the big difference at the peak value. Therefore, the simultaneous dynamic fracture criteria is more accurate and effective to simulate the dynamic crack velocity in the impact case than the one using the constant fracture energy criteria in PD. Similar results can be observed in the Figure 4 . 15 and Figure 4 . 17 for lamina with fiber in 15º and 30º directions. To investigate the dynamic mixed - mode fracture, further experimental study and the corresponding PD modeling will be conducted. 64 Figure 4 . 12 Crack propagation process (First row) in the composite beam with fiber in 0º with r espect to the loading direction and the corr esponding strain energy density (Second row) . Figure 4 . 13 Crack velocities in a lamina with fiber s in 0º orientation . Comparison of the experimental result [24] , PD simulations with ordinary fracture energy and the fitted simultaneous dynamic fracture energy. 65 Figure 4 . 14 Crack propagation process (First row) in the composite beam with fiber in 15 º with r espect to the loading direction and the corr esponding strain energy density (Second row) . Figure 4 . 15 Crack velocities in a lamina with fiber s in 15 º orientation . Comparison of the experimental result [24] , PD simulations with ordinary fracture energy and the fitted simultaneous dynamic fracture energy. 66 Figure 4 . 16 Crack propagation process (First row) in the composite beam with fiber in 30 º with r espect to the loading direction and the corr esponding strain energy density (Second row) . Figure 4 . 17 Crack velocities in a lamina with fiber s in 30 º orientation . Comparison of the experi mental result [24] , PD simulations with ordinary fracture energy and the fitted simultaneous dynamic fractu re energy. 67 Figure 4 . 18 Percent of error of peak velocity value calculated with PD by using ordinary fracture energy and by using the fitted simultaneous d ynamic fracture energy for fracture along fibers in 0º, 15º, and 30º orientations. And the percentage of mode II fracture in the corresponding fracture patterns [24] . The crack velocity peak value calculated from PD by using both ordinary toughness and the fitted dynamic toughness has been compared to that from the experimental results [24] . The error of the simulated peak crack velocity in composites with fiber in 0º, 15º, and 30º is shown in Figure 4 . 18 . The corresponding percentage of mode II fracture over mode I and mode II fracture from the experiments is also shown in Figure 4 . 18 . Based on the comparison in Figure 4 . 18 , the peak crack velocity error calculated with the ordinary PD modeling is as hig h as 70% ~ 80%, which is about 60% higher than the error calculated with the fitted simultaneous dynamic PD modeling. By applying the instantaneous dynamic fracture toughness to PD failure simulation, the peak crack velocity error can be brought down by ab out 87%, 79%, and 73% for dynamic fracture in 68 composites with fiber in 0º, 15º, and 30º respectively, compared to the results by using ordinary PD modeling. Therefore, to predict the high - speed dynamic fracture process more accurately, it is recommended to apply the simultaneous dynamic fracture toughness to the bond failure criteria in PD. The peak crack velocity error differs in composites with different fiber orientations as described in Figure 4 . 18 . Error for fracture in 0º dir ection is about 10% lower than that for fracture in 15º, and 30º directions, which is consistent with the percentage of mode II fracture in composites with different fiber orientations as shown in Figure 4 . 18 . The bond failure cri teria were defined with the mode I fracture toughness in this paper, so the error is much smaller for fracture in 0º which has the majority of mode I fracture. One possible reason for the larger errors for fracture in 15 º and 30º directions can be due to t he use of the orthogonal grid pattern in this study. Further experiments and PD simulations of mixed modes dynamic fracture in directions like 15º, 30º , and 45º are to be conducted. 4.6 Conclusion A new bond - based peridynamic model with continuous ma terial co nstant ( meso - modulus ) has been developed for unidirectional composites by using a homogenization method. Impact fracture can be simulated in the lamina with fiber oriented not only in grid - friendly directions 0º, 45º , and 90 º but also in such arbitrary dir ections as 15º or 30º. A simultaneous crack velocity related dynamic strain energy release rate was extracted from fitted experimental results. By applying the simultaneous dynamic fracture energy formulations into the failure criteria (critical stretch) i n the PD model , the calculated 69 dynamic fracture process and crack velocity match better with the experimental results than the ones which use a constant fracture energy. 70 Chapter 5. Peridynamic M odeling of I mpact - induced D amage E volution and Delamination i n L aminated C omposite M aterials In this chapter , the two - dimensional bond - based peridynamic model for orthotropic composites has been extended into three - dimensional for laminated composites. In the model, both the critical bond stretch and critical bond compression have been used to describe the damage of the intralayer and interface of laminated composites . The proposed PD model is then employed to study the damage of a laminated composite plate subjected to out - of - plane impact loading. The laminates hav e different fiber layouts [ varies from 0º, 15º, 30º, 45º, 60º, and 75º. The matrix and intralayer damage initiation and propagation, as well as the interlayer delamination , have been simulated with the PD model. A consistent trend of the damage and delamination patterns to the experiment al results [47] h ave been predicted and discussed . 5.1 The PD Model for Laminated Composites 5.1.1 Micromodulus of bonds A t wo - dimensional peridynamic model for a lamina [77] has been extended into three - dimensional for laminated composite. In the model, the assumption was defined for intralayers as shown in Eq. ( 5 - 1 ) , where is the off - Material constant is dependent on , The details of the description and calculation of continuous micromodulus can be found Chapter 4, section 4.2 , an d in papers [77,78] . The micromodulus of interlayer bonds are defined the s ame as that of the matrix bonds. 71 ( 5 - 1 ) 5.1.2 Failure criteria The failure of intralayer bonds and interlayer bonds has been defined in this sec ti on . F or the intralayer bonds in an orthotr opic lamina, fiber matrix as shown in Figure 5 . 1 . Critical stretch for bonds in the fiber direction was defined as . Critical stretch for bonds in the transverse direction is set as stretches . The critical stretch is equalized with the longitudinal ultimate teinsile strain . The critical stretch is equalized with the transve rse ultimate teinsile strain . Equations of and are presented in Eq. ( 5 - 2) . The corresponding critical compression of material bonds are defined as and , which can be equalized to the longitudinal ultimate com pressive strain and transverse ultimate compressive strain seperatly, as shown in Eq. ( 5 - 3). For bonds in all other directions, the critical tensile strech and compresion are defined as the fourth order of sinusoidal equations o f critical stretch in longitudinal and transverse directio n, as shown in Eq. ( 5 - 4) and Eq. ( 5 - 5). The interlayer bonds failure criteria are set as the same as the matrix bonds. The interlayer critical stretch and compression are defined as the same as that of the intralayer bonds in transverse direction. For each intralayer material point, it interacts with both the in - plane material points and the material points in the a djacent layer within the horizon, as shown in Figure 5 . 2 . The damage of the intralayer material points is set as the ratio of broken bonds to the 72 initially connected bonds . For each intralayer material point, the ratio of the broken bonds crossing the interlayer to all the original bonds crossing the interlay er is defined as the interlayer damage. Figure 5 . 1 Peridynamic failure criteria. Critical stretch of bonds in fiber direction and transverse direction . ( 5 - 2 ) ( 5 - 3 ) ( 5 - 4 ) ( 5 - 5 ) 73 Figure 5 . 2 The in tralayer material bond interacts with the material points in the adjacent layer within the horizon. 5.1.3 Problem description The PD model is employed to study the impact - induced delamination and matrix damage in a laminated composite plate as shown in Figure 5 . 3 . The laminated plate is with the fiber orientation [90 / / plate is 90 mm long, 90 mm wide, and 1.5 mm thick. The composite laminate is made of E - glass fibers and SC - 15 epoxy. The composite plate is fixed in between two clamp plates as the bound ary condition. The impact loading is added to the center of the top surface of the composite plate. The material properties and the ultimate tensile/compressive strain of the E - glass/epoxy lamina are listed in Table 5 - 1 and Table 5 - 2 . [69,79] 74 Table 5 - 1 Table 5 - 2 Figure 5 . 3 The impact on the top surface of t he laminated composite plate. 75 Figure 5 . 4 Impact algorithm. (a) Body force added to the material point at the center of the top layer. (b) Body force. The laminate plate is discretized into orthogonal PD cubic material points with a unit length size of 0.5 mm . Each material point has a horizon with the size of = 3.2*dx as suggested by Hu [67] and Silling . [19] The time step for the PD explicit integration is set as dt = 1 x 10 - 5 s based on the stability analysis in the paper . [19] I mpact loadi ng is added to the center of the top surface of the plate. To simulate the impact condition , a time - dependent body force was added to the material point in the center of the top layer as shown in Figure 5 . 4 (a). To simulate the im pact force magnitude and time period , the body force was set as shown in Figure 5 . 4 (b). The laminated plate has a fixed boundary condition in PD modeling. The material points within a horizon of the boundary were defined with zer o displacement s in x, y and z directions for all the time steps. 5.2 Results and discussion T he impact - induced matrix /intralayer damage and delamination have been simu la ted with the PD model for laminates with lamina layout s of [90/0/90], [90/15/90], [90/30/9 0], 76 [90/45/90], [90/60/90], and [90/75/90]. The intralayer damage, delamination, and out - of - plane displacement contours are shown in the following figures . The detail laminate damage initiation and evolution process will be discussed in the laminates [90/0 /90] . The damage patterns of the composites after certain time of loading are also discussed for the laminates [90/15/90], [90/30/90], [90/45/90], [90/60/90], and [90/75/90]. 5.2.1 Impact - induce d intralayer damage and delamination process in laminates [90/0/90 ] The PD simulated matrix damage and delamination in the laminates [90/0/90] are shown in Figure 5 . 5 , Figure 5 . 6 , and Figure 5 . 7 . Damage initiation in the laminates was capture d after 150 µ s of the loading, as shown in Figure 5 . 5 . The matrix damage initiated from the second layer, in which the fiber is in the direction of 0º. The damage d area is in a small peanut shape along the fiber direction in this layer . Seldom damage occurred in the top and third layers. The initiation of the matrix damage is due to the shear loading rather than compression or tension in the top and bottom surface of the plate. The delamination initiated in the first interlayer (th e interlayer between the top layer and the second layer), shown as the delamination 1 in Figure 5 . 5 . No delamination occurred in the second interlayer. The out - of - plane displacement of the plate is shown in Figure 5 . 5 . 77 Figure 5 . 5 Intralayer damage, delamination , and displacement field of the laminates [90/0/90] at damage initiation , after loading for 15 0 s . Figure 5 . 6 Intralayer damage, delamination, and displacement field of the laminates [90/0/90] after loading for 25 0 s . 78 Figure 5 . 7 Intralayer damage, delamination , and displacement field of the laminat es [90/0/90] after loading for 350 s . The intralayer damage and delamination of the laminates [90/0/90] at 25 0 after the impact loading is shown in Figure 5 . 6 . The out - of - plane displacement value is getting larger than the one at the damage initiation. Also, the deformation area is getting bigger a round the impact center, especially in 0º direction, resulting in a peanut - shaped region along the 0º direction. The displacement field is symmetric by both x and y - axis crossing the plate center. Matrix damage occurred in all the three layers. In the top layer, the damage at the center has an ellipse shape along the fiber direction in 9 0º, the damage in the second layer is a larger ellipse - shaped area along the fiber direction in 0º. The matrix damage in the bottom layer is extended in both 0º and 9 0º. Matrix damage areas in the middle and bottom layers are bigger than that in the top layer , as shown in Figure 5 . 6 . Delamination for the first interlayer is typi cally in peanut shape and has a similar area to 79 that of the damage in the second layer. Delamination shape in the second interlayer has a shape and area close to the damage in the bottom layer. The intralayer damage and delamination of the laminates [90/0/ 90] 35 0 after the impact loading are shown in Figure 5 . 7 . The out - of - plane displacement value and area are getting even larger than that at the damage initiation. Both matrix damage and delamination areas increased. The matrix damage areas in the top and bottom layers have the similar sizes and shape of the ellipse along the fiber direction in 9 0º. The matrix damage in the second layer has a larger size , and with the shape of the ellipse along the fiber direction in 0º. Delamina tion in the first interlayer has a typical peanut shape along the fiber direction of the second lamina (in 0º). It also has a similar area size to the damage d area of the second lamina. Delamination in the second interlayer is also getting bigger and has a similar shape to that of the damage in the bottom layer. 5.2.2 Impact - induce d intralayer damage and delamination in laminates [90/15/90], [90/30/90], [90/ 45 /90], [90/60/90], and [90/75/90]. The PD simulated intralayer damage, delamination, and displacement fie ld of the laminates [90/ 15 /90] after loading for 250 s is shown in Figure 5 . 8 . All the damage and delamination areas are centrosymmetric due to the impact loa d ing at the center of the plate. In the top layer, the damage at the center has an ellipse shape along the fiber direction i n 9 0º. The damage in the second layer is a larger ellipse - like - shaped area along the fiber direction in 15º. The damage d areas in the top layer and second layer are larger than that of the bottom layer, which has a smaller ellipse shape along the direction close to but not precise ly in 9 0º (fiber direction). The intralayer damage in both top and bottom layer contain part extended in 15 º, the fiber direction for the second lamina. Delamination 80 for the first interlayer is typically centrosymmetric in an obliq ue peanut shape and has a similar area and orientation to that of the damage in the second layer. Delamination shape in the second interlayer has a smaller area and with the shape close to the damage in the bottom layer. The out - of - plane displacement field is also with an ellipse shape along the direction of 15 º, and with an area slightly bigger than the damage d area. The PD simulated Intralayer damage, delamination, and displacement field of the laminates [90/ 30 /90] after loading for 300 s is shown in Figure 5 . 9 . Similar to the damage patterns of the laminates [90/ 15 /90] , The intralayer damage areas have the peanut ellipse shape and orientation in the in - plane fiber directions. The delaminations are a s have the si milar shape and size to the damage d area in the corresponding lower layer. Figure 5 . 8 Intralayer damage, delamination, and displacement field of the laminates [90/ 15 /90] after loading for 25 0 s . 81 Figure 5 . 9 Intralayer damage , delamination , and displacement field of the laminates [90/ 30 /90] after loading for 300 s . Figure 5 . 10 Intralayer damage, delamination , and displacement field of the laminates [90/ 45 /90] after loading for 300 s . 82 The PD simulated intralayer damage, delamination, and displacement fields of the laminates [90/ 45 /90] after loading for 300 s , laminates [90/ 60 /90] after loading for 250 s , a nd laminates [90/ 75 /90] after loading for 300 s are shown in Figure 5 . 10 , Figure 5 . 11 , and Figure 5 . 12 respectively. The contours of the layer damages, delaminations, and the ou t - of - plane displacements have the similar patterns for these three laminates. Firstly, all the damage and delamination areas are centrosymmetric due to the impact loa d ing at the center of the plate. In the top layer, the damage areas have the oblique peanu t shape along the direction between the two fiber directions of the top layer in 90º and the second layer in 45º, 60º, and 75º. The damage areas in the second layer have the most significan t size and orientations along the fiber direction in 45º, 60º, and 75º. The damage areas in the top layer and second layer are larger than that of the bottom layer, which has the similar shapes, sizes, and orientations to the damage areas in the top layers . The delamination areas have similar shapes, sizes, and orientatio ns to the damage areas in the corresponding layer below the in t erlayer . The out - of - plane displacement fields are also with the peanut shape along the directions of intralayer damage orientations , and with an area slightly bigger than the damage areas. For all the laminates, the intralayer damage areas have the peanut shape and the orientations in exactly or close to the fiber directions. The delamination areas also have the peanut shapes, with sizes and orientations close to that of the corresponding layer below. The out - of - plane displacement fields have the areas size covering all the intralayer and interlayer damage sizes, and a shape covering all the intralayer and interlayer damage shapes. More damage evolution process of the laminates [90/15/90], [90/3 0/90], [90/45/90], [90/60/90], and [90/75/90] will be presented in Appendix D. 83 Figure 5 . 11 Intralayer damage , delamination, and displacement field of the laminates [90/ 60 /90] after loading for 25 0 s . Figure 5 . 12 Intralayer damage, delamination, and displacement field of the laminates [90/ 75 /90] after loading for 25 0 s . 84 5.3 Conclusion The bond based peridynamic model for unidirectional lamina has been exte nded for the composite laminates, with the definition of micromodulus and failure criteria for interlayer bonds. Out - of - plane impact on the [ 90/ /90] laminate composites has been simulated with the peridynamic model. Matrix damage in each layer and delamin ation in the interlayers have been simulated with the PD modeling. 85 Chapter 6. A n Experimental S tudy of the In - plane Tensile Properties and the Out - of - plane Low - velocity Impact Damage Process of the U nidirectional L aminate, T wo - dimensiona l W oven L aminate, and the Q uasi - three - dimensional C arbon F iber C omposite Material s 6.1 Introduction Delamination is o ne of the most common failure mode s of the laminated composite s in application . To explore the composite materials with hi gher delamination resistant capability , the qu asi - three - dimensional (Q3D) composite structure has been introduced [80] . In the Q3D fiber structure, the fibers from each layer are woven into those in the adjacent layers (above and below). As a result, the multiple layers phy sically attached to each other through the thickness direction [80] . The Q3D woven fiber layers are physically interlocked and held together as one three - dimensionally woven structure. But the Q3D woven structure is different fr om the three - dimensional (3D) weaves that with the fiber yarns specifically weaved in the thickness direction. Due to the step - by - step interlocking through the thickness, the fiber yarns in the Q3D structure can be maintained as flat as possible to keep an effective in - plane stiffness as the laminates. The study [80] on bi - axial Q3D woven structure ([ 0/90 ]) shows that bi - impact - induced damage, higher specific energy absorption, lower impact - induced str uctural degradation, and competitive in - plane properties than the laminated [80] In this study, the Q3D woven structure has the tri - axial fiber orientation in 0 º, 6 0 º, and - 6 0 º. Compared to the bi - axial Q3D woven structure in 0 º and 9 0 º, the current tri - axial 86 Q3D structure is supposed to have a greater in - plane isotropy and a better resistance to delamination [81] . The three composite structures are the unidirectional laminate (UDL) in [0/60/ - 60] 4 , the two - dimensional plane w eav ed laminate (2DW laminate) in [ 0/60/ - 60 ] 4 , and the quasi - three - dimensional composite structure (Q3D) in [0/60/ - 60] 4 . The study aims at identifying the potential advantage of the advanced Q3D composite structure in delamination, and the competitive other material properties, such as the in - plane stiffness and strength. Firstly, the composite panels were fabricated with the vacuum injection process with the curing at 122 ºC for 4 hours. The tensile specimens along orientations at 0 º, 15º, 30º, and 90º were prepared and tested according to the ASTM D3039 standard. Secondly, the unidirectional tensile testing of the specimens was conducted with the MTS machine, combining the three - dimensional Digital Image Correlation (3D DIC) method, from which the loading - d eformation relations of the tensile testing have been extracted. Then, been calculated for the three composite structures. The quasi - isotropic material properties and the orientation influence on the tensile strength of the composite structures have been analyzed and discussed. Moreover, the out - of - plane low - velocity impact testing was conducted for the three composite plate samples, combining with the 3D DIC analysis. The impact loading, deformation, and the impact penetration damage process will be discussed. 87 6.2 Fabrication of the composite plaques 6.2.1 The fiber structures preparation The Q3D fiber structure can be braided with the braiding machine at CVRC, MSU, as shown i n Figure 6 . 1 (a). The machine braids the fibers into a closed cylindrical tube - shaped fabric as shown in Figure 6 . 1 (b). The braided Q3D structure has the fibers oriented in three directions as shown in Figure 6 . 1 (c), the 0 º direction, and the other two symmetric directions, where can be set as the desired angle by operating the braiding machine. Figure 6 . 1 Quasi - three - dimensiona l (Q3D) composite structure. (a) The braiding machine, (b) the braiding process, and (c) the braided Q3D glass fiber structure. The carbon fibers are provided by the Ford Motor Company. The carbon fiber is the DOWAKASA 24K A - 42, with the tensile modulus of 240 GPa, and the tensile strength of 4200 MPa. The fiber tows were used to prepare the UDL structure and braid the 2DW and the Q3D structures, which is shown in Figure 6 . 2 . Each of the three composite structures has a twelve ply of fibers, repetitively orientating in three different directions as - 60º, 0º, and 60º. The UDL fiber structure is as shown in 88 Figure 6 . 2 (a), with the unidirectional fiber ply orientation [0/60/ - 60] 4 . The 2DW structure is a layup of four woven fabrics, each is braided with the fibers in - 60º, 0º, and 60º directions. The 2DW has a fiber ply orientation [ 0/60/ - 60 ] 4 , as shown in Figure 6 . 2 (b). The Q3D structure is directly braided with the twelve plies rep etitively in the orientation of directions as - 60º, 0º, and 60º ( [0/60/ - 60] 4 ), as shown in Figure 6 . 2 (c). Each layer was braided interlocked with the adjacent layer, resulting in an integral fabric structure. Figure 6 . 2 The carbon fiber braided structures. (a) UD laminate, (b) 2DW laminate, and (c) Q3D fabric. 6.2.2 Fabrication process of the composite plaques The resin system used for the composite fabrication is the SC - 15 epoxy, w hich is used by being mixed with the hardener (part B). The curing of the resin is conducted at 122 ºC for 4 hours as suggested in the study [82] . The vacuum infusion setup and process are shown in Figure 6 . 3 . Firstly, a flat aluminum fla t plate mold is prepared with its surface cleaned 89 and polished with the sandpaper . Then mold surface is coated with a thin layer of the release agent, for composite plaques releasing after the curing. A releasing cloth is then placed on the surface of the plate mold, with the fabrics directly on the cloth as shown in Figure 6 . 3 (a). Another release cloth is then placed on the fabric with a drainage grid placed on it as shown in Figure 6 . 3 (b). The sealing ta pe with the inlet and the outlet is then placed around the fabrics. A vacuum bag is then placed on the sealing tape to seal the fabrics inside. The inlet is for resin flowing through the whole fabrics as the infusion, the outlet is connected to a vacuum pu mp to suck the air inside the sealing space. Once the inlet is closed, the vacuum pump is turned on and the air inside is sucked out. After the pressure inside reaches the negative standard atmosphere air pressure, with no leaking for the whole sealing bag , the inlet is put into the bucket with resin. The resin is sucked into the vacuum bag and goes through the fabrics. Once the resin reaches the other end of the fabrics, both the inlet and the outlet will be closed. The whole plate is then placed into the oven, curing at 122 ºC for 4 hours [82] . Figure 6 . 3 The vacuum infusion process. (a) the layout of the releasing cloth and the fabric, (b) vacuum infusion setup . 90 6.3 In - plane quasi - static tensile testing In - plane mate rial properties have been studied with the unidirectional tensile testing. The tensile loading conditions and specimen preparation were conducted according to the ASTM D3039 [83] . Details from the specimen prepa ration to the tensile failure analysis are discussed below. 6.3.1 Specimen preparation Figure 6 . 4 The three different composite plaques The cured composite plates are shown in Figure 6 . 4 . The UDL plate has the smoothest surface with the smallest surface roughness compared to the 2DW and Q3D composite. The difference in composite plaques surfaces is caused by the different fiber structures layout and interlayer fiber tows interlock betwe en laminate fiber layout and woven fiber structures. The three composite plaques have the similar thickness around 2 mm. The volume fractions of the composite plaques were calculated. The three composites have the same volume of fibers per each unit of t he in - plane area. Because they have the 91 same layers and tows amount of the carbon fibers resulted from the same braiding template, even though the fiber tows layout and braiding patterns are different. The fiber volume per unit in - plane area then was calcu lated as dividing the fiber mass by the area. The thickness of the composite panels was measured as the composite material volume per unit in - plane area. The volume fraction for the UD L , 2DW, and Q3D plaques are then calculated as 42.8%, 44.9%, and 39.6% r espectively. The specimens were cut in different orientations in the three composite structures/plaques. The 0º direction specimen is cut along the fibers in 0º orientation in the composite plaques. Other specimens were cut with angles of 15º, 30º, and 90º to the 0º direction, as shown in Figure 6 . 4 (b). The specimen preparation is shown in Figure 6 . 5 , according to were cut f rom a glass - fiber laminated composite plate and mounted to both ends and sides of the specimens with the epoxy glue as adhesive. The specimens are ready for tensile testing after the curing of the epoxy glue. 92 Figure 6 . 5 specimens 6.3.2 Tensile testing set up The unidirectional tensile testing system set up is shown in Figure 6 . 6 . Specimens were placed in between the sample clamps of the MTS tensile machine, as shown in Figure 6 . 6 . The MTS machine is connected to the data collection and machine control system as shown in Figure 6 . 6 . The tensile testing is set as displacement loading with the loading rate of 0.2 mm/min, unti l the tensile failure of the specimens. The MTS machine recorded the tensile loading/force history for each tensile testing. The light source is used to generate light projected to the surface of the specimens during the tensile testing. Two cameras were supported with the frames and placed in front of the specimen, facing the specimen surface perpendicularly, to catch the loading surface deformation during the tensile process for DIC analysis. 93 Figure 6 . 6 Tensile testing setup with MTS machine 6.3.3 Tensile testing load - displacement curve The deformation process of the specimen is extracted with the 3D DIC analysis. The tensile deformation process was recorded with the two cameras with the figure save frequency of 3/s. The save d figures are then imputed into the 3D DIC software ARAMIS from the GOM Correlate . The deformation/strain fields of the sample surface were calculated with the software. The unidirectional deformation/strain (strain in the loading direction) history is the n extracted. For example, the DIC extracted longitudinal strain (strain y) fields of the UDL 0º specimen at a different time during the tensile process is shown in Figure 6 . 7 . The strain field was calculated through the whole speci men surface. To exclude the local constraint effect from the specimen clamps of the MTS machine, only the central 1/3 of the strain field contour was used. The average value of the central part of strain contour was calculated with ARAMIS (DIC software), a s the strain value at 94 the corresponding time step. The original strain field on the specimen surface is zero before any tensile loading added to the specimen, shown as the first figure in Figure 6 . 7 . As the loading increased, the average strain contour value increased, as marked in the contour bars in Figure 6 . 7 . Figure 6 . 7 DIC extracted longitudinal strain field for the UDL laminate 0 º specimen By combini ng the DIC extracted strain history with the loading history recorded form the MTS machine, t he tensile testing loading - displacement curves for the three different composite specimens were extracted. For example, the loading - displacement curves of the tens ile of UDL specimens with different orientations are shown in Figure 6 . 8 . All curves start from the linear elastic stage to the nonlinear stage, then end as the tensile failure of the specimens. 95 Figure 6 . 8 . Tensile testing loading - displacement curves The curves for all the specimens have very similar linear elastic deformation stages, which is because of the homogeneously quasi - isotropic effect due to the fiber orientations in - 60 º, 0º, and 60º. The similar linear elastic stage brings out the similar elastic properties for all the specimens, detail elastic properties will be discussed in the results part. For specimens with fiber orientations in 0º and 15º, the curves tend to be a more linear - failure trend, with the nonlinear part not as obvious as that of the curves for the specimens with fiber orientations in 30º and 90º. This is due to the different amount of fiber effect compared to the matrix effect for the specimens with diff erent fiber orientation. For specimens in 0º and 15º, more fibers are either along or close to the longitudinal specimen orientation, therefore, fibers effect counts more for the specimens 96 during the longitudinal tensile testing. While for the specimens in 30º and 90º, fibers layout is further to the longitudinal specimen orientation, as a result, the fiber effect counts less, the matrix effect counts more, which results in the more nonlinear deformation before the failure. The tensile failure loading for specimens in 0º is much higher than the specimens in all other orientations, which is also because there is more fiber effect in tensile strength for the specimens in 0º. 6.3.4 Figure 6 . 9 . The tensile modulus of the three composite specimens with different fiber orientations The tensile modulus for the three composite samples in different orientations is shown in Figure 6 . 9 . The composite structures have the similar value of tensile modulus around 35 97 GPa, for samples in different orientations, such as in 0º, 15º, 30º, and 90º as shown in the figure . The consistent tensile modulus value in different orientations is because of the in - plane quasi - isotro pic property of the composites. The composites have the fiber orientations evenly orientating in - 60º, 0º, and 60º directions for 12 layers, which results in the average distribution of the orthotropic in - plane property of the unidirectional fiber tows, ev entually causes the in - plane quasi - isotropic property. For the three different composite structures, tensile modulus values are close to each other in any one of the orientations. The difference in fiber layout structures did not bring much difference in t he tensile modulus of the composites. slight difference in value may be caused by the different sample quality resulted from the manufacturing process. Detail influence on the material properties caused by the manufacturing process can be further discussed . Figure 6 . 10 . modulus, the Poisson ratio values have little difference between the different composite orientations are also similar, which further reflects the quasi - isotropic elastic material properties of th e composite structures. 98 Figure 6 . 10 Tensile modulus of the three composite specimens with different fiber orientations 6.3.5 Tensile strength The tensile strength of the three composite samples in different or ientations is shown in Figure 6 . 11 . For the three different composite structures, the tensile strength values are very close for specimens in the same orientation, which is because they have the same fiber layout orientations even the fiber layout structures are different. Moreover, the major contribution to the tensile strength is from the fiber since the fibers have the much higher tensile strength compared to the epoxy matrix. Therefore, the same fibers amount and orientations i n the specimens resulted in the similar tensile strength. Slight difference in strength can be caused by the different out - of - plane fiber undulation and woven structure, also could be caused by the different specimen thickness, fiber volume ratios resulted from the fabrication process. 99 Figure 6 . 11 The tensile strength of the three composite specimens with different fiber orientations For either one of the three composite structures, the tensile strength o f the specimens varies in different orientations. Specimens in 0º direction have the highest tensile strength around 600 MPa. The specimens in all other directions have the much lower average tensile strength. Specimens in 15º direction have the tensile st rength of about 360 MPa, while the specimens in 30º and 90º directions have almost the same level of tensile strength around 330 MPa. The difference in tensile strength is because the specimens have the biggest amount of fibers along the longitudinal speci men direction (loading direction). The carbon fiber has a much higher tensile strength than the matrix materials. Therefore, the more fibers orient in or close to the longitudinal direction of the specimens, the higher tensile strength the specimens have. This property of the tensile strength changing according to the specimen cut orientations is different from that of the elastic 100 the homogenized meso - scale propertie s, which is because the composite materials were treated as a quasi - homogeneous fiber - matrix - mixed media. The composites have the same amount of carbon fibers in 0 º, 6 0 º, and - 6 0 º orientations, three evenly distributed orientations among the plane, which r esults in the quasi - isotropic in - plane properties in a homogenized way. 101 Figure 6 . 12 Specimens failure of the UDL, 2DW, and the Q3D composites in different orientations 102 6.3.6 Tensile failure analysis of UD L , 2DW, and Q3D composite materials specimens The tensile failure specimens of the UDL, 2DW, and the Q3D composite materials in different orientations are shown in the Figure 6 . 12 . For all three composite materials, the failure patt erns between specimens in different orientations are different. The specimens in 0 º direction failed with the severest specimen breakages. The specimens break through the whole sample with even two failure cross sections and a huge part of the fiber tows b reakage and peeling off. The specimens broke with the strongest sound at the failure. The failure cross sections are along the lateral direction. The specimens in 15 º broke also with rough failure cross sections since the fibers orientation in 15 º and 45 º , which is close to the longitudinal direction. The specimens in 30 º and 90 º have the similar failure patterns because the fiber orientations are the same due to the symmetry of the fiber orientations in [0/60/ - 60] composites. The specimens broke with the c leaner failure cross sections, which is because there are four layers of fibers along the 90 º direction with only matrix material failure, then less sever e fiber breakage and pooling out occurred. The failure cross sections align ed in the lateral direction , which is because of the axisymmetric fiber orientations along the longitudinal direction. In the UDL specimens, failure cross sections came with both severe fibers breakage and the pealing out in between laminates, because the weak interlayer strength c aused the delamination. Less delamination occurred in the 2DW specimens because there are only three interlayers in the specimen. while for the Q3D specimens, the adjacent fiber tows interlock with each other, which results in less delamination at the tens ile failure, as shown in the Figure 6 . 12 . Because of the interlayer fiber bonding difference among the three composite structures, for specimens in the same orientation, the UDL specimens 103 have the roughest failure cross - section , t he 2DW specimens have cleaner and the Q3D specimens have the cleanest failure cross sections. 6.4 Out - of - plane low - velocity impact testing Out - of - plane low - velocity impact testing was conducted to study the impact failure mechanism and the impact resistance of the three different composite structures. 6.4.1 Impact specimen preparation Three composite plaques were fabricated as shown in Figure 6 . 13 . They are the UDL, 2DW, and the Q3D composite plaques. Each composite plaque was then cut into four 4*4 inches plate samples as shown in the Figure 6 . 13 . The plate samples were used for the impact testing, with one surface painted with white flat painting as background and with black dot painting for the DIC analysis. The painting dots sizes and distribution density are determined according to the camera resolution and the DIC analysis requirements. More details about the DIC analysis experimental settings can be found in the paper [63,84] . Figure 6 . 13 The three composite plaques with the corresponding painted impact plate samples 104 6.4.2 Impact testing setup Figure 6 . 14 Out - of - plane impact testing setup The impact testing settings are described in this section as shown in the Figure 6 . 14 . The setup includes the Dynatup machine and the 3D DIC settings. The machine is the Instron Dyn atup 9250 system, driven by the air system with a pressure of around 90 Psi. The impactor/drop weight has a weight of 17 kg, including the load cell attached to the bottom of the drop weight. The load cell has a tub with the tub head diameter of 0.5 " is sh own in the figure. Right below the load cell tub is the sample clamps. The sample clamps have the circular hole in the center with the diameter of 3 " is shown in the figure. During the impact testing, the impactor is raised to a height as the setting for t he impact 105 energy or impact velocity or just impact height. The samples are placed in between the clamps and will be clamped tight by the machine after the fire button clicked. The impactor then drops freely and hit the top surface of the sample to generate the impact on the sample. The Dynatup is connected to a control system as shown in the figure. The impact loading history and the displacement of the impactor after the drop are recorded with the load cell and the laser sensors separately and saved into the control system. The other part of t he setting is the DIC system, which includes two high - speed cameras, a light source, and the video control system as shown in the Figure 6 . 14 . The two cameras were placed in front of the Dynatup machine, with a focus angle of 20º according to the 3D DIC image quality requirements. [65] To catch the bottom surface image of the specimen that placed in th e horizontal direction, a mirror with an angle of 45º was placed right below the specimen. Therefore, the bottom surface of the specimen can be reflected and caught by the cameras placed in the horizontal plane. The lateral view of the mirror settings is shown in the Figure 6 . 14 . The light source is placed in between the cameras. With the help of the reflecting mirror, the light can be reflected and projected on the bottom surface of the specimen, as the light source of the camera s. The frequency of the high - speed cameras can reach as high as 20000 / s , with a time step between two adjacent images as short as 5 0 , which ensures the cameras catch the detail deformation and failure process during the impact process with a very high accuracy. The cameras were controlled by the Photron FASTCAM Viewer system installed in the laptop connected as shown in the figure. T he whole 3D DIC settings were purchased from the Trillion I nc. 106 6.4.3 Impact loading, deflection, and energy absorption of the three composites Figure 6 . 15 The impact loading and deflection history The drop weig ht (17 kg) was set to impact on all the composite plate specimens at the impact velocity of 1 m/s. The impact loading, energy absorption, and the impactor displacement histories were recorded by the Dynatup automatically during the testing. The impactor di splacement was treated as the specimen center deflection since the specimen plate is thin. The impact loading history and the deflection history of the three composite specimens center point are shown in the Figure 6 . 15 . The loadi ng time duration is about 15 ms , which includes several impact stages. The different impact stages can be obviously distinguished as follows based on the loading curve roughness in the figure: 107 (1) The contact process of the impactor and the sample top surface. The impact loading starts to rise from zero, then fluctuates slightly due to the further full contact of the impactor and the sample surface, which is not smoothly flat because of the fiber layout and woven undulation. The less fluctuation of the loading curve of the impact on the UDL sample exactly reflects that the UDL sample has a smoother surface with less roughness compared to the other two woven structure. (2) The elastic stable loading process. The loading curves are relatively smooth, rise immediately till 1000 N. (3) The damage evolution process. The damage initiates with a sudden severe fluctuation of the loading curves around 2.5 ms . The curves fluctuate more and more with the damage evolution. The UDL curve fluctuates much stronger than that of the 2DW and Q3D composites, which reflects the severe delamination and the in - plane fibers - matrix debonding in the UDL during the damage evolution. While for the 2DW and the Q3D, the fiber tows were woven together, therefore less delamination and debonding with sm aller damage area occurred in the damage evolution process. (4) Peak loading. The loading curves reach the peak value around 4 ms during this stage. The peak loadings are between 2000 N and 2500 N for the three composites. The curves drop suddenly with a huge magnitude right after the peak value, which reflects the unstable failure of the composites during the impact loading process. The unstable failure could be the fiber breakage due to the complex combination of the bending and shear loading from the impacto r. 108 (5) Penetration process. The penetration process starts with the unstable sudden failure of the composite plates. The loading continues to rise up after the unstable drop, which reflects the late stage of the impact penetration. Bigger contact areas occ ur during the penetration due to the spherical impactor head shape, more materials damage occurred. Which results in a continuous loading with a high average value but very strong fluctuation. Figure 6 . 16 The impact Loading and absorbed energy to deflection The impact loading and absorbed energy change according to the deflection of the composite plate sample center point are shown in the Figure 6 . 16 . The different impact stages discussed above are also marked in the Figure 6 . 16 . 109 6.4.4 3D DIC analysis setting in GOM and calibration with the loading - displacement curve . Figure 6 . 17 C alibration of the DIC deflection curves with Dynatup machine recorded data in the 3D DIC testing were arranged as described in the Figure 6 . 17 . To get the proper image quality for the DIC analysis with GOM, t he resolution of the image is set as 512*512 pixels. In GOM software, the facet is set with a size of about 3~7 black dots. While each black dot is set with about 3~7 pixels size. The images taken at the same time with the left and right cameras were impor ted into the GOM software. By using the images and the 3D DIC calibration file, the GOM software then calculated 110 the data of the displacements, strain, impact velocity and energy of the composite plate samples during the impact. All the data calculated and extracted from the GOM software is the dynamic data, with the time step of 50 s . The DIC analysis results were calibrated by comparing the calculated impact point deflection in the vertical direction and the impactor displacement history recorded from the DYNATUP machine. The deflection history of the plate sample center point (imp act point) and the impactor displacement history curves for the three composite samples are shown in the Figure 6 . 17 . Form the figures, the overlap as a good match of the DIC calculated deflection curves and the Impactor displacem ent curves can be obviously observed, for the beginning stage, when the DIC date is capable to be extracted with no severe damage to break the paintings in the image. The good match between the DIC data and the machine data is taken as an effective calibra tion of the DIC analysis date. More detail analysis from the DIC date will be discussed in the later sections. 6.4.5 A comparison study of the impact failure mechanism of the three composite structures The damaged composite plate samples after the impact of 1 m/ s from the 17 kg impactor are shown in the Figure 6 . 18 . The impact damage areas on the frontal surfaces have the similar sizes for the UDL, 2DW, and the Q3D composite plates. While the damages in the back surfaces are different as shown in the Figure 6 . 18 . Long strips of the fibers were peeled off in the ULD plate in the - 60 º degree, resulting in a much larger damage area in the UDL than the 2DW and the Q3D plates, which have the similar sizes of the damag e areas. This is because that in the UDL plate, each layer of the lamina was connected to the other layer s with the matrix interlayer. While for the other two composite structures, 111 there are fibers interlocks between layers. Especially for the Q3D composit e, the interlock of fiber tows was designed between each adjacent layer. Figure 6 . 18 impact failure on both sides of the samples For detail understanding of the impact damage process besides the impact fa ilure of the composite plats as discussed above, the bottom surface damage morphologies are presented in the Figure 6 . 19 . The images present the bottom surfaces of the UDL, 2DW and the Q3D composite plats at 5 ms and 8 ms separate ly. The first row of images can be treated as the initiation of the damage because at 5 ms the peak loading just dropped and the penetration of the impactor in the plates initiated. The damages patterns are different. For the UDL plate, the lamina fiber to w in the - 60 º degree started to be peeled off from the plate. For the 2DW plate, the damage initiates with a much smaller area but also align 112 - 60 º degree as the fiber orientation in the bottom surface of the plate. For the Q3D plate, the damage initiation pattern is different than the other two plates. The damage initiation pattern is shown in the figure as a star shape break up with the broken point in the center, and the radial damage propagation evenly to the edge side of the plate, which is similar to t hat of a brittle isotropic plate. As the damage evolution to the impact penetration to 8 ms , the impactor surface almost penetrated through the plate samples. The damages in the three composite plates increase in areas with the consistent damage pattern co mpared to the damage initiation at 5 ms . The bottom layer fiber strips were totally peeled off in the center area along the - 60 º degree for the UDL plate. The damage areas in the 2DW and Q3D plates also increased by size with the consistent damage orientat ion and pattern as that at 5 ms . The damage pattern of the Q3D plate is still similar to that of a brittle isotropic plate, which reflects that the Q3D plate has the highest local quasi - static material properties compared to the other two. Even though all the three composite plates were tested as quasi - isotropic in the global average meso - scale specimens in the tensile testing. The global quasi - isotropic elastic material properties of the three composites are due to the same amount of the fibers in the aver aged - counted orientations in - 60º, 0º, and 60º degrees. While the different local material properties in the damage initiation and evolution are introduced by the different fiber constructions between each adjacent layers of the composites. The damage evol ution process reflected the local quasi - isotropic material properties difference in the different composite structures. 113 Figure 6 . 19 The impact failure on the bottom surface of the samples at different tim e steps To further study the damage evolution mechanism difference in the three different composite plates, the major strain contours with orientation arrows of the bottom surfaces were calculated from the 3D DIC analysis, as shown in the Figure 6 . 20 . The first row of contours presents the major strain with orientation arrows for the three composite plates at 3.8 ms , when the loading is almost the peak loading as shown in the Figure 6 . 20 . The second row of contours presents the major strain at 4.5 ms , when the loading curves just drop right after the peak value. the major strain fields on the three composite plate surfaces are different. 114 Figure 6 . 20 The DI C analysis of major strain on the bottom surfaces of the samples at different time steps At the peak loading, the UDL composite has the largest major strain area with the over - scale high strain marked as dark red as shown in the Figure 6 . 20 . The high major strain field has a shape with the convex orientation along the fiber direction of the bottom layer as shown in the figure. The 2DW plate has the smaller major strain area and the Q3D has the smallest area. The orientation of the 2DW plate high strain area is still obvious along the fiber orientation. But for the Q3D plate, the high major strain field has a centrosymmetric star - shaped area, which is similar to that of the isotropic material plate. 115 The difference in high major strai n fields shape reflects the different local quasi - isotropic material properties of the three composite structures. The difference in major strain fields increases severely right after the peak loading when the impact - induced penetration occurs. The impact penetration brings different damage patterns to the three composite plate samples, especially on the bottom surface as shown in the second row of contours in the Figure 6 . 20 . For the UDL plate, the penetration causes the peeling - off of the bottom surface layer, with the enlarged high major strain field along the fiber orientation more obviously as a strip - shape. The major strain area on the 2DW plate also increases, with the similar aspect ratio along the fiber direction. For the Q3D plate, the major strain field increases with the area but still with the star - shaped centrosymmetric shape. The major strain direction is marked with the arrows for all the composite plates. The arrows orientate almost in the tangential direction of t he circular plate, vertical to boundary curve of the high strain fields. W hen subjected to the same impact energy, the UDL plate has the largest average damage area, the Q3D has the smallest average damage area, the 2DW is in between. Therefore, the Q3D p late has the biggest ratio of impact energy over damage area , which means under the same area of material damage, the Q3D composite can absorb higher impact energy than the UDL and 2DW composites due to the woven structure difference. To further verify the energy absorption capability of the three composite materials, multi - modes interlayer fracture/delamination testing can be performed in the future. 116 6.4.6 The impact failure process Figure 6 . 21 The DIC analys is of out - of - plane displacement fields during the impact process The DIC analyzed impact displacement and strain fields for the three composites at different time steps are illustrated in the Figure 6 . 21 ~23. The impact deformation and damage process of the composite plates, and the corresponding comparisons between each composite structure can be studied from the figures. 117 Figure 6 . 22 The DIC analysis of displacement fields in x - direction during the impact process The out - of - plane displacement (displacement z) fields of the three composite plates from DIC analysis are shown in the Figure 6 . 21 , at the time steps from 1 ms to 5 ms . For all three composite plates, the displacement z increases with the time steps. The center point of the plate has the biggest displacement increase because the impact position is the center of the plate. The displacement at the edge boundary is almost zero because the edge boun dary is fixed with the clamps during the whole impact testing. The displacement contours are centrosymmetric in early impact stage such as before 3 ms , which reflects the homogenized quasi - isotropic in - plane material property of the three composite materia ls at the macro scale. After the peak loading, the contours changed. The displacement contour for Q3D plate still kept almost centrosymmetric. While for the 118 2DW and UDL plates, the displacement distribution around the impact center align s with a trend to t he fiber orientation of the bottom layer in - 60 º . The difference reflects that the Q3D composite material keeps both homogenized global and the local quasi - isotropic material property due to the fiber interlock structure between each adjacent layer. The 2D - isotropic material property. The displacement x fields of the three composite plates from DIC analysis are shown in the Figure 6 . 22 , at the time steps from 1 ms to 5 ms . The displacement x fields are different from the corresponding out - of - plane displacement fields. The contours of UDL plate present obvious orientation along the fiber orientation of the bottom surface layer, which reflects the material o rthotropy of the surface layer. For the Q3D plate, the displacement x contours keep almost axisymmetric through the whole process, which reflects the quasi - isotropic material property. The 2DW plate is in between, it has the axisymmetric contours before th e peak loading, as the elastic deformation process. The contours later than the peak loading have the orientation along the surface layer fiber direction, which also reflects the local orthotropic property of the damage pattern. The convex contour orientat ion is less obvious than that of the UDL plate contours but essentially different from that of the Q3D plate that has the highest quasi - isotropic material property in both the elastic deformation stage and the material damage stage. 119 Figure 6 . 23 The DIC analysis of major strain fields during the impact process The major strain fields of the three composite plates from DIC analysis are shown in the Figure 6 . 23 , at the time steps from 1 ms to 5 ms . The major strain increases as the time step too. The UDL plate reaches the earliest high major strain (at 3 ms ) at the center of the plate as shown in the figure. The major strain contours on the UDL and 2DW plates have the orientation along to the fiber direction of the surface layer. The major strain of the Q3D plate is close to that of the isotropic material plate. Similar results were discussed in the Figure 6 . 20 . 120 Figure 6 . 24 The DIC analysis of shear angle (in - plane shear strain) fields during the impact process The in - plane shear angle (engineering shear strain) fields of the three composite plates from DIC analysis are shown in the Figure 6 . 24 , at the time steps from 1 ms to 5 ms . Similarly, the shear strain contour of the UDL plate reaches the earliest high value at the center position with the severest orientation along the fiber direction in the surface layer. The contours on the 2 DW and Q3D plates have smaller areas and less orthotropic orientation compared to that of the UDL plate. The detail contours of the shear strain evolution process are shown in the Figure 6 . 24 . 121 6.5 Conclusion Conclusions for in - plan e tensile testing: 1. Quasi - isotropic in - plane material properties including tensile modulus and unidirectional quasi - static tensile testing. Tensile strength decreased rapidly with the s pecimens oriented from 0º to other directions ( 15º, 30º, and 90º ) as defined in the composite plaques. 2. For the three different composite structures in the same orientation, the in - plane antitatively. 3. The influence of the different fiber structures on the composite material performance is the failure mechanism during the tensile loading process. For UDL specimens, more interlayer delamination can be found at the failure cross - section other than the fiber breakage. The Q3D specimens have the smoother failure cross section with less roughness of fiber tows peeling off, which is because the fiber layers are weaved together as an integral plaque with less interlayer failure. Conclusions for ou t - of - plane low - velocity impact testing: 1. The impact stiffness and impact strength are quantitatively similar for the three composite plate samples, with the impactor of 17 kg and impact velocity of 1 m/s. 2. The different fiber layout structures influence the composite impact performance s. Firstly, the different damage and failure mechanism s during the impact process. The UDL plate has the delamination on the bottom surface layer because there is no strengthening for the interlayer in the UDL plate. The penetra tion on the Q3D 122 plate is similar to that of the isotropic material plate because the Q3D plate has the best homogeneous property through - thickness direction due to the adjacent layer fiber tows interlock. Secondly, The ratio of the impact energy over the a verage damage area reflects that the Q3D composite has higher impact energy absorption capability per unit damage area than the UDL and the 2DW composites due to the fibers - interlock between each adjacent layer . 123 Chapter 7. Conclusion and Outlook 7.1 Conclusion an d Contributions In this work, the impact fracture, impact damage , and failure process ha ve been investigated with experimental testing and peridynamic modeling and simulation, for fiber - reinforced composite materials. The specific conclusions and contribu tions are: 1) Identified the influence of the impact energy/loading on the fracture process, and the feasibility of 2D PD modeling in capturing the different impact fracture The impact - induced dynamic fracture initiation and propagation in single - edge - notched PMMA beams have been analyzed. C rack velocities have the similar trend, they rise from a lower value, then reach the peak value, and then decrease till fracture. Peak velocity of the fracture in beam subjected to bigger impact loading is higher than that in the beam under smaller impact loading . The PD simulated crack velocities can match the experimental results basically at thi s velocity range below 300 m/s. The PD simulated crack velocity deviates from the experimental results around the peak value s . Th e deviation increases as the experimental peak velocity increases. The simulation of higher crack velocity in different materials need s to be further investigated . 2) Developed the meso - scale PD model for orthotropic composite materials, proposed the homogeni zation PD modeling method , captured the impact fracture process with the fitted dynamic failure criteria in the PD model. A new bond - based peridynamic model with the contin uous material constants has been developed for orthotropic composites by using a hom ogenization method. Impact fracture 124 patterns can be simulated in the unidirectional lamina with fiber oriented not only in grid - friendly directions 0º, 45º , and 90 º but also in such arbitrary directions as 15º or 30º. A simultaneous crack velocity related dynamic strain energy release rate was extracted from fitted experimental results. By applying the simultaneous dynamic fracture energy formulations into the failure criteria in the PD model , the calculated dynamic fracture process and crack velocity match more accurately with the experimental results than the ones which use a constant fracture energy. The homogenization method can be applied to develop meso - scale PD models for other composite materials/structures by combining the trength in continuum mechanics system. 3) Extended the PD model for laminated composite materials, studied the impact delamination and planer damage of the laminates. P eridynamic model for orthotropic lamina has been extended for the composite laminates. By a pplying the micromodulus and orientation - dependent failure criteria to PD, the out - of - plane impact damage process in the [ varies from 15º, 35º, 45º, 60º, to 75º . Both the m atrix damage and delamination in the com posite laminates have been simulated with effective patterns compared to previous experimental studies. The model can be further developed and employed to simulate multi - modes fracture and failure in laminated composite materials. 4) A s ystematic study of a n mechanical properties and impact resistant potentials through design, fabrication, and impact testing combining high - speed 3D DIC method. 125 Quasi - isotropic in - ratio have been identified for the composite structures (UDL, 2DW, and Q3D) through unidirectional quasi - static tensile testing. Tensile strength decreased rapidly with the specimens oriented from 0º, 15º, 30º, to 90º directions as defined in the composit e plaques. For the three different composite structures in the same orientation, the in - plane The Q3D specimens have the smoother failure cross section with less roughness o f fiber tows peeling off because the fiber layers are weaved together as an integral plaque with less interlayer failure. The influence of the different fiber structures on the composite impact performance is the damage and failure mechanism during the imp act process. The UDL plate has the delamination on the bottom surface layer because there is no strengthening for the interlayer in the UDL plate. The penetration on the Q3D plate is similar to that of the isotropic material plate because the Q3D plate has the best homogeneous property through thickness direction. Moreover, the Q3D composite has higher impact energy absorption capability per unit damage area than the UDL and the 2DW composites due to the fiber tows interlock between each adjacent layer. The experimental study p ropose s the method of material dynamic properties characterization, and the design idea of novel composite materials with high - stiffness , high - strength , and high - damage enduring potentials . 126 7.2 Outlook of Future Work The homogenization m ethod was used to develop the meso - scale PD model for composites by directly link ing the laminate theory. Similarly, the method can be applied to develop the PD model for other composites with complex fiber structures, such as the woven fabric composites. The development of the PD model for the plane woven composites and other fiber composite structures will be conducted in the future. More numerical studies with different modeling variables and simulations parameters will be conducted with the PD model an d its extended versions, for obtaining effective numerical simulations of the composites in different mechanical or multi - physical loading situations. More detail impact algorithm will be added to the 3D modeling of impact on laminates. More experimental characterization of damage and delamination process will be performed to quantitatively verify the modeling results of intralayer damage and delamination evolution process. The experimental method of impact combining DIC (with high - speed camera) can be app lied to more composite structures in different mechanical situations. For example, the compression after impact experiments (CAI) of the laminated composites. The compression buckling and shear failing can be captured with the high - speed camera with DIC in both the lateral and the frontal directions, to capture the in - plane and out - of - plane impact failure process of composites in the same time. The comprehensive details of the material dynamic failure process can help on understanding the material propertie s and 127 providing significant knowledge for the development of numerical modeling and simulation. The exploring of the advantages of the Q3D composite structures can be extended with different testing. Such as the quasi - static bending, the open mode fracture testing etc. Future experimental investigations will be conducted to identify more mechanical advantages of the Q3D composite structure, for further fiber - reinforced composite materials design on the demanding of light - weighting applications in industrial areas. 128 APPENDICES 129 Ap p endix A Numerical Flowchart of a Peridynamic Program The explicit numerical flowchart of the PD program is shown in the Figure A. 1 . Figure A. 1 The flowchart of a peridynamic program 130 Appendix B Impact algorithm in the Peridynamic modeling The impact algorithm in the 2D peridynamic modeling is shown in Figure B . 1 . Figure B . 1 Impact alg orithm in the PD modeling. For the impact algorithm in PD simulation, we set a spherical projectile with the certain size and mass toward the plate /beam according to the experimental conditions . The impactor is defined as a rigid body. As shown in Figure B . 1 (a), the impactor moves towards the sample in the beginning. Once the impactor contacts the sample, it penetrates inside and overlaps with the material point s as shown in Figure B . 1 (b) . To model the 131 rig id impact, t he point s are forced to move to the surface of the impactor at the closest path Figure B . 1 (c) . Thus, the contact surface is defined between the impactor and the sample at the current time step. Displacements of points at the sample surface area result in the corresponding bond forces, which interact with the impactor explicitly. Similar impact algorithm is used in the peridynamics as described by Madenci [34] . 132 Appendix C Original notch definition in the Peridynamic modeling The definition of the original notch at the center of the bottom surface of the beam in the 2D pe ridynamic modeling is shown in Figure C . 1 . All the material bonds crossing the original notch/crack are defined as broken, which generates the different damage ratios of the material points around the original notch as shown below . Figure C . 1 Impact algorithm in the PD modeling. 133 Appendix D PD Simulated L aminates D amage Evolution P rocess The damage evolution process simulated with the PD model are shown as figures in this appendix. T he laminate s have the fiber layout of [90/15/90], [90/30/90], [90/45/90], [90/60/90], and [90/75/90] . The laminates are subjected to the mimic im pa ct loa d ing as described in Chapter 5 . Figure D . 1 Intralayer damage, delamination, and displacement field of the laminates [90/ 15 /90] at damage initiation, after loading for 150 s . 134 Figure D . 2 Intralayer damage, delamination, and displacement field of the laminates [90/ 15 /90] at after loading for 3 50 s . 135 Figure D . 3 Intralayer damage, delamination, and displacement field of the laminates [90/ 30 /90] after loading for 200 s . Figure D . 4 Intralayer damage, delamination, and displacement field o f the laminates [ 90 / 30 /90] at after loading for 400 s . 136 Figure D . 5 Intralayer damage, delamination, and displacement field of the laminates [90/ 45 /90] after loading for 200 s . Figure D . 6 I ntralayer damage, delamination, and displacement field of the laminates [90/ 45 /90] after loading for 400 s . 137 Figure D . 7 Intralayer damage, delamination, and displacement field of the laminates [90/ 60 /90] at damage initiatio n, after loading for 150 s . Figure D . 8 Intralayer damage , delamination, and displacement field of the laminates [90/ 60 /90] at after loading for 350 s . 138 Figure D . 9 Intralayer damage, delamin ation, and displacement field of the laminates [90/ 75/ 90] at damage initiation, after loading for 150 s . Figure D . 10 Intralayer damage, delamination, and displacement field of the laminates [90/ 75 /90] at after loading for 350 s . 139 BIBLIOGRAPHY 140 BIBLIOGRAPHY [1] P.K. Mallick, Fiber - Reinforced Composites: Materials, Manufacturing, and Design, Third Edition, CRC Press, 2007. [2] A.P. M ouritz, M.K. Bannister, P.J. Falzon, K.H. Leong, Review of applications for advanced three - dimensional fibre textile composites, Compos. Part Appl. Sci. Manuf. 30 (1999) 1445 1461. doi :10.1016/S1359 - 835X(99)00034 - 2. [3] C.G. Davila, P.P. Camanho, C.A. Ros e, Failure Criteria for FRP Laminates, J. Compos. Mater. 39 (2005) 323 345. doi:10.1177/0021998305046452. [4] J. - K. Kim, M. - L. Sham, Impact and delamination failure of woven - fabric composites, Compos. Sci. Technol. 60 (2000) 745 761. doi:10.1016/S0266 - 353 8(99)00166 - 9. [5] M.O.W. Richardson, M.J. Wisheart, Review of low - velocity impact properties of composite materials, Compos. Part Appl. Sci. Manuf. 27 (1996) 1123 1131. doi :10.1016/1359 - 835X(96)00074 - 7. [6] W.J. Cantwell, J. Morton, The impact resistance of composite materials a review, Composites. 22 (1991) 347 362. doi :10.1016/0010 - 4361(91)90549 - V. [7] C. ZWEBEN, Tensile failure of fiber composites., AIAA J. 6 (1968) 2325 2331. doi:10.2514/3.4990. [8] S.T. Pinho, C.G. Davila, P.P. Camanho, L. Iannuc ci, P. Robinson, Failure Models and Criteria for FRP Under In - Plane or Three - Dimensional Stress States Including Shear Non - Linearity, 2005. http://ntrs.nasa.gov/search.jsp?R=20050110223 (accessed October 10, 2016). [9] C.R. Schultheisz, A.M. Waas, Compres sive failure of composites, part I: Testing and micromechanical theories, Prog. Aerosp. Sci. 32 (1996) 1 42. doi:10.1016/0376 - 0421(94)00002 - 3. [10] S. Kyriakides, R. Arseculeratne, E.J. Perry, K.M. Liechti, On the compressive failure of fiber reinforced c omposites, Int. J. Solids Struct. 32 (1995) 689 738. doi:10.1016/0020 - 7683(94)00157 - R. [11] D.W. Scott, J.S. Lai, A. - H. Zureick, Creep Behavior of Fiber - Reinforced Polymeric Composites: A Review of the Technical Literature, J. Reinf. Plast. Compos. 14 (19 95) 588 617. doi:10.1177/073168449501400603. [12] J. Degrieck, W. Van Paepegem, Fatigue damage modeling of fibre - reinforced composite materials: Review, Appl. Mech. Rev. 54 (2001) 279 300. doi:10.1115/1.1381395. 141 [13] F. Hou, S. Hong, Characterization of R - curve behavior of translaminar crack growth in cross - ply composite laminates using digital image correlation, Eng. Fract. Mech. 117 (2014) 51 70. doi:10.1016/j.engfracmech.2014.01.010. [14] X. Li, S.R. Hallett, M.R. Wisnom, Numerical investigation of pro gressive damage and the effect of layup in overheight compact tension tests, Compos. Part Appl. Sci. Manuf. 43 (2012) 2137 2150. doi:10.1016/j.compositesa.2012.03.002. [15] S. Li, M.D. Thouless, A.M. Waas, J.A. Schroeder, P.D. Zavattieri, Use of a cohesiv e - zone model to analyze the fracture of a fiber - reinforced polymer matrix composite, Compos. Sci. Technol. 65 (2005) 537 549. doi:10.1016/j.compscitech.2004.08.004. [16] E. Oterkus, E. Madenci, O. Weckner, S. Silling, P. Bogert, A. Tessler, Combined finit e element and peridynamic analyses for predicting failure in a stiffened composite curved panel with a central slot, Compos. Struct. 94 (2012) 839 850. doi:10.1016/j.compstruct.2011.07.019. [17] B. Kilic, E. Madenci, Coupling of peridynamic theory and the finite element method, J. Mech. Mater. Struct. 5 (2010) 707 733. doi:10.2140/jomms.2010.5.707. [18] S.A. Silling, Reformulation of elasticity theory for discontinuities and long - range forces, J. Mech. Phys. Solids. 48 (2000) 175 209. doi:10.1016/S0022 - 50 96(99)00029 - 0. [19] S.A. Silling, E. Askari, A meshfree method based on the peridynamic model of solid mechanics, Comput. Struct. 83 (2005) 1526 1535. doi:10.1016/j.compstruc.2004.11.026. [20] S.A. Silling, M. Epton, O. Weckner, J. Xu, E. Askari, Peridyn amic States and Constitutive Modeling, J. Elast. 88 (2007) 151 184. doi:10.1007/s10659 - 007 - 9125 - 1. [21] ASTM, Composite Materials: Fatigue and Fracture, Fourth Volume, in: n.d. [22] M.M. Shokrieh, M.J. Omidi, Tension behavior of unidirectional glass/epox y composites under different strain rates, Compos. Struct. 88 (2009) 595 601. doi:10.1016/j.compstruct.2008.06.012. [23] M.M. Shokrieh, M.J. Omidi, Investigation of strain rate effects on in - plane shear properties of glass/epoxy composites, Compos. Struct . 91 (2009) 95 102. doi:10.1016/j.compstruct.2009.04.035. [24] D. Lee, H. Tippur, M. Kirugulige, P. Bogert, Experimental Study of Dynamic Crack Growth in Unidirectional Graphite/Epoxy Composites using Digital Image 142 Correlation Method and High - Speed Photog raphy, J. Compos. Mater. (2009). doi:10.1177/0021998309342139. [25] D. Lee, H. Tippur, P. Bogert, Quasi - static and dynamic fracture of graphite/epoxy composites: An optical study of loading - rate effects, Compos. Part B Eng. 41 (2010) 462 474. doi:10.1016/ j.compositesb.2010.05.007. [26] D. Lee, H.V. Tippur, B.J. Jensen, P.B. Bogert, Tensile and Fracture Characterization of PETI - 5 and IM7/PETI - 5 Graphite/Epoxy Composites Under Quasi - Static and Dynamic Loading Conditions, J. Eng. Mater. Technol. 133 (2011) 0 21015 021015. doi:10.1115/1.4003487. [27] D. Lee, H. Tippur, P. Bogert, Dynamic fracture of graphite/epoxy composites stiffened by buffer strips: An experimental study, Compos. Struct. 94 (2012) 3538 3545. doi:10.1016/j.compstruct.2012.05.032. [28] R.M. Sencu, Z.J. Yang, Y.C. Wang, 5 - From micro to macro: Simulating crack propagation in carbon fibre composites, in: P.W.R. Beaumont, C.S. Hodzic (Eds.), Struct. Integr. Durab. Adv. Compos., Woodhead Publishing, 2015: pp. 105 124. http://www.sciencedirect.co m/science/article/pii/B9780081001370000055 (accessed September 14, 2015). [29] S.R. Hallett, B.G. Green, W.G. Jiang, M.R. Wisnom, An experimental and numerical investigation into the damage mechanisms in notched composites, Compos. Part Appl. Sci. Manuf. 40 (2009) 613 624. doi:10.1016/j.compositesa.2009.02.021. [30] experimental/numerical investigation into the main driving force for crack propagation in uni - directional fibre - re inforced composite laminae, Compos. Struct. 107 (2014) 119 130. doi:10.1016/j.compstruct.2013.05.039. [31] E.J. Pineda, A.M. Waas, A Thermodynamically - Based Mesh Objective Work Potential Theory for Predicting Intralaminar Progressive Damage and Failure in Fiber - Reinforced Laminates, in: Honolulu, HI, United States, 2012. http://ntrs.nasa.gov/search.jsp?R=20120012876 (accessed May 8, 2016). [32] B. Kilic, A. Agwai, E. Madenci, Peridynamic theory for progressive damage prediction in center - cracked composite laminates, Compos. Struct. 90 (2009) 141 151. doi:10.1016/j.compstruct.2009.02.015. [33] E. Oterkus, E. Madenci, Peridynamic analysis of fiber - reinforced composite materials, J. Mech. Mater. Struct. 7 (2012) 45 84. doi:10.2140/jomms.2012.7.45. [34] E. M adenci, E. Oterkus, Peridynamic Theory and Its Applications, Springer, New York, 2014. 143 [35] W. Hu, Y.D. Ha, F. Bobaru, Peridynamic model for dynamic fracture in unidirectional fiber - reinforced composites, Comput. Methods Appl. Mech. Eng. 217 220 (2012) 24 7 261. doi:10.1016/j.cma.2012.01.016. [36] Q. Bing, C.T. Sun, Modeling and testing strain rate - dependent compressive strength of carbon/epoxy composites, Compos. Sci. Technol. 65 (2005) 2481 2491. doi:10.1016/j.compscitech.2005.06.012. [37] J. Tsai, C.T. Sun, Constitutive model for high strain rate response of polymeric composites, Compos. Sci. Technol. 62 (2002) 1289 1297. doi:10.1016/S0266 - 3538(02)00064 - 7. [38] Z. Kutlu, F. - K. Chang, Modeling Compression Failure of laminated Composites Containing Multi ple Through - the - Width Delaminations, J. Compos. Mater. 26 (1992) 350 387. doi:10.1177/002199839202600303. [39] P.F. Liu, J.Y. Zheng, Recent developments on damage modeling and finite element analysis for composite laminates: A review, Mater. Des. 31 (2010 ) 3825 3834. doi:10.1016/j.matdes.2010.03.031. [40] B. Kilic, E. Madenci, Structural stability and failure analysis using peridynamic theory, Int. J. Non - Linear Mech. 44 (2009) 845 854. doi:10.1016/j.ijnonlinmec.2009.05.007. [41] D. Huang, Q. Zhang, P. Q iao, Damage and progressive failure of concrete structures using non - local peridynamic modeling, Sci. China Technol. Sci. 54 (2011) 591 596. doi:10.1007/s11431 - 011 - 4306 - 3. [42] F. Shen, Q. Zhang, D. Huang, Damage and Failure Process of Concrete Structure under Uniaxial Compression Based on Peridynamics Modeling, Math. Probl. Eng. 2013 (2013) e631074. doi:10.1155/2013/631074. [43] Y. Wang, X. Zhou, X. Xu, Numerical simulation of propagation and coalescence of flaws in rock materials under compressive loads using the extended non - ordinary state - based peridynamics, Eng. Fract. Mech. 163 (2016) 248 273. doi:10.1016/j.engfracmech.2016.06.013. [44] Y.L. Hu, E. Madenci, Bond - based peridynamic modeling of composite laminates with arbitrary fiber orientation and s tacking sequence, Compos. Struct. 153 (2016) 139 175. doi:10.1016/j.compstruct.2016.05.063. [45] S. Agrawal, K.K. Singh, P.K. Sarkar, Impact damage on fibre - reinforced polymer matrix composite A review, J. Compos. Mater. (2013) 0021998312472217. doi:10. 1177/0021998312472217. 144 [46] S. Abrate, Impact on Laminated Composite Materials, Appl. Mech. Rev. 44 (1991) 155 190. doi:10.1115/1.3119500. [47] H.Y. Choi, F. - K. Chang, A Model for Predicting Damage in Graphite/Epoxy Laminated Composites Resulting from Lo w - Velocity Point Impact, J. Compos. Mater. 26 (1992) 2134 2169. doi:10.1177/002199839202601408. [48] H.Y. Choi, H. - Y.T. Wu, F. - K. Chang, A New Approach toward Understanding Damage Mechanisms and Mechanics of Laminated Composites Due to Low - Velocity Impact : Part II Analysis, J. Compos. Mater. 25 (1991) 1012 1038. doi:10.1177/002199839102500804. [49] O.T. Topac, B. Tasdemir, B. Gozluklu, E. Gurses, D. Coker, Experimental and Computational Investigation of Out - of - Plane Low Velocity Impact Behavior of CFRP Co mposite Plates, in: SpringerLink, Springer International Publishing, n.d.: pp. 9 16. doi:10.1007/978 - 3 - 319 - 21611 - 9_2. [50] S. Takeda, S. Minakuchi, Y. Okabe, N. Takeda, Delamination monitoring of laminated composites subjected to low - velocity impact using small - diameter FBG sensors, Compos. Part Appl. Sci. Manuf. 36 (2005) 903 908. doi:10.1016/j.compositesa.2004.12.005. [51] C. Bouvet, S. Rivallant, J.J. Barrau, Low velocity impact modeling in composite laminates capturing permanent indentation, Compos. S ci. Technol. 72 (2012) 1977 1988. doi:10.1016/j.compscitech.2012.08.019. [52] D. Feng, F. Aymerich, Damage prediction in composite sandwich panels subjected to low - velocity impact, Compos. Part Appl. Sci. Manuf. 52 (2013) 12 22. doi:10.1016/j.compositesa. 2013.04.010. [53] R. Karakuzu, E. Erbil, M. Aktas, Impact characterization of glass/epoxy composite plates: An experimental and numerical study, Compos. Part B Eng. 41 (2010) 388 395. doi:10.1016/j.compositesb.2010.02.003. [54] V. Tita, J. de Carvalho, D . Vandepitte, Failure analysis of low velocity impact on thin composite laminates: Experimental and numerical approaches, Compos. Struct. 83 (2008) 413 428. doi:10.1016/j.compstruct.2007.06.003. [55] A. Turon, P.P. Camanho, J. Costa, C.G. Dávila, A damage model for the simulation of delamination in advanced composites under variable - mode loading, Mech. Mater. 38 (2006) 1072 1089. doi:10.1016/j.mechmat.2005.10.003. [56] J.P. Hou, N. Petrinic, C. Ruiz, S.R. Hallett, Prediction of impact damage in composite p lates, Compos. Sci. Technol. 60 (2000) 273 281. doi:10.1016/S0266 - 3538(99)00126 - 8. 145 [57] J.P. Hou, N. Petrinic, C. Ruiz, A delamination criterion for laminated composites under low - velocity impact, Compos. Sci. Technol. 61 (2001) 2069 2074. doi:10.1016/S02 66 - 3538(01)00128 - 2. [58] C. Bouvet, B. Castanié, M. Bizeul, J. - J. Barrau, Low velocity impact modelling in laminate composite panels with discrete interface elements, Int. J. Solids Struct. 46 (2009) 2809 2821. doi:10.1016/j.ijsolstr.2009.03.010. [59] Y. Shi, T. Swait, C. Soutis, Modelling damage evolution in composite laminates subjected to low velocity impact, Compos. Struct. 94 (2012) 2902 2913. doi :10.1016/j.compstruct.2012.03.039. [60] L. Maio, E. Monaco, F. Ricci, L. Lecce, Simulation of low veloci ty impact on composite laminates with progressive failure analysis, Compos. Struct. 103 (2013) 75 85. doi:10.1016/j.compstruct.2013.02.027. [61] J. Xu, S. Silling, Peridynamic Analysis of Impact Damage in Composite Laminates, J. Aerosp. Eng. 21 (2008) 187 194. doi:10.1061/(ASCE)0893 - 1321(2008)21:3(187). [62] T.C. Chu, W.F. Ranson, M.A. Sutton, Applications of digital - image - correlation techniques to experimental mechanics, Exp. Mech. 25 (1985) 232 244. doi:10.1007/BF02325092. [63] F. Hild, S. Roux, Digita l Image Correlation: from Displacement Measurement to Identification of Elastic Properties a Review, Strain. 42 (2006) 69 80. doi:10.1111/j.1475 - 1305.2006.00258.x. [64] S. Roux, J. Réthoré, F. Hild, Digital image correlation and fracture: an advanced te chnique for estimating stress intensity factors of 2D and 3D cracks, J. Phys. Appl. Phys. 42 (2009) 214004. doi:10.1088/0022 - 3727/42/21/214004. [65] GOM Training Webinar - 2D Digital Image Correlation with GOM Correlate, 2016. https://www.youtube.com/watc h?v=pGXuXg7dRlo. [66] Y.D. Ha, F. Bobaru, Studies of dynamic crack propagation and crack branching with peridynamics, Int. J. Fract. 162 (2010) 229 244. doi :10.1007/s10704 - 010 - 9442 - 4. [67] Y. Hu, Y. Yu, H. Wang, Peridynamic analytical method for progress ive damage in notched composite laminates, Compos. Struct. 108 (2014) 801 810. doi :10.1016/j.compstruct.2013.10.018. [68] M. Ghajari, L. Iannucci, P. Curtis, A peridynamic material model for the analysis of dynamic crack propagation in orthotropic media, C omput. Methods Appl. Mech. Eng. 276 (2014) 431 452. doi :10.1016/j.cma.2014.04.002. 146 [69] T. Jia, DEVELOPMENT AND APPLICATIONS OF NEW PERIDYNAMIC MODELS, Michigan State University, 2012. [70] N. Liu, D. Liu, W. Zhou, Peridynamic modelling of impact damage in three - point bending beam with offset notch, Appl. Math. Mech. (2016) 1 12. doi:10.1007/s10483 - 017 - 2158 - 6. [71] J.N. Reddy, Mechanics of Laminated Composite Plates and Shells: Theory a nd Analysis, Second Edition, CRC Press, 2004. [72] D. Dipasquale, G. Sarego, M. Zaccariotto, U. Galvanetto, Dependence of crack paths on the orientation of regular 2D peridynamic grids, Eng. Fract. Mech. 160 (2016) 248 263. doi:10.1016/j.engfracmech.2016. 03.022. [73] K. Takahashi, K. Arakawa, Dependence of crack acceleration on the dynamic stress - intensity factor in polymers, Exp. Mech. 27 (1987) 195 199. doi:10.1007/BF02319474. [74] A.S. Kobayashi, M. Ramulu, M.S. Dadkhah, K. - H. Yang, B.S.J. Kang, Dynam ic fracture toughness, Int. J. Fract. 30 (1986) 275 285. doi:10.1007/BF00019707. [75] J. Krafft, G. Irwin, Crack - Velocity Considerations, in: ASTM Committee E - 24 (Ed.), Fract. Toughness Test. Its Appl., ASTM International, 100 Barr Harbor Drive, PO Box C7 00, West Conshohocken, PA 19428 - 2959, 1965: pp. 114 - 114 16. http://www.astm.org/doiLink.cgi?STP26587S (accessed April 13, 2016). [76] A.S. Kobayashi, S. Mall, Dynamic fracture toughness of Homalite - 100, Exp. Mech. 18 (1978) 11 18. doi :10.1007/BF02326552. [77] W. Zhou, D. Liu, N. Liu, Analyzing dynamic fracture process in fiber - reinforced composite materials with a peridynamic model, Eng. Fract. Mech. 178 (2017) 60 76. doi :10.1016/j.engfracmech.2017.04.022. [78] W. Zhou, and D. Liu, A Peridynamic Model f or Analyzing Crack Propagation in Unidirectional Composite Lamina, Proc. Am. Soc. Compos. Thirty - First Tech. Conf. (2016). [79] D. Hartman, M.E. Greenwood, D.M. Miller, High Strength Glass Fibers, agy , 2006. [80] K. Rosario, D. Liu, Assessment of Quasi - T hree - Dimensional Composites - with Discussions on Fiber Straining and Weaving Effectiveness, J. Compos. Mater. 44 (2010) 2953 2973. doi :10.1177/0021998310371538. [81] C. Atas, D. Liu, Impact response of woven composites with small weaving angles, Int. J. I mpact Eng. 35 (2008) 80 97. doi:10.1016/j.ijimpeng.2006.12.004. 147 [82] R. Jensen, A. Forster, C. Copeland, J. Dibelka, Cure Schedule Evaluation of SC15 and SC79 Low - Viscosity Epoxy VARTM Resins, Army Research Laboratory, 2005. [83] Standard Test Method for Tensile Properties of Polymer Matrix Composite Materials, ASTM D3039, 17. [84] H. Schreier, J. - J. Orteu, M.A. Sutton, Image Correlation for Shape, Motion and Deformation Measurements, Springer US, n.d. doi :10.1007/978 - 0 - 387 - 78747 - 3.